TPTP Problem File: ITP224^3.p
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%------------------------------------------------------------------------------
% File : ITP224^3 : TPTP v8.2.0. Released v8.1.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_Member 00088_002463
% 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 : 0065_VEBT_Member_00088_002463 [Des22]
% Status : ContradictoryAxioms
% Rating : 0.80 v8.2.0, 0.77 v8.1.0
% Syntax : Number of formulae : 11168 (6432 unt; 910 typ; 0 def)
% Number of atoms : 25417 (12104 equ; 0 cnn)
% Maximal formula atoms : 71 ( 2 avg)
% Number of connectives : 105768 (2280 ~; 477 |;1431 &;93705 @)
% ( 0 <=>;7875 =>; 0 <=; 0 <~>)
% Maximal formula depth : 39 ( 5 avg)
% Number of types : 66 ( 65 usr)
% Number of type conns : 3286 (3286 >; 0 *; 0 +; 0 <<)
% Number of symbols : 848 ( 845 usr; 53 con; 0-8 aty)
% Number of variables : 22938 (2145 ^;20220 !; 573 ?;22938 :)
% SPC : TH0_CAX_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 18:30:18.398
%------------------------------------------------------------------------------
% Could-be-implicit typings (65)
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thf(ty_n_t__Nat__Onat,type,
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% Explicit typings (845)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
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thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
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thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
archim8280529875227126926d_real: real > int ).
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bNF_re4695409256820837752l_real: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( real > real > real ) > $o ).
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bNF_re3023117138289059399t_real: ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_Eo_001_Eo,type,
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bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
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bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).
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bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).
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bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
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 ).
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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__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Int__Oint_J,type,
bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).
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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,
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thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
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thf(sy_c_Complex_Ocomplex_OIm,type,
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thf(sy_c_Complex_Ocomplex_ORe,type,
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thf(sy_c_List_Oreplicate_001_Eo,type,
replicate_o: nat > $o > list_o ).
thf(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
replicate_int: nat > int > list_int ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
replicate_real: nat > real > list_real ).
thf(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
replicate_set_nat: nat > set_nat > list_set_nat ).
thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_List_Oupto,type,
upto: int > int > list_int ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
case_nat_o: $o > ( nat > $o ) > nat > $o ).
thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
case_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
thf(sy_c_Nat_Onat_Opred,type,
pred: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
semiring_1_Nats_int: set_int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
semiri4939895301339042750nteger: nat > code_integer ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
semiri8010041392384452111omplex: nat > complex ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
semiri681578069525770553at_rat: nat > rat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
size_size_list_o: list_o > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
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size_s3451745648224563538omplex: list_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Nat__Bijection_Olist__encode,type,
nat_list_encode: list_nat > nat ).
thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
nat_list_encode_rel: list_nat > list_nat > $o ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: product_prod_nat_nat > nat ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > set_nat ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: set_nat > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_NthRoot_Osqrt,type,
sqrt: real > real ).
thf(sy_c_Num_OBitM,type,
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thf(sy_c_Num_Oinc,type,
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neg_nu8804712462038260780nteger: code_integer > code_integer ).
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neg_nu7009210354673126013omplex: complex > complex ).
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thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
neg_numeral_dbl_rat: rat > rat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
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neg_nu6511756317524482435omplex: complex > complex ).
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neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
neg_nu3179335615603231917ec_rat: rat > rat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
neg_nu6075765906172075777c_real: real > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
neg_nu5831290666863070958nteger: code_integer > code_integer ).
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neg_nu8557863876264182079omplex: complex > complex ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
neg_nu5219082963157363817nc_rat: rat > rat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
neg_numeral_sub_int: num > num > int ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Num_Osqr,type,
sqr: num > num ).
thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
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ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_eq_set_num: set_num > set_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
ord_less_eq_set_rat: set_rat > set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
ord_max_int: int > int > int ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
ord_min_int: int > int > int ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
order_mono_nat_nat: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
order_5726023648592871131at_nat: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
order_7092887310737990675l_real: ( real > real ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
top_top_set_int: set_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
power_8256067586552552935nteger: code_integer > nat > code_integer ).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
power_power_complex: complex > nat > complex ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
power_power_rat: rat > nat > rat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
product_Pair_o_o: $o > $o > product_prod_o_o ).
thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
product_Pair_o_int: $o > int > product_prod_o_int ).
thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
product_Pair_o_nat: $o > nat > product_prod_o_nat ).
thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).
thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
product_Pair_nat_num: nat > num > product_prod_nat_num ).
thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
product_Pair_num_num: num > num > product_prod_num_num ).
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).
thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).
thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
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produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
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produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
produc8211389475949308722nt_int: ( int > int > int ) > product_prod_int_int > int ).
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produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_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,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
product_fst_int_int: product_prod_int_int > int ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
product_snd_int_int: product_prod_int_int > int ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Rat_OAbs__Rat,type,
abs_Rat: product_prod_int_int > rat ).
thf(sy_c_Rat_OFract,type,
fract: int > int > rat ).
thf(sy_c_Rat_OFrct,type,
frct: product_prod_int_int > rat ).
thf(sy_c_Rat_ORep__Rat,type,
rep_Rat: rat > product_prod_int_int ).
thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
field_5140801741446780682s_real: set_real ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
field_7254667332652039916t_real: rat > real ).
thf(sy_c_Rat_Onormalize,type,
normalize: product_prod_int_int > product_prod_int_int ).
thf(sy_c_Rat_Oof__int,type,
of_int: int > rat ).
thf(sy_c_Rat_Opcr__rat,type,
pcr_rat: product_prod_int_int > rat > $o ).
thf(sy_c_Rat_Opositive,type,
positive: rat > $o ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > product_prod_int_int ).
thf(sy_c_Rat_Oratrel,type,
ratrel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Real_Opcr__real,type,
pcr_real: ( nat > rat ) > real > $o ).
thf(sy_c_Real_Opositive,type,
positive2: real > $o ).
thf(sy_c_Real_Ovanishes,type,
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thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
real_V2521375963428798218omplex: set_complex ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
real_V1022390504157884413omplex: complex > real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
real_V7735802525324610683m_real: real > real ).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
real_V1803761363581548252l_real: real > real ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
real_V1485227260804924795R_real: real > real > real ).
thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
algebr932160517623751201me_int: int > int > $o ).
thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
algebr934650988132801477me_nat: nat > nat > $o ).
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
divide_divide_rat: rat > rat > rat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
dvd_dvd_complex: complex > complex > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
dvd_dvd_rat: rat > rat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
modulo_modulo_int: int > int > int ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
zero_n356916108424825756nteger: $o > code_integer ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
zero_n1201886186963655149omplex: $o > complex ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
zero_n2684676970156552555ol_int: $o > int ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
zero_n2687167440665602831ol_nat: $o > nat ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
zero_n2052037380579107095ol_rat: $o > rat ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
zero_n3304061248610475627l_real: $o > real ).
thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
suminf_int: ( nat > int ) > int ).
thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
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thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osummable_001t__Int__Oint,type,
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thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
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thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
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thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osums_001t__Int__Oint,type,
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thf(sy_c_Set_OCollect_001t__Int__Oint,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
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thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_OPow_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
image_nat_char: ( nat > char ) > set_nat > set_char ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
image_char_nat: ( char > nat ) > set_char > set_nat ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
set_or7049704709247886629st_num: num > num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
set_or633870826150836451st_rat: rat > rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
set_ord_atLeast_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
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__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_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or890127255671739683et_nat: set_nat > set_set_nat ).
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_Ochar_Osize__char,type,
size_char: char > nat ).
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__Member_OVEBT__internal_Obit__concat,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
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_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_Opred__nat,type,
pred_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_fChoice_001t__Real__Oreal,type,
fChoice_real: ( real > $o ) > real ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Code____Numeral__Ointeger,type,
member_Code_integer: code_integer > set_Code_integer > $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_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $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_info____,type,
info: option4927543243414619207at_nat ).
thf(sy_v_m,type,
m: nat ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_na,type,
na: nat ).
thf(sy_v_summary____,type,
summary: vEBT_VEBT ).
thf(sy_v_ta____,type,
ta: vEBT_VEBT ).
thf(sy_v_treeList____,type,
treeList: list_VEBT_VEBT ).
% Relevant facts (10208)
thf(fact_0_Node_OIH_I2_J,axiom,
~ ( vEBT_invar_vebt @ summary @ zero_zero_nat ) ).
% Node.IH(2)
thf(fact_1__092_060open_062t_A_092_060in_062_Aset_AtreeList_092_060close_062,axiom,
member_VEBT_VEBT @ ta @ ( set_VEBT_VEBT2 @ treeList ) ).
% \<open>t \<in> set treeList\<close>
thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_O_At_A_092_060in_062_Aset_AtreeList_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [T: vEBT_VEBT] :
~ ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ treeList ) ) ).
% \<open>\<And>thesis. (\<And>t. t \<in> set treeList \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_3__092_060open_062invar__vebt_At_An_092_060close_062,axiom,
vEBT_invar_vebt @ ta @ n ).
% \<open>invar_vebt t n\<close>
thf(fact_4_Node_OIH_I1_J,axiom,
! [X3a: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3a @ ( set_VEBT_VEBT2 @ treeList ) )
=> ~ ( vEBT_invar_vebt @ X3a @ zero_zero_nat ) ) ).
% Node.IH(1)
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_O_Ainvar__vebt_At_An_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [N: nat] :
~ ( vEBT_invar_vebt @ ta @ N ) ).
% \<open>\<And>thesis. (\<And>n. invar_vebt t n \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6_Node_Oprems,axiom,
vEBT_invar_vebt @ ( vEBT_Node @ info @ deg @ treeList @ summary ) @ zero_zero_nat ).
% Node.prems
thf(fact_7_sum__power2__eq__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_8_sum__power2__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_9_sum__power2__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_10_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_11_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_12_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_13_power2__eq__iff__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_14_power2__eq__iff__nonneg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_15_power2__eq__iff__nonneg,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_16_power2__eq__iff__nonneg,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_17_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_18_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_19_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_20_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_21_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_22_set__n__deg__not__0,axiom,
! [TreeList: list_VEBT_VEBT,N2: nat,M: nat] :
( ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ) ).
% set_n_deg_not_0
thf(fact_23_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_24_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_25_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
= zero_zero_real ) ).
% power_zero_numeral
thf(fact_26_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
= zero_zero_complex ) ).
% power_zero_numeral
thf(fact_27_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
= zero_zero_rat ) ).
% power_zero_numeral
thf(fact_28_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_29_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_30_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_31_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_32_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_33_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_34_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_35_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_36_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_37_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_38_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_39_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_40_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_41_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_42_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_43_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_44_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_45_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_46_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_47_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_48_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_49_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_50_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_51_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_52_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_53_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_54_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_55_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_56_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_57_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_58_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_59_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_60_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_61_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_62_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_63_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_64_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_65_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_66_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_67_add__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add_0
thf(fact_68_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_69_add__0,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add_0
thf(fact_70_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_71_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_72_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_73_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_74_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_75_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_76_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_77_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_78_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_79_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_80_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_81_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_82_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_83_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_84_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_85_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_86_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_87_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_88_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_89_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_90_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_91_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_92_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_93_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_94_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_95_double__zero__sym,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( plus_plus_rat @ A @ A ) )
= ( A = zero_zero_rat ) ) ).
% double_zero_sym
thf(fact_96_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_97_add_Oright__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.right_neutral
thf(fact_98_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_99_add_Oright__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.right_neutral
thf(fact_100_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_101_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_102_power__one,axiom,
! [N2: nat] :
( ( power_power_nat @ one_one_nat @ N2 )
= one_one_nat ) ).
% power_one
thf(fact_103_power__one,axiom,
! [N2: nat] :
( ( power_power_int @ one_one_int @ N2 )
= one_one_int ) ).
% power_one
thf(fact_104_power__one,axiom,
! [N2: nat] :
( ( power_power_real @ one_one_real @ N2 )
= one_one_real ) ).
% power_one
thf(fact_105_power__one,axiom,
! [N2: nat] :
( ( power_power_complex @ one_one_complex @ N2 )
= one_one_complex ) ).
% power_one
thf(fact_106_power__one,axiom,
! [N2: nat] :
( ( power_power_rat @ one_one_rat @ N2 )
= one_one_rat ) ).
% power_one
thf(fact_107_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_108_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_109_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_110_power__one__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_111_power__one__right,axiom,
! [A: rat] :
( ( power_power_rat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_112_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_113_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_114_one__reorient,axiom,
! [X: rat] :
( ( one_one_rat = X )
= ( X = one_one_rat ) ) ).
% one_reorient
thf(fact_115_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_116_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_117_one__le__power,axiom,
! [A: rat,N2: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_118_one__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_119_one__le__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_120_one__le__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_121_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_122_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_123_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_124_power__0,axiom,
! [A: complex] :
( ( power_power_complex @ A @ zero_zero_nat )
= one_one_complex ) ).
% power_0
thf(fact_125_power__0,axiom,
! [A: rat] :
( ( power_power_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% power_0
thf(fact_126_power__le__one,axiom,
! [A: rat,N2: 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 @ N2 ) @ one_one_rat ) ) ) ).
% power_le_one
thf(fact_127_power__le__one,axiom,
! [A: nat,N2: 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 @ N2 ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_128_power__le__one,axiom,
! [A: int,N2: 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 @ N2 ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_129_power__le__one,axiom,
! [A: real,N2: 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 @ N2 ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_130_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= one_one_nat ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_131_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= one_one_int ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_132_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N2 )
= one_one_real ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N2 )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_133_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N2 )
= one_one_complex ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N2 )
= zero_zero_complex ) ) ) ).
% power_0_left
thf(fact_134_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N2 )
= one_one_rat ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N2 )
= zero_zero_rat ) ) ) ).
% power_0_left
thf(fact_135_power__increasing,axiom,
! [N2: nat,N3: nat,A: rat] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_136_power__increasing,axiom,
! [N2: nat,N3: nat,A: nat] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_137_power__increasing,axiom,
! [N2: nat,N3: nat,A: int] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_138_power__increasing,axiom,
! [N2: nat,N3: nat,A: real] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_139_power__decreasing,axiom,
! [N2: nat,N3: nat,A: rat] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_140_power__decreasing,axiom,
! [N2: nat,N3: nat,A: nat] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_141_power__decreasing,axiom,
! [N2: nat,N3: nat,A: int] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_142_power__decreasing,axiom,
! [N2: nat,N3: nat,A: real] :
( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_143_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_144_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_145_one__power2,axiom,
( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real ) ).
% one_power2
thf(fact_146_one__power2,axiom,
( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_complex ) ).
% one_power2
thf(fact_147_one__power2,axiom,
( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_rat ) ).
% one_power2
thf(fact_148_mem__Collect__eq,axiom,
! [A: complex,P: complex > $o] :
( ( member_complex @ A @ ( collect_complex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_149_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_150_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_151_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_152_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_153_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_154_Collect__mem__eq,axiom,
! [A2: set_complex] :
( ( collect_complex
@ ^ [X3: complex] : ( member_complex @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_155_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_156_Collect__mem__eq,axiom,
! [A2: set_list_nat] :
( ( collect_list_nat
@ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_157_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_158_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_159_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X3: int] : ( member_int @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_160_Collect__cong,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X2: real] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_real @ P )
= ( collect_real @ Q ) ) ) ).
% Collect_cong
thf(fact_161_Collect__cong,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X2: list_nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_list_nat @ P )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_162_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X2: set_nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_163_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_164_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_165_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_166_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_167_zero__reorient,axiom,
! [X: rat] :
( ( zero_zero_rat = X )
= ( X = zero_zero_rat ) ) ).
% zero_reorient
thf(fact_168_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_169_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_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_182_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B2: real] : ( plus_plus_real @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_183_add_Ocommute,axiom,
( plus_plus_rat
= ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_184_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_185_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_186_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_187_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_188_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_189_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_190_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_191_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_192_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_193_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_194_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_195_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_196_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_197_group__cancel_Oadd2,axiom,
! [B3: rat,K: rat,B: rat,A: rat] :
( ( B3
= ( plus_plus_rat @ K @ B ) )
=> ( ( plus_plus_rat @ A @ B3 )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_198_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_199_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_200_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_201_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_202_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_203_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_204_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_205_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_rat @ I @ K )
= ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_206_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_207_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_208_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_209_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_210_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_211_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_212_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_213_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_214_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_215_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_216_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_217_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_218_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_219_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_220_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_221_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C2: nat] :
( B2
= ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_222_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_223_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_224_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_225_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_226_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_227_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_228_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_229_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_230_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_231_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_232_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_233_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_234_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_235_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( ord_less_eq_rat @ K @ L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_236_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_237_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_238_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_239_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( ord_less_eq_rat @ K @ L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_240_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_241_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_242_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_243_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_244_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_245_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_246_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_247_add_Ogroup__left__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add.group_left_neutral
thf(fact_248_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_249_add_Ogroup__left__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add.group_left_neutral
thf(fact_250_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_251_add_Ocomm__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.comm_neutral
thf(fact_252_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_253_add_Ocomm__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.comm_neutral
thf(fact_254_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_255_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_256_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_257_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_258_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_259_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_260_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_261_power__not__zero,axiom,
! [A: nat,N2: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N2 )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_262_power__not__zero,axiom,
! [A: int,N2: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N2 )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_263_power__not__zero,axiom,
! [A: real,N2: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N2 )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_264_power__not__zero,axiom,
! [A: complex,N2: nat] :
( ( A != zero_zero_complex )
=> ( ( power_power_complex @ A @ N2 )
!= zero_zero_complex ) ) ).
% power_not_zero
thf(fact_265_power__not__zero,axiom,
! [A: rat,N2: nat] :
( ( A != zero_zero_rat )
=> ( ( power_power_rat @ A @ N2 )
!= zero_zero_rat ) ) ).
% power_not_zero
thf(fact_266_add__nonpos__eq__0__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ( ( plus_plus_rat @ X @ Y )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_267_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_268_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_269_add__nonpos__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_270_add__nonneg__eq__0__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ( plus_plus_rat @ X @ Y )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_271_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_272_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_273_add__nonneg__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_274_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_275_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_276_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_277_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_278_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_279_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_280_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_281_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_282_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_283_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_284_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_285_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_286_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_287_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_288_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_289_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_290_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_291_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_292_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_293_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_294_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_295_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_296_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_297_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_298_zero__le__power,axiom,
! [A: rat,N2: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_299_zero__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_300_zero__le__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_301_zero__le__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_302_power__mono,axiom,
! [A: rat,B: rat,N2: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_303_power__mono,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_304_power__mono,axiom,
! [A: int,B: int,N2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_305_power__mono,axiom,
! [A: real,B: real,N2: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_306_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_307_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_308_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real ) ).
% zero_power2
thf(fact_309_zero__power2,axiom,
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_complex ) ).
% zero_power2
thf(fact_310_zero__power2,axiom,
( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_rat ) ).
% zero_power2
thf(fact_311_power2__nat__le__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% power2_nat_le_imp_le
thf(fact_312_power2__nat__le__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% power2_nat_le_eq_le
thf(fact_313_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_314_power2__le__imp__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_315_power2__le__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_316_power2__le__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_317_power2__le__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_318_power2__eq__imp__eq,axiom,
! [X: rat,Y: rat] :
( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_319_power2__eq__imp__eq,axiom,
! [X: nat,Y: nat] :
( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_320_power2__eq__imp__eq,axiom,
! [X: int,Y: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_321_power2__eq__imp__eq,axiom,
! [X: real,Y: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_322_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_323_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_324_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_325_sum__power2__le__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_326_sum__power2__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_327_sum__power2__le__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_328_sum__power2__ge__zero,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_329_sum__power2__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_330_sum__power2__ge__zero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_331_one__add__one,axiom,
( ( plus_plus_complex @ one_one_complex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_332_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_333_one__add__one,axiom,
( ( plus_plus_rat @ one_one_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_334_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_335_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_336_invar__vebt_Ointros_I2_J,axiom,
! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M = N2 )
=> ( ( Deg
= ( plus_plus_nat @ N2 @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_337_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N2 ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_338_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_339_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_340_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_341_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_342_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ one_one_complex )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_343_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
= ( numeral_numeral_real @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_344_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
= ( numeral_numeral_rat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_345_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_346_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_347_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_348_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_349_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_350_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_351_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_one_complex
= ( numera6690914467698888265omplex @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_352_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_one_real
= ( numeral_numeral_real @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_353_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_one_rat
= ( numeral_numeral_rat @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_354_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_355_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_one_int
= ( numeral_numeral_int @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_356_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numera6690914467698888265omplex @ N2 )
= one_one_complex )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_357_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numeral_numeral_real @ N2 )
= one_one_real )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_358_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numeral_numeral_rat @ N2 )
= one_one_rat )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_359_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numeral_numeral_nat @ N2 )
= one_one_nat )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_360_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numeral_numeral_int @ N2 )
= one_one_int )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_361_nat__induct2,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N: nat] :
( ( P @ N )
=> ( P @ ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct2
thf(fact_362_exp__add__not__zero__imp__right,axiom,
! [M: nat,N2: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_right
thf(fact_363_exp__add__not__zero__imp__right,axiom,
! [M: nat,N2: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_right
thf(fact_364_exp__add__not__zero__imp__left,axiom,
! [M: nat,N2: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_left
thf(fact_365_exp__add__not__zero__imp__left,axiom,
! [M: nat,N2: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_left
thf(fact_366_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_367_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_368_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numera6690914467698888265omplex @ M )
= ( numera6690914467698888265omplex @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_369_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_370_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numeral_numeral_rat @ M )
= ( numeral_numeral_rat @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_371_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_372_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_373_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_374_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_375_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_376_add__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_377_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_378_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_379_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_380_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_381_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_382_add__numeral__left,axiom,
! [V: num,W: num,Z: complex] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_383_add__numeral__left,axiom,
! [V: num,W: num,Z: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_384_add__numeral__left,axiom,
! [V: num,W: num,Z: rat] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_385_add__numeral__left,axiom,
! [V: num,W: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_386_add__numeral__left,axiom,
! [V: num,W: num,Z: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_387_numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% numeral_le_iff
thf(fact_388_numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% numeral_le_iff
thf(fact_389_numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% numeral_le_iff
thf(fact_390_numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% numeral_le_iff
thf(fact_391_not__exp__less__eq__0__int,axiom,
! [N2: nat] :
~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_int ) ).
% not_exp_less_eq_0_int
thf(fact_392_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_393_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_394_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_395_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_396_add__One__commute,axiom,
! [N2: num] :
( ( plus_plus_num @ one @ N2 )
= ( plus_plus_num @ N2 @ one ) ) ).
% add_One_commute
thf(fact_397_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_398_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_399_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_400_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_401_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
| ( ord_less_eq_nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_402_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_403_size__neq__size__imp__neq,axiom,
! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ X )
!= ( size_s6755466524823107622T_VEBT @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_404_size__neq__size__imp__neq,axiom,
! [X: list_o,Y: list_o] :
( ( ( size_size_list_o @ X )
!= ( size_size_list_o @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_405_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_406_size__neq__size__imp__neq,axiom,
! [X: list_int,Y: list_int] :
( ( ( size_size_list_int @ X )
!= ( size_size_list_int @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_407_size__neq__size__imp__neq,axiom,
! [X: num,Y: num] :
( ( ( size_size_num @ X )
!= ( size_size_num @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_408_le__numeral__extra_I3_J,axiom,
ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% le_numeral_extra(3)
thf(fact_409_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_410_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_411_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_412_zero__neq__numeral,axiom,
! [N2: num] :
( zero_zero_complex
!= ( numera6690914467698888265omplex @ N2 ) ) ).
% zero_neq_numeral
thf(fact_413_zero__neq__numeral,axiom,
! [N2: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N2 ) ) ).
% zero_neq_numeral
thf(fact_414_zero__neq__numeral,axiom,
! [N2: num] :
( zero_zero_rat
!= ( numeral_numeral_rat @ N2 ) ) ).
% zero_neq_numeral
thf(fact_415_zero__neq__numeral,axiom,
! [N2: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N2 ) ) ).
% zero_neq_numeral
thf(fact_416_zero__neq__numeral,axiom,
! [N2: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N2 ) ) ).
% zero_neq_numeral
thf(fact_417_le__numeral__extra_I4_J,axiom,
ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% le_numeral_extra(4)
thf(fact_418_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_419_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_420_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_421_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_422_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_423_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_424_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_425_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_426_add__eq__self__zero,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= M )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_427_add__leE,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_428_le__add1,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).
% le_add1
thf(fact_429_le__add2,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).
% le_add2
thf(fact_430_add__leD1,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% add_leD1
thf(fact_431_add__leD2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_432_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N: nat] :
( L
= ( plus_plus_nat @ K @ N ) ) ) ).
% le_Suc_ex
thf(fact_433_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_434_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_435_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_436_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_437_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_438_zero__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).
% zero_le_numeral
thf(fact_439_zero__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% zero_le_numeral
thf(fact_440_zero__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).
% zero_le_numeral
thf(fact_441_zero__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).
% zero_le_numeral
thf(fact_442_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).
% not_numeral_le_zero
thf(fact_443_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_444_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_445_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_446_one__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) ) ).
% one_le_numeral
thf(fact_447_one__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% one_le_numeral
thf(fact_448_one__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).
% one_le_numeral
thf(fact_449_one__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N2 ) ) ).
% one_le_numeral
thf(fact_450_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_451_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_452_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_453_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_454_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_455_numeral__Bit0,axiom,
! [N2: num] :
( ( numera6690914467698888265omplex @ ( bit0 @ N2 ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_456_numeral__Bit0,axiom,
! [N2: num] :
( ( numeral_numeral_real @ ( bit0 @ N2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_457_numeral__Bit0,axiom,
! [N2: num] :
( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_458_numeral__Bit0,axiom,
! [N2: num] :
( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_459_numeral__Bit0,axiom,
! [N2: num] :
( ( numeral_numeral_int @ ( bit0 @ N2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_460_numeral__One,axiom,
( ( numera6690914467698888265omplex @ one )
= one_one_complex ) ).
% numeral_One
thf(fact_461_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_462_numeral__One,axiom,
( ( numeral_numeral_rat @ one )
= one_one_rat ) ).
% numeral_One
thf(fact_463_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_464_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_465_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_466_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_467_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_468_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_469_semiring__norm_I68_J,axiom,
! [N2: num] : ( ord_less_eq_num @ one @ N2 ) ).
% semiring_norm(68)
thf(fact_470_semiring__norm_I71_J,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% semiring_norm(71)
thf(fact_471_semiring__norm_I6_J,axiom,
! [M: num,N2: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( bit0 @ ( plus_plus_num @ M @ N2 ) ) ) ).
% semiring_norm(6)
thf(fact_472_enat__ord__number_I1_J,axiom,
! [M: num,N2: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% enat_ord_number(1)
thf(fact_473_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_474_semiring__norm_I83_J,axiom,
! [N2: num] :
( one
!= ( bit0 @ N2 ) ) ).
% semiring_norm(83)
thf(fact_475_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_476_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_477_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_478_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_479__092_060open_0620_A_060_Alength_AtreeList_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ treeList ) ).
% \<open>0 < length treeList\<close>
thf(fact_480_invar__vebt_Ointros_I3_J,axiom,
! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M
= ( suc @ N2 ) )
=> ( ( Deg
= ( plus_plus_nat @ N2 @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_481_even__odd__cases,axiom,
! [X: nat] :
( ! [N: nat] :
( X
!= ( plus_plus_nat @ N @ N ) )
=> ~ ! [N: nat] :
( X
!= ( plus_plus_nat @ N @ ( suc @ N ) ) ) ) ).
% even_odd_cases
thf(fact_482_deg__not__0,axiom,
! [T2: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% deg_not_0
thf(fact_483_semiring__norm_I87_J,axiom,
! [M: num,N2: num] :
( ( ( bit0 @ M )
= ( bit0 @ N2 ) )
= ( M = N2 ) ) ).
% semiring_norm(87)
thf(fact_484_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_485_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_486_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_487_numeral__less__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
= ( ord_less_num @ M @ N2 ) ) ).
% numeral_less_iff
thf(fact_488_numeral__less__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
= ( ord_less_num @ M @ N2 ) ) ).
% numeral_less_iff
thf(fact_489_numeral__less__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
= ( ord_less_num @ M @ N2 ) ) ).
% numeral_less_iff
thf(fact_490_numeral__less__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
= ( ord_less_num @ M @ N2 ) ) ).
% numeral_less_iff
thf(fact_491_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_492_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_493_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_494_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_495_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_496_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_497_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_498_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_499_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_500_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_501_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_502_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_503_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_504_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_505_Suc__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_eq
thf(fact_506_Suc__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_507_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_508_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_509_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_510_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_511_Suc__le__mono,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N2 @ M ) ) ).
% Suc_le_mono
thf(fact_512_add__Suc__right,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ M @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% add_Suc_right
thf(fact_513_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_514_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_515_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_516_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_517_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_518_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_519_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_520_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_521_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_522_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_523_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_524_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_525_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_526_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_527_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_528_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_529_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_530_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_531_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_532_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_533_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_534_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_535_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_536_power__inject__exp,axiom,
! [A: real,M: nat,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M )
= ( power_power_real @ A @ N2 ) )
= ( M = N2 ) ) ) ).
% power_inject_exp
thf(fact_537_power__inject__exp,axiom,
! [A: rat,M: nat,N2: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ( power_power_rat @ A @ M )
= ( power_power_rat @ A @ N2 ) )
= ( M = N2 ) ) ) ).
% power_inject_exp
thf(fact_538_power__inject__exp,axiom,
! [A: nat,M: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N2 ) )
= ( M = N2 ) ) ) ).
% power_inject_exp
thf(fact_539_power__inject__exp,axiom,
! [A: int,M: nat,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N2 ) )
= ( M = N2 ) ) ) ).
% power_inject_exp
thf(fact_540_power__0__Suc,axiom,
! [N2: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_541_power__0__Suc,axiom,
! [N2: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N2 ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_542_power__0__Suc,axiom,
! [N2: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N2 ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_543_power__0__Suc,axiom,
! [N2: nat] :
( ( power_power_complex @ zero_zero_complex @ ( suc @ N2 ) )
= zero_zero_complex ) ).
% power_0_Suc
thf(fact_544_power__0__Suc,axiom,
! [N2: nat] :
( ( power_power_rat @ zero_zero_rat @ ( suc @ N2 ) )
= zero_zero_rat ) ).
% power_0_Suc
thf(fact_545_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_546_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_547_power__Suc0__right,axiom,
! [A: real] :
( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_548_power__Suc0__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_549_power__Suc0__right,axiom,
! [A: rat] :
( ( power_power_rat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_550_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_551_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_552_add__gr__0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_553_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_554_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_555_power__Suc__0,axiom,
! [N2: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_556_nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_557_one__less__numeral__iff,axiom,
! [N2: num] :
( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
= ( ord_less_num @ one @ N2 ) ) ).
% one_less_numeral_iff
thf(fact_558_one__less__numeral__iff,axiom,
! [N2: num] :
( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
= ( ord_less_num @ one @ N2 ) ) ).
% one_less_numeral_iff
thf(fact_559_one__less__numeral__iff,axiom,
! [N2: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
= ( ord_less_num @ one @ N2 ) ) ).
% one_less_numeral_iff
thf(fact_560_one__less__numeral__iff,axiom,
! [N2: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
= ( ord_less_num @ one @ N2 ) ) ).
% one_less_numeral_iff
thf(fact_561_power__strict__increasing__iff,axiom,
! [B: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_562_power__strict__increasing__iff,axiom,
! [B: rat,X: nat,Y: nat] :
( ( ord_less_rat @ one_one_rat @ B )
=> ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_563_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_564_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_565_power__eq__0__iff,axiom,
! [A: nat,N2: nat] :
( ( ( power_power_nat @ A @ N2 )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_566_power__eq__0__iff,axiom,
! [A: int,N2: nat] :
( ( ( power_power_int @ A @ N2 )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_567_power__eq__0__iff,axiom,
! [A: real,N2: nat] :
( ( ( power_power_real @ A @ N2 )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_568_power__eq__0__iff,axiom,
! [A: complex,N2: nat] :
( ( ( power_power_complex @ A @ N2 )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_569_power__eq__0__iff,axiom,
! [A: rat,N2: nat] :
( ( ( power_power_rat @ A @ N2 )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_570_Suc__numeral,axiom,
! [N2: num] :
( ( suc @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% Suc_numeral
thf(fact_571_power__strict__decreasing__iff,axiom,
! [B: real,M: nat,N2: 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 @ N2 ) )
= ( ord_less_nat @ N2 @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_572_power__strict__decreasing__iff,axiom,
! [B: rat,M: nat,N2: 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 @ N2 ) )
= ( ord_less_nat @ N2 @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_573_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N2: 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 @ N2 ) )
= ( ord_less_nat @ N2 @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_574_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N2: 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 @ N2 ) )
= ( ord_less_nat @ N2 @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_575_power__increasing__iff,axiom,
! [B: rat,X: nat,Y: nat] :
( ( ord_less_rat @ one_one_rat @ B )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_576_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_577_power__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_578_power__increasing__iff,axiom,
! [B: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_579_power__mono__iff,axiom,
! [A: rat,B: rat,N2: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_580_power__mono__iff,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_581_power__mono__iff,axiom,
! [A: int,B: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_582_power__mono__iff,axiom,
! [A: real,B: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_583_add__2__eq__Suc_H,axiom,
! [N2: nat] :
( ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N2 ) ) ) ).
% add_2_eq_Suc'
thf(fact_584_add__2__eq__Suc,axiom,
! [N2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
= ( suc @ ( suc @ N2 ) ) ) ).
% add_2_eq_Suc
thf(fact_585_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_586_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_587_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_588_power__decreasing__iff,axiom,
! [B: rat,M: nat,N2: 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 @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_589_power__decreasing__iff,axiom,
! [B: nat,M: nat,N2: 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 @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_590_power__decreasing__iff,axiom,
! [B: int,M: nat,N2: 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 @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_591_power__decreasing__iff,axiom,
! [B: real,M: nat,N2: 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 @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_592_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_593_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_594_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_595_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_596_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_597_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_598_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_599_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_600_i0__lb,axiom,
! [N2: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N2 ) ).
% i0_lb
thf(fact_601_ile0__eq,axiom,
! [N2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N2 @ zero_z5237406670263579293d_enat )
= ( N2 = zero_z5237406670263579293d_enat ) ) ).
% ile0_eq
thf(fact_602_not__less__less__Suc__eq,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_603_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_604_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_605_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N: nat] :
( ~ ( P @ N )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_606_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( P @ M3 ) )
=> ( P @ N ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_607_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_608_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_609_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_610_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less_nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_611_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_612_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_613_Suc__less__SucD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_SucD
thf(fact_614_less__antisym,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
=> ( M = N2 ) ) ) ).
% less_antisym
thf(fact_615_Suc__less__eq2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_616_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_617_not__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_618_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less_nat @ M @ N2 )
| ( ord_less_nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_619_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_620_less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ).
% less_Suc_eq
thf(fact_621_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ N2 )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_622_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_623_less__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_624_less__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M @ N2 )
=> ( M = N2 ) ) ) ).
% less_SucE
thf(fact_625_Suc__lessI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ( suc @ M )
!= N2 )
=> ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_626_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_627_Suc__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_lessD
thf(fact_628_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_629_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N2: nat,M: nat] :
( ! [N: nat] : ( ord_less_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M ) )
= ( ord_less_nat @ N2 @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_630_lift__Suc__mono__less__iff,axiom,
! [F: nat > rat,N2: nat,M: nat] :
( ! [N: nat] : ( ord_less_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_rat @ ( F @ N2 ) @ ( F @ M ) )
= ( ord_less_nat @ N2 @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_631_lift__Suc__mono__less__iff,axiom,
! [F: nat > num,N2: nat,M: nat] :
( ! [N: nat] : ( ord_less_num @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_num @ ( F @ N2 ) @ ( F @ M ) )
= ( ord_less_nat @ N2 @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_632_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M: nat] :
( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M ) )
= ( ord_less_nat @ N2 @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_633_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N2: nat,M: nat] :
( ! [N: nat] : ( ord_less_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M ) )
= ( ord_less_nat @ N2 @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_634_lift__Suc__mono__less,axiom,
! [F: nat > real,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_nat @ N2 @ N5 )
=> ( ord_less_real @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_635_lift__Suc__mono__less,axiom,
! [F: nat > rat,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_nat @ N2 @ N5 )
=> ( ord_less_rat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_636_lift__Suc__mono__less,axiom,
! [F: nat > num,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_num @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_nat @ N2 @ N5 )
=> ( ord_less_num @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_637_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_nat @ N2 @ N5 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_638_lift__Suc__mono__less,axiom,
! [F: nat > int,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_nat @ N2 @ N5 )
=> ( ord_less_int @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_639_less__Suc__eq__0__disj,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_640_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_641_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_642_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M2: nat] :
( N2
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_643_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_644_le__imp__less__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_645_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_646_less__Suc__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_647_le__less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% le_less_Suc_eq
thf(fact_648_Suc__le__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_lessD
thf(fact_649_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( ord_less_nat @ N @ J )
=> ( ( P @ ( suc @ N ) )
=> ( P @ N ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_650_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( ord_less_nat @ N @ J )
=> ( ( P @ N )
=> ( P @ ( suc @ N ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_651_Suc__le__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_eq
thf(fact_652_Suc__leI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_leI
thf(fact_653_less__imp__Suc__add,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_654_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
? [K2: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_655_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_656_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_657_less__natE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ! [Q2: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_658_power__gt1,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N2 ) ) ) ) ).
% power_gt1
thf(fact_659_power__gt1,axiom,
! [A: rat,N2: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) ) ) ).
% power_gt1
thf(fact_660_power__gt1,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) ) ) ).
% power_gt1
thf(fact_661_power__gt1,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N2 ) ) ) ) ).
% power_gt1
thf(fact_662_power__less__imp__less__exp,axiom,
! [A: real,M: nat,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_663_power__less__imp__less__exp,axiom,
! [A: rat,M: nat,N2: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_664_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_665_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_666_power__strict__increasing,axiom,
! [N2: nat,N3: nat,A: real] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_667_power__strict__increasing,axiom,
! [N2: nat,N3: nat,A: rat] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_668_power__strict__increasing,axiom,
! [N2: nat,N3: nat,A: nat] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_669_power__strict__increasing,axiom,
! [N2: nat,N3: nat,A: int] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_670_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N2 )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_671_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ N )
=> ( P @ ( suc @ N ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_672_power__gt__expt,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).
% power_gt_expt
thf(fact_673_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_674_gr__implies__not__zero,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_675_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_676_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_677_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_678_less__numeral__extra_I3_J,axiom,
~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% less_numeral_extra(3)
thf(fact_679_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_680_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_681_power__divide,axiom,
! [A: complex,B: complex,N2: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N2 )
= ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N2 ) @ ( power_power_complex @ B @ N2 ) ) ) ).
% power_divide
thf(fact_682_power__divide,axiom,
! [A: real,B: real,N2: nat] :
( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N2 )
= ( divide_divide_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).
% power_divide
thf(fact_683_power__divide,axiom,
! [A: rat,B: rat,N2: nat] :
( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N2 )
= ( divide_divide_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ).
% power_divide
thf(fact_684_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_685_less__numeral__extra_I4_J,axiom,
~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% less_numeral_extra(4)
thf(fact_686_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_687_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_688_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_689_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_690_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_691_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_692_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_693_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_694_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_695_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_696_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_697_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_698_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_699_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_700_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_701_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_702_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_703_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_704_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_705_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_706_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_707_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_708_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_709_add__mono__thms__linordered__field_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( K = L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_710_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_711_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_712_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_713_add__mono__thms__linordered__field_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( ord_less_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_714_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_715_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_716_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_717_add__mono__thms__linordered__field_I5_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( ord_less_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_718_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_719_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_720_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_721_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_722_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_723_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_724_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N: nat] :
( ( P @ ( suc @ N ) )
=> ( P @ N ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_725_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N2: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] :
( ( P @ X2 @ Y2 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y2 ) ) )
=> ( P @ M @ N2 ) ) ) ) ).
% diff_induct
thf(fact_726_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N: nat] :
( ( P @ N )
=> ( P @ ( suc @ N ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_727_vebt__buildup_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ( ( X
!= ( suc @ zero_zero_nat ) )
=> ~ ! [Va: nat] :
( X
!= ( suc @ ( suc @ Va ) ) ) ) ) ).
% vebt_buildup.cases
thf(fact_728_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_729_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_730_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_731_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_732_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_733_transitive__stepwise__le,axiom,
! [M: nat,N2: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y2: nat,Z2: nat] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z2 )
=> ( R @ X2 @ Z2 ) ) )
=> ( ! [N: nat] : ( R @ N @ ( suc @ N ) )
=> ( R @ M @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_734_nat__induct__at__least,axiom,
! [M: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ M )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ N )
=> ( P @ ( suc @ N ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_735_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
=> ( P @ M3 ) )
=> ( P @ N ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_736_not__less__eq__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).
% not_less_eq_eq
thf(fact_737_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_738_le__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M @ N2 )
| ( M
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_739_Suc__le__D,axiom,
! [N2: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
=> ? [M5: nat] :
( M6
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_740_le__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_741_le__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M @ N2 )
=> ( M
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_742_Suc__leD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% Suc_leD
thf(fact_743_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ~ ( P @ N )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_744_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_745_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_746_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_747_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_748_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_749_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_750_add__Suc__shift,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N2 )
= ( plus_plus_nat @ M @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_751_add__Suc,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N2 )
= ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% add_Suc
thf(fact_752_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_753_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_754_le__neq__implies__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( M != N2 )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_755_less__or__eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_756_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_nat @ M2 @ N4 )
| ( M2 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_757_less__imp__le__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_imp_le_nat
thf(fact_758_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
& ( M2 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_759_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_760_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_761_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_762_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_763_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_764_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_765_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_766_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_767_power__Suc__less__one,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ one_one_real ) ) ) ).
% power_Suc_less_one
thf(fact_768_power__Suc__less__one,axiom,
! [A: rat,N2: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ one_one_rat ) ) ) ).
% power_Suc_less_one
thf(fact_769_power__Suc__less__one,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_770_power__Suc__less__one,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_771_power__strict__decreasing,axiom,
! [N2: nat,N3: nat,A: real] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_772_power__strict__decreasing,axiom,
! [N2: nat,N3: nat,A: rat] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_773_power__strict__decreasing,axiom,
! [N2: nat,N3: nat,A: nat] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_774_power__strict__decreasing,axiom,
! [N2: nat,N3: nat,A: int] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_775_one__less__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ) ).
% one_less_power
thf(fact_776_one__less__power,axiom,
! [A: rat,N2: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% one_less_power
thf(fact_777_one__less__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% one_less_power
thf(fact_778_one__less__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ) ).
% one_less_power
thf(fact_779_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_780_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_781_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_782_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_783_power__strict__mono,axiom,
! [A: rat,B: rat,N2: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ) ).
% power_strict_mono
thf(fact_784_power__strict__mono,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ) ).
% power_strict_mono
thf(fact_785_power__strict__mono,axiom,
! [A: int,B: int,N2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ) ).
% power_strict_mono
thf(fact_786_power__strict__mono,axiom,
! [A: real,B: real,N2: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ) ).
% power_strict_mono
thf(fact_787_less__2__cases__iff,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N2 = zero_zero_nat )
| ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_788_less__2__cases,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N2 = zero_zero_nat )
| ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_789_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_790_divide__numeral__1,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_791_divide__numeral__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_792_divide__numeral__1,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_793_power__one__over,axiom,
! [A: complex,N2: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N2 )
= ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N2 ) ) ) ).
% power_one_over
thf(fact_794_power__one__over,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N2 )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% power_one_over
thf(fact_795_power__one__over,axiom,
! [A: rat,N2: nat] :
( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N2 )
= ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% power_one_over
thf(fact_796_not__numeral__less__zero,axiom,
! [N2: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).
% not_numeral_less_zero
thf(fact_797_not__numeral__less__zero,axiom,
! [N2: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).
% not_numeral_less_zero
thf(fact_798_not__numeral__less__zero,axiom,
! [N2: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_799_not__numeral__less__zero,axiom,
! [N2: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_800_zero__less__numeral,axiom,
! [N2: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).
% zero_less_numeral
thf(fact_801_zero__less__numeral,axiom,
! [N2: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).
% zero_less_numeral
thf(fact_802_zero__less__numeral,axiom,
! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% zero_less_numeral
thf(fact_803_zero__less__numeral,axiom,
! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).
% zero_less_numeral
thf(fact_804_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_805_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_806_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_807_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_808_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_809_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_810_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_811_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_812_add__mono__thms__linordered__field_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( ord_less_eq_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_813_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_814_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_815_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_816_add__mono__thms__linordered__field_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( ord_less_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_817_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_818_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_819_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_820_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_821_less__numeral__extra_I1_J,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% less_numeral_extra(1)
thf(fact_822_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_823_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_824_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_825_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_826_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_827_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_828_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_829_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_830_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_831_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_832_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_833_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_834_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_835_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_836_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_837_not__numeral__less__one,axiom,
! [N2: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ one_one_real ) ).
% not_numeral_less_one
thf(fact_838_not__numeral__less__one,axiom,
! [N2: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat ) ).
% not_numeral_less_one
thf(fact_839_not__numeral__less__one,axiom,
! [N2: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_840_not__numeral__less__one,axiom,
! [N2: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_841_zero__less__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% zero_less_power
thf(fact_842_zero__less__power,axiom,
! [A: rat,N2: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% zero_less_power
thf(fact_843_zero__less__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% zero_less_power
thf(fact_844_zero__less__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% zero_less_power
thf(fact_845_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_set_nat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_846_lift__Suc__antimono__le,axiom,
! [F: nat > rat,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_rat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_847_lift__Suc__antimono__le,axiom,
! [F: nat > num,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_848_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_849_lift__Suc__antimono__le,axiom,
! [F: nat > int,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_850_lift__Suc__antimono__le,axiom,
! [F: nat > real,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_real @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_851_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_852_lift__Suc__mono__le,axiom,
! [F: nat > rat,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_eq_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_853_lift__Suc__mono__le,axiom,
! [F: nat > num,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_eq_num @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_854_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_855_lift__Suc__mono__le,axiom,
! [F: nat > int,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_eq_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_856_lift__Suc__mono__le,axiom,
! [F: nat > real,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_857_one__is__add,axiom,
! [M: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N2 ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_858_add__is__1,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_859_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_860_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_861_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_862_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_863_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_864_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_865_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M5: nat,N: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_866_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_867_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_868_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_869_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_870_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_871_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_872_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_873_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_874_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_875_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_876_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_877_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_878_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_879_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_880_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_881_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_882_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_883_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_884_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_885_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_886_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_887_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_888_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_889_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_890_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_891_power__le__imp__le__base,axiom,
! [A: rat,N2: nat,B: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ ( power_power_rat @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_892_power__le__imp__le__base,axiom,
! [A: nat,N2: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ ( power_power_nat @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_893_power__le__imp__le__base,axiom,
! [A: int,N2: nat,B: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ ( power_power_int @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_894_power__le__imp__le__base,axiom,
! [A: real,N2: nat,B: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ ( power_power_real @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_895_power__inject__base,axiom,
! [A: rat,N2: nat,B: rat] :
( ( ( power_power_rat @ A @ ( suc @ N2 ) )
= ( power_power_rat @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_896_power__inject__base,axiom,
! [A: nat,N2: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N2 ) )
= ( power_power_nat @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_897_power__inject__base,axiom,
! [A: int,N2: nat,B: int] :
( ( ( power_power_int @ A @ ( suc @ N2 ) )
= ( power_power_int @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_898_power__inject__base,axiom,
! [A: real,N2: nat,B: real] :
( ( ( power_power_real @ A @ ( suc @ N2 ) )
= ( power_power_real @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_899_power__less__imp__less__base,axiom,
! [A: rat,N2: nat,B: rat] :
( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_900_power__less__imp__less__base,axiom,
! [A: nat,N2: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_901_power__less__imp__less__base,axiom,
! [A: int,N2: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_902_power__less__imp__less__base,axiom,
! [A: real,N2: nat,B: real] :
( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_903_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_904_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_905_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= zero_zero_int ) ) ).
% zero_power
thf(fact_906_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_real @ zero_zero_real @ N2 )
= zero_zero_real ) ) ).
% zero_power
thf(fact_907_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_complex @ zero_zero_complex @ N2 )
= zero_zero_complex ) ) ).
% zero_power
thf(fact_908_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_rat @ zero_zero_rat @ N2 )
= zero_zero_rat ) ) ).
% zero_power
thf(fact_909_nat__one__le__power,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N2 ) ) ) ).
% nat_one_le_power
thf(fact_910_power__Suc__le__self,axiom,
! [A: rat,N2: 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 @ N2 ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_911_power__Suc__le__self,axiom,
! [A: nat,N2: 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 @ N2 ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_912_power__Suc__le__self,axiom,
! [A: int,N2: 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 @ N2 ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_913_power__Suc__le__self,axiom,
! [A: real,N2: 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 @ N2 ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_914_power__le__imp__le__exp,axiom,
! [A: rat,M: nat,N2: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_915_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_916_power__le__imp__le__exp,axiom,
! [A: int,M: nat,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_917_power__le__imp__le__exp,axiom,
! [A: real,M: nat,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_918_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_919_power__eq__imp__eq__base,axiom,
! [A: rat,N2: nat,B: rat] :
( ( ( power_power_rat @ A @ N2 )
= ( power_power_rat @ B @ N2 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_920_power__eq__imp__eq__base,axiom,
! [A: nat,N2: nat,B: nat] :
( ( ( power_power_nat @ A @ N2 )
= ( power_power_nat @ B @ N2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_921_power__eq__imp__eq__base,axiom,
! [A: int,N2: nat,B: int] :
( ( ( power_power_int @ A @ N2 )
= ( power_power_int @ B @ N2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_922_power__eq__imp__eq__base,axiom,
! [A: real,N2: nat,B: real] :
( ( ( power_power_real @ A @ N2 )
= ( power_power_real @ B @ N2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_923_power__eq__iff__eq__base,axiom,
! [N2: nat,A: rat,B: rat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ( power_power_rat @ A @ N2 )
= ( power_power_rat @ B @ N2 ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_924_power__eq__iff__eq__base,axiom,
! [N2: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N2 )
= ( power_power_nat @ B @ N2 ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_925_power__eq__iff__eq__base,axiom,
! [N2: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N2 )
= ( power_power_int @ B @ N2 ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_926_power__eq__iff__eq__base,axiom,
! [N2: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( power_power_real @ A @ N2 )
= ( power_power_real @ B @ N2 ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_927_self__le__power,axiom,
! [A: rat,N2: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% self_le_power
thf(fact_928_self__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% self_le_power
thf(fact_929_self__le__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% self_le_power
thf(fact_930_self__le__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% self_le_power
thf(fact_931_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_932_less__exp,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% less_exp
thf(fact_933_Suc__nat__number__of__add,axiom,
! [V: num,N2: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N2 ) ) ).
% Suc_nat_number_of_add
thf(fact_934_div__exp__eq,axiom,
! [A: nat,M: nat,N2: 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 ) ) @ N2 ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% div_exp_eq
thf(fact_935_div__exp__eq,axiom,
! [A: int,M: nat,N2: 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 ) ) @ N2 ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% div_exp_eq
thf(fact_936_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_937_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_938_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_939_power2__less__imp__less,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_rat @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_940_power2__less__imp__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_nat @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_941_power2__less__imp__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_int @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_942_power2__less__imp__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_real @ X @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_943_sum__power2__gt__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_944_sum__power2__gt__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_rat )
| ( Y != zero_zero_rat ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_945_sum__power2__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_946_not__sum__power2__lt__zero,axiom,
! [X: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% not_sum_power2_lt_zero
thf(fact_947_not__sum__power2__lt__zero,axiom,
! [X: rat,Y: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% not_sum_power2_lt_zero
thf(fact_948_not__sum__power2__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% not_sum_power2_lt_zero
thf(fact_949_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 )
=> ? [N: 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 ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_950_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 )
=> ? [N: 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 ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_951_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_952_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_953_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_954_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_955_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_956_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_957_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_958_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_959_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_960_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_961_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_962_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_963_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_964_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_965_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_966_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_967_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_968_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_969_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_970_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_971_i0__less,axiom,
! [N2: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 )
= ( N2 != zero_z5237406670263579293d_enat ) ) ).
% i0_less
thf(fact_972_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_973_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_974_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_975_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_976_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_977_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_978_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_979_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_980_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_981_division__ring__divide__zero,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% division_ring_divide_zero
thf(fact_982_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_983_division__ring__divide__zero,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% division_ring_divide_zero
thf(fact_984_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_985_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_986_semiring__norm_I78_J,axiom,
! [M: num,N2: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less_num @ M @ N2 ) ) ).
% semiring_norm(78)
thf(fact_987_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_988_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_989_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_990_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_991_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_992_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_993_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_994_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_995_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_996_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_997_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_998_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_999_divide__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% divide_self
thf(fact_1000_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_1001_divide__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% divide_self
thf(fact_1002_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_1003_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_1004_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_1005_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_1006_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_1007_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_1008_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_1009_div__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( divide_divide_nat @ M @ N2 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1010_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1011_semiring__norm_I76_J,axiom,
! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).
% semiring_norm(76)
thf(fact_1012_enat__ord__number_I2_J,axiom,
! [M: num,N2: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% enat_ord_number(2)
thf(fact_1013_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_1014_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_1015_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_1016_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_1017_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_1018_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_1019_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_1020_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_1021_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_1022_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_1023_zero__one__enat__neq_I1_J,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_one_enat_neq(1)
thf(fact_1024_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1025_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_1026_iadd__is__0,axiom,
! [M: extended_enat,N2: extended_enat] :
( ( ( plus_p3455044024723400733d_enat @ M @ N2 )
= zero_z5237406670263579293d_enat )
= ( ( M = zero_z5237406670263579293d_enat )
& ( N2 = zero_z5237406670263579293d_enat ) ) ) ).
% iadd_is_0
thf(fact_1027_not__iless0,axiom,
! [N2: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N2 @ zero_z5237406670263579293d_enat ) ).
% not_iless0
thf(fact_1028_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1029_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1030_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1031_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_1032_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1033_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1034_linordered__field__no__ub,axiom,
! [X4: real] :
? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_1035_linordered__field__no__ub,axiom,
! [X4: rat] :
? [X_1: rat] : ( ord_less_rat @ X4 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_1036_linordered__field__no__lb,axiom,
! [X4: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X4 ) ).
% linordered_field_no_lb
thf(fact_1037_linordered__field__no__lb,axiom,
! [X4: rat] :
? [Y2: rat] : ( ord_less_rat @ Y2 @ X4 ) ).
% linordered_field_no_lb
thf(fact_1038_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_1039_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_1040_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_1041_div__le__mono,axiom,
! [M: nat,N2: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).
% div_le_mono
thf(fact_1042_div__le__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).
% div_le_dividend
thf(fact_1043_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_1044_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_1045_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_1046_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_1047_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_1048_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_1049_divide__nonneg__nonneg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1050_divide__nonneg__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1051_divide__nonneg__nonpos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1052_divide__nonneg__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1053_divide__nonpos__nonneg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1054_divide__nonpos__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1055_divide__nonpos__nonpos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1056_divide__nonpos__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1057_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_1058_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_1059_divide__neg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_1060_divide__neg__neg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_1061_divide__neg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_1062_divide__neg__pos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_neg_pos
thf(fact_1063_divide__pos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_1064_divide__pos__neg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_pos_neg
thf(fact_1065_divide__pos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_1066_divide__pos__pos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_1067_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_1068_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_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_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_1079_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_1080_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N2: nat] :
( ( ( divide_divide_nat @ M @ N2 )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N2 )
| ( N2 = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1081_Suc__div__le__mono,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ ( divide_divide_nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_div_le_mono
thf(fact_1082_field__le__epsilon,axiom,
! [X: rat,Y: rat] :
( ! [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
=> ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E ) ) )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% field_le_epsilon
thf(fact_1083_field__le__epsilon,axiom,
! [X: real,Y: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_epsilon
thf(fact_1084_frac__le,axiom,
! [Y: rat,X: rat,W: rat,Z: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W )
=> ( ( ord_less_eq_rat @ W @ Z )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% frac_le
thf(fact_1085_frac__le,axiom,
! [Y: real,X: real,W: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% frac_le
thf(fact_1086_frac__less,axiom,
! [X: rat,Y: rat,W: rat,Z: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W )
=> ( ( ord_less_eq_rat @ W @ Z )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% frac_less
thf(fact_1087_frac__less,axiom,
! [X: real,Y: real,W: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% frac_less
thf(fact_1088_frac__less2,axiom,
! [X: rat,Y: rat,W: rat,Z: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W )
=> ( ( ord_less_rat @ W @ Z )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% frac_less2
thf(fact_1089_frac__less2,axiom,
! [X: real,Y: real,W: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_real @ W @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% frac_less2
thf(fact_1090_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_1091_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_1092_divide__nonneg__neg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonneg_neg
thf(fact_1093_divide__nonneg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_1094_divide__nonneg__pos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1095_divide__nonneg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1096_divide__nonpos__neg,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1097_divide__nonpos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1098_divide__nonpos__pos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonpos_pos
thf(fact_1099_divide__nonpos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_1100_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_1101_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_1102_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_1103_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_1104_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_1105_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_1106_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_1107_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_1108_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_1109_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_1110_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_1111_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_1112_numeral__Bit0__div__2,axiom,
! [N2: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N2 ) ) ).
% numeral_Bit0_div_2
thf(fact_1113_numeral__Bit0__div__2,axiom,
! [N2: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N2 ) ) ).
% numeral_Bit0_div_2
thf(fact_1114_div__le__mono2,axiom,
! [M: nat,N2: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1115_div__greater__zero__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N2 ) )
= ( ( ord_less_eq_nat @ N2 @ M )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% div_greater_zero_iff
thf(fact_1116_div__eq__dividend__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N2 )
= M )
= ( N2 = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1117_div__less__dividend,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1118_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_1119_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_1120_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_1121_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_1122_div__2__gt__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_1123_Suc__n__div__2__gt__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_1124_buildup__gives__valid,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N2 ) @ N2 ) ) ).
% buildup_gives_valid
thf(fact_1125_div__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% div_self
thf(fact_1126_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_1127_div__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% div_self
thf(fact_1128_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_1129_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_1130_field__less__half__sum,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% field_less_half_sum
thf(fact_1131_field__less__half__sum,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% field_less_half_sum
thf(fact_1132_zdiv__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_1133_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_1134_div__by__1,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ one_one_complex )
= A ) ).
% div_by_1
thf(fact_1135_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_1136_div__by__1,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ one_one_rat )
= A ) ).
% div_by_1
thf(fact_1137_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_1138_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_1139_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_1140_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_1141_div__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% div_0
thf(fact_1142_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_1143_div__0,axiom,
! [A: rat] :
( ( divide_divide_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% div_0
thf(fact_1144_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_1145_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_1146_div__by__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% div_by_0
thf(fact_1147_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_1148_div__by__0,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% div_by_0
thf(fact_1149_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_1150_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_1151_high__bound__aux,axiom,
! [Ma: nat,N2: nat,M: nat] :
( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
=> ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% high_bound_aux
thf(fact_1152_length__pos__if__in__set,axiom,
! [X: complex,Xs: list_complex] :
( ( member_complex @ X @ ( set_complex2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1153_length__pos__if__in__set,axiom,
! [X: real,Xs: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1154_length__pos__if__in__set,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1155_length__pos__if__in__set,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1156_length__pos__if__in__set,axiom,
! [X: $o,Xs: list_o] :
( ( member_o @ X @ ( set_o2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1157_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1158_length__pos__if__in__set,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1159_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X3: nat,N4: nat] : ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% high_def
thf(fact_1160_enat__less__induct,axiom,
! [P: extended_enat > $o,N2: extended_enat] :
( ! [N: extended_enat] :
( ! [M3: extended_enat] :
( ( ord_le72135733267957522d_enat @ M3 @ N )
=> ( P @ M3 ) )
=> ( P @ N ) )
=> ( P @ N2 ) ) ).
% enat_less_induct
thf(fact_1161_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1162_linorder__neqE__linordered__idom,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
=> ( ~ ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1163_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1164_realpow__pos__nth2,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ ( suc @ N2 ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_1165_subset__code_I1_J,axiom,
! [Xs: list_complex,B3: set_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ B3 )
= ( ! [X3: complex] :
( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ( member_complex @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_1166_subset__code_I1_J,axiom,
! [Xs: list_real,B3: set_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B3 )
= ( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( member_real @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_1167_subset__code_I1_J,axiom,
! [Xs: list_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B3 )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( member_set_nat @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_1168_subset__code_I1_J,axiom,
! [Xs: list_VEBT_VEBT,B3: set_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B3 )
= ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( member_VEBT_VEBT @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_1169_subset__code_I1_J,axiom,
! [Xs: list_int,B3: set_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B3 )
= ( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( member_int @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_1170_subset__code_I1_J,axiom,
! [Xs: list_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B3 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_1171_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1172_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1173_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1174_neq__if__length__neq,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
!= ( size_s6755466524823107622T_VEBT @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_1175_neq__if__length__neq,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( size_size_list_o @ Xs )
!= ( size_size_list_o @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_1176_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_1177_neq__if__length__neq,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
!= ( size_size_list_int @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_1178_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs2: list_VEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ Xs2 )
= N2 ) ).
% Ex_list_of_length
thf(fact_1179_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs2: list_o] :
( ( size_size_list_o @ Xs2 )
= N2 ) ).
% Ex_list_of_length
thf(fact_1180_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N2 ) ).
% Ex_list_of_length
thf(fact_1181_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs2: list_int] :
( ( size_size_list_int @ Xs2 )
= N2 ) ).
% Ex_list_of_length
thf(fact_1182_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1183_field__lbound__gt__zero,axiom,
! [D1: rat,D2: rat] :
( ( ord_less_rat @ zero_zero_rat @ D1 )
=> ( ( ord_less_rat @ zero_zero_rat @ D2 )
=> ? [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
& ( ord_less_rat @ E @ D1 )
& ( ord_less_rat @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1184_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_1185_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_1186_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
% zero_neq_one
thf(fact_1187_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1188_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_1189_two__realpow__ge__one,axiom,
! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).
% two_realpow_ge_one
thf(fact_1190_realpow__pos__nth__unique,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
& ( ( power_power_real @ X2 @ N2 )
= A )
& ! [Y3: real] :
( ( ( ord_less_real @ zero_zero_real @ Y3 )
& ( ( power_power_real @ Y3 @ N2 )
= A ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1191_realpow__pos__nth,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ N2 )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1192_length__induct,axiom,
! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
( ! [Xs2: list_VEBT_VEBT] :
( ! [Ys2: list_VEBT_VEBT] :
( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_1193_length__induct,axiom,
! [P: list_o > $o,Xs: list_o] :
( ! [Xs2: list_o] :
( ! [Ys2: list_o] :
( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_1194_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_1195_length__induct,axiom,
! [P: list_int > $o,Xs: list_int] :
( ! [Xs2: list_int] :
( ! [Ys2: list_int] :
( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_1196_zdiv__mono1,axiom,
! [A: int,A4: int,B: int] :
( ( ord_less_eq_int @ A @ A4 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A4 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1197_zdiv__mono2,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1198_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1199_zdiv__mono1__neg,axiom,
! [A: int,A4: int,B: int] :
( ( ord_less_eq_int @ A @ A4 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1200_zdiv__mono2__neg,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1201_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1202_div__positive__int,axiom,
! [L: int,K: int] :
( ( ord_less_eq_int @ L @ K )
=> ( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% div_positive_int
thf(fact_1203_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1204_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1205_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1206_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1207_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1208_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1209_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_1210_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X: nat,N2: nat,M: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_1211_not__one__le__zero,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_le_zero
thf(fact_1212_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1213_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1214_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_1215_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_1216_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_1217_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_1218_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_1219_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_1220_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_1221_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_1222_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_1223_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_1224_zero__less__one,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one
thf(fact_1225_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1226_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_1227_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_1228_not__one__less__zero,axiom,
~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_less_zero
thf(fact_1229_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1230_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_1231_add__less__zeroD,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_1232_add__less__zeroD,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
=> ( ( ord_less_rat @ X @ zero_zero_rat )
| ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).
% add_less_zeroD
thf(fact_1233_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_1234_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_1235_less__add__one,axiom,
! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% less_add_one
thf(fact_1236_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1237_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_1238_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_1239_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_1240_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_1241_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_1242_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_1243_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_1244_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_1245_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_1246_discrete,axiom,
( ord_less_nat
= ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).
% discrete
thf(fact_1247_discrete,axiom,
( ord_less_int
= ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).
% discrete
thf(fact_1248_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_1249_zero__less__two,axiom,
ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% zero_less_two
thf(fact_1250_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1251_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1252_high__inv,axiom,
! [X: nat,N2: nat,Y: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X ) @ N2 )
= Y ) ) ).
% high_inv
thf(fact_1253_buildup__nothing__in__leaf,axiom,
! [N2: nat,X: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X ) ).
% buildup_nothing_in_leaf
thf(fact_1254_dbl__simps_I3_J,axiom,
( ( neg_nu7009210354673126013omplex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_1255_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_1256_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_1257_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_1258_option_Osize_I3_J,axiom,
( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_1259_option_Osize_I3_J,axiom,
( ( size_size_option_num @ none_num )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_1260_even__succ__div__exp,axiom,
! [A: code_integer,N2: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
= ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_1261_even__succ__div__exp,axiom,
! [A: nat,N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_1262_even__succ__div__exp,axiom,
! [A: int,N2: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_1263_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_1264_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_1265_mult__zero__left,axiom,
! [A: complex] :
( ( times_times_complex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_1266_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_1267_mult__zero__left,axiom,
! [A: rat] :
( ( times_times_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% mult_zero_left
thf(fact_1268_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1269_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1270_mult__zero__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_1271_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_1272_mult__zero__right,axiom,
! [A: rat] :
( ( times_times_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% mult_zero_right
thf(fact_1273_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1274_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1275_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_1276_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_1277_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_1278_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_1279_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_1280_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_1281_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_1282_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_1283_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_1284_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_1285_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_1286_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_1287_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_1288_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_1289_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_1290_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
= ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1291_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1292_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
= ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1293_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1294_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1295_numeral__times__numeral,axiom,
! [M: num,N2: num] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) )
= ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ).
% numeral_times_numeral
thf(fact_1296_numeral__times__numeral,axiom,
! [M: num,N2: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ).
% numeral_times_numeral
thf(fact_1297_numeral__times__numeral,axiom,
! [M: num,N2: num] :
( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
= ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ).
% numeral_times_numeral
thf(fact_1298_numeral__times__numeral,axiom,
! [M: num,N2: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ).
% numeral_times_numeral
thf(fact_1299_numeral__times__numeral,axiom,
! [M: num,N2: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ).
% numeral_times_numeral
thf(fact_1300_mult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% mult_1
thf(fact_1301_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_1302_mult__1,axiom,
! [A: rat] :
( ( times_times_rat @ one_one_rat @ A )
= A ) ).
% mult_1
thf(fact_1303_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1304_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_1305_mult_Oright__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.right_neutral
thf(fact_1306_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_1307_mult_Oright__neutral,axiom,
! [A: rat] :
( ( times_times_rat @ A @ one_one_rat )
= A ) ).
% mult.right_neutral
thf(fact_1308_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1309_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_1310_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_1311_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_1312_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_1313_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_1314_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_1315_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_1316_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_1317_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_1318_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_1319_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_1320_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_1321_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_1322_dvd__0__right,axiom,
! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% dvd_0_right
thf(fact_1323_dvd__0__right,axiom,
! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% dvd_0_right
thf(fact_1324_dvd__0__right,axiom,
! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% dvd_0_right
thf(fact_1325_dvd__0__right,axiom,
! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% dvd_0_right
thf(fact_1326_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_1327_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_1328_dvd__0__left__iff,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
= ( A = zero_z3403309356797280102nteger ) ) ).
% dvd_0_left_iff
thf(fact_1329_dvd__0__left__iff,axiom,
! [A: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A )
= ( A = zero_zero_complex ) ) ).
% dvd_0_left_iff
thf(fact_1330_dvd__0__left__iff,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
= ( A = zero_zero_real ) ) ).
% dvd_0_left_iff
thf(fact_1331_dvd__0__left__iff,axiom,
! [A: rat] :
( ( dvd_dvd_rat @ zero_zero_rat @ A )
= ( A = zero_zero_rat ) ) ).
% dvd_0_left_iff
thf(fact_1332_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_1333_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_1334_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_1335_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_1336_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_1337_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_1338_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_1339_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_1340_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_1341_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_1342_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_1343_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_1344_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_1345_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_1346_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_1347_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_1348_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_1349_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1350_mult__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1351_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1352_mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1353_mult__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1354_nat__mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1355_nat__1__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N2 ) )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1356_dbl__simps_I2_J,axiom,
( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% dbl_simps(2)
thf(fact_1357_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_real @ zero_zero_real )
= zero_zero_real ) ).
% dbl_simps(2)
thf(fact_1358_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% dbl_simps(2)
thf(fact_1359_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_1360_bit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H: nat,L2: nat,D3: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D3 ) ) @ L2 ) ) ) ).
% bit_concat_def
thf(fact_1361_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_1362_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_1363_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_1364_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_1365_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_1366_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_1367_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_1368_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_1369_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_1370_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_1371_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_1372_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_1373_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_1374_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_1375_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_1376_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_1377_sum__squares__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1378_sum__squares__eq__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1379_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1380_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_1381_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_1382_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_1383_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_1384_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_1385_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_1386_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_1387_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_1388_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_1389_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_1390_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_1391_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_1392_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_1393_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_1394_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_1395_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_1396_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_1397_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_1398_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_1399_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_1400_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_1401_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_1402_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_1403_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_1404_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_1405_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_1406_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_1407_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_1408_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_1409_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_1410_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_1411_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_1412_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_1413_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_1414_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_1415_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_1416_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_1417_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_1418_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_1419_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_1420_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_1421_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_1422_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_1423_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_1424_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_1425_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_1426_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_1427_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_1428_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_1429_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_1430_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_1431_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_1432_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_1433_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_1434_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_1435_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_1436_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_1437_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_1438_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_1439_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_1440_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_1441_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_1442_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_1443_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_1444_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_1445_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_1446_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_1447_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_1448_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_1449_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_1450_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_1451_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_1452_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_1453_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_1454_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_1455_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_1456_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_1457_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_1458_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_1459_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_1460_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_1461_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_1462_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_1463_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_1464_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_1465_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_1466_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_1467_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_1468_mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1469_one__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N2 ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1470_mult__less__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1471_nat__0__less__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1472_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1473_mult__Suc__right,axiom,
! [M: nat,N2: nat] :
( ( times_times_nat @ M @ ( suc @ N2 ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N2 ) ) ) ).
% mult_Suc_right
thf(fact_1474_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M @ N2 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1475_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
= ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_1476_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_1477_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_1478_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_1479_divide__le__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_1480_divide__le__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_1481_le__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_1482_le__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_1483_divide__eq__eq__numeral1_I1_J,axiom,
! [B: complex,W: num,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
= A )
= ( ( ( ( numera6690914467698888265omplex @ W )
!= zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
& ( ( ( numera6690914467698888265omplex @ W )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1484_divide__eq__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
= A )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1485_divide__eq__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
= A )
= ( ( ( ( numeral_numeral_rat @ W )
!= zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
& ( ( ( numeral_numeral_rat @ W )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1486_eq__divide__eq__numeral1_I1_J,axiom,
! [A: complex,B: complex,W: num] :
( ( A
= ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
= ( ( ( ( numera6690914467698888265omplex @ W )
!= zero_zero_complex )
=> ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
= B ) )
& ( ( ( numera6690914467698888265omplex @ W )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1487_eq__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( A
= ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
= B ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1488_eq__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( ( A
= ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
= ( ( ( ( numeral_numeral_rat @ W )
!= zero_zero_rat )
=> ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
= B ) )
& ( ( ( numeral_numeral_rat @ W )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1489_divide__less__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_1490_divide__less__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_1491_less__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_1492_less__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
= ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_1493_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_1494_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_1495_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_1496_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_1497_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_1498_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_1499_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_1500_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_1501_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_1502_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_1503_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_1504_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_1505_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_1506_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_1507_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_1508_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_1509_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_1510_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_1511_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_1512_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_1513_even__Suc,axiom,
! [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% even_Suc
thf(fact_1514_even__Suc__Suc__iff,axiom,
! [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N2 ) ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% even_Suc_Suc_iff
thf(fact_1515_one__le__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_1516_mult__le__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1517_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1518_div__mult__self__is__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N2 ) @ N2 )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1519_div__mult__self1__is__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M ) @ N2 )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1520_power__add__numeral,axiom,
! [A: complex,M: num,N2: num] :
( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N2 ) ) )
= ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% power_add_numeral
thf(fact_1521_power__add__numeral,axiom,
! [A: real,M: num,N2: num] :
( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N2 ) ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% power_add_numeral
thf(fact_1522_power__add__numeral,axiom,
! [A: rat,M: num,N2: num] :
( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N2 ) ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% power_add_numeral
thf(fact_1523_power__add__numeral,axiom,
! [A: nat,M: num,N2: num] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% power_add_numeral
thf(fact_1524_power__add__numeral,axiom,
! [A: int,M: num,N2: num] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% power_add_numeral
thf(fact_1525_power__add__numeral2,axiom,
! [A: complex,M: num,N2: num,B: complex] :
( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
= ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_1526_power__add__numeral2,axiom,
! [A: real,M: num,N2: num,B: real] :
( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
= ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_1527_power__add__numeral2,axiom,
! [A: rat,M: num,N2: num,B: rat] :
( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
= ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_1528_power__add__numeral2,axiom,
! [A: nat,M: num,N2: num,B: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
= ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_1529_power__add__numeral2,axiom,
! [A: int,M: num,N2: num,B: int] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
= ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_1530_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_1531_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_1532_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_1533_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_1534_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_1535_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_1536_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_1537_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_1538_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_1539_odd__Suc__div__two,axiom,
! [N2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% odd_Suc_div_two
thf(fact_1540_even__Suc__div__two,axiom,
! [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_Suc_div_two
thf(fact_1541_zero__le__power__eq__numeral,axiom,
! [A: rat,W: num] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_1542_zero__le__power__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_1543_zero__le__power__eq__numeral,axiom,
! [A: real,W: num] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_1544_power__less__zero__eq,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% power_less_zero_eq
thf(fact_1545_power__less__zero__eq,axiom,
! [A: rat,N2: nat] :
( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% power_less_zero_eq
thf(fact_1546_power__less__zero__eq,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq
thf(fact_1547_power__less__zero__eq__numeral,axiom,
! [A: real,W: num] :
( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_1548_power__less__zero__eq__numeral,axiom,
! [A: rat,W: num] :
( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_1549_power__less__zero__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_1550_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_1551_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_1552_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_1553_zero__less__power__eq__numeral,axiom,
! [A: real,W: num] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( ( numeral_numeral_nat @ W )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A != zero_zero_real ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_1554_zero__less__power__eq__numeral,axiom,
! [A: rat,W: num] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( ( numeral_numeral_nat @ W )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A != zero_zero_rat ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_1555_zero__less__power__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
= ( ( ( numeral_numeral_nat @ W )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_1556_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_1557_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_1558_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_1559_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_1560_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_1561_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_1562_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_1563_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_1564_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_1565_even__power,axiom,
! [A: code_integer,N2: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% even_power
thf(fact_1566_even__power,axiom,
! [A: nat,N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N2 ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% even_power
thf(fact_1567_even__power,axiom,
! [A: int,N2: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N2 ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% even_power
thf(fact_1568_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_1569_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_1570_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_1571_power__le__zero__eq__numeral,axiom,
! [A: rat,W: num] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_rat @ A @ zero_zero_rat ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A = zero_zero_rat ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_1572_power__le__zero__eq__numeral,axiom,
! [A: int,W: num] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_1573_power__le__zero__eq__numeral,axiom,
! [A: real,W: num] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( ord_less_eq_real @ A @ zero_zero_real ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
& ( A = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_1574_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N ) ) ) ).
% real_arch_pow
thf(fact_1575_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_1576_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N: nat] : ( ord_less_real @ ( power_power_real @ X @ N ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1577_dvdE,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ~ ! [K3: code_integer] :
( A
!= ( times_3573771949741848930nteger @ B @ K3 ) ) ) ).
% dvdE
thf(fact_1578_dvdE,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ~ ! [K3: real] :
( A
!= ( times_times_real @ B @ K3 ) ) ) ).
% dvdE
thf(fact_1579_dvdE,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ~ ! [K3: rat] :
( A
!= ( times_times_rat @ B @ K3 ) ) ) ).
% dvdE
thf(fact_1580_dvdE,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ~ ! [K3: nat] :
( A
!= ( times_times_nat @ B @ K3 ) ) ) ).
% dvdE
thf(fact_1581_dvdE,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ~ ! [K3: int] :
( A
!= ( times_times_int @ B @ K3 ) ) ) ).
% dvdE
thf(fact_1582_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_1583_dvdI,axiom,
! [A: real,B: real,K: real] :
( ( A
= ( times_times_real @ B @ K ) )
=> ( dvd_dvd_real @ B @ A ) ) ).
% dvdI
thf(fact_1584_dvdI,axiom,
! [A: rat,B: rat,K: rat] :
( ( A
= ( times_times_rat @ B @ K ) )
=> ( dvd_dvd_rat @ B @ A ) ) ).
% dvdI
thf(fact_1585_dvdI,axiom,
! [A: nat,B: nat,K: nat] :
( ( A
= ( times_times_nat @ B @ K ) )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% dvdI
thf(fact_1586_dvdI,axiom,
! [A: int,B: int,K: int] :
( ( A
= ( times_times_int @ B @ K ) )
=> ( dvd_dvd_int @ B @ A ) ) ).
% dvdI
thf(fact_1587_dvd__def,axiom,
( dvd_dvd_Code_integer
= ( ^ [B2: code_integer,A3: code_integer] :
? [K2: code_integer] :
( A3
= ( times_3573771949741848930nteger @ B2 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_1588_dvd__def,axiom,
( dvd_dvd_real
= ( ^ [B2: real,A3: real] :
? [K2: real] :
( A3
= ( times_times_real @ B2 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_1589_dvd__def,axiom,
( dvd_dvd_rat
= ( ^ [B2: rat,A3: rat] :
? [K2: rat] :
( A3
= ( times_times_rat @ B2 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_1590_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B2: nat,A3: nat] :
? [K2: nat] :
( A3
= ( times_times_nat @ B2 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_1591_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B2: int,A3: int] :
? [K2: int] :
( A3
= ( times_times_int @ B2 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_1592_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_1593_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_1594_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_1595_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_1596_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_1597_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_1598_dvd__refl,axiom,
! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% dvd_refl
thf(fact_1599_dvd__refl,axiom,
! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).
% dvd_refl
thf(fact_1600_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_1601_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_1602_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_1603_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_1604_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_1605_dvd__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_trans
thf(fact_1606_dvd__trans,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ C )
=> ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_trans
thf(fact_1607_dvd__trans,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ B @ C )
=> ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% dvd_trans
thf(fact_1608_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_1609_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_1610_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_1611_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_1612_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_1613_dvd__triv__left,axiom,
! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).
% dvd_triv_left
thf(fact_1614_dvd__triv__left,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% dvd_triv_left
thf(fact_1615_dvd__triv__left,axiom,
! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_1616_dvd__triv__left,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_1617_dvd__triv__left,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% dvd_triv_left
thf(fact_1618_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_1619_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_1620_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_1621_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_1622_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_1623_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_1624_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_1625_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_1626_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_1627_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_1628_dvd__triv__right,axiom,
! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).
% dvd_triv_right
thf(fact_1629_dvd__triv__right,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% dvd_triv_right
thf(fact_1630_dvd__triv__right,axiom,
! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_1631_dvd__triv__right,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_1632_dvd__triv__right,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% dvd_triv_right
thf(fact_1633_complete__real,axiom,
! [S2: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S2 )
=> ( ? [Z3: real] :
! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ord_less_eq_real @ X2 @ Z3 ) )
=> ? [Y2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Y2 ) )
& ! [Z3: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ord_less_eq_real @ X2 @ Z3 ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ) ) ) ).
% complete_real
thf(fact_1634_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_1635_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_1636_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_1637_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_1638_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_1639_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_1640_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_1641_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_1642_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_1643_mult_Ocommute,axiom,
( times_times_rat
= ( ^ [A3: rat,B2: rat] : ( times_times_rat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_1644_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_1645_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_1646_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_1647_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_1648_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_1649_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_1650_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_1651_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_1652_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_1653_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_1654_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_1655_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_1656_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_1657_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_1658_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_1659_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_1660_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_1661_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_1662_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_1663_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_1664_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_1665_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_1666_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_1667_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_1668_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_1669_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_1670_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_1671_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_1672_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_1673_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_1674_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_1675_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_1676_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_1677_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_1678_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_1679_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_1680_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_1681_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_1682_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_1683_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_1684_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_1685_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_1686_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_1687_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_1688_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_1689_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_1690_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_1691_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_1692_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_1693_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_1694_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_1695_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_1696_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_1697_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_1698_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_1699_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_1700_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_1701_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_1702_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_1703_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_1704_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_1705_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_1706_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_1707_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_1708_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_1709_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_1710_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_1711_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_1712_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_1713_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_1714_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_1715_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_1716_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_1717_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_1718_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_1719_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_1720_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_1721_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_1722_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_1723_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% dvd_mult_cancel
thf(fact_1724_dvd__field__iff,axiom,
( dvd_dvd_complex
= ( ^ [A3: complex,B2: complex] :
( ( A3 = zero_zero_complex )
=> ( B2 = zero_zero_complex ) ) ) ) ).
% dvd_field_iff
thf(fact_1725_dvd__field__iff,axiom,
( dvd_dvd_real
= ( ^ [A3: real,B2: real] :
( ( A3 = zero_zero_real )
=> ( B2 = zero_zero_real ) ) ) ) ).
% dvd_field_iff
thf(fact_1726_dvd__field__iff,axiom,
( dvd_dvd_rat
= ( ^ [A3: rat,B2: rat] :
( ( A3 = zero_zero_rat )
=> ( B2 = zero_zero_rat ) ) ) ) ).
% dvd_field_iff
thf(fact_1727_dvd__0__left,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
=> ( A = zero_z3403309356797280102nteger ) ) ).
% dvd_0_left
thf(fact_1728_dvd__0__left,axiom,
! [A: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A )
=> ( A = zero_zero_complex ) ) ).
% dvd_0_left
thf(fact_1729_dvd__0__left,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
=> ( A = zero_zero_real ) ) ).
% dvd_0_left
thf(fact_1730_dvd__0__left,axiom,
! [A: rat] :
( ( dvd_dvd_rat @ zero_zero_rat @ A )
=> ( A = zero_zero_rat ) ) ).
% dvd_0_left
thf(fact_1731_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_1732_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_1733_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_1734_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_1735_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_1736_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_1737_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_1738_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_1739_one__dvd,axiom,
! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% one_dvd
thf(fact_1740_one__dvd,axiom,
! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% one_dvd
thf(fact_1741_one__dvd,axiom,
! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% one_dvd
thf(fact_1742_one__dvd,axiom,
! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% one_dvd
thf(fact_1743_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_1744_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_1745_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_1746_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_1747_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_1748_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_1749_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_1750_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_1751_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_1752_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_1753_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_1754_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_1755_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_1756_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_1757_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_1758_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_1759_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_1760_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_1761_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_1762_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_1763_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_1764_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_1765_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_1766_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_1767_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_1768_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_1769_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_1770_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_1771_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_1772_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_1773_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_1774_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_1775_dvd__power__same,axiom,
! [X: code_integer,Y: code_integer,N2: nat] :
( ( dvd_dvd_Code_integer @ X @ Y )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N2 ) @ ( power_8256067586552552935nteger @ Y @ N2 ) ) ) ).
% dvd_power_same
thf(fact_1776_dvd__power__same,axiom,
! [X: nat,Y: nat,N2: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) ) ) ).
% dvd_power_same
thf(fact_1777_dvd__power__same,axiom,
! [X: int,Y: int,N2: nat] :
( ( dvd_dvd_int @ X @ Y )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) ) ) ).
% dvd_power_same
thf(fact_1778_dvd__power__same,axiom,
! [X: real,Y: real,N2: nat] :
( ( dvd_dvd_real @ X @ Y )
=> ( dvd_dvd_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) ) ) ).
% dvd_power_same
thf(fact_1779_dvd__power__same,axiom,
! [X: complex,Y: complex,N2: nat] :
( ( dvd_dvd_complex @ X @ Y )
=> ( dvd_dvd_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) ) ) ).
% dvd_power_same
thf(fact_1780_dvd__power__same,axiom,
! [X: rat,Y: rat,N2: nat] :
( ( dvd_dvd_rat @ X @ Y )
=> ( dvd_dvd_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) ) ) ).
% dvd_power_same
thf(fact_1781_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_1782_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_1783_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_1784_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_1785_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_1786_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_1787_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_1788_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_1789_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_1790_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_1791_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_1792_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_1793_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_1794_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_1795_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_1796_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_1797_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_1798_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_1799_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_1800_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_1801_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_1802_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_1803_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_1804_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_1805_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_1806_mult_Ocomm__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.comm_neutral
thf(fact_1807_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_1808_mult_Ocomm__neutral,axiom,
! [A: rat] :
( ( times_times_rat @ A @ one_one_rat )
= A ) ).
% mult.comm_neutral
thf(fact_1809_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_1810_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_1811_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_1812_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_1813_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_1814_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_1815_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_1816_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_1817_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_1818_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_1819_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_1820_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_1821_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_1822_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_1823_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_1824_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_1825_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_1826_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_1827_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_1828_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_1829_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_1830_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_1831_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_1832_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_1833_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_1834_combine__common__factor,axiom,
! [A: real,E2: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_1835_combine__common__factor,axiom,
! [A: rat,E2: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_1836_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_1837_combine__common__factor,axiom,
! [A: int,E2: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_1838_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_1839_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_1840_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_1841_divide__divide__times__eq,axiom,
! [X: complex,Y: complex,Z: complex,W: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_1842_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_1843_divide__divide__times__eq,axiom,
! [X: rat,Y: rat,Z: rat,W: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
= ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_1844_times__divide__times__eq,axiom,
! [X: complex,Y: complex,Z: complex,W: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_1845_times__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_1846_times__divide__times__eq,axiom,
! [X: rat,Y: rat,Z: rat,W: rat] :
( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
= ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_1847_power__commutes,axiom,
! [A: complex,N2: nat] :
( ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A )
= ( times_times_complex @ A @ ( power_power_complex @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_1848_power__commutes,axiom,
! [A: real,N2: nat] :
( ( times_times_real @ ( power_power_real @ A @ N2 ) @ A )
= ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_1849_power__commutes,axiom,
! [A: rat,N2: nat] :
( ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ A )
= ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_1850_power__commutes,axiom,
! [A: nat,N2: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_1851_power__commutes,axiom,
! [A: int,N2: nat] :
( ( times_times_int @ ( power_power_int @ A @ N2 ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_1852_power__mult__distrib,axiom,
! [A: complex,B: complex,N2: nat] :
( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N2 )
= ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ ( power_power_complex @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_1853_power__mult__distrib,axiom,
! [A: real,B: real,N2: nat] :
( ( power_power_real @ ( times_times_real @ A @ B ) @ N2 )
= ( times_times_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_1854_power__mult__distrib,axiom,
! [A: rat,B: rat,N2: nat] :
( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N2 )
= ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_1855_power__mult__distrib,axiom,
! [A: nat,B: nat,N2: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N2 )
= ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_1856_power__mult__distrib,axiom,
! [A: int,B: int,N2: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N2 )
= ( times_times_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_1857_power__commuting__commutes,axiom,
! [X: complex,Y: complex,N2: nat] :
( ( ( times_times_complex @ X @ Y )
= ( times_times_complex @ Y @ X ) )
=> ( ( times_times_complex @ ( power_power_complex @ X @ N2 ) @ Y )
= ( times_times_complex @ Y @ ( power_power_complex @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_1858_power__commuting__commutes,axiom,
! [X: real,Y: real,N2: nat] :
( ( ( times_times_real @ X @ Y )
= ( times_times_real @ Y @ X ) )
=> ( ( times_times_real @ ( power_power_real @ X @ N2 ) @ Y )
= ( times_times_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_1859_power__commuting__commutes,axiom,
! [X: rat,Y: rat,N2: nat] :
( ( ( times_times_rat @ X @ Y )
= ( times_times_rat @ Y @ X ) )
=> ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ Y )
= ( times_times_rat @ Y @ ( power_power_rat @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_1860_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N2: nat] :
( ( ( times_times_nat @ X @ Y )
= ( times_times_nat @ Y @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_1861_power__commuting__commutes,axiom,
! [X: int,Y: int,N2: nat] :
( ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_1862_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( M = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_1863_power__mult,axiom,
! [A: nat,M: nat,N2: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N2 ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N2 ) ) ).
% power_mult
thf(fact_1864_power__mult,axiom,
! [A: int,M: nat,N2: nat] :
( ( power_power_int @ A @ ( times_times_nat @ M @ N2 ) )
= ( power_power_int @ ( power_power_int @ A @ M ) @ N2 ) ) ).
% power_mult
thf(fact_1865_power__mult,axiom,
! [A: real,M: nat,N2: nat] :
( ( power_power_real @ A @ ( times_times_nat @ M @ N2 ) )
= ( power_power_real @ ( power_power_real @ A @ M ) @ N2 ) ) ).
% power_mult
thf(fact_1866_power__mult,axiom,
! [A: complex,M: nat,N2: nat] :
( ( power_power_complex @ A @ ( times_times_nat @ M @ N2 ) )
= ( power_power_complex @ ( power_power_complex @ A @ M ) @ N2 ) ) ).
% power_mult
thf(fact_1867_power__mult,axiom,
! [A: rat,M: nat,N2: nat] :
( ( power_power_rat @ A @ ( times_times_nat @ M @ N2 ) )
= ( power_power_rat @ ( power_power_rat @ A @ M ) @ N2 ) ) ).
% power_mult
thf(fact_1868_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1869_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( M = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1870_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1871_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1872_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1873_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1874_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1875_add__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1876_add__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_1877_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1878_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1879_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1880_div__mult2__eq,axiom,
! [M: nat,N2: nat,Q3: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N2 @ Q3 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N2 ) @ Q3 ) ) ).
% div_mult2_eq
thf(fact_1881_is__unitE,axiom,
! [A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ~ ( ( A != zero_z3403309356797280102nteger )
=> ! [B5: code_integer] :
( ( B5 != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B5 @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
= B5 )
=> ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B5 )
= A )
=> ( ( ( times_3573771949741848930nteger @ A @ B5 )
= one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ C @ A )
!= ( times_3573771949741848930nteger @ C @ B5 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_1882_is__unitE,axiom,
! [A: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [B5: nat] :
( ( B5 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B5 @ one_one_nat )
=> ( ( ( divide_divide_nat @ one_one_nat @ A )
= B5 )
=> ( ( ( divide_divide_nat @ one_one_nat @ B5 )
= A )
=> ( ( ( times_times_nat @ A @ B5 )
= one_one_nat )
=> ( ( divide_divide_nat @ C @ A )
!= ( times_times_nat @ C @ B5 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_1883_is__unitE,axiom,
! [A: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [B5: int] :
( ( B5 != zero_zero_int )
=> ( ( dvd_dvd_int @ B5 @ one_one_int )
=> ( ( ( divide_divide_int @ one_one_int @ A )
= B5 )
=> ( ( ( divide_divide_int @ one_one_int @ B5 )
= A )
=> ( ( ( times_times_int @ A @ B5 )
= one_one_int )
=> ( ( divide_divide_int @ C @ A )
!= ( times_times_int @ C @ B5 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_1884_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_1885_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_1886_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_1887_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_1888_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_1889_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_1890_evenE,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ~ ! [B5: code_integer] :
( A
!= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) ) ) ).
% evenE
thf(fact_1891_evenE,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ! [B5: nat] :
( A
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) ) ) ).
% evenE
thf(fact_1892_evenE,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ! [B5: int] :
( A
!= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) ) ) ).
% evenE
thf(fact_1893_dvd__mult__cancel2,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M ) @ M )
= ( N2 = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_1894_dvd__mult__cancel1,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N2 ) @ M )
= ( N2 = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_1895_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_1896_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_1897_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_1898_not__is__unit__0,axiom,
~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% not_is_unit_0
thf(fact_1899_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_1900_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_1901_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_1902_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_1903_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_1904_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_1905_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_1906_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_1907_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_1908_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_1909_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_1910_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_1911_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_1912_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_1913_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_1914_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_1915_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_1916_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_1917_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_1918_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_1919_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_1920_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_1921_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_1922_div__power,axiom,
! [B: code_integer,A: code_integer,N2: nat] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N2 )
= ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ) ).
% div_power
thf(fact_1923_div__power,axiom,
! [B: nat,A: nat,N2: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N2 )
= ( divide_divide_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).
% div_power
thf(fact_1924_div__power,axiom,
! [B: int,A: int,N2: nat] :
( ( dvd_dvd_int @ B @ A )
=> ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N2 )
= ( divide_divide_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% div_power
thf(fact_1925_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_1926_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_1927_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_1928_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_1929_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_1930_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_1931_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_1932_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_1933_zero__le__square,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% zero_le_square
thf(fact_1934_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_1935_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_1936_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_1937_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_1938_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_1939_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_1940_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_1941_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_1942_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_1943_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_1944_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_1945_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_1946_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_1947_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_1948_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_1949_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_1950_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_1951_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_1952_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_1953_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_1954_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_1955_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_1956_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_1957_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_1958_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_1959_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_1960_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_1961_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_1962_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_1963_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_1964_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_1965_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_1966_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_1967_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_1968_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_1969_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_1970_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_1971_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_1972_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_1973_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_1974_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_1975_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_1976_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_1977_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_1978_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_1979_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_1980_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_1981_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_1982_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_1983_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_1984_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_1985_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_1986_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_1987_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_1988_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_1989_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_1990_not__square__less__zero,axiom,
! [A: rat] :
~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% not_square_less_zero
thf(fact_1991_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_1992_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_1993_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_1994_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_1995_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_1996_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_1997_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_1998_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_1999_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_2000_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_2001_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_2002_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_2003_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_2004_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_2005_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_2006_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_2007_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_2008_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_2009_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_2010_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_2011_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_2012_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_2013_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_2014_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_2015_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_2016_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_2017_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_2018_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_2019_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_2020_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_2021_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_2022_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_2023_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_2024_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_2025_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_2026_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_2027_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_2028_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_2029_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_2030_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_2031_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_2032_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_2033_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_2034_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_2035_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_2036_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_2037_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_2038_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_2039_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_2040_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_2041_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_2042_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_2043_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_2044_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_2045_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_2046_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_2047_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_2048_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_2049_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_2050_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_2051_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_2052_dvd__power__le,axiom,
! [X: code_integer,Y: code_integer,N2: nat,M: nat] :
( ( dvd_dvd_Code_integer @ X @ Y )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N2 ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_2053_dvd__power__le,axiom,
! [X: nat,Y: nat,N2: nat,M: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_2054_dvd__power__le,axiom,
! [X: int,Y: int,N2: nat,M: nat] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_2055_dvd__power__le,axiom,
! [X: real,Y: real,N2: nat,M: nat] :
( ( dvd_dvd_real @ X @ Y )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dvd_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_2056_dvd__power__le,axiom,
! [X: complex,Y: complex,N2: nat,M: nat] :
( ( dvd_dvd_complex @ X @ Y )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dvd_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_2057_dvd__power__le,axiom,
! [X: rat,Y: rat,N2: nat,M: nat] :
( ( dvd_dvd_rat @ X @ Y )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dvd_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_2058_power__le__dvd,axiom,
! [A: code_integer,N2: nat,B: code_integer,M: nat] :
( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ B )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_2059_power__le__dvd,axiom,
! [A: nat,N2: nat,B: nat,M: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ B )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_2060_power__le__dvd,axiom,
! [A: int,N2: nat,B: int,M: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ B )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_2061_power__le__dvd,axiom,
! [A: real,N2: nat,B: real,M: nat] :
( ( dvd_dvd_real @ ( power_power_real @ A @ N2 ) @ B )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_2062_power__le__dvd,axiom,
! [A: complex,N2: nat,B: complex,M: nat] :
( ( dvd_dvd_complex @ ( power_power_complex @ A @ N2 ) @ B )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_2063_power__le__dvd,axiom,
! [A: rat,N2: nat,B: rat,M: nat] :
( ( dvd_dvd_rat @ ( power_power_rat @ A @ N2 ) @ B )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_rat @ ( power_power_rat @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_2064_le__imp__power__dvd,axiom,
! [M: nat,N2: nat,A: code_integer] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).
% le_imp_power_dvd
thf(fact_2065_le__imp__power__dvd,axiom,
! [M: nat,N2: nat,A: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).
% le_imp_power_dvd
thf(fact_2066_le__imp__power__dvd,axiom,
! [M: nat,N2: nat,A: int] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).
% le_imp_power_dvd
thf(fact_2067_le__imp__power__dvd,axiom,
! [M: nat,N2: nat,A: real] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).
% le_imp_power_dvd
thf(fact_2068_le__imp__power__dvd,axiom,
! [M: nat,N2: nat,A: complex] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).
% le_imp_power_dvd
thf(fact_2069_le__imp__power__dvd,axiom,
! [M: nat,N2: nat,A: rat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ).
% le_imp_power_dvd
thf(fact_2070_less__1__mult,axiom,
! [M: real,N2: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N2 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_2071_less__1__mult,axiom,
! [M: rat,N2: rat] :
( ( ord_less_rat @ one_one_rat @ M )
=> ( ( ord_less_rat @ one_one_rat @ N2 )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_2072_less__1__mult,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_2073_less__1__mult,axiom,
! [M: int,N2: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N2 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_2074_frac__eq__eq,axiom,
! [Y: complex,Z: complex,X: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ X @ Y )
= ( divide1717551699836669952omplex @ W @ Z ) )
= ( ( times_times_complex @ X @ Z )
= ( times_times_complex @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2075_frac__eq__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W @ Z ) )
= ( ( times_times_real @ X @ Z )
= ( times_times_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2076_frac__eq__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ( divide_divide_rat @ X @ Y )
= ( divide_divide_rat @ W @ Z ) )
= ( ( times_times_rat @ X @ Z )
= ( times_times_rat @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2077_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_2078_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_2079_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_2080_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_2081_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_2082_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_2083_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_2084_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_2085_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_2086_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_2087_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_2088_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_2089_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_2090_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_2091_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_2092_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_2093_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_2094_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_2095_mult__numeral__1,axiom,
! [A: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_2096_mult__numeral__1,axiom,
! [A: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_2097_mult__numeral__1,axiom,
! [A: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_2098_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_2099_mult__numeral__1,axiom,
! [A: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_2100_mult__numeral__1__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_2101_mult__numeral__1__right,axiom,
! [A: real] :
( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_2102_mult__numeral__1__right,axiom,
! [A: rat] :
( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_2103_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_2104_mult__numeral__1__right,axiom,
! [A: int] :
( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_2105_nat__dvd__not__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N2 )
=> ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_2106_left__right__inverse__power,axiom,
! [X: complex,Y: complex,N2: nat] :
( ( ( times_times_complex @ X @ Y )
= one_one_complex )
=> ( ( times_times_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) )
= one_one_complex ) ) ).
% left_right_inverse_power
thf(fact_2107_left__right__inverse__power,axiom,
! [X: real,Y: real,N2: nat] :
( ( ( times_times_real @ X @ Y )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_2108_left__right__inverse__power,axiom,
! [X: rat,Y: rat,N2: nat] :
( ( ( times_times_rat @ X @ Y )
= one_one_rat )
=> ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
= one_one_rat ) ) ).
% left_right_inverse_power
thf(fact_2109_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N2: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_2110_left__right__inverse__power,axiom,
! [X: int,Y: int,N2: nat] :
( ( ( times_times_int @ X @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_2111_power__Suc2,axiom,
! [A: complex,N2: nat] :
( ( power_power_complex @ A @ ( suc @ N2 ) )
= ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A ) ) ).
% power_Suc2
thf(fact_2112_power__Suc2,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ A @ ( suc @ N2 ) )
= ( times_times_real @ ( power_power_real @ A @ N2 ) @ A ) ) ).
% power_Suc2
thf(fact_2113_power__Suc2,axiom,
! [A: rat,N2: nat] :
( ( power_power_rat @ A @ ( suc @ N2 ) )
= ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ A ) ) ).
% power_Suc2
thf(fact_2114_power__Suc2,axiom,
! [A: nat,N2: nat] :
( ( power_power_nat @ A @ ( suc @ N2 ) )
= ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A ) ) ).
% power_Suc2
thf(fact_2115_power__Suc2,axiom,
! [A: int,N2: nat] :
( ( power_power_int @ A @ ( suc @ N2 ) )
= ( times_times_int @ ( power_power_int @ A @ N2 ) @ A ) ) ).
% power_Suc2
thf(fact_2116_power__Suc,axiom,
! [A: complex,N2: nat] :
( ( power_power_complex @ A @ ( suc @ N2 ) )
= ( times_times_complex @ A @ ( power_power_complex @ A @ N2 ) ) ) ).
% power_Suc
thf(fact_2117_power__Suc,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ A @ ( suc @ N2 ) )
= ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).
% power_Suc
thf(fact_2118_power__Suc,axiom,
! [A: rat,N2: nat] :
( ( power_power_rat @ A @ ( suc @ N2 ) )
= ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ).
% power_Suc
thf(fact_2119_power__Suc,axiom,
! [A: nat,N2: nat] :
( ( power_power_nat @ A @ ( suc @ N2 ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).
% power_Suc
thf(fact_2120_power__Suc,axiom,
! [A: int,N2: nat] :
( ( power_power_int @ A @ ( suc @ N2 ) )
= ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).
% power_Suc
thf(fact_2121_zdvd__antisym__nonneg,axiom,
! [M: int,N2: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ N2 )
=> ( ( dvd_dvd_int @ M @ N2 )
=> ( ( dvd_dvd_int @ N2 @ M )
=> ( M = N2 ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_2122_zdvd__not__zless,axiom,
! [M: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N2 )
=> ~ ( dvd_dvd_int @ N2 @ M ) ) ) ).
% zdvd_not_zless
thf(fact_2123_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_2124_power__add,axiom,
! [A: complex,M: nat,N2: nat] :
( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N2 ) )
= ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).
% power_add
thf(fact_2125_power__add,axiom,
! [A: real,M: nat,N2: nat] :
( ( power_power_real @ A @ ( plus_plus_nat @ M @ N2 ) )
= ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).
% power_add
thf(fact_2126_power__add,axiom,
! [A: rat,M: nat,N2: nat] :
( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N2 ) )
= ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ).
% power_add
thf(fact_2127_power__add,axiom,
! [A: nat,M: nat,N2: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N2 ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).
% power_add
thf(fact_2128_power__add,axiom,
! [A: int,M: nat,N2: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M @ N2 ) )
= ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).
% power_add
thf(fact_2129_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_2130_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_2131_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( M = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_2132_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_2133_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_2134_mult__Suc,axiom,
! [M: nat,N2: nat] :
( ( times_times_nat @ ( suc @ M ) @ N2 )
= ( plus_plus_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ).
% mult_Suc
thf(fact_2135_dbl__def,axiom,
( neg_numeral_dbl_real
= ( ^ [X3: real] : ( plus_plus_real @ X3 @ X3 ) ) ) ).
% dbl_def
thf(fact_2136_dbl__def,axiom,
( neg_numeral_dbl_rat
= ( ^ [X3: rat] : ( plus_plus_rat @ X3 @ X3 ) ) ) ).
% dbl_def
thf(fact_2137_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X3: int] : ( plus_plus_int @ X3 @ X3 ) ) ) ).
% dbl_def
thf(fact_2138_mult__eq__self__implies__10,axiom,
! [M: nat,N2: nat] :
( ( M
= ( times_times_nat @ M @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_2139_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N2: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N2 ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_2140_times__div__less__eq__dividend,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_2141_div__times__less__eq__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_2142_oddE,axiom,
! [A: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ~ ! [B5: code_integer] :
( A
!= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) @ one_one_Code_integer ) ) ) ).
% oddE
thf(fact_2143_oddE,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ! [B5: nat] :
( A
!= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) @ one_one_nat ) ) ) ).
% oddE
thf(fact_2144_oddE,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ! [B5: int] :
( A
!= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) @ one_one_int ) ) ) ).
% oddE
thf(fact_2145_power__odd__eq,axiom,
! [A: complex,N2: nat] :
( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
= ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_2146_power__odd__eq,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
= ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_2147_power__odd__eq,axiom,
! [A: rat,N2: nat] :
( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
= ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_2148_power__odd__eq,axiom,
! [A: nat,N2: nat] :
( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
= ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_2149_power__odd__eq,axiom,
! [A: int,N2: nat] :
( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
= ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_2150_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_2151_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_2152_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_2153_even__numeral,axiom,
! [N2: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) ) ).
% even_numeral
thf(fact_2154_even__numeral,axiom,
! [N2: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ).
% even_numeral
thf(fact_2155_even__numeral,axiom,
! [N2: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ).
% even_numeral
thf(fact_2156_is__unit__power__iff,axiom,
! [A: code_integer,N2: nat] :
( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ one_one_Code_integer )
= ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
| ( N2 = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_2157_is__unit__power__iff,axiom,
! [A: nat,N2: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_2158_is__unit__power__iff,axiom,
! [A: int,N2: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
| ( N2 = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_2159_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_2160_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_2161_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_2162_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_2163_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_2164_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_2165_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_2166_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_2167_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_2168_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_2169_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_2170_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_2171_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_2172_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_2173_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_2174_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_2175_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_2176_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_2177_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_2178_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_2179_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_2180_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_2181_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_2182_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_2183_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_2184_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_2185_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_2186_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_2187_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_2188_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_2189_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_2190_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_2191_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_2192_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_2193_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_2194_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_2195_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_2196_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_2197_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_2198_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_2199_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_2200_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_2201_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_2202_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_2203_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_2204_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_2205_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_2206_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_2207_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_2208_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_2209_mult__left__le__one__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ Y @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_2210_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_2211_mult__left__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_2212_mult__right__le__one__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ Y @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_2213_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_2214_mult__right__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_2215_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_2216_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_2217_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_2218_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_2219_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_2220_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_2221_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_2222_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_2223_sum__squares__le__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_2224_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_2225_sum__squares__le__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_2226_sum__squares__ge__zero,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_2227_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_2228_sum__squares__ge__zero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_2229_sum__squares__gt__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
= ( ( X != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_2230_sum__squares__gt__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
= ( ( X != zero_zero_rat )
| ( Y != zero_zero_rat ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_2231_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_2232_not__sum__squares__lt__zero,axiom,
! [X: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_2233_not__sum__squares__lt__zero,axiom,
! [X: rat,Y: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).
% not_sum_squares_lt_zero
thf(fact_2234_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_2235_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_2236_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_2237_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_2238_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_2239_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_2240_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_2241_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_2242_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_2243_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_2244_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_2245_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_2246_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_2247_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_2248_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_2249_mult__imp__div__pos__less,axiom,
! [Y: real,X: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y ) )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_2250_mult__imp__div__pos__less,axiom,
! [Y: rat,X: rat,Z: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y ) )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_2251_mult__imp__less__div__pos,axiom,
! [Y: real,Z: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X )
=> ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_2252_mult__imp__less__div__pos,axiom,
! [Y: rat,Z: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X )
=> ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_2253_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_2254_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_2255_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_2256_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_2257_divide__eq__eq__numeral_I1_J,axiom,
! [B: complex,C: complex,W: num] :
( ( ( divide1717551699836669952omplex @ B @ C )
= ( numera6690914467698888265omplex @ W ) )
= ( ( ( C != zero_zero_complex )
=> ( B
= ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( ( numera6690914467698888265omplex @ W )
= zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_2258_divide__eq__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( ( ( divide_divide_real @ B @ C )
= ( numeral_numeral_real @ W ) )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_2259_divide__eq__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ( divide_divide_rat @ B @ C )
= ( numeral_numeral_rat @ W ) )
= ( ( ( C != zero_zero_rat )
=> ( B
= ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W )
= zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_2260_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: complex,C: complex] :
( ( ( numera6690914467698888265omplex @ W )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
= B ) )
& ( ( C = zero_zero_complex )
=> ( ( numera6690914467698888265omplex @ W )
= zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_2261_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( ( ( numeral_numeral_real @ W )
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_2262_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ( numeral_numeral_rat @ W )
= ( divide_divide_rat @ B @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
= B ) )
& ( ( C = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W )
= zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_2263_add__divide__eq__if__simps_I2_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_2264_add__divide__eq__if__simps_I2_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_2265_add__divide__eq__if__simps_I2_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_2266_add__divide__eq__if__simps_I1_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_2267_add__divide__eq__if__simps_I1_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_2268_add__divide__eq__if__simps_I1_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_2269_add__frac__eq,axiom,
! [Y: complex,Z: complex,X: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_2270_add__frac__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_2271_add__frac__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_2272_add__frac__num,axiom,
! [Y: complex,X: complex,Z: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_2273_add__frac__num,axiom,
! [Y: real,X: real,Z: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_2274_add__frac__num,axiom,
! [Y: rat,X: rat,Z: rat] :
( ( Y != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_2275_add__num__frac,axiom,
! [Y: complex,Z: complex,X: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_2276_add__num__frac,axiom,
! [Y: real,Z: real,X: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_2277_add__num__frac,axiom,
! [Y: rat,Z: rat,X: rat] :
( ( Y != zero_zero_rat )
=> ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y ) )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_2278_add__divide__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_2279_add__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_2280_add__divide__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_2281_divide__add__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_2282_divide__add__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_2283_divide__add__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_2284_power__less__power__Suc,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% power_less_power_Suc
thf(fact_2285_power__less__power__Suc,axiom,
! [A: rat,N2: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% power_less_power_Suc
thf(fact_2286_power__less__power__Suc,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_less_power_Suc
thf(fact_2287_power__less__power__Suc,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_less_power_Suc
thf(fact_2288_power__gt1__lemma,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% power_gt1_lemma
thf(fact_2289_power__gt1__lemma,axiom,
! [A: rat,N2: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% power_gt1_lemma
thf(fact_2290_power__gt1__lemma,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_gt1_lemma
thf(fact_2291_power__gt1__lemma,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_gt1_lemma
thf(fact_2292_dvd__imp__le,axiom,
! [K: nat,N2: nat] :
( ( dvd_dvd_nat @ K @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% dvd_imp_le
thf(fact_2293_zdvd__imp__le,axiom,
! [Z: int,N2: int] :
( ( dvd_dvd_int @ Z @ N2 )
=> ( ( ord_less_int @ zero_zero_int @ N2 )
=> ( ord_less_eq_int @ Z @ N2 ) ) ) ).
% zdvd_imp_le
thf(fact_2294_n__less__n__mult__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_2295_n__less__m__mult__n,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_2296_one__less__mult,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_2297_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_2298_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_2299_div__less__iff__less__mult,axiom,
! [Q3: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N2 )
= ( ord_less_nat @ M @ ( times_times_nat @ N2 @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_2300_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_2301_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M7 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_2302_even__zero,axiom,
dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% even_zero
thf(fact_2303_even__zero,axiom,
dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% even_zero
thf(fact_2304_even__zero,axiom,
dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% even_zero
thf(fact_2305_odd__one,axiom,
~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% odd_one
thf(fact_2306_odd__one,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% odd_one
thf(fact_2307_odd__one,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% odd_one
thf(fact_2308_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_2309_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_2310_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_2311_bit__eq__rec,axiom,
( ( ^ [Y5: code_integer,Z4: code_integer] : Y5 = Z4 )
= ( ^ [A3: code_integer,B2: code_integer] :
( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
= ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
& ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( divide6298287555418463151nteger @ B2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_2312_bit__eq__rec,axiom,
( ( ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
= ( ^ [A3: nat,B2: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
& ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_2313_bit__eq__rec,axiom,
( ( ^ [Y5: int,Z4: int] : Y5 = Z4 )
= ( ^ [A3: int,B2: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
& ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ B2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_2314_dvd__power__iff,axiom,
! [X: code_integer,M: nat,N2: nat] :
( ( X != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N2 ) )
= ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
| ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% dvd_power_iff
thf(fact_2315_dvd__power__iff,axiom,
! [X: nat,M: nat,N2: nat] :
( ( X != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N2 ) )
= ( ( dvd_dvd_nat @ X @ one_one_nat )
| ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% dvd_power_iff
thf(fact_2316_dvd__power__iff,axiom,
! [X: int,M: nat,N2: nat] :
( ( X != zero_zero_int )
=> ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N2 ) )
= ( ( dvd_dvd_int @ X @ one_one_int )
| ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% dvd_power_iff
thf(fact_2317_dvd__power,axiom,
! [N2: nat,X: code_integer] :
( ( ( ord_less_nat @ zero_zero_nat @ N2 )
| ( X = one_one_Code_integer ) )
=> ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N2 ) ) ) ).
% dvd_power
thf(fact_2318_dvd__power,axiom,
! [N2: nat,X: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N2 )
| ( X = one_one_nat ) )
=> ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N2 ) ) ) ).
% dvd_power
thf(fact_2319_dvd__power,axiom,
! [N2: nat,X: int] :
( ( ( ord_less_nat @ zero_zero_nat @ N2 )
| ( X = one_one_int ) )
=> ( dvd_dvd_int @ X @ ( power_power_int @ X @ N2 ) ) ) ).
% dvd_power
thf(fact_2320_dvd__power,axiom,
! [N2: nat,X: real] :
( ( ( ord_less_nat @ zero_zero_nat @ N2 )
| ( X = one_one_real ) )
=> ( dvd_dvd_real @ X @ ( power_power_real @ X @ N2 ) ) ) ).
% dvd_power
thf(fact_2321_dvd__power,axiom,
! [N2: nat,X: complex] :
( ( ( ord_less_nat @ zero_zero_nat @ N2 )
| ( X = one_one_complex ) )
=> ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N2 ) ) ) ).
% dvd_power
thf(fact_2322_dvd__power,axiom,
! [N2: nat,X: rat] :
( ( ( ord_less_nat @ zero_zero_nat @ N2 )
| ( X = one_one_rat ) )
=> ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N2 ) ) ) ).
% dvd_power
thf(fact_2323_field__le__mult__one__interval,axiom,
! [X: rat,Y: rat] :
( ! [Z2: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z2 )
=> ( ( ord_less_rat @ Z2 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ Y ) ) )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_2324_field__le__mult__one__interval,axiom,
! [X: real,Y: real] :
( ! [Z2: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ Z2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ Y ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_2325_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_2326_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_2327_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_2328_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_2329_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_2330_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_2331_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_2332_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_2333_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_2334_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_2335_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_2336_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_2337_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_2338_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_2339_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_2340_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_2341_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_2342_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_2343_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_2344_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_2345_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_2346_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_2347_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_2348_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_2349_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_2350_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_2351_mult__imp__le__div__pos,axiom,
! [Y: rat,Z: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X )
=> ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_2352_mult__imp__le__div__pos,axiom,
! [Y: real,Z: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X )
=> ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_2353_mult__imp__div__pos__le,axiom,
! [Y: rat,X: rat,Z: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_le
thf(fact_2354_mult__imp__div__pos__le,axiom,
! [Y: real,X: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_le
thf(fact_2355_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_2356_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_2357_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_2358_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_2359_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_2360_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_2361_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_2362_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_2363_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_2364_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_2365_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_2366_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_2367_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_2368_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_2369_convex__bound__le,axiom,
! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
( ( ord_less_eq_rat @ X @ A )
=> ( ( ord_less_eq_rat @ Y @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V )
=> ( ( ( plus_plus_rat @ U @ V )
= one_one_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_2370_convex__bound__le,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_2371_convex__bound__le,axiom,
! [X: real,A: real,Y: real,U: real,V: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ( ord_less_eq_real @ Y @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_2372_divide__less__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_2373_divide__less__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_2374_less__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ 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 @ W ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_2375_less__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ 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 @ W ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_2376_power__Suc__less,axiom,
! [A: real,N2: 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 @ N2 ) ) @ ( power_power_real @ A @ N2 ) ) ) ) ).
% power_Suc_less
thf(fact_2377_power__Suc__less,axiom,
! [A: rat,N2: 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 @ N2 ) ) @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% power_Suc_less
thf(fact_2378_power__Suc__less,axiom,
! [A: nat,N2: 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 @ N2 ) ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_Suc_less
thf(fact_2379_power__Suc__less,axiom,
! [A: int,N2: 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 @ N2 ) ) @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_Suc_less
thf(fact_2380_mult__2,axiom,
! [Z: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_complex @ Z @ Z ) ) ).
% mult_2
thf(fact_2381_mult__2,axiom,
! [Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2
thf(fact_2382_mult__2,axiom,
! [Z: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_rat @ Z @ Z ) ) ).
% mult_2
thf(fact_2383_mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2
thf(fact_2384_mult__2,axiom,
! [Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2
thf(fact_2385_mult__2__right,axiom,
! [Z: complex] :
( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
= ( plus_plus_complex @ Z @ Z ) ) ).
% mult_2_right
thf(fact_2386_mult__2__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2_right
thf(fact_2387_mult__2__right,axiom,
! [Z: rat] :
( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
= ( plus_plus_rat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_2388_mult__2__right,axiom,
! [Z: nat] :
( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_2389_mult__2__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2_right
thf(fact_2390_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_2391_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_2392_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_2393_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_2394_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_2395_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_2396_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_2397_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_2398_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_2399_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_2400_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_2401_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_2402_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_2403_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_2404_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_2405_Suc__double__not__eq__double,axiom,
! [M: nat,N2: nat] :
( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% Suc_double_not_eq_double
thf(fact_2406_double__not__eq__Suc__double,axiom,
! [M: nat,N2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% double_not_eq_Suc_double
thf(fact_2407_power__even__eq,axiom,
! [A: nat,N2: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_2408_power__even__eq,axiom,
! [A: int,N2: nat] :
( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_2409_power__even__eq,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_2410_power__even__eq,axiom,
! [A: complex,N2: nat] :
( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_complex @ ( power_power_complex @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_2411_power__even__eq,axiom,
! [A: rat,N2: nat] :
( ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_rat @ ( power_power_rat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_2412_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N2: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
=> ( ( ord_less_nat @ one_one_nat @ I )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% power_dvd_imp_le
thf(fact_2413_div__nat__eqI,axiom,
! [N2: nat,Q3: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q3 ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q3 ) ) )
=> ( ( divide_divide_nat @ M @ N2 )
= Q3 ) ) ) ).
% div_nat_eqI
thf(fact_2414_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N2 @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N2 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_2415_split__div,axiom,
! [P: nat > $o,M: nat,N2: nat] :
( ( P @ ( divide_divide_nat @ M @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N2 != zero_zero_nat )
=> ! [I3: nat,J3: nat] :
( ( ord_less_nat @ J3 @ N2 )
=> ( ( M
= ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J3 ) )
=> ( P @ I3 ) ) ) ) ) ) ).
% split_div
thf(fact_2416_dividend__less__div__times,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) ) ) ) ).
% dividend_less_div_times
thf(fact_2417_dividend__less__times__div,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_2418_power__mono__odd,axiom,
! [N2: nat,A: rat,B: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).
% power_mono_odd
thf(fact_2419_power__mono__odd,axiom,
! [N2: nat,A: int,B: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% power_mono_odd
thf(fact_2420_power__mono__odd,axiom,
! [N2: nat,A: real,B: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% power_mono_odd
thf(fact_2421_convex__bound__lt,axiom,
! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
( ( ord_less_rat @ X @ A )
=> ( ( ord_less_rat @ Y @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V )
=> ( ( ( plus_plus_rat @ U @ V )
= one_one_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_2422_convex__bound__lt,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X @ A )
=> ( ( ord_less_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_2423_convex__bound__lt,axiom,
! [X: real,A: real,Y: real,U: real,V: real] :
( ( ord_less_real @ X @ A )
=> ( ( ord_less_real @ Y @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_2424_divide__le__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ 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 @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_2425_divide__le__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ 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 @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_2426_le__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ 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 @ W ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_2427_le__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ 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 @ W ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_2428_odd__pos,axiom,
! [N2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% odd_pos
thf(fact_2429_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% dvd_power_iff_le
thf(fact_2430_num_Osize__gen_I1_J,axiom,
( ( size_num @ one )
= zero_zero_nat ) ).
% num.size_gen(1)
thf(fact_2431_split__div_H,axiom,
! [P: nat > $o,M: nat,N2: nat] :
( ( P @ ( divide_divide_nat @ M @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q4 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_2432_zero__le__power__eq,axiom,
! [A: rat,N2: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_2433_zero__le__power__eq,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_2434_zero__le__power__eq,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_2435_zero__le__odd__power,axiom,
! [N2: nat,A: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% zero_le_odd_power
thf(fact_2436_zero__le__odd__power,axiom,
! [N2: nat,A: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% zero_le_odd_power
thf(fact_2437_zero__le__odd__power,axiom,
! [N2: nat,A: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% zero_le_odd_power
thf(fact_2438_zero__le__even__power,axiom,
! [N2: nat,A: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% zero_le_even_power
thf(fact_2439_zero__le__even__power,axiom,
! [N2: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% zero_le_even_power
thf(fact_2440_zero__le__even__power,axiom,
! [N2: nat,A: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% zero_le_even_power
thf(fact_2441_power2__sum,axiom,
! [X: complex,Y: complex] :
( ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_2442_power2__sum,axiom,
! [X: real,Y: real] :
( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_2443_power2__sum,axiom,
! [X: rat,Y: rat] :
( ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_2444_power2__sum,axiom,
! [X: nat,Y: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_2445_power2__sum,axiom,
! [X: int,Y: int] :
( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_2446_zero__le__even__power_H,axiom,
! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% zero_le_even_power'
thf(fact_2447_zero__le__even__power_H,axiom,
! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% zero_le_even_power'
thf(fact_2448_zero__le__even__power_H,axiom,
! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% zero_le_even_power'
thf(fact_2449_nat__bit__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ! [N: nat] :
( ( P @ N )
=> ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_bit_induct
thf(fact_2450_zero__less__power__eq,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
= ( ( N2 = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( A != zero_zero_real ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_2451_zero__less__power__eq,axiom,
! [A: rat,N2: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
= ( ( N2 = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( A != zero_zero_rat ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_2452_zero__less__power__eq,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
= ( ( N2 = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_2453_sum__squares__bound,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_squares_bound
thf(fact_2454_sum__squares__bound,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_squares_bound
thf(fact_2455_odd__0__le__power__imp__0__le,axiom,
! [A: rat,N2: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_2456_odd__0__le__power__imp__0__le,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_2457_odd__0__le__power__imp__0__le,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_2458_odd__power__less__zero,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_real ) ) ).
% odd_power_less_zero
thf(fact_2459_odd__power__less__zero,axiom,
! [A: rat,N2: nat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_rat ) ) ).
% odd_power_less_zero
thf(fact_2460_odd__power__less__zero,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_int ) ) ).
% odd_power_less_zero
thf(fact_2461_power__le__zero__eq,axiom,
! [A: rat,N2: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
= ( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( ord_less_eq_rat @ A @ zero_zero_rat ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( A = zero_zero_rat ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_2462_power__le__zero__eq,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_2463_power__le__zero__eq,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
= ( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( ord_less_eq_real @ A @ zero_zero_real ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( A = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_2464_arith__geo__mean,axiom,
! [U: rat,X: rat,Y: rat] :
( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_rat @ X @ Y ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_2465_arith__geo__mean,axiom,
! [U: real,X: real,Y: real] :
( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ X @ Y ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_2466_pow__divides__pow__iff,axiom,
! [N2: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_2467_pow__divides__pow__iff,axiom,
! [N2: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_2468_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_2469_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_2470_low__inv,axiom,
! [X: nat,N2: nat,Y: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X ) @ N2 )
= X ) ) ).
% low_inv
thf(fact_2471_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_2472_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_2473_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T3: vEBT_VEBT,X3: nat] :
( ( vEBT_V5719532721284313246member @ T3 @ X3 )
| ( vEBT_VEBT_membermima @ T3 @ X3 ) ) ) ) ).
% both_member_options_def
thf(fact_2474_div2__even__ext__nat,axiom,
! [X: nat,Y: nat] :
( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
=> ( X = Y ) ) ) ).
% div2_even_ext_nat
thf(fact_2475_unity__coeff__ex,axiom,
! [P: code_integer > $o,L: code_integer] :
( ( ? [X3: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X3 ) ) )
= ( ? [X3: code_integer] :
( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X3 @ zero_z3403309356797280102nteger ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_2476_unity__coeff__ex,axiom,
! [P: complex > $o,L: complex] :
( ( ? [X3: complex] : ( P @ ( times_times_complex @ L @ X3 ) ) )
= ( ? [X3: complex] :
( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X3 @ zero_zero_complex ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_2477_unity__coeff__ex,axiom,
! [P: real > $o,L: real] :
( ( ? [X3: real] : ( P @ ( times_times_real @ L @ X3 ) ) )
= ( ? [X3: real] :
( ( dvd_dvd_real @ L @ ( plus_plus_real @ X3 @ zero_zero_real ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_2478_unity__coeff__ex,axiom,
! [P: rat > $o,L: rat] :
( ( ? [X3: rat] : ( P @ ( times_times_rat @ L @ X3 ) ) )
= ( ? [X3: rat] :
( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X3 @ zero_zero_rat ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_2479_unity__coeff__ex,axiom,
! [P: nat > $o,L: nat] :
( ( ? [X3: nat] : ( P @ ( times_times_nat @ L @ X3 ) ) )
= ( ? [X3: nat] :
( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X3 @ zero_zero_nat ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_2480_unity__coeff__ex,axiom,
! [P: int > $o,L: int] :
( ( ? [X3: int] : ( P @ ( times_times_int @ L @ X3 ) ) )
= ( ? [X3: int] :
( ( dvd_dvd_int @ L @ ( plus_plus_int @ X3 @ zero_zero_int ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_2481_mult__le__cancel__iff1,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
= ( ord_less_eq_rat @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_2482_mult__le__cancel__iff1,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_2483_mult__le__cancel__iff1,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_2484_mult__le__cancel__iff2,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
= ( ord_less_eq_rat @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_2485_mult__le__cancel__iff2,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_2486_mult__le__cancel__iff2,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_2487_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D4: nat,X2: nat,Y2: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( times_times_nat @ A @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ D4 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_2488_buildup__nothing__in__min__max,axiom,
! [N2: nat,X: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X ) ).
% buildup_nothing_in_min_max
thf(fact_2489_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_2490_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_2491_unset__bit__nonnegative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_2492_set__bit__nonnegative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_2493_unset__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% unset_bit_negative_int_iff
thf(fact_2494_set__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% set_bit_negative_int_iff
thf(fact_2495_semiring__norm_I13_J,axiom,
! [M: num,N2: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N2 ) ) ) ) ).
% semiring_norm(13)
thf(fact_2496_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_2497_semiring__norm_I12_J,axiom,
! [N2: num] :
( ( times_times_num @ one @ N2 )
= N2 ) ).
% semiring_norm(12)
thf(fact_2498_num__double,axiom,
! [N2: num] :
( ( times_times_num @ ( bit0 @ one ) @ N2 )
= ( bit0 @ N2 ) ) ).
% num_double
thf(fact_2499_power__mult__numeral,axiom,
! [A: nat,M: num,N2: num] :
( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% power_mult_numeral
thf(fact_2500_power__mult__numeral,axiom,
! [A: int,M: num,N2: num] :
( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% power_mult_numeral
thf(fact_2501_power__mult__numeral,axiom,
! [A: real,M: num,N2: num] :
( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% power_mult_numeral
thf(fact_2502_power__mult__numeral,axiom,
! [A: complex,M: num,N2: num] :
( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% power_mult_numeral
thf(fact_2503_power__mult__numeral,axiom,
! [A: rat,M: num,N2: num] :
( ( power_power_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% power_mult_numeral
thf(fact_2504_dvd__antisym,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd_nat @ M @ N2 )
=> ( ( dvd_dvd_nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% dvd_antisym
thf(fact_2505_zdvd__mono,axiom,
! [K: int,M: int,T2: int] :
( ( K != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T2 )
= ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T2 ) ) ) ) ).
% zdvd_mono
thf(fact_2506_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_2507_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_2508_zdvd__mult__cancel,axiom,
! [K: int,M: int,N2: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N2 ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M @ N2 ) ) ) ).
% zdvd_mult_cancel
thf(fact_2509_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_2510_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_2511_zdvd__reduce,axiom,
! [K: int,N2: int,M: int] :
( ( dvd_dvd_int @ K @ ( plus_plus_int @ N2 @ ( times_times_int @ K @ M ) ) )
= ( dvd_dvd_int @ K @ N2 ) ) ).
% zdvd_reduce
thf(fact_2512_zdvd__period,axiom,
! [A: int,D: int,X: int,T2: int,C: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T2 ) )
= ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T2 ) ) ) ) ).
% zdvd_period
thf(fact_2513_imult__is__0,axiom,
! [M: extended_enat,N2: extended_enat] :
( ( ( times_7803423173614009249d_enat @ M @ N2 )
= zero_z5237406670263579293d_enat )
= ( ( M = zero_z5237406670263579293d_enat )
| ( N2 = zero_z5237406670263579293d_enat ) ) ) ).
% imult_is_0
thf(fact_2514_unset__bit__less__eq,axiom,
! [N2: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ K ) ).
% unset_bit_less_eq
thf(fact_2515_set__bit__greater__eq,axiom,
! [K: int,N2: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N2 @ K ) ) ).
% set_bit_greater_eq
thf(fact_2516_div__mult2__numeral__eq,axiom,
! [A: nat,K: num,L: num] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_2517_div__mult2__numeral__eq,axiom,
! [A: int,K: num,L: num] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_2518_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_2519_enat__0__less__mult__iff,axiom,
! [M: extended_enat,N2: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N2 ) )
= ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
& ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 ) ) ) ).
% enat_0_less_mult_iff
thf(fact_2520_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_2521_minf_I7_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ~ ( ord_less_real @ T2 @ X4 ) ) ).
% minf(7)
thf(fact_2522_minf_I7_J,axiom,
! [T2: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ~ ( ord_less_rat @ T2 @ X4 ) ) ).
% minf(7)
thf(fact_2523_minf_I7_J,axiom,
! [T2: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z2 )
=> ~ ( ord_less_num @ T2 @ X4 ) ) ).
% minf(7)
thf(fact_2524_minf_I7_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ~ ( ord_less_nat @ T2 @ X4 ) ) ).
% minf(7)
thf(fact_2525_minf_I7_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ~ ( ord_less_int @ T2 @ X4 ) ) ).
% minf(7)
thf(fact_2526_minf_I5_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ord_less_real @ X4 @ T2 ) ) ).
% minf(5)
thf(fact_2527_minf_I5_J,axiom,
! [T2: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ( ord_less_rat @ X4 @ T2 ) ) ).
% minf(5)
thf(fact_2528_minf_I5_J,axiom,
! [T2: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z2 )
=> ( ord_less_num @ X4 @ T2 ) ) ).
% minf(5)
thf(fact_2529_minf_I5_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ord_less_nat @ X4 @ T2 ) ) ).
% minf(5)
thf(fact_2530_minf_I5_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ord_less_int @ X4 @ T2 ) ) ).
% minf(5)
thf(fact_2531_minf_I4_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(4)
thf(fact_2532_minf_I4_J,axiom,
! [T2: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(4)
thf(fact_2533_minf_I4_J,axiom,
! [T2: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(4)
thf(fact_2534_minf_I4_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(4)
thf(fact_2535_minf_I4_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(4)
thf(fact_2536_minf_I3_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(3)
thf(fact_2537_minf_I3_J,axiom,
! [T2: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(3)
thf(fact_2538_minf_I3_J,axiom,
! [T2: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(3)
thf(fact_2539_minf_I3_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(3)
thf(fact_2540_minf_I3_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( X4 != T2 ) ) ).
% minf(3)
thf(fact_2541_minf_I2_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q5: real > $o] :
( ? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z3 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z3 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q5 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_2542_minf_I2_J,axiom,
! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q5: rat > $o] :
( ? [Z3: rat] :
! [X2: rat] :
( ( ord_less_rat @ X2 @ Z3 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: rat] :
! [X2: rat] :
( ( ord_less_rat @ X2 @ Z3 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q5 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_2543_minf_I2_J,axiom,
! [P: num > $o,P2: num > $o,Q: num > $o,Q5: num > $o] :
( ? [Z3: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z3 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z3 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q5 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_2544_minf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q5: nat > $o] :
( ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z3 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z3 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q5 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_2545_minf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q5: int > $o] :
( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q5 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_2546_minf_I1_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q5: real > $o] :
( ? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z3 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z3 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q5 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_2547_minf_I1_J,axiom,
! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q5: rat > $o] :
( ? [Z3: rat] :
! [X2: rat] :
( ( ord_less_rat @ X2 @ Z3 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: rat] :
! [X2: rat] :
( ( ord_less_rat @ X2 @ Z3 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q5 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_2548_minf_I1_J,axiom,
! [P: num > $o,P2: num > $o,Q: num > $o,Q5: num > $o] :
( ? [Z3: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z3 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z3 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q5 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_2549_minf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q5: nat > $o] :
( ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z3 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z3 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q5 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_2550_minf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q5: int > $o] :
( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q5 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_2551_pinf_I7_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ord_less_real @ T2 @ X4 ) ) ).
% pinf(7)
thf(fact_2552_pinf_I7_J,axiom,
! [T2: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ( ord_less_rat @ T2 @ X4 ) ) ).
% pinf(7)
thf(fact_2553_pinf_I7_J,axiom,
! [T2: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ Z2 @ X4 )
=> ( ord_less_num @ T2 @ X4 ) ) ).
% pinf(7)
thf(fact_2554_pinf_I7_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ord_less_nat @ T2 @ X4 ) ) ).
% pinf(7)
thf(fact_2555_pinf_I7_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ord_less_int @ T2 @ X4 ) ) ).
% pinf(7)
thf(fact_2556_pinf_I5_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ~ ( ord_less_real @ X4 @ T2 ) ) ).
% pinf(5)
thf(fact_2557_pinf_I5_J,axiom,
! [T2: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ~ ( ord_less_rat @ X4 @ T2 ) ) ).
% pinf(5)
thf(fact_2558_pinf_I5_J,axiom,
! [T2: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ Z2 @ X4 )
=> ~ ( ord_less_num @ X4 @ T2 ) ) ).
% pinf(5)
thf(fact_2559_pinf_I5_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T2 ) ) ).
% pinf(5)
thf(fact_2560_pinf_I5_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ~ ( ord_less_int @ X4 @ T2 ) ) ).
% pinf(5)
thf(fact_2561_pinf_I4_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(4)
thf(fact_2562_pinf_I4_J,axiom,
! [T2: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(4)
thf(fact_2563_pinf_I4_J,axiom,
! [T2: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(4)
thf(fact_2564_pinf_I4_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(4)
thf(fact_2565_pinf_I4_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(4)
thf(fact_2566_pinf_I3_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(3)
thf(fact_2567_pinf_I3_J,axiom,
! [T2: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(3)
thf(fact_2568_pinf_I3_J,axiom,
! [T2: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(3)
thf(fact_2569_pinf_I3_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(3)
thf(fact_2570_pinf_I3_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( X4 != T2 ) ) ).
% pinf(3)
thf(fact_2571_pinf_I2_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q5: real > $o] :
( ? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q5 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_2572_pinf_I2_J,axiom,
! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q5: rat > $o] :
( ? [Z3: rat] :
! [X2: rat] :
( ( ord_less_rat @ Z3 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: rat] :
! [X2: rat] :
( ( ord_less_rat @ Z3 @ X2 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q5 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_2573_pinf_I2_J,axiom,
! [P: num > $o,P2: num > $o,Q: num > $o,Q5: num > $o] :
( ? [Z3: num] :
! [X2: num] :
( ( ord_less_num @ Z3 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: num] :
! [X2: num] :
( ( ord_less_num @ Z3 @ X2 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q5 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_2574_pinf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q5: nat > $o] :
( ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z3 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z3 @ X2 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q5 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_2575_pinf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q5: int > $o] :
( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q5 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_2576_pinf_I1_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q5: real > $o] :
( ? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: real] :
! [X2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q5 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_2577_pinf_I1_J,axiom,
! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q5: rat > $o] :
( ? [Z3: rat] :
! [X2: rat] :
( ( ord_less_rat @ Z3 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: rat] :
! [X2: rat] :
( ( ord_less_rat @ Z3 @ X2 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q5 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_2578_pinf_I1_J,axiom,
! [P: num > $o,P2: num > $o,Q: num > $o,Q5: num > $o] :
( ? [Z3: num] :
! [X2: num] :
( ( ord_less_num @ Z3 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: num] :
! [X2: num] :
( ( ord_less_num @ Z3 @ X2 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q5 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_2579_pinf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q5: nat > $o] :
( ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z3 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z3 @ X2 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q5 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_2580_pinf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q5: int > $o] :
( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ( ( Q @ X2 )
= ( Q5 @ X2 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q5 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_2581_pos__zmult__eq__1__iff,axiom,
! [M: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N2 )
= one_one_int )
= ( ( M = one_one_int )
& ( N2 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_2582_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_2583_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_2584_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_2585_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_2586_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_2587_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_2588_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_2589_q__pos__lemma,axiom,
! [B4: int,Q6: int,R3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q6 ) @ R3 ) )
=> ( ( ord_less_int @ R3 @ B4 )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ord_less_eq_int @ zero_zero_int @ Q6 ) ) ) ) ).
% q_pos_lemma
thf(fact_2590_zdiv__mono2__lemma,axiom,
! [B: int,Q3: int,R4: int,B4: int,Q6: int,R3: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R4 )
= ( plus_plus_int @ ( times_times_int @ B4 @ Q6 ) @ R3 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q6 ) @ R3 ) )
=> ( ( ord_less_int @ R3 @ B4 )
=> ( ( 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 @ Q3 @ Q6 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_2591_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q3: int,R4: int,B4: int,Q6: int,R3: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R4 )
= ( plus_plus_int @ ( times_times_int @ B4 @ Q6 ) @ R3 ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q6 ) @ R3 ) @ zero_zero_int )
=> ( ( ord_less_int @ R4 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R3 )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ Q6 @ Q3 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_2592_unique__quotient__lemma,axiom,
! [B: int,Q6: int,R3: int,Q3: int,R4: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R4 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R3 )
=> ( ( ord_less_int @ R3 @ B )
=> ( ( ord_less_int @ R4 @ B )
=> ( ord_less_eq_int @ Q6 @ Q3 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_2593_unique__quotient__lemma__neg,axiom,
! [B: int,Q6: int,R3: int,Q3: int,R4: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R4 ) )
=> ( ( ord_less_eq_int @ R4 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R4 )
=> ( ( ord_less_int @ B @ R3 )
=> ( ord_less_eq_int @ Q3 @ Q6 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_2594_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_2595_split__zdiv,axiom,
! [P: int > $o,N2: int,K: int] :
( ( P @ ( divide_divide_int @ N2 @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I3: int,J3: int] :
( ( ( ord_less_int @ K @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_zdiv
thf(fact_2596_int__div__neg__eq,axiom,
! [A: int,B: int,Q3: int,R4: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R4 ) )
=> ( ( ord_less_eq_int @ R4 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R4 )
=> ( ( divide_divide_int @ A @ B )
= Q3 ) ) ) ) ).
% int_div_neg_eq
thf(fact_2597_int__div__pos__eq,axiom,
! [A: int,B: int,Q3: int,R4: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R4 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R4 )
=> ( ( ord_less_int @ R4 @ B )
=> ( ( divide_divide_int @ A @ B )
= Q3 ) ) ) ) ).
% int_div_pos_eq
thf(fact_2598_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_2599_pinf_I6_J,axiom,
! [T2: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ~ ( ord_less_eq_rat @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_2600_pinf_I6_J,axiom,
! [T2: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ Z2 @ X4 )
=> ~ ( ord_less_eq_num @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_2601_pinf_I6_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_2602_pinf_I6_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_2603_pinf_I6_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ~ ( ord_less_eq_real @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_2604_pinf_I8_J,axiom,
! [T2: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ( ord_less_eq_rat @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_2605_pinf_I8_J,axiom,
! [T2: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ Z2 @ X4 )
=> ( ord_less_eq_num @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_2606_pinf_I8_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ord_less_eq_nat @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_2607_pinf_I8_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ord_less_eq_int @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_2608_pinf_I8_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ord_less_eq_real @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_2609_minf_I6_J,axiom,
! [T2: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ( ord_less_eq_rat @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_2610_minf_I6_J,axiom,
! [T2: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z2 )
=> ( ord_less_eq_num @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_2611_minf_I6_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ord_less_eq_nat @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_2612_minf_I6_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ord_less_eq_int @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_2613_minf_I6_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ord_less_eq_real @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_2614_minf_I8_J,axiom,
! [T2: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ~ ( ord_less_eq_rat @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_2615_minf_I8_J,axiom,
! [T2: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z2 )
=> ~ ( ord_less_eq_num @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_2616_minf_I8_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ~ ( ord_less_eq_nat @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_2617_minf_I8_J,axiom,
! [T2: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ~ ( ord_less_eq_int @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_2618_minf_I8_J,axiom,
! [T2: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ~ ( ord_less_eq_real @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_2619_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N: nat] :
( X
!= ( suc @ N ) ) ) ).
% list_decode.cases
thf(fact_2620_dvd__pos__nat,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( dvd_dvd_nat @ M @ N2 )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_2621_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
= ( P @ B5 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_2622_imp__le__cong,axiom,
! [X: int,X5: int,P: $o,P2: $o] :
( ( X = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_2623_conj__le__cong,axiom,
! [X: int,X5: int,P: $o,P2: $o] :
( ( X = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_2624_bezout__add__nat,axiom,
! [A: nat,B: nat] :
? [D4: nat,X2: nat,Y2: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( ( times_times_nat @ A @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ D4 ) )
| ( ( times_times_nat @ B @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y2 ) @ D4 ) ) ) ) ).
% bezout_add_nat
thf(fact_2625_bezout__lemma__nat,axiom,
! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ D @ B )
=> ( ( ( ( times_times_nat @ A @ X )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
| ( ( times_times_nat @ B @ X )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
=> ? [X2: nat,Y2: nat] :
( ( dvd_dvd_nat @ D @ A )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
& ( ( ( times_times_nat @ A @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y2 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y2 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_2626_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X: nat,N2: nat,M: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_2627_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_2628_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_2629_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_2630_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_2631_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_2632_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_2633_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_2634_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_2635_mult__less__iff1,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_2636_mult__less__iff1,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
= ( ord_less_rat @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_2637_mult__less__iff1,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_2638_minf_I10_J,axiom,
! [D: code_integer,S: code_integer] :
? [Z2: code_integer] :
! [X4: code_integer] :
( ( ord_le6747313008572928689nteger @ X4 @ Z2 )
=> ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) )
= ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_2639_minf_I10_J,axiom,
! [D: real,S: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_2640_minf_I10_J,axiom,
! [D: rat,S: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_2641_minf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_2642_minf_I10_J,axiom,
! [D: int,S: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_2643_minf_I9_J,axiom,
! [D: code_integer,S: code_integer] :
? [Z2: code_integer] :
! [X4: code_integer] :
( ( ord_le6747313008572928689nteger @ X4 @ Z2 )
=> ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) )
= ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ).
% minf(9)
thf(fact_2644_minf_I9_J,axiom,
! [D: real,S: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) ) ) ).
% minf(9)
thf(fact_2645_minf_I9_J,axiom,
! [D: rat,S: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) ) ) ).
% minf(9)
thf(fact_2646_minf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ).
% minf(9)
thf(fact_2647_minf_I9_J,axiom,
! [D: int,S: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ).
% minf(9)
thf(fact_2648_pinf_I10_J,axiom,
! [D: code_integer,S: code_integer] :
? [Z2: code_integer] :
! [X4: code_integer] :
( ( ord_le6747313008572928689nteger @ Z2 @ X4 )
=> ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) )
= ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_2649_pinf_I10_J,axiom,
! [D: real,S: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_2650_pinf_I10_J,axiom,
! [D: rat,S: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_2651_pinf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_2652_pinf_I10_J,axiom,
! [D: int,S: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_2653_pinf_I9_J,axiom,
! [D: code_integer,S: code_integer] :
? [Z2: code_integer] :
! [X4: code_integer] :
( ( ord_le6747313008572928689nteger @ Z2 @ X4 )
=> ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) )
= ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ).
% pinf(9)
thf(fact_2654_pinf_I9_J,axiom,
! [D: real,S: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) ) ) ).
% pinf(9)
thf(fact_2655_pinf_I9_J,axiom,
! [D: rat,S: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) ) ) ).
% pinf(9)
thf(fact_2656_pinf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ).
% pinf(9)
thf(fact_2657_pinf_I9_J,axiom,
! [D: int,S: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ).
% pinf(9)
thf(fact_2658_both__member__options__ding,axiom,
! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N2 )
=> ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_ding
thf(fact_2659_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X3: nat,N4: nat] : ( modulo_modulo_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% low_def
thf(fact_2660_triangle__def,axiom,
( nat_triangle
= ( ^ [N4: nat] : ( divide_divide_nat @ ( times_times_nat @ N4 @ ( suc @ N4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% triangle_def
thf(fact_2661_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_2662_set__decode__Suc,axiom,
! [N2: nat,X: nat] :
( ( member_nat @ ( suc @ N2 ) @ ( nat_set_decode @ X ) )
= ( member_nat @ N2 @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_2663_add__scale__eq__noteq,axiom,
! [R4: complex,A: complex,B: complex,C: complex,D: complex] :
( ( R4 != zero_zero_complex )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_complex @ A @ ( times_times_complex @ R4 @ C ) )
!= ( plus_plus_complex @ B @ ( times_times_complex @ R4 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_2664_add__scale__eq__noteq,axiom,
! [R4: real,A: real,B: real,C: real,D: real] :
( ( R4 != zero_zero_real )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_real @ A @ ( times_times_real @ R4 @ C ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R4 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_2665_add__scale__eq__noteq,axiom,
! [R4: rat,A: rat,B: rat,C: rat,D: rat] :
( ( R4 != zero_zero_rat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_rat @ A @ ( times_times_rat @ R4 @ C ) )
!= ( plus_plus_rat @ B @ ( times_times_rat @ R4 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_2666_add__scale__eq__noteq,axiom,
! [R4: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R4 != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R4 @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R4 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_2667_add__scale__eq__noteq,axiom,
! [R4: int,A: int,B: int,C: int,D: int] :
( ( R4 != zero_zero_int )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R4 @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R4 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_2668_even__succ__mod__exp,axiom,
! [A: nat,N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_2669_even__succ__mod__exp,axiom,
! [A: int,N2: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
= ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_2670_even__succ__mod__exp,axiom,
! [A: code_integer,N2: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_2671_even__mult__exp__div__exp__iff,axiom,
! [A: code_integer,M: nat,N2: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
= ( ( ord_less_nat @ N2 @ M )
| ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
= zero_z3403309356797280102nteger )
| ( ( ord_less_eq_nat @ M @ N2 )
& ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_2672_even__mult__exp__div__exp__iff,axiom,
! [A: nat,M: nat,N2: 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 ) ) @ N2 ) ) )
= ( ( ord_less_nat @ N2 @ M )
| ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
= zero_zero_nat )
| ( ( ord_less_eq_nat @ M @ N2 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_2673_even__mult__exp__div__exp__iff,axiom,
! [A: int,M: nat,N2: 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 ) ) @ N2 ) ) )
= ( ( ord_less_nat @ N2 @ M )
| ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
= zero_zero_int )
| ( ( ord_less_eq_nat @ M @ N2 )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_2674_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numera6690914467698888265omplex @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_2675_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_2676_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_rat @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_2677_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_2678_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_2679_inthall,axiom,
! [Xs: list_complex,P: complex > $o,N2: nat] :
( ! [X2: complex] :
( ( member_complex @ X2 @ ( set_complex2 @ Xs ) )
=> ( P @ X2 ) )
=> ( ( ord_less_nat @ N2 @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( P @ ( nth_complex @ Xs @ N2 ) ) ) ) ).
% inthall
thf(fact_2680_inthall,axiom,
! [Xs: list_real,P: real > $o,N2: nat] :
( ! [X2: real] :
( ( member_real @ X2 @ ( set_real2 @ Xs ) )
=> ( P @ X2 ) )
=> ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
=> ( P @ ( nth_real @ Xs @ N2 ) ) ) ) ).
% inthall
thf(fact_2681_inthall,axiom,
! [Xs: list_set_nat,P: set_nat > $o,N2: nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( P @ X2 ) )
=> ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( P @ ( nth_set_nat @ Xs @ N2 ) ) ) ) ).
% inthall
thf(fact_2682_inthall,axiom,
! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N2: nat] :
( ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P @ X2 ) )
=> ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).
% inthall
thf(fact_2683_inthall,axiom,
! [Xs: list_o,P: $o > $o,N2: nat] :
( ! [X2: $o] :
( ( member_o @ X2 @ ( set_o2 @ Xs ) )
=> ( P @ X2 ) )
=> ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
=> ( P @ ( nth_o @ Xs @ N2 ) ) ) ) ).
% inthall
thf(fact_2684_inthall,axiom,
! [Xs: list_nat,P: nat > $o,N2: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) )
=> ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ N2 ) ) ) ) ).
% inthall
thf(fact_2685_inthall,axiom,
! [Xs: list_int,P: int > $o,N2: nat] :
( ! [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ( P @ X2 ) )
=> ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
=> ( P @ ( nth_int @ Xs @ N2 ) ) ) ) ).
% inthall
thf(fact_2686_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_2687_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_2688_mod__mod__trivial,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mod_trivial
thf(fact_2689_real__divide__square__eq,axiom,
! [R4: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R4 @ A ) @ ( times_times_real @ R4 @ R4 ) )
= ( divide_divide_real @ A @ R4 ) ) ).
% real_divide_square_eq
thf(fact_2690_diff__self,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% diff_self
thf(fact_2691_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_2692_diff__self,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ A )
= zero_zero_rat ) ).
% diff_self
thf(fact_2693_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_2694_diff__0__right,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_0_right
thf(fact_2695_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_2696_diff__0__right,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ zero_zero_rat )
= A ) ).
% diff_0_right
thf(fact_2697_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_2698_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_2699_diff__zero,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_zero
thf(fact_2700_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_2701_diff__zero,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ zero_zero_rat )
= A ) ).
% diff_zero
thf(fact_2702_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_2703_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_2704_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_2705_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_2706_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ A )
= zero_zero_rat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_2707_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_2708_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_2709_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_2710_add__diff__cancel,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_2711_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_2712_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_2713_diff__add__cancel,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_2714_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_2715_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_2716_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_2717_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_2718_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_2719_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_2720_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_2721_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_2722_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_2723_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_2724_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_2725_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_2726_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_2727_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_2728_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_2729_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_2730_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_2731_bits__mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_2732_bits__mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_mod_0
thf(fact_2733_bits__mod__0,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
= zero_z3403309356797280102nteger ) ).
% bits_mod_0
thf(fact_2734_mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mod_0
thf(fact_2735_mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mod_0
thf(fact_2736_mod__0,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
= zero_z3403309356797280102nteger ) ).
% mod_0
thf(fact_2737_mod__by__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ zero_zero_nat )
= A ) ).
% mod_by_0
thf(fact_2738_mod__by__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ zero_zero_int )
= A ) ).
% mod_by_0
thf(fact_2739_mod__by__0,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
= A ) ).
% mod_by_0
thf(fact_2740_mod__self,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ A )
= zero_zero_nat ) ).
% mod_self
thf(fact_2741_mod__self,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ A )
= zero_zero_int ) ).
% mod_self
thf(fact_2742_mod__self,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ A )
= zero_z3403309356797280102nteger ) ).
% mod_self
thf(fact_2743_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_2744_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_2745_mod__add__self1,axiom,
! [B: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_add_self1
thf(fact_2746_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_2747_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_2748_mod__add__self2,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_add_self2
thf(fact_2749_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_2750_minus__mod__self2,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% minus_mod_self2
thf(fact_2751_diff__Suc__Suc,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_2752_Suc__diff__diff,axiom,
! [M: nat,N2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_2753_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_2754_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_2755_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_2756_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_2757_mod__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( modulo_modulo_nat @ M @ N2 )
= M ) ) ).
% mod_less
thf(fact_2758_triangle__0,axiom,
( ( nat_triangle @ zero_zero_nat )
= zero_zero_nat ) ).
% triangle_0
thf(fact_2759_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_2760_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_2761_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_2762_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_2763_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_2764_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_2765_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_2766_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_2767_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_2768_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_2769_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_2770_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_2771_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_2772_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_2773_diff__numeral__special_I9_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% diff_numeral_special(9)
thf(fact_2774_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_2775_diff__numeral__special_I9_J,axiom,
( ( minus_minus_rat @ one_one_rat @ one_one_rat )
= zero_zero_rat ) ).
% diff_numeral_special(9)
thf(fact_2776_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_2777_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_2778_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_2779_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_2780_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_2781_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_2782_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_2783_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_2784_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_2785_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_2786_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_2787_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_2788_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_2789_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_2790_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_2791_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_2792_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_2793_bits__mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_2794_bits__mod__by__1,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% bits_mod_by_1
thf(fact_2795_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_2796_mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_2797_mod__by__1,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% mod_by_1
thf(fact_2798_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_2799_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_2800_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_2801_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_2802_mod__div__trivial,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
= zero_z3403309356797280102nteger ) ).
% mod_div_trivial
thf(fact_2803_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_2804_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_2805_bits__mod__div__trivial,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
= zero_z3403309356797280102nteger ) ).
% bits_mod_div_trivial
thf(fact_2806_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_2807_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_2808_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_2809_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_2810_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_2811_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_2812_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_2813_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_2814_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_2815_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_2816_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_2817_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_2818_dvd__imp__mod__0,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( modulo_modulo_nat @ B @ A )
= zero_zero_nat ) ) ).
% dvd_imp_mod_0
thf(fact_2819_dvd__imp__mod__0,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( modulo_modulo_int @ B @ A )
= zero_zero_int ) ) ).
% dvd_imp_mod_0
thf(fact_2820_dvd__imp__mod__0,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( modulo364778990260209775nteger @ B @ A )
= zero_z3403309356797280102nteger ) ) ).
% dvd_imp_mod_0
thf(fact_2821_zero__less__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% zero_less_diff
thf(fact_2822_diff__is__0__eq,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% diff_is_0_eq
thf(fact_2823_diff__is__0__eq_H,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_2824_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_2825_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_2826_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_2827_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_2828_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_2829_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( modulo_modulo_int @ K @ L )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_2830_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( modulo_modulo_int @ K @ L )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_2831_triangle__Suc,axiom,
! [N2: nat] :
( ( nat_triangle @ ( suc @ N2 ) )
= ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).
% triangle_Suc
thf(fact_2832_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_2833_zmod__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_2834_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_2835_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_2836_Suc__mod__mult__self4,axiom,
! [N2: nat,K: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N2 @ K ) @ M ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_mod_mult_self4
thf(fact_2837_Suc__mod__mult__self3,axiom,
! [K: nat,N2: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N2 ) @ M ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_mod_mult_self3
thf(fact_2838_Suc__mod__mult__self2,axiom,
! [M: nat,N2: nat,K: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N2 @ K ) ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_mod_mult_self2
thf(fact_2839_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N2 ) ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_mod_mult_self1
thf(fact_2840_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_2841_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_2842_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_2843_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_2844_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_2845_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_2846_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_2847_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_2848_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_2849_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_2850_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_2851_Suc__times__numeral__mod__eq,axiom,
! [K: num,N2: nat] :
( ( ( numeral_numeral_nat @ K )
!= one_one_nat )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N2 ) ) @ ( numeral_numeral_nat @ K ) )
= one_one_nat ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_2852_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_2853_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_2854_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_2855_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_2856_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_2857_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_2858_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_2859_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_2860_not__mod2__eq__Suc__0__eq__0,axiom,
! [N2: nat] :
( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= ( suc @ zero_zero_nat ) )
= ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% not_mod2_eq_Suc_0_eq_0
thf(fact_2861_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_2862_odd__Suc__minus__one,axiom,
! [N2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% odd_Suc_minus_one
thf(fact_2863_even__diff__nat,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) )
= ( ( ord_less_nat @ M @ N2 )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% even_diff_nat
thf(fact_2864_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_2865_semiring__parity__class_Oeven__mask__iff,axiom,
! [N2: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) )
= ( N2 = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_2866_semiring__parity__class_Oeven__mask__iff,axiom,
! [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_2867_semiring__parity__class_Oeven__mask__iff,axiom,
! [N2: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
= ( N2 = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_2868_odd__two__times__div__two__nat,axiom,
! [N2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_2869_dvd__minus__mod,axiom,
! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% dvd_minus_mod
thf(fact_2870_dvd__minus__mod,axiom,
! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).
% dvd_minus_mod
thf(fact_2871_dvd__minus__mod,axiom,
! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% dvd_minus_mod
thf(fact_2872_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_2873_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_2874_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_2875_diff__eq__diff__eq,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ( minus_minus_rat @ A @ B )
= ( minus_minus_rat @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_2876_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_2877_diff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_2878_diff__right__commute,axiom,
! [A: rat,C: rat,B: rat] :
( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
= ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_2879_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_2880_diff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_2881_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_2882_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_2883_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_2884_mod__diff__cong,axiom,
! [A: int,C: int,A4: int,B: int,B4: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A4 @ 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 @ A4 @ B4 ) @ C ) ) ) ) ).
% mod_diff_cong
thf(fact_2885_mod__diff__cong,axiom,
! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ A4 @ C ) )
=> ( ( ( modulo364778990260209775nteger @ B @ C )
= ( modulo364778990260209775nteger @ B4 @ C ) )
=> ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
= ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A4 @ B4 ) @ C ) ) ) ) ).
% mod_diff_cong
thf(fact_2886_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_2887_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_2888_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_2889_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_2890_le__mod__geq,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( modulo_modulo_nat @ M @ N2 )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).
% le_mod_geq
thf(fact_2891_mod__geq,axiom,
! [M: nat,N2: nat] :
( ~ ( ord_less_nat @ M @ N2 )
=> ( ( modulo_modulo_nat @ M @ N2 )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).
% mod_geq
thf(fact_2892_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M2: nat,N4: nat] : ( if_nat @ ( ord_less_nat @ M2 @ N4 ) @ M2 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N4 ) @ N4 ) ) ) ) ).
% mod_if
thf(fact_2893_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_2894_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_2895_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_2896_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_2897_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_2898_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_2899_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_2900_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_2901_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_2902_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_2903_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_2904_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_2905_mod__eq__dvd__iff__nat,axiom,
! [N2: nat,M: nat,Q3: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( ( modulo_modulo_nat @ M @ Q3 )
= ( modulo_modulo_nat @ N2 @ Q3 ) )
= ( dvd_dvd_nat @ Q3 @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_2906_modulo__nat__def,axiom,
( modulo_modulo_nat
= ( ^ [M2: nat,N4: nat] : ( minus_minus_nat @ M2 @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N4 ) @ N4 ) ) ) ) ).
% modulo_nat_def
thf(fact_2907_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_2908_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_2909_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_2910_mod__mult__cong,axiom,
! [A: nat,C: nat,A4: nat,B: nat,B4: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A4 @ 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 @ A4 @ B4 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_2911_mod__mult__cong,axiom,
! [A: int,C: int,A4: int,B: int,B4: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A4 @ 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 @ A4 @ B4 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_2912_mod__mult__cong,axiom,
! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ A4 @ C ) )
=> ( ( ( modulo364778990260209775nteger @ B @ C )
= ( modulo364778990260209775nteger @ B4 @ C ) )
=> ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A4 @ B4 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_2913_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_2914_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_2915_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_2916_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_2917_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_2918_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_2919_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_2920_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_2921_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_2922_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_2923_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_2924_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_2925_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_2926_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_2927_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_2928_mod__add__cong,axiom,
! [A: nat,C: nat,A4: nat,B: nat,B4: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A4 @ 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 @ A4 @ B4 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_2929_mod__add__cong,axiom,
! [A: int,C: int,A4: int,B: int,B4: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A4 @ 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 @ A4 @ B4 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_2930_mod__add__cong,axiom,
! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ A4 @ C ) )
=> ( ( ( modulo364778990260209775nteger @ B @ C )
= ( modulo364778990260209775nteger @ B4 @ C ) )
=> ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A4 @ B4 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_2931_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_2932_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_2933_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_2934_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_2935_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_2936_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_2937_power__mod,axiom,
! [A: nat,B: nat,N2: nat] :
( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N2 ) @ B )
= ( modulo_modulo_nat @ ( power_power_nat @ A @ N2 ) @ B ) ) ).
% power_mod
thf(fact_2938_power__mod,axiom,
! [A: int,B: int,N2: nat] :
( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N2 ) @ B )
= ( modulo_modulo_int @ ( power_power_int @ A @ N2 ) @ B ) ) ).
% power_mod
thf(fact_2939_power__mod,axiom,
! [A: code_integer,B: code_integer,N2: nat] :
( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N2 ) @ B )
= ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ B ) ) ).
% power_mod
thf(fact_2940_dvd__mod__iff,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
= ( dvd_dvd_nat @ C @ A ) ) ) ).
% dvd_mod_iff
thf(fact_2941_dvd__mod__iff,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
= ( dvd_dvd_int @ C @ A ) ) ) ).
% dvd_mod_iff
thf(fact_2942_dvd__mod__iff,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
= ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% dvd_mod_iff
thf(fact_2943_dvd__mod__imp__dvd,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( dvd_dvd_nat @ C @ A ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_2944_dvd__mod__imp__dvd,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
=> ( ( dvd_dvd_int @ C @ B )
=> ( dvd_dvd_int @ C @ A ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_2945_dvd__mod__imp__dvd,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
=> ( ( dvd_dvd_Code_integer @ C @ B )
=> ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_2946_dvd__mod,axiom,
! [K: nat,M: nat,N2: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N2 )
=> ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ).
% dvd_mod
thf(fact_2947_dvd__mod,axiom,
! [K: int,M: int,N2: int] :
( ( dvd_dvd_int @ K @ M )
=> ( ( dvd_dvd_int @ K @ N2 )
=> ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N2 ) ) ) ) ).
% dvd_mod
thf(fact_2948_dvd__mod,axiom,
! [K: code_integer,M: code_integer,N2: code_integer] :
( ( dvd_dvd_Code_integer @ K @ M )
=> ( ( dvd_dvd_Code_integer @ K @ N2 )
=> ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N2 ) ) ) ) ).
% dvd_mod
thf(fact_2949_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_2950_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_2951_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_2952_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_2953_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_2954_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_2955_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_2956_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_2957_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_2958_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_2959_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_2960_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_2961_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_2962_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_2963_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_2964_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: complex,Z4: complex] : Y5 = Z4 )
= ( ^ [A3: complex,B2: complex] :
( ( minus_minus_complex @ A3 @ B2 )
= zero_zero_complex ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_2965_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: real,Z4: real] : Y5 = Z4 )
= ( ^ [A3: real,B2: real] :
( ( minus_minus_real @ A3 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_2966_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: rat,Z4: rat] : Y5 = Z4 )
= ( ^ [A3: rat,B2: rat] :
( ( minus_minus_rat @ A3 @ B2 )
= zero_zero_rat ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_2967_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: int,Z4: int] : Y5 = Z4 )
= ( ^ [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_2968_mod__Suc__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N2 ) ) ).
% mod_Suc_Suc_eq
thf(fact_2969_mod__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% mod_Suc_eq
thf(fact_2970_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_2971_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_2972_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_2973_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_2974_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_2975_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_2976_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_2977_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_2978_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_2979_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_2980_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_2981_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_2982_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_2983_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_2984_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_2985_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_2986_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_2987_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_2988_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_2989_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_2990_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_2991_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_2992_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_2993_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_2994_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_2995_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_2996_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_2997_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_2998_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_2999_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_3000_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_3001_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_3002_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_3003_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_3004_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_3005_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_3006_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_3007_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_3008_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_3009_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_3010_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_3011_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_3012_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_3013_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_3014_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_3015_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_3016_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_3017_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_3018_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_3019_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_3020_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_3021_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_3022_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_3023_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_3024_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_3025_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_3026_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_3027_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_3028_zmod__eq__0D,axiom,
! [M: int,D: int] :
( ( ( modulo_modulo_int @ M @ D )
= zero_zero_int )
=> ? [Q2: int] :
( M
= ( times_times_int @ D @ Q2 ) ) ) ).
% zmod_eq_0D
thf(fact_3029_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_3030_mod__less__eq__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ M ) ).
% mod_less_eq_dividend
thf(fact_3031_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_3032_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_3033_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_3034_dvd__diff,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( ( dvd_dvd_Code_integer @ X @ Y )
=> ( ( dvd_dvd_Code_integer @ X @ Z )
=> ( dvd_dvd_Code_integer @ X @ ( minus_8373710615458151222nteger @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_3035_dvd__diff,axiom,
! [X: real,Y: real,Z: real] :
( ( dvd_dvd_real @ X @ Y )
=> ( ( dvd_dvd_real @ X @ Z )
=> ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_3036_dvd__diff,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( dvd_dvd_rat @ X @ Y )
=> ( ( dvd_dvd_rat @ X @ Z )
=> ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_3037_dvd__diff,axiom,
! [X: int,Y: int,Z: int] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( dvd_dvd_int @ X @ Z )
=> ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_3038_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_3039_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_3040_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N: nat] :
( ( P @ ( suc @ N ) )
=> ( P @ N ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_3041_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_3042_diffs0__imp__equal,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M )
= zero_zero_nat )
=> ( M = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_3043_diff__less__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_3044_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_3045_eq__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_3046_le__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_3047_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_3048_diff__le__mono,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_3049_diff__le__self,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).
% diff_le_self
thf(fact_3050_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_3051_diff__le__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_3052_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_3053_diff__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M @ N2 ) ) ).
% diff_cancel2
thf(fact_3054_diff__add__inverse,axiom,
! [N2: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
= M ) ).
% diff_add_inverse
thf(fact_3055_diff__add__inverse2,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
= M ) ).
% diff_add_inverse2
thf(fact_3056_diff__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_3057_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_3058_dvd__diff__nat,axiom,
! [K: nat,M: nat,N2: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N2 )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% dvd_diff_nat
thf(fact_3059_mod__nat__eqI,axiom,
! [R4: nat,N2: nat,M: nat] :
( ( ord_less_nat @ R4 @ N2 )
=> ( ( ord_less_eq_nat @ R4 @ M )
=> ( ( dvd_dvd_nat @ N2 @ ( minus_minus_nat @ M @ R4 ) )
=> ( ( modulo_modulo_nat @ M @ N2 )
= R4 ) ) ) ) ).
% mod_nat_eqI
thf(fact_3060_subset__decode__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% subset_decode_imp_le
thf(fact_3061_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_3062_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_3063_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_3064_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_3065_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_3066_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_3067_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_3068_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_3069_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_3070_cong__exp__iff__simps_I9_J,axiom,
! [M: num,Q3: num,N2: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_3071_cong__exp__iff__simps_I9_J,axiom,
! [M: num,Q3: num,N2: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_3072_cong__exp__iff__simps_I9_J,axiom,
! [M: num,Q3: num,N2: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_3073_cong__exp__iff__simps_I4_J,axiom,
! [M: num,N2: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ one ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_3074_cong__exp__iff__simps_I4_J,axiom,
! [M: num,N2: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ one ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_3075_cong__exp__iff__simps_I4_J,axiom,
! [M: num,N2: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_3076_mod__eqE,axiom,
! [A: int,C: int,B: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ B @ C ) )
=> ~ ! [D4: int] :
( B
!= ( plus_plus_int @ A @ ( times_times_int @ C @ D4 ) ) ) ) ).
% mod_eqE
thf(fact_3077_mod__eqE,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ B @ C ) )
=> ~ ! [D4: code_integer] :
( B
!= ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D4 ) ) ) ) ).
% mod_eqE
thf(fact_3078_mod__eq__0__iff__dvd,axiom,
! [A: nat,B: nat] :
( ( ( modulo_modulo_nat @ A @ B )
= zero_zero_nat )
= ( dvd_dvd_nat @ B @ A ) ) ).
% mod_eq_0_iff_dvd
thf(fact_3079_mod__eq__0__iff__dvd,axiom,
! [A: int,B: int] :
( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
= ( dvd_dvd_int @ B @ A ) ) ).
% mod_eq_0_iff_dvd
thf(fact_3080_mod__eq__0__iff__dvd,axiom,
! [A: code_integer,B: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ B )
= zero_z3403309356797280102nteger )
= ( dvd_dvd_Code_integer @ B @ A ) ) ).
% mod_eq_0_iff_dvd
thf(fact_3081_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_nat
= ( ^ [A3: nat,B2: nat] :
( ( modulo_modulo_nat @ B2 @ A3 )
= zero_zero_nat ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_3082_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_int
= ( ^ [A3: int,B2: int] :
( ( modulo_modulo_int @ B2 @ A3 )
= zero_zero_int ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_3083_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_Code_integer
= ( ^ [A3: code_integer,B2: code_integer] :
( ( modulo364778990260209775nteger @ B2 @ A3 )
= zero_z3403309356797280102nteger ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_3084_mod__0__imp__dvd,axiom,
! [A: nat,B: nat] :
( ( ( modulo_modulo_nat @ A @ B )
= zero_zero_nat )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% mod_0_imp_dvd
thf(fact_3085_mod__0__imp__dvd,axiom,
! [A: int,B: int] :
( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% mod_0_imp_dvd
thf(fact_3086_mod__0__imp__dvd,axiom,
! [A: code_integer,B: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ B )
= zero_z3403309356797280102nteger )
=> ( dvd_dvd_Code_integer @ B @ A ) ) ).
% mod_0_imp_dvd
thf(fact_3087_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_3088_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_3089_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_3090_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_VEBT_VEBT,Z4: list_VEBT_VEBT] : Y5 = Z4 )
= ( ^ [Xs3: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs3 )
= ( size_s6755466524823107622T_VEBT @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
=> ( ( nth_VEBT_VEBT @ Xs3 @ I3 )
= ( nth_VEBT_VEBT @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_3091_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_o,Z4: list_o] : Y5 = Z4 )
= ( ^ [Xs3: list_o,Ys3: list_o] :
( ( ( size_size_list_o @ Xs3 )
= ( size_size_list_o @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs3 ) )
=> ( ( nth_o @ Xs3 @ I3 )
= ( nth_o @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_3092_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_nat,Z4: list_nat] : Y5 = Z4 )
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I3 )
= ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_3093_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_int,Z4: list_int] : Y5 = Z4 )
= ( ^ [Xs3: list_int,Ys3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs3 ) )
=> ( ( nth_int @ Xs3 @ I3 )
= ( nth_int @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_3094_Skolem__list__nth,axiom,
! [K: nat,P: nat > vEBT_VEBT > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X6: vEBT_VEBT] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs3: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_VEBT_VEBT @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_3095_Skolem__list__nth,axiom,
! [K: nat,P: nat > $o > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X6: $o] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs3: list_o] :
( ( ( size_size_list_o @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_o @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_3096_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X6: nat] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_3097_Skolem__list__nth,axiom,
! [K: nat,P: nat > int > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X6: int] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_int @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_3098_nth__equalityI,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_s6755466524823107622T_VEBT @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( nth_VEBT_VEBT @ Xs @ I2 )
= ( nth_VEBT_VEBT @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_3099_nth__equalityI,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
=> ( ( nth_o @ Xs @ I2 )
= ( nth_o @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_3100_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_3101_nth__equalityI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ Xs @ I2 )
= ( nth_int @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_3102_le__iff__diff__le__0,axiom,
( ord_less_eq_rat
= ( ^ [A3: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B2 ) @ zero_zero_rat ) ) ) ).
% le_iff_diff_le_0
thf(fact_3103_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_3104_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_3105_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_3106_less__iff__diff__less__0,axiom,
( ord_less_rat
= ( ^ [A3: rat,B2: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B2 ) @ zero_zero_rat ) ) ) ).
% less_iff_diff_less_0
thf(fact_3107_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_3108_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_3109_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_3110_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_3111_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_3112_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_3113_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_3114_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_3115_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_3116_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_3117_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_3118_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_3119_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_3120_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_3121_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_3122_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_3123_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_3124_add__le__imp__le__diff,axiom,
! [I: rat,K: rat,N2: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
=> ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N2 @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_3125_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_3126_add__le__imp__le__diff,axiom,
! [I: int,K: int,N2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N2 @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_3127_add__le__imp__le__diff,axiom,
! [I: real,K: real,N2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
=> ( ord_less_eq_real @ I @ ( minus_minus_real @ N2 @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_3128_add__le__add__imp__diff__le,axiom,
! [I: rat,K: rat,N2: rat,J: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
=> ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
=> ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
=> ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ N2 @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_3129_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N2: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_3130_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N2: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
=> ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
=> ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N2 @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_3131_add__le__add__imp__diff__le,axiom,
! [I: real,K: real,N2: real,J: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
=> ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
=> ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N2 @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_3132_mod__Suc,axiom,
! [M: nat,N2: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
= N2 )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
!= N2 )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
= ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ).
% mod_Suc
thf(fact_3133_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_3134_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_3135_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_3136_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_3137_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_3138_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_3139_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_3140_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_3141_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_3142_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_3143_mod__induct,axiom,
! [P: nat > $o,N2: nat,P3: nat,M: nat] :
( ( P @ N2 )
=> ( ( ord_less_nat @ N2 @ P3 )
=> ( ( ord_less_nat @ M @ P3 )
=> ( ! [N: nat] :
( ( ord_less_nat @ N @ P3 )
=> ( ( P @ N )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N ) @ P3 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_3144_mod__less__divisor,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).
% mod_less_divisor
thf(fact_3145_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N2: nat] :
( ! [M5: nat] : ( P @ M5 @ zero_zero_nat )
=> ( ! [M5: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ N @ ( modulo_modulo_nat @ M5 @ N ) )
=> ( P @ M5 @ N ) ) )
=> ( P @ M @ N2 ) ) ) ).
% gcd_nat_induct
thf(fact_3146_eq__add__iff1,axiom,
! [A: real,E2: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_3147_eq__add__iff1,axiom,
! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_3148_eq__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_3149_eq__add__iff2,axiom,
! [A: real,E2: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( C
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_3150_eq__add__iff2,axiom,
! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( C
= ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_3151_eq__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_3152_square__diff__square__factored,axiom,
! [X: real,Y: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_3153_square__diff__square__factored,axiom,
! [X: rat,Y: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
= ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_3154_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_3155_mult__diff__mult,axiom,
! [X: real,Y: real,A: real,B: real] :
( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_3156_mult__diff__mult,axiom,
! [X: rat,Y: rat,A: rat,B: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X @ Y ) @ ( times_times_rat @ A @ B ) )
= ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_3157_mult__diff__mult,axiom,
! [X: int,Y: int,A: int,B: int] :
( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_3158_mod__Suc__le__divisor,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N2 ) ) @ N2 ) ).
% mod_Suc_le_divisor
thf(fact_3159_mod__eq__0D,axiom,
! [M: nat,D: nat] :
( ( ( modulo_modulo_nat @ M @ D )
= zero_zero_nat )
=> ? [Q2: nat] :
( M
= ( times_times_nat @ D @ Q2 ) ) ) ).
% mod_eq_0D
thf(fact_3160_zmod__le__nonneg__dividend,axiom,
! [M: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% zmod_le_nonneg_dividend
thf(fact_3161_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_3162_neg__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% neg_mod_bound
thf(fact_3163_nat__mod__eq__iff,axiom,
! [X: nat,N2: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X @ N2 )
= ( modulo_modulo_nat @ Y @ N2 ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus_nat @ X @ ( times_times_nat @ N2 @ Q1 ) )
= ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_3164_Suc__diff__Suc,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_3165_diff__less__Suc,axiom,
! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_3166_diff__less,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).
% diff_less
thf(fact_3167_Suc__diff__le,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
= ( suc @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_3168_less__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_3169_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_3170_diff__add__0,axiom,
! [N2: nat,M: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_3171_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_3172_add__diff__inverse__nat,axiom,
! [M: nat,N2: nat] :
( ~ ( ord_less_nat @ M @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M @ N2 ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_3173_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_3174_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_3175_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_3176_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_3177_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_3178_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ M @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_3179_dvd__minus__self,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) )
= ( ( ord_less_nat @ N2 @ M )
| ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% dvd_minus_self
thf(fact_3180_dvd__diffD,axiom,
! [K: nat,M: nat,N2: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
=> ( ( dvd_dvd_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_3181_dvd__diffD1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
=> ( ( dvd_dvd_nat @ K @ M )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dvd_nat @ K @ N2 ) ) ) ) ).
% dvd_diffD1
thf(fact_3182_less__eq__dvd__minus,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( dvd_dvd_nat @ M @ N2 )
= ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_3183_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one )
= X ) ).
% pow.simps(1)
thf(fact_3184_mult__exp__mod__exp__eq,axiom,
! [M: nat,N2: nat,A: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_3185_mult__exp__mod__exp__eq,axiom,
! [M: nat,N2: nat,A: int] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
= ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_3186_mult__exp__mod__exp__eq,axiom,
! [M: nat,N2: nat,A: code_integer] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
= ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_3187_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_3188_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_3189_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_3190_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_3191_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_3192_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_3193_cong__exp__iff__simps_I2_J,axiom,
! [N2: num,Q3: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
= zero_zero_nat )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q3 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(2)
thf(fact_3194_cong__exp__iff__simps_I2_J,axiom,
! [N2: num,Q3: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
= zero_zero_int )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q3 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(2)
thf(fact_3195_cong__exp__iff__simps_I2_J,axiom,
! [N2: num,Q3: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
= zero_z3403309356797280102nteger )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q3 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(2)
thf(fact_3196_cong__exp__iff__simps_I1_J,axiom,
! [N2: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ one ) )
= zero_zero_nat ) ).
% cong_exp_iff_simps(1)
thf(fact_3197_cong__exp__iff__simps_I1_J,axiom,
! [N2: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ one ) )
= zero_zero_int ) ).
% cong_exp_iff_simps(1)
thf(fact_3198_cong__exp__iff__simps_I1_J,axiom,
! [N2: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) )
= zero_z3403309356797280102nteger ) ).
% cong_exp_iff_simps(1)
thf(fact_3199_cong__exp__iff__simps_I8_J,axiom,
! [M: num,Q3: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_3200_cong__exp__iff__simps_I8_J,axiom,
! [M: num,Q3: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_3201_cong__exp__iff__simps_I8_J,axiom,
! [M: num,Q3: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
!= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_3202_cong__exp__iff__simps_I6_J,axiom,
! [Q3: num,N2: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_3203_cong__exp__iff__simps_I6_J,axiom,
! [Q3: num,N2: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_3204_cong__exp__iff__simps_I6_J,axiom,
! [Q3: num,N2: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
!= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_3205_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_3206_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_3207_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_3208_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_3209_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_3210_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_3211_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_3212_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_3213_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_3214_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_3215_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_3216_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_3217_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_3218_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_3219_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_3220_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_3221_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_3222_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_3223_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_3224_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_3225_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_3226_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_3227_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_3228_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_3229_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_3230_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_3231_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_3232_nth__mem,axiom,
! [N2: nat,Xs: list_complex] :
( ( ord_less_nat @ N2 @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( member_complex @ ( nth_complex @ Xs @ N2 ) @ ( set_complex2 @ Xs ) ) ) ).
% nth_mem
thf(fact_3233_nth__mem,axiom,
! [N2: nat,Xs: list_real] :
( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
=> ( member_real @ ( nth_real @ Xs @ N2 ) @ ( set_real2 @ Xs ) ) ) ).
% nth_mem
thf(fact_3234_nth__mem,axiom,
! [N2: nat,Xs: list_set_nat] :
( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( member_set_nat @ ( nth_set_nat @ Xs @ N2 ) @ ( set_set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_3235_nth__mem,axiom,
! [N2: nat,Xs: list_VEBT_VEBT] :
( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N2 ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% nth_mem
thf(fact_3236_nth__mem,axiom,
! [N2: nat,Xs: list_o] :
( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
=> ( member_o @ ( nth_o @ Xs @ N2 ) @ ( set_o2 @ Xs ) ) ) ).
% nth_mem
thf(fact_3237_nth__mem,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N2 ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_3238_nth__mem,axiom,
! [N2: nat,Xs: list_int] :
( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
=> ( member_int @ ( nth_int @ Xs @ N2 ) @ ( set_int2 @ Xs ) ) ) ).
% nth_mem
thf(fact_3239_list__ball__nth,axiom,
! [N2: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_3240_list__ball__nth,axiom,
! [N2: nat,Xs: list_o,P: $o > $o] :
( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ ( set_o2 @ Xs ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_o @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_3241_list__ball__nth,axiom,
! [N2: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_nat @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_3242_list__ball__nth,axiom,
! [N2: nat,Xs: list_int,P: int > $o] :
( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_int @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_3243_in__set__conv__nth,axiom,
! [X: complex,Xs: list_complex] :
( ( member_complex @ X @ ( set_complex2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) )
& ( ( nth_complex @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_3244_in__set__conv__nth,axiom,
! [X: real,Xs: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
& ( ( nth_real @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_3245_in__set__conv__nth,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs ) )
& ( ( nth_set_nat @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_3246_in__set__conv__nth,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
& ( ( nth_VEBT_VEBT @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_3247_in__set__conv__nth,axiom,
! [X: $o,Xs: list_o] :
( ( member_o @ X @ ( set_o2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
& ( ( nth_o @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_3248_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_3249_in__set__conv__nth,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
& ( ( nth_int @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_3250_all__nth__imp__all__set,axiom,
! [Xs: list_complex,P: complex > $o,X: complex] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( P @ ( nth_complex @ Xs @ I2 ) ) )
=> ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_3251_all__nth__imp__all__set,axiom,
! [Xs: list_real,P: real > $o,X: real] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
=> ( P @ ( nth_real @ Xs @ I2 ) ) )
=> ( ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_3252_all__nth__imp__all__set,axiom,
! [Xs: list_set_nat,P: set_nat > $o,X: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( P @ ( nth_set_nat @ Xs @ I2 ) ) )
=> ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_3253_all__nth__imp__all__set,axiom,
! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) )
=> ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_3254_all__nth__imp__all__set,axiom,
! [Xs: list_o,P: $o > $o,X: $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
=> ( P @ ( nth_o @ Xs @ I2 ) ) )
=> ( ( member_o @ X @ ( set_o2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_3255_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_3256_all__nth__imp__all__set,axiom,
! [Xs: list_int,P: int > $o,X: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( P @ ( nth_int @ Xs @ I2 ) ) )
=> ( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_3257_all__set__conv__all__nth,axiom,
! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_3258_all__set__conv__all__nth,axiom,
! [Xs: list_o,P: $o > $o] :
( ( ! [X3: $o] :
( ( member_o @ X3 @ ( set_o2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
=> ( P @ ( nth_o @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_3259_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_3260_all__set__conv__all__nth,axiom,
! [Xs: list_int,P: int > $o] :
( ( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( P @ ( nth_int @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_3261_ordered__ring__class_Ole__add__iff1,axiom,
! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_3262_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_3263_ordered__ring__class_Ole__add__iff1,axiom,
! [A: real,E2: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_3264_ordered__ring__class_Ole__add__iff2,axiom,
! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_3265_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_3266_ordered__ring__class_Ole__add__iff2,axiom,
! [A: real,E2: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_3267_less__add__iff2,axiom,
! [A: real,E2: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% less_add_iff2
thf(fact_3268_less__add__iff2,axiom,
! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% less_add_iff2
thf(fact_3269_less__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% less_add_iff2
thf(fact_3270_less__add__iff1,axiom,
! [A: real,E2: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_3271_less__add__iff1,axiom,
! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_3272_less__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_3273_add__divide__eq__if__simps_I4_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_3274_add__divide__eq__if__simps_I4_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_3275_add__divide__eq__if__simps_I4_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_3276_diff__frac__eq,axiom,
! [Y: complex,Z: complex,X: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_3277_diff__frac__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_3278_diff__frac__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_3279_diff__divide__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_3280_diff__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_3281_diff__divide__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_3282_divide__diff__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_3283_divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_3284_divide__diff__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
= ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_3285_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_3286_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_3287_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_3288_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_3289_mod__le__divisor,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).
% mod_le_divisor
thf(fact_3290_inf__period_I3_J,axiom,
! [D: code_integer,D5: code_integer,T2: code_integer] :
( ( dvd_dvd_Code_integer @ D @ D5 )
=> ! [X4: code_integer,K4: code_integer] :
( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ T2 ) )
= ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K4 @ D5 ) ) @ T2 ) ) ) ) ).
% inf_period(3)
thf(fact_3291_inf__period_I3_J,axiom,
! [D: real,D5: real,T2: real] :
( ( dvd_dvd_real @ D @ D5 )
=> ! [X4: real,K4: real] :
( ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ T2 ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X4 @ ( times_times_real @ K4 @ D5 ) ) @ T2 ) ) ) ) ).
% inf_period(3)
thf(fact_3292_inf__period_I3_J,axiom,
! [D: rat,D5: rat,T2: rat] :
( ( dvd_dvd_rat @ D @ D5 )
=> ! [X4: rat,K4: rat] :
( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ T2 ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K4 @ D5 ) ) @ T2 ) ) ) ) ).
% inf_period(3)
thf(fact_3293_inf__period_I3_J,axiom,
! [D: int,D5: int,T2: int] :
( ( dvd_dvd_int @ D @ D5 )
=> ! [X4: int,K4: int] :
( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T2 ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X4 @ ( times_times_int @ K4 @ D5 ) ) @ T2 ) ) ) ) ).
% inf_period(3)
thf(fact_3294_inf__period_I4_J,axiom,
! [D: code_integer,D5: code_integer,T2: code_integer] :
( ( dvd_dvd_Code_integer @ D @ D5 )
=> ! [X4: code_integer,K4: code_integer] :
( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ T2 ) ) )
= ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K4 @ D5 ) ) @ T2 ) ) ) ) ) ).
% inf_period(4)
thf(fact_3295_inf__period_I4_J,axiom,
! [D: real,D5: real,T2: real] :
( ( dvd_dvd_real @ D @ D5 )
=> ! [X4: real,K4: real] :
( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ T2 ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X4 @ ( times_times_real @ K4 @ D5 ) ) @ T2 ) ) ) ) ) ).
% inf_period(4)
thf(fact_3296_inf__period_I4_J,axiom,
! [D: rat,D5: rat,T2: rat] :
( ( dvd_dvd_rat @ D @ D5 )
=> ! [X4: rat,K4: rat] :
( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ T2 ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K4 @ D5 ) ) @ T2 ) ) ) ) ) ).
% inf_period(4)
thf(fact_3297_inf__period_I4_J,axiom,
! [D: int,D5: int,T2: int] :
( ( dvd_dvd_int @ D @ D5 )
=> ! [X4: int,K4: int] :
( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T2 ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X4 @ ( times_times_int @ K4 @ D5 ) ) @ T2 ) ) ) ) ) ).
% inf_period(4)
thf(fact_3298_mod__greater__zero__iff__not__dvd,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N2 ) )
= ( ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_3299_mod__eq__nat1E,axiom,
! [M: nat,Q3: nat,N2: nat] :
( ( ( modulo_modulo_nat @ M @ Q3 )
= ( modulo_modulo_nat @ N2 @ Q3 ) )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ~ ! [S3: nat] :
( M
!= ( plus_plus_nat @ N2 @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_3300_mod__eq__nat2E,axiom,
! [M: nat,Q3: nat,N2: nat] :
( ( ( modulo_modulo_nat @ M @ Q3 )
= ( modulo_modulo_nat @ N2 @ Q3 ) )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ~ ! [S3: nat] :
( N2
!= ( plus_plus_nat @ M @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_3301_nat__mod__eq__lemma,axiom,
! [X: nat,N2: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X @ N2 )
= ( modulo_modulo_nat @ Y @ N2 ) )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ? [Q2: nat] :
( X
= ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_3302_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_3303_neg__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% neg_mod_sign
thf(fact_3304_zmod__trivial__iff,axiom,
! [I: int,K: int] :
( ( ( modulo_modulo_int @ I @ K )
= I )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zmod_trivial_iff
thf(fact_3305_pos__mod__conj,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
& ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% pos_mod_conj
thf(fact_3306_neg__mod__conj,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
& ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% neg_mod_conj
thf(fact_3307_mod__mult2__eq,axiom,
! [M: nat,N2: nat,Q3: nat] :
( ( modulo_modulo_nat @ M @ ( times_times_nat @ N2 @ Q3 ) )
= ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N2 ) @ Q3 ) ) @ ( modulo_modulo_nat @ M @ N2 ) ) ) ).
% mod_mult2_eq
thf(fact_3308_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_3309_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 ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_3310_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 ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_3311_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_3312_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_3313_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_3314_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_3315_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_3316_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_3317_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_3318_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_3319_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_3320_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_3321_even__mod__4__div__2,axiom,
! [N2: nat] :
( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( suc @ zero_zero_nat ) )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_mod_4_div_2
thf(fact_3322_frac__le__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
= ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% frac_le_eq
thf(fact_3323_frac__le__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% frac_le_eq
thf(fact_3324_frac__less__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% frac_less_eq
thf(fact_3325_frac__less__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
= ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% frac_less_eq
thf(fact_3326_even__even__mod__4__iff,axiom,
! [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% even_even_mod_4_iff
thf(fact_3327_power2__commute,axiom,
! [X: complex,Y: complex] :
( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_complex @ ( minus_minus_complex @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_3328_power2__commute,axiom,
! [X: real,Y: real] :
( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ ( minus_minus_real @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_3329_power2__commute,axiom,
! [X: rat,Y: rat] :
( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ ( minus_minus_rat @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_3330_power2__commute,axiom,
! [X: int,Y: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ ( minus_minus_int @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_3331_split__mod,axiom,
! [P: nat > $o,M: nat,N2: nat] :
( ( P @ ( modulo_modulo_nat @ M @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
=> ( P @ M ) )
& ( ( N2 != zero_zero_nat )
=> ! [I3: nat,J3: nat] :
( ( ord_less_nat @ J3 @ N2 )
=> ( ( M
= ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_3332_power__diff,axiom,
! [A: complex,N2: nat,M: nat] :
( ( A != zero_zero_complex )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N2 ) )
= ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ) ) ).
% power_diff
thf(fact_3333_power__diff,axiom,
! [A: real,N2: nat,M: nat] :
( ( A != zero_zero_real )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N2 ) )
= ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% power_diff
thf(fact_3334_power__diff,axiom,
! [A: rat,N2: nat,M: nat] :
( ( A != zero_zero_rat )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N2 ) )
= ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% power_diff
thf(fact_3335_power__diff,axiom,
! [A: nat,N2: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) )
= ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% power_diff
thf(fact_3336_power__diff,axiom,
! [A: int,N2: nat,M: nat] :
( ( A != zero_zero_int )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) )
= ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% power_diff
thf(fact_3337_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( modulo_modulo_int @ K @ L )
= ( plus_plus_int @ K @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_3338_int__mod__pos__eq,axiom,
! [A: int,B: int,Q3: int,R4: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R4 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R4 )
=> ( ( ord_less_int @ R4 @ B )
=> ( ( modulo_modulo_int @ A @ B )
= R4 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_3339_int__mod__neg__eq,axiom,
! [A: int,B: int,Q3: int,R4: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R4 ) )
=> ( ( ord_less_eq_int @ R4 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R4 )
=> ( ( modulo_modulo_int @ A @ B )
= R4 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_3340_split__zmod,axiom,
! [P: int > $o,N2: int,K: int] :
( ( P @ ( modulo_modulo_int @ N2 @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ N2 ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ J3 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I3: int,J3: int] :
( ( ( ord_less_int @ K @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_3341_mod__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
= ( ( dvd_dvd_int @ L @ K )
| ( ( L = zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ K ) )
| ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% mod_int_pos_iff
thf(fact_3342_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_3343_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_3344_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_3345_div__if,axiom,
( divide_divide_nat
= ( ^ [M2: nat,N4: nat] :
( if_nat
@ ( ( ord_less_nat @ M2 @ N4 )
| ( N4 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N4 ) @ N4 ) ) ) ) ) ).
% div_if
thf(fact_3346_div__geq,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( ord_less_nat @ M @ N2 )
=> ( ( divide_divide_nat @ M @ N2 )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% div_geq
thf(fact_3347_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N4: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_3348_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_less_add_iff1
thf(fact_3349_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_3350_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N4: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_3351_dvd__minus__add,axiom,
! [Q3: nat,N2: nat,R4: nat,M: nat] :
( ( ord_less_eq_nat @ Q3 @ N2 )
=> ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R4 @ M ) )
=> ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ Q3 ) )
= ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R4 @ M ) @ Q3 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_3352_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_3353_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_3354_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_3355_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_3356_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_3357_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_3358_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_3359_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_3360_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_3361_scaling__mono,axiom,
! [U: rat,V: rat,R4: rat,S: rat] :
( ( ord_less_eq_rat @ U @ V )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ R4 )
=> ( ( ord_less_eq_rat @ R4 @ S )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R4 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% scaling_mono
thf(fact_3362_scaling__mono,axiom,
! [U: real,V: real,R4: real,S: real] :
( ( ord_less_eq_real @ U @ V )
=> ( ( ord_less_eq_real @ zero_zero_real @ R4 )
=> ( ( ord_less_eq_real @ R4 @ S )
=> ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R4 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% scaling_mono
thf(fact_3363_Suc__times__mod__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N2 ) ) @ M )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_3364_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N2: nat,M: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
!= zero_zero_nat ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_3365_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N2: nat,M: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
!= zero_zero_int ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_3366_power__diff__power__eq,axiom,
! [A: nat,N2: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
= ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ N2 @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
= ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_3367_power__diff__power__eq,axiom,
! [A: int,N2: nat,M: nat] :
( ( A != zero_zero_int )
=> ( ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
= ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ N2 @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
= ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_3368_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_3369_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_3370_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_3371_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_3372_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_3373_split__neg__lemma,axiom,
! [K: int,P: int > int > $o,N2: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
= ( ! [I3: int,J3: int] :
( ( ( ord_less_int @ K @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_3374_split__pos__lemma,axiom,
! [K: int,P: int > int > $o,N2: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
= ( ! [I3: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_3375_power__minus__mult,axiom,
! [N2: nat,A: complex] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
= ( power_power_complex @ A @ N2 ) ) ) ).
% power_minus_mult
thf(fact_3376_power__minus__mult,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
= ( power_power_real @ A @ N2 ) ) ) ).
% power_minus_mult
thf(fact_3377_power__minus__mult,axiom,
! [N2: nat,A: rat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
= ( power_power_rat @ A @ N2 ) ) ) ).
% power_minus_mult
thf(fact_3378_power__minus__mult,axiom,
! [N2: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N2 ) ) ) ).
% power_minus_mult
thf(fact_3379_power__minus__mult,axiom,
! [N2: nat,A: int] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
= ( power_power_int @ A @ N2 ) ) ) ).
% power_minus_mult
thf(fact_3380_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) ) ) ) ).
% diff_le_diff_pow
thf(fact_3381_le__div__geq,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( divide_divide_nat @ M @ N2 )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% le_div_geq
thf(fact_3382_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_3383_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_3384_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_3385_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_3386_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_3387_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_3388_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_3389_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_3390_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_3391_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_3392_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_3393_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_3394_div__exp__mod__exp__eq,axiom,
! [A: nat,N2: nat,M: nat] :
( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( 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 @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_3395_div__exp__mod__exp__eq,axiom,
! [A: int,N2: nat,M: nat] :
( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( 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 @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_3396_div__exp__mod__exp__eq,axiom,
! [A: code_integer,N2: nat,M: nat] :
( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
= ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_3397_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_3398_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_3399_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_3400_add__0__iff,axiom,
! [B: complex,A: complex] :
( ( B
= ( plus_plus_complex @ B @ A ) )
= ( A = zero_zero_complex ) ) ).
% add_0_iff
thf(fact_3401_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_3402_add__0__iff,axiom,
! [B: rat,A: rat] :
( ( B
= ( plus_plus_rat @ B @ A ) )
= ( A = zero_zero_rat ) ) ).
% add_0_iff
thf(fact_3403_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_3404_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_3405_crossproduct__eq,axiom,
! [W: real,Y: real,X: real,Z: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_3406_crossproduct__eq,axiom,
! [W: rat,Y: rat,X: rat,Z: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z ) )
= ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_3407_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_3408_crossproduct__eq,axiom,
! [W: int,Y: int,X: int,Z: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_3409_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_3410_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_3411_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_3412_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_3413_power2__diff,axiom,
! [X: complex,Y: complex] :
( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_diff
thf(fact_3414_power2__diff,axiom,
! [X: real,Y: real] :
( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_diff
thf(fact_3415_power2__diff,axiom,
! [X: rat,Y: rat] :
( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_diff
thf(fact_3416_power2__diff,axiom,
! [X: int,Y: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_diff
thf(fact_3417_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_3418_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_3419_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_3420_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_3421_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_3422_unset__bit__Suc,axiom,
! [N2: nat,A: code_integer] :
( ( bit_se8260200283734997820nteger @ ( suc @ N2 ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_3423_unset__bit__Suc,axiom,
! [N2: nat,A: int] :
( ( bit_se4203085406695923979it_int @ ( suc @ N2 ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_3424_unset__bit__Suc,axiom,
! [N2: nat,A: nat] :
( ( bit_se4205575877204974255it_nat @ ( suc @ N2 ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_3425_set__bit__Suc,axiom,
! [N2: nat,A: code_integer] :
( ( bit_se2793503036327961859nteger @ ( suc @ N2 ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_3426_set__bit__Suc,axiom,
! [N2: nat,A: int] :
( ( bit_se7879613467334960850it_int @ ( suc @ N2 ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_3427_set__bit__Suc,axiom,
! [N2: nat,A: nat] :
( ( bit_se7882103937844011126it_nat @ ( suc @ N2 ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_3428_even__mask__div__iff_H,axiom,
! [M: nat,N2: 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 ) ) @ N2 ) ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% even_mask_div_iff'
thf(fact_3429_even__mask__div__iff_H,axiom,
! [M: nat,N2: 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 ) ) @ N2 ) ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% even_mask_div_iff'
thf(fact_3430_even__mask__div__iff_H,axiom,
! [M: nat,N2: 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 ) ) @ N2 ) ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% even_mask_div_iff'
thf(fact_3431_even__mask__div__iff,axiom,
! [M: nat,N2: 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 ) ) @ N2 ) ) )
= ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
= zero_z3403309356797280102nteger )
| ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% even_mask_div_iff
thf(fact_3432_even__mask__div__iff,axiom,
! [M: nat,N2: 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 ) ) @ N2 ) ) )
= ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
= zero_zero_nat )
| ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% even_mask_div_iff
thf(fact_3433_even__mask__div__iff,axiom,
! [M: nat,N2: 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 ) ) @ N2 ) ) )
= ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
= zero_zero_int )
| ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% even_mask_div_iff
thf(fact_3434_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_3435_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_3436_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_3437_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N4: nat,TreeList2: list_VEBT_VEBT,X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ N4 ) ) @ ( vEBT_VEBT_low @ X3 @ N4 ) ) ) ) ).
% in_children_def
thf(fact_3438_verit__le__mono__div,axiom,
! [A2: nat,B3: nat,N2: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N2 )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B3 @ N2 )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B3 @ N2 ) ) ) ) ).
% verit_le_mono_div
thf(fact_3439_verit__le__mono__div__int,axiom,
! [A2: int,B3: int,N2: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ N2 )
=> ( ord_less_eq_int
@ ( plus_plus_int @ ( divide_divide_int @ A2 @ N2 )
@ ( if_int
@ ( ( modulo_modulo_int @ B3 @ N2 )
= zero_zero_int )
@ one_one_int
@ zero_zero_int ) )
@ ( divide_divide_int @ B3 @ N2 ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_3440_div__mod__decomp,axiom,
! [A2: nat,N2: nat] :
( A2
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A2 @ N2 ) ) ) ).
% div_mod_decomp
thf(fact_3441_div__less__mono,axiom,
! [A2: nat,B3: nat,N2: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( modulo_modulo_nat @ A2 @ N2 )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B3 @ N2 )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A2 @ N2 ) @ ( divide_divide_nat @ B3 @ N2 ) ) ) ) ) ) ).
% div_less_mono
thf(fact_3442_flip__bit__Suc,axiom,
! [N2: nat,A: code_integer] :
( ( bit_se1345352211410354436nteger @ ( suc @ N2 ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_3443_flip__bit__Suc,axiom,
! [N2: nat,A: int] :
( ( bit_se2159334234014336723it_int @ ( suc @ N2 ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_3444_flip__bit__Suc,axiom,
! [N2: nat,A: nat] :
( ( bit_se2161824704523386999it_nat @ ( suc @ N2 ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_3445_div__mod__decomp__int,axiom,
! [A2: int,N2: int] :
( A2
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_int @ A2 @ N2 ) ) ) ).
% div_mod_decomp_int
thf(fact_3446_Leaf__0__not,axiom,
! [A: $o,B: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).
% Leaf_0_not
thf(fact_3447_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_3448_VEBT_Oinject_I2_J,axiom,
! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
( ( ( vEBT_Leaf @ X21 @ X222 )
= ( vEBT_Leaf @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% VEBT.inject(2)
thf(fact_3449_idiff__0,axiom,
! [N2: extended_enat] :
( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N2 )
= zero_z5237406670263579293d_enat ) ).
% idiff_0
thf(fact_3450_idiff__0__right,axiom,
! [N2: extended_enat] :
( ( minus_3235023915231533773d_enat @ N2 @ zero_z5237406670263579293d_enat )
= N2 ) ).
% idiff_0_right
thf(fact_3451_flip__bit__nonnegative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% flip_bit_nonnegative_int_iff
thf(fact_3452_flip__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% flip_bit_negative_int_iff
thf(fact_3453_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_3454_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_3455_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_3456_VEBT_Oexhaust,axiom,
! [Y: vEBT_VEBT] :
( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
( Y
!= ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X212: $o,X223: $o] :
( Y
!= ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% VEBT.exhaust
thf(fact_3457_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_3458_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_3459_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_3460_zdvd__zdiffD,axiom,
! [K: int,M: int,N2: int] :
( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N2 ) )
=> ( ( dvd_dvd_int @ K @ N2 )
=> ( dvd_dvd_int @ K @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_3461_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_3462_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_3463_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_3464_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ zero_zero_nat )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_3465_add__diff__assoc__enat,axiom,
! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Z @ Y )
=> ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z ) )
= ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z ) ) ) ).
% add_diff_assoc_enat
thf(fact_3466_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_3467_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_3468_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_3469_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_3470_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_3471_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_3472_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_3473_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_3474_verit__comp__simplify1_I2_J,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_3475_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_3476_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_3477_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_3478_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_3479_mod__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( modulo_modulo_int @ K @ L )
= ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_3480_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_3481_verit__comp__simplify1_I1_J,axiom,
! [A: rat] :
~ ( ord_less_rat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_3482_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_3483_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_3484_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_3485_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int,K3: int] :
( ( P1 @ X2 )
= ( P1 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ( ( P @ X2 )
= ( P1 @ X2 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_3486_plusinfinity,axiom,
! [D: int,P2: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int,K3: int] :
( ( P2 @ X2 )
= ( P2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [X_12: int] : ( P2 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_3487_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_3488_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_3489_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_3490_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_3491_even__diff__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% even_diff_iff
thf(fact_3492_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% div_pos_geq
thf(fact_3493_verit__comp__simplify1_I3_J,axiom,
! [B4: rat,A4: rat] :
( ( ~ ( ord_less_eq_rat @ B4 @ A4 ) )
= ( ord_less_rat @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_3494_verit__comp__simplify1_I3_J,axiom,
! [B4: num,A4: num] :
( ( ~ ( ord_less_eq_num @ B4 @ A4 ) )
= ( ord_less_num @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_3495_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A4: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A4 ) )
= ( ord_less_nat @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_3496_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A4: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A4 ) )
= ( ord_less_int @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_3497_verit__comp__simplify1_I3_J,axiom,
! [B4: real,A4: real] :
( ( ~ ( ord_less_eq_real @ B4 @ A4 ) )
= ( ord_less_real @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_3498_verit__sum__simplify,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% verit_sum_simplify
thf(fact_3499_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_3500_verit__sum__simplify,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% verit_sum_simplify
thf(fact_3501_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_3502_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_3503_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_3504_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_3505_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_3506_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_3507_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_3508_zdiv__mono__strict,axiom,
! [A2: int,B3: int,N2: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ N2 )
=> ( ( ( modulo_modulo_int @ A2 @ N2 )
= zero_zero_int )
=> ( ( ( modulo_modulo_int @ B3 @ N2 )
= zero_zero_int )
=> ( ord_less_int @ ( divide_divide_int @ A2 @ N2 ) @ ( divide_divide_int @ B3 @ N2 ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_3509_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_3510_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_3511_concat__bit__Suc,axiom,
! [N2: nat,K: int,L: int] :
( ( bit_concat_bit @ ( suc @ N2 ) @ K @ L )
= ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_3512_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A5: real,B5: real,C3: real] :
( ( P @ A5 @ B5 )
=> ( ( P @ B5 @ C3 )
=> ( ( ord_less_eq_real @ A5 @ B5 )
=> ( ( ord_less_eq_real @ B5 @ C3 )
=> ( P @ A5 @ C3 ) ) ) ) )
=> ( ! [X2: real] :
( ( ord_less_eq_real @ A @ X2 )
=> ( ( ord_less_eq_real @ X2 @ B )
=> ? [D6: real] :
( ( ord_less_real @ zero_zero_real @ D6 )
& ! [A5: real,B5: real] :
( ( ( ord_less_eq_real @ A5 @ X2 )
& ( ord_less_eq_real @ X2 @ B5 )
& ( ord_less_real @ ( minus_minus_real @ B5 @ A5 ) @ D6 ) )
=> ( P @ A5 @ B5 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_3513_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_3514_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_3515_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_3516_signed__take__bit__Suc,axiom,
! [N2: nat,A: code_integer] :
( ( bit_ri6519982836138164636nteger @ ( suc @ N2 ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_3517_signed__take__bit__Suc,axiom,
! [N2: nat,A: int] :
( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_3518_take__bit__rec,axiom,
( bit_se1745604003318907178nteger
= ( ^ [N4: nat,A3: code_integer] : ( if_Code_integer @ ( N4 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_3519_take__bit__rec,axiom,
( bit_se2923211474154528505it_int
= ( ^ [N4: nat,A3: int] : ( if_int @ ( N4 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_3520_take__bit__rec,axiom,
( bit_se2925701944663578781it_nat
= ( ^ [N4: nat,A3: nat] : ( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_3521_take__bit__of__0,axiom,
! [N2: nat] :
( ( bit_se2923211474154528505it_int @ N2 @ zero_zero_int )
= zero_zero_int ) ).
% take_bit_of_0
thf(fact_3522_take__bit__of__0,axiom,
! [N2: nat] :
( ( bit_se2925701944663578781it_nat @ N2 @ zero_zero_nat )
= zero_zero_nat ) ).
% take_bit_of_0
thf(fact_3523_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_3524_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
= ( P
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_3525_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_3526_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_3527_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_3528_of__bool__eq_I1_J,axiom,
( ( zero_n1201886186963655149omplex @ $false )
= zero_zero_complex ) ).
% of_bool_eq(1)
thf(fact_3529_of__bool__eq_I1_J,axiom,
( ( zero_n3304061248610475627l_real @ $false )
= zero_zero_real ) ).
% of_bool_eq(1)
thf(fact_3530_of__bool__eq_I1_J,axiom,
( ( zero_n2052037380579107095ol_rat @ $false )
= zero_zero_rat ) ).
% of_bool_eq(1)
thf(fact_3531_of__bool__eq_I1_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $false )
= zero_zero_nat ) ).
% of_bool_eq(1)
thf(fact_3532_of__bool__eq_I1_J,axiom,
( ( zero_n2684676970156552555ol_int @ $false )
= zero_zero_int ) ).
% of_bool_eq(1)
thf(fact_3533_of__bool__eq_I1_J,axiom,
( ( zero_n356916108424825756nteger @ $false )
= zero_z3403309356797280102nteger ) ).
% of_bool_eq(1)
thf(fact_3534_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n1201886186963655149omplex @ P )
= zero_zero_complex )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_3535_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n3304061248610475627l_real @ P )
= zero_zero_real )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_3536_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n2052037380579107095ol_rat @ P )
= zero_zero_rat )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_3537_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P )
= zero_zero_nat )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_3538_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n2684676970156552555ol_int @ P )
= zero_zero_int )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_3539_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n356916108424825756nteger @ P )
= zero_z3403309356797280102nteger )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_3540_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_3541_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_3542_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_3543_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_3544_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_3545_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n1201886186963655149omplex @ P )
= one_one_complex )
= P ) ).
% of_bool_eq_1_iff
thf(fact_3546_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n3304061248610475627l_real @ P )
= one_one_real )
= P ) ).
% of_bool_eq_1_iff
thf(fact_3547_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2052037380579107095ol_rat @ P )
= one_one_rat )
= P ) ).
% of_bool_eq_1_iff
thf(fact_3548_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P )
= one_one_nat )
= P ) ).
% of_bool_eq_1_iff
thf(fact_3549_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2684676970156552555ol_int @ P )
= one_one_int )
= P ) ).
% of_bool_eq_1_iff
thf(fact_3550_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n356916108424825756nteger @ P )
= one_one_Code_integer )
= P ) ).
% of_bool_eq_1_iff
thf(fact_3551_of__bool__eq_I2_J,axiom,
( ( zero_n1201886186963655149omplex @ $true )
= one_one_complex ) ).
% of_bool_eq(2)
thf(fact_3552_of__bool__eq_I2_J,axiom,
( ( zero_n3304061248610475627l_real @ $true )
= one_one_real ) ).
% of_bool_eq(2)
thf(fact_3553_of__bool__eq_I2_J,axiom,
( ( zero_n2052037380579107095ol_rat @ $true )
= one_one_rat ) ).
% of_bool_eq(2)
thf(fact_3554_of__bool__eq_I2_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $true )
= one_one_nat ) ).
% of_bool_eq(2)
thf(fact_3555_of__bool__eq_I2_J,axiom,
( ( zero_n2684676970156552555ol_int @ $true )
= one_one_int ) ).
% of_bool_eq(2)
thf(fact_3556_of__bool__eq_I2_J,axiom,
( ( zero_n356916108424825756nteger @ $true )
= one_one_Code_integer ) ).
% of_bool_eq(2)
thf(fact_3557_signed__take__bit__of__0,axiom,
! [N2: nat] :
( ( bit_ri631733984087533419it_int @ N2 @ zero_zero_int )
= zero_zero_int ) ).
% signed_take_bit_of_0
thf(fact_3558_concat__bit__0,axiom,
! [K: int,L: int] :
( ( bit_concat_bit @ zero_zero_nat @ K @ L )
= L ) ).
% concat_bit_0
thf(fact_3559_concat__bit__of__zero__2,axiom,
! [N2: nat,K: int] :
( ( bit_concat_bit @ N2 @ K @ zero_zero_int )
= ( bit_se2923211474154528505it_int @ N2 @ K ) ) ).
% concat_bit_of_zero_2
thf(fact_3560_take__bit__of__Suc__0,axiom,
! [N2: nat] :
( ( bit_se2925701944663578781it_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% take_bit_of_Suc_0
thf(fact_3561_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_3562_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_3563_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_3564_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_3565_zero__less__of__bool__iff,axiom,
! [P: $o] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
= P ) ).
% zero_less_of_bool_iff
thf(fact_3566_take__bit__0,axiom,
! [A: int] :
( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
= zero_zero_int ) ).
% take_bit_0
thf(fact_3567_take__bit__0,axiom,
! [A: nat] :
( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% take_bit_0
thf(fact_3568_take__bit__Suc__1,axiom,
! [N2: nat] :
( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ one_one_int )
= one_one_int ) ).
% take_bit_Suc_1
thf(fact_3569_take__bit__Suc__1,axiom,
! [N2: nat] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ one_one_nat )
= one_one_nat ) ).
% take_bit_Suc_1
thf(fact_3570_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_3571_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_3572_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_3573_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_3574_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_3575_take__bit__numeral__1,axiom,
! [L: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ one_one_int )
= one_one_int ) ).
% take_bit_numeral_1
thf(fact_3576_take__bit__numeral__1,axiom,
! [L: num] :
( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ one_one_nat )
= one_one_nat ) ).
% take_bit_numeral_1
thf(fact_3577_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_3578_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_3579_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_3580_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_3581_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_3582_Suc__0__mod__eq,axiom,
! [N2: nat] :
( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( zero_n2687167440665602831ol_nat
@ ( N2
!= ( suc @ zero_zero_nat ) ) ) ) ).
% Suc_0_mod_eq
thf(fact_3583_signed__take__bit__Suc__1,axiom,
! [N2: nat] :
( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ one_one_int )
= one_one_int ) ).
% signed_take_bit_Suc_1
thf(fact_3584_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_3585_concat__bit__nonnegative__iff,axiom,
! [N2: nat,K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N2 @ K @ L ) )
= ( ord_less_eq_int @ zero_zero_int @ L ) ) ).
% concat_bit_nonnegative_iff
thf(fact_3586_concat__bit__negative__iff,axiom,
! [N2: nat,K: int,L: int] :
( ( ord_less_int @ ( bit_concat_bit @ N2 @ K @ L ) @ zero_zero_int )
= ( ord_less_int @ L @ zero_zero_int ) ) ).
% concat_bit_negative_iff
thf(fact_3587_take__bit__of__1__eq__0__iff,axiom,
! [N2: nat] :
( ( ( bit_se2923211474154528505it_int @ N2 @ one_one_int )
= zero_zero_int )
= ( N2 = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_3588_take__bit__of__1__eq__0__iff,axiom,
! [N2: nat] :
( ( ( bit_se2925701944663578781it_nat @ N2 @ one_one_nat )
= zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_3589_signed__take__bit__Suc__bit0,axiom,
! [N2: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_3590_odd__of__bool__self,axiom,
! [P3: $o] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P3 ) ) )
= P3 ) ).
% odd_of_bool_self
thf(fact_3591_odd__of__bool__self,axiom,
! [P3: $o] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P3 ) ) )
= P3 ) ).
% odd_of_bool_self
thf(fact_3592_odd__of__bool__self,axiom,
! [P3: $o] :
( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P3 ) ) )
= P3 ) ).
% odd_of_bool_self
thf(fact_3593_take__bit__of__1,axiom,
! [N2: nat] :
( ( bit_se1745604003318907178nteger @ N2 @ one_one_Code_integer )
= ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% take_bit_of_1
thf(fact_3594_take__bit__of__1,axiom,
! [N2: nat] :
( ( bit_se2923211474154528505it_int @ N2 @ one_one_int )
= ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% take_bit_of_1
thf(fact_3595_take__bit__of__1,axiom,
! [N2: nat] :
( ( bit_se2925701944663578781it_nat @ N2 @ one_one_nat )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% take_bit_of_1
thf(fact_3596_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_3597_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_3598_of__bool__half__eq__0,axiom,
! [B: $o] :
( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) ).
% of_bool_half_eq_0
thf(fact_3599_even__take__bit__eq,axiom,
! [N2: nat,A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N2 @ A ) )
= ( ( N2 = zero_zero_nat )
| ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_take_bit_eq
thf(fact_3600_even__take__bit__eq,axiom,
! [N2: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
= ( ( N2 = zero_zero_nat )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_take_bit_eq
thf(fact_3601_even__take__bit__eq,axiom,
! [N2: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N2 @ A ) )
= ( ( N2 = zero_zero_nat )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_take_bit_eq
thf(fact_3602_take__bit__Suc__0,axiom,
! [A: code_integer] :
( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
= ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_0
thf(fact_3603_take__bit__Suc__0,axiom,
! [A: int] :
( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
= ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_0
thf(fact_3604_take__bit__Suc__0,axiom,
! [A: nat] :
( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
= ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_0
thf(fact_3605_one__div__2__pow__eq,axiom,
! [N2: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_3606_one__div__2__pow__eq,axiom,
! [N2: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
= ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_3607_one__div__2__pow__eq,axiom,
! [N2: nat] :
( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
= ( zero_n356916108424825756nteger @ ( N2 = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_3608_bits__1__div__exp,axiom,
! [N2: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_3609_bits__1__div__exp,axiom,
! [N2: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
= ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_3610_bits__1__div__exp,axiom,
! [N2: nat] :
( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
= ( zero_n356916108424825756nteger @ ( N2 = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_3611_take__bit__of__exp,axiom,
! [M: nat,N2: nat] :
( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
= ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N2 @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% take_bit_of_exp
thf(fact_3612_take__bit__of__exp,axiom,
! [M: nat,N2: nat] :
( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N2 @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% take_bit_of_exp
thf(fact_3613_take__bit__of__exp,axiom,
! [M: nat,N2: nat] :
( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N2 @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% take_bit_of_exp
thf(fact_3614_take__bit__of__2,axiom,
! [N2: nat] :
( ( bit_se1745604003318907178nteger @ N2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% take_bit_of_2
thf(fact_3615_take__bit__of__2,axiom,
! [N2: nat] :
( ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_of_2
thf(fact_3616_take__bit__of__2,axiom,
! [N2: nat] :
( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% take_bit_of_2
thf(fact_3617_one__mod__2__pow__eq,axiom,
! [N2: nat] :
( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% one_mod_2_pow_eq
thf(fact_3618_one__mod__2__pow__eq,axiom,
! [N2: nat] :
( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
= ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% one_mod_2_pow_eq
thf(fact_3619_one__mod__2__pow__eq,axiom,
! [N2: nat] :
( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
= ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% one_mod_2_pow_eq
thf(fact_3620_take__bit__diff,axiom,
! [N2: nat,K: int,L: int] :
( ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ L ) ) )
= ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ K @ L ) ) ) ).
% take_bit_diff
thf(fact_3621_signed__take__bit__diff,axiom,
! [N2: nat,K: int,L: int] :
( ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L ) ) )
= ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ K @ L ) ) ) ).
% signed_take_bit_diff
thf(fact_3622_signed__take__bit__take__bit,axiom,
! [M: nat,N2: nat,A: int] :
( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
= ( if_int_int @ ( ord_less_eq_nat @ N2 @ M ) @ ( bit_se2923211474154528505it_int @ N2 ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).
% signed_take_bit_take_bit
thf(fact_3623_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [N2: nat,A: int,B: int] :
( ( ( bit_ri631733984087533419it_int @ N2 @ A )
= ( bit_ri631733984087533419it_int @ N2 @ B ) )
= ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A )
= ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ B ) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
thf(fact_3624_of__bool__eq__iff,axiom,
! [P3: $o,Q3: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P3 )
= ( zero_n2687167440665602831ol_nat @ Q3 ) )
= ( P3 = Q3 ) ) ).
% of_bool_eq_iff
thf(fact_3625_of__bool__eq__iff,axiom,
! [P3: $o,Q3: $o] :
( ( ( zero_n2684676970156552555ol_int @ P3 )
= ( zero_n2684676970156552555ol_int @ Q3 ) )
= ( P3 = Q3 ) ) ).
% of_bool_eq_iff
thf(fact_3626_of__bool__eq__iff,axiom,
! [P3: $o,Q3: $o] :
( ( ( zero_n356916108424825756nteger @ P3 )
= ( zero_n356916108424825756nteger @ Q3 ) )
= ( P3 = Q3 ) ) ).
% of_bool_eq_iff
thf(fact_3627_concat__bit__eq__iff,axiom,
! [N2: nat,K: int,L: int,R4: int,S: int] :
( ( ( bit_concat_bit @ N2 @ K @ L )
= ( bit_concat_bit @ N2 @ R4 @ S ) )
= ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
= ( bit_se2923211474154528505it_int @ N2 @ R4 ) )
& ( L = S ) ) ) ).
% concat_bit_eq_iff
thf(fact_3628_concat__bit__take__bit__eq,axiom,
! [N2: nat,B: int] :
( ( bit_concat_bit @ N2 @ ( bit_se2923211474154528505it_int @ N2 @ B ) )
= ( bit_concat_bit @ N2 @ B ) ) ).
% concat_bit_take_bit_eq
thf(fact_3629_take__bit__add,axiom,
! [N2: nat,A: int,B: int] :
( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ A ) @ ( bit_se2923211474154528505it_int @ N2 @ B ) ) )
= ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ A @ B ) ) ) ).
% take_bit_add
thf(fact_3630_take__bit__add,axiom,
! [N2: nat,A: nat,B: nat] :
( ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N2 @ A ) @ ( bit_se2925701944663578781it_nat @ N2 @ B ) ) )
= ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ A @ B ) ) ) ).
% take_bit_add
thf(fact_3631_take__bit__tightened,axiom,
! [N2: nat,A: int,B: int,M: nat] :
( ( ( bit_se2923211474154528505it_int @ N2 @ A )
= ( bit_se2923211474154528505it_int @ N2 @ B ) )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( ( bit_se2923211474154528505it_int @ M @ A )
= ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).
% take_bit_tightened
thf(fact_3632_take__bit__tightened,axiom,
! [N2: nat,A: nat,B: nat,M: nat] :
( ( ( bit_se2925701944663578781it_nat @ N2 @ A )
= ( bit_se2925701944663578781it_nat @ N2 @ B ) )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( ( bit_se2925701944663578781it_nat @ M @ A )
= ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).
% take_bit_tightened
thf(fact_3633_take__bit__signed__take__bit,axiom,
! [M: nat,N2: nat,A: int] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N2 @ A ) )
= ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).
% take_bit_signed_take_bit
thf(fact_3634_take__bit__nat__less__eq__self,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M ) ).
% take_bit_nat_less_eq_self
thf(fact_3635_take__bit__tightened__less__eq__nat,axiom,
! [M: nat,N2: nat,Q3: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q3 ) @ ( bit_se2925701944663578781it_nat @ N2 @ Q3 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_3636_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_3637_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_3638_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_3639_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_3640_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_3641_take__bit__mult,axiom,
! [N2: nat,K: int,L: int] :
( ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ L ) ) )
= ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ K @ L ) ) ) ).
% take_bit_mult
thf(fact_3642_signed__take__bit__mult,axiom,
! [N2: nat,K: int,L: int] :
( ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L ) ) )
= ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ K @ L ) ) ) ).
% signed_take_bit_mult
thf(fact_3643_signed__take__bit__add,axiom,
! [N2: nat,K: int,L: int] :
( ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L ) ) )
= ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ K @ L ) ) ) ).
% signed_take_bit_add
thf(fact_3644_concat__bit__assoc,axiom,
! [N2: nat,K: int,M: nat,L: int,R4: int] :
( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M @ L @ R4 ) )
= ( bit_concat_bit @ ( plus_plus_nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L ) @ R4 ) ) ).
% concat_bit_assoc
thf(fact_3645_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_3646_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_3647_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_3648_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_3649_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_3650_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_3651_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_3652_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_3653_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_3654_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% of_bool_less_eq_one
thf(fact_3655_split__of__bool__asm,axiom,
! [P: complex > $o,P3: $o] :
( ( P @ ( zero_n1201886186963655149omplex @ P3 ) )
= ( ~ ( ( P3
& ~ ( P @ one_one_complex ) )
| ( ~ P3
& ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_3656_split__of__bool__asm,axiom,
! [P: real > $o,P3: $o] :
( ( P @ ( zero_n3304061248610475627l_real @ P3 ) )
= ( ~ ( ( P3
& ~ ( P @ one_one_real ) )
| ( ~ P3
& ~ ( P @ zero_zero_real ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_3657_split__of__bool__asm,axiom,
! [P: rat > $o,P3: $o] :
( ( P @ ( zero_n2052037380579107095ol_rat @ P3 ) )
= ( ~ ( ( P3
& ~ ( P @ one_one_rat ) )
| ( ~ P3
& ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_3658_split__of__bool__asm,axiom,
! [P: nat > $o,P3: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P3 ) )
= ( ~ ( ( P3
& ~ ( P @ one_one_nat ) )
| ( ~ P3
& ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_3659_split__of__bool__asm,axiom,
! [P: int > $o,P3: $o] :
( ( P @ ( zero_n2684676970156552555ol_int @ P3 ) )
= ( ~ ( ( P3
& ~ ( P @ one_one_int ) )
| ( ~ P3
& ~ ( P @ zero_zero_int ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_3660_split__of__bool__asm,axiom,
! [P: code_integer > $o,P3: $o] :
( ( P @ ( zero_n356916108424825756nteger @ P3 ) )
= ( ~ ( ( P3
& ~ ( P @ one_one_Code_integer ) )
| ( ~ P3
& ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_3661_split__of__bool,axiom,
! [P: complex > $o,P3: $o] :
( ( P @ ( zero_n1201886186963655149omplex @ P3 ) )
= ( ( P3
=> ( P @ one_one_complex ) )
& ( ~ P3
=> ( P @ zero_zero_complex ) ) ) ) ).
% split_of_bool
thf(fact_3662_split__of__bool,axiom,
! [P: real > $o,P3: $o] :
( ( P @ ( zero_n3304061248610475627l_real @ P3 ) )
= ( ( P3
=> ( P @ one_one_real ) )
& ( ~ P3
=> ( P @ zero_zero_real ) ) ) ) ).
% split_of_bool
thf(fact_3663_split__of__bool,axiom,
! [P: rat > $o,P3: $o] :
( ( P @ ( zero_n2052037380579107095ol_rat @ P3 ) )
= ( ( P3
=> ( P @ one_one_rat ) )
& ( ~ P3
=> ( P @ zero_zero_rat ) ) ) ) ).
% split_of_bool
thf(fact_3664_split__of__bool,axiom,
! [P: nat > $o,P3: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P3 ) )
= ( ( P3
=> ( P @ one_one_nat ) )
& ( ~ P3
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_of_bool
thf(fact_3665_split__of__bool,axiom,
! [P: int > $o,P3: $o] :
( ( P @ ( zero_n2684676970156552555ol_int @ P3 ) )
= ( ( P3
=> ( P @ one_one_int ) )
& ( ~ P3
=> ( P @ zero_zero_int ) ) ) ) ).
% split_of_bool
thf(fact_3666_split__of__bool,axiom,
! [P: code_integer > $o,P3: $o] :
( ( P @ ( zero_n356916108424825756nteger @ P3 ) )
= ( ( P3
=> ( P @ one_one_Code_integer ) )
& ( ~ P3
=> ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% split_of_bool
thf(fact_3667_of__bool__def,axiom,
( zero_n1201886186963655149omplex
= ( ^ [P4: $o] : ( if_complex @ P4 @ one_one_complex @ zero_zero_complex ) ) ) ).
% of_bool_def
thf(fact_3668_of__bool__def,axiom,
( zero_n3304061248610475627l_real
= ( ^ [P4: $o] : ( if_real @ P4 @ one_one_real @ zero_zero_real ) ) ) ).
% of_bool_def
thf(fact_3669_of__bool__def,axiom,
( zero_n2052037380579107095ol_rat
= ( ^ [P4: $o] : ( if_rat @ P4 @ one_one_rat @ zero_zero_rat ) ) ) ).
% of_bool_def
thf(fact_3670_of__bool__def,axiom,
( zero_n2687167440665602831ol_nat
= ( ^ [P4: $o] : ( if_nat @ P4 @ one_one_nat @ zero_zero_nat ) ) ) ).
% of_bool_def
thf(fact_3671_of__bool__def,axiom,
( zero_n2684676970156552555ol_int
= ( ^ [P4: $o] : ( if_int @ P4 @ one_one_int @ zero_zero_int ) ) ) ).
% of_bool_def
thf(fact_3672_of__bool__def,axiom,
( zero_n356916108424825756nteger
= ( ^ [P4: $o] : ( if_Code_integer @ P4 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% of_bool_def
thf(fact_3673_take__bit__tightened__less__eq__int,axiom,
! [M: nat,N2: nat,K: int] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% take_bit_tightened_less_eq_int
thf(fact_3674_take__bit__int__less__eq__self__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% take_bit_int_less_eq_self_iff
thf(fact_3675_take__bit__nonnegative,axiom,
! [N2: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ).
% take_bit_nonnegative
thf(fact_3676_take__bit__decr__eq,axiom,
! [N2: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N2 @ K )
!= zero_zero_int )
=> ( ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ K @ one_one_int ) )
= ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ one_one_int ) ) ) ).
% take_bit_decr_eq
thf(fact_3677_take__bit__int__greater__self__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% take_bit_int_greater_self_iff
thf(fact_3678_not__take__bit__negative,axiom,
! [N2: nat,K: int] :
~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ zero_zero_int ) ).
% not_take_bit_negative
thf(fact_3679_take__bit__unset__bit__eq,axiom,
! [N2: nat,M: nat,A: int] :
( ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se4203085406695923979it_int @ M @ A ) )
= ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
& ( ~ ( ord_less_eq_nat @ N2 @ M )
=> ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se4203085406695923979it_int @ M @ A ) )
= ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_3680_take__bit__unset__bit__eq,axiom,
! [N2: nat,M: nat,A: nat] :
( ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se4205575877204974255it_nat @ M @ A ) )
= ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
& ( ~ ( ord_less_eq_nat @ N2 @ M )
=> ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se4205575877204974255it_nat @ M @ A ) )
= ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_3681_take__bit__set__bit__eq,axiom,
! [N2: nat,M: nat,A: int] :
( ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se7879613467334960850it_int @ M @ A ) )
= ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
& ( ~ ( ord_less_eq_nat @ N2 @ M )
=> ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se7879613467334960850it_int @ M @ A ) )
= ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_3682_take__bit__set__bit__eq,axiom,
! [N2: nat,M: nat,A: nat] :
( ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se7882103937844011126it_nat @ M @ A ) )
= ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
& ( ~ ( ord_less_eq_nat @ N2 @ M )
=> ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se7882103937844011126it_nat @ M @ A ) )
= ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_3683_take__bit__flip__bit__eq,axiom,
! [N2: nat,M: nat,A: int] :
( ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se2159334234014336723it_int @ M @ A ) )
= ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
& ( ~ ( ord_less_eq_nat @ N2 @ M )
=> ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se2159334234014336723it_int @ M @ A ) )
= ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_3684_take__bit__flip__bit__eq,axiom,
! [N2: nat,M: nat,A: nat] :
( ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se2161824704523386999it_nat @ M @ A ) )
= ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
& ( ~ ( ord_less_eq_nat @ N2 @ M )
=> ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se2161824704523386999it_nat @ M @ A ) )
= ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_3685_signed__take__bit__eq__take__bit__shift,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N4: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_3686_take__bit__Suc__bit0,axiom,
! [N2: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_3687_take__bit__Suc__bit0,axiom,
! [N2: nat,K: num] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_3688_take__bit__eq__mod,axiom,
( bit_se1745604003318907178nteger
= ( ^ [N4: nat,A3: code_integer] : ( modulo364778990260209775nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% take_bit_eq_mod
thf(fact_3689_take__bit__eq__mod,axiom,
( bit_se2923211474154528505it_int
= ( ^ [N4: nat,A3: int] : ( modulo_modulo_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% take_bit_eq_mod
thf(fact_3690_take__bit__eq__mod,axiom,
( bit_se2925701944663578781it_nat
= ( ^ [N4: nat,A3: nat] : ( modulo_modulo_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% take_bit_eq_mod
thf(fact_3691_take__bit__nat__eq__self,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
=> ( ( bit_se2925701944663578781it_nat @ N2 @ M )
= M ) ) ).
% take_bit_nat_eq_self
thf(fact_3692_take__bit__nat__less__exp,axiom,
! [N2: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% take_bit_nat_less_exp
thf(fact_3693_take__bit__nat__eq__self__iff,axiom,
! [N2: nat,M: nat] :
( ( ( bit_se2925701944663578781it_nat @ N2 @ M )
= M )
= ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% take_bit_nat_eq_self_iff
thf(fact_3694_take__bit__nat__def,axiom,
( bit_se2925701944663578781it_nat
= ( ^ [N4: nat,M2: nat] : ( modulo_modulo_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% take_bit_nat_def
thf(fact_3695_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_3696_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_3697_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_3698_take__bit__int__less__exp,axiom,
! [N2: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).
% take_bit_int_less_exp
thf(fact_3699_take__bit__int__def,axiom,
( bit_se2923211474154528505it_int
= ( ^ [N4: nat,K2: int] : ( modulo_modulo_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% take_bit_int_def
thf(fact_3700_signed__take__bit__int__less__exp,axiom,
! [N2: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).
% signed_take_bit_int_less_exp
thf(fact_3701_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_3702_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_3703_take__bit__eq__0__iff,axiom,
! [N2: nat,A: code_integer] :
( ( ( bit_se1745604003318907178nteger @ N2 @ A )
= zero_z3403309356797280102nteger )
= ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ A ) ) ).
% take_bit_eq_0_iff
thf(fact_3704_take__bit__eq__0__iff,axiom,
! [N2: nat,A: int] :
( ( ( bit_se2923211474154528505it_int @ N2 @ A )
= zero_zero_int )
= ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ A ) ) ).
% take_bit_eq_0_iff
thf(fact_3705_take__bit__eq__0__iff,axiom,
! [N2: nat,A: nat] :
( ( ( bit_se2925701944663578781it_nat @ N2 @ A )
= zero_zero_nat )
= ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ A ) ) ).
% take_bit_eq_0_iff
thf(fact_3706_take__bit__nat__less__self__iff,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M ) ) ).
% take_bit_nat_less_self_iff
thf(fact_3707_bits__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [A5: nat] :
( ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A5 )
=> ( P @ A5 ) )
=> ( ! [A5: nat,B5: $o] :
( ( P @ A5 )
=> ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A5 )
=> ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
=> ( P @ A ) ) ) ).
% bits_induct
thf(fact_3708_bits__induct,axiom,
! [P: int > $o,A: int] :
( ! [A5: int] :
( ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A5 )
=> ( P @ A5 ) )
=> ( ! [A5: int,B5: $o] :
( ( P @ A5 )
=> ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A5 )
=> ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
=> ( P @ A ) ) ) ).
% bits_induct
thf(fact_3709_bits__induct,axiom,
! [P: code_integer > $o,A: code_integer] :
( ! [A5: code_integer] :
( ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= A5 )
=> ( P @ A5 ) )
=> ( ! [A5: code_integer,B5: $o] :
( ( P @ A5 )
=> ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= A5 )
=> ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
=> ( P @ A ) ) ) ).
% bits_induct
thf(fact_3710_take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
= ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_3711_take__bit__int__less__self__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).
% take_bit_int_less_self_iff
thf(fact_3712_take__bit__incr__eq,axiom,
! [N2: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N2 @ K )
!= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
=> ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ K @ one_one_int ) )
= ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ) ).
% take_bit_incr_eq
thf(fact_3713_signed__take__bit__int__less__eq,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
=> ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) ) ) ).
% signed_take_bit_int_less_eq
thf(fact_3714_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
= ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_3715_signed__take__bit__int__less__self__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ K )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_3716_take__bit__int__less__eq,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_3717_exp__mod__exp,axiom,
! [M: nat,N2: nat] :
( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% exp_mod_exp
thf(fact_3718_exp__mod__exp,axiom,
! [M: nat,N2: nat] :
( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% exp_mod_exp
thf(fact_3719_exp__mod__exp,axiom,
! [M: nat,N2: nat] :
( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
= ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% exp_mod_exp
thf(fact_3720_take__bit__int__eq__self__iff,axiom,
! [N2: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N2 @ K )
= K )
= ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_3721_take__bit__int__eq__self,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
=> ( ( bit_se2923211474154528505it_int @ N2 @ K )
= K ) ) ) ).
% take_bit_int_eq_self
thf(fact_3722_take__bit__Suc,axiom,
! [N2: nat,A: code_integer] :
( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ A )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% take_bit_Suc
thf(fact_3723_take__bit__Suc,axiom,
! [N2: nat,A: int] :
( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% take_bit_Suc
thf(fact_3724_take__bit__Suc,axiom,
! [N2: nat,A: nat] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ A )
= ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% take_bit_Suc
thf(fact_3725_take__bit__int__greater__eq,axiom,
! [K: int,N2: nat] :
( ( ord_less_int @ K @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% take_bit_int_greater_eq
thf(fact_3726_stable__imp__take__bit__eq,axiom,
! [A: code_integer,N2: nat] :
( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= A )
=> ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se1745604003318907178nteger @ N2 @ A )
= zero_z3403309356797280102nteger ) )
& ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se1745604003318907178nteger @ N2 @ A )
= ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_3727_stable__imp__take__bit__eq,axiom,
! [A: int,N2: nat] :
( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A )
=> ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se2923211474154528505it_int @ N2 @ A )
= zero_zero_int ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se2923211474154528505it_int @ N2 @ A )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_3728_stable__imp__take__bit__eq,axiom,
! [A: nat,N2: nat] :
( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se2925701944663578781it_nat @ N2 @ A )
= zero_zero_nat ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se2925701944663578781it_nat @ N2 @ A )
= ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_3729_exp__div__exp__eq,axiom,
! [M: nat,N2: nat] :
( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( times_times_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= zero_zero_nat )
& ( ord_less_eq_nat @ N2 @ M ) ) )
@ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% exp_div_exp_eq
thf(fact_3730_exp__div__exp__eq,axiom,
! [M: nat,N2: nat] :
( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
= ( times_times_int
@ ( zero_n2684676970156552555ol_int
@ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
!= zero_zero_int )
& ( ord_less_eq_nat @ N2 @ M ) ) )
@ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% exp_div_exp_eq
thf(fact_3731_exp__div__exp__eq,axiom,
! [M: nat,N2: nat] :
( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
= ( times_3573771949741848930nteger
@ ( zero_n356916108424825756nteger
@ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
!= zero_z3403309356797280102nteger )
& ( ord_less_eq_nat @ N2 @ M ) ) )
@ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% exp_div_exp_eq
thf(fact_3732_signed__take__bit__rec,axiom,
( bit_ri6519982836138164636nteger
= ( ^ [N4: nat,A3: code_integer] : ( if_Code_integer @ ( N4 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_3733_signed__take__bit__rec,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N4: nat,A3: int] : ( if_int @ ( N4 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_3734_invar__vebt_Ocases,axiom,
! [A1: vEBT_VEBT,A22: nat] :
( ( vEBT_invar_vebt @ A1 @ A22 )
=> ( ( ? [A5: $o,B5: $o] :
( A1
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( A22
!= ( suc @ zero_zero_nat ) ) )
=> ( ! [TreeList3: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M5 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
=> ( ( M5 = N )
=> ( ( Deg2
= ( plus_plus_nat @ N @ M5 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList3: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M5 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
=> ( ( M5
= ( suc @ N ) )
=> ( ( Deg2
= ( plus_plus_nat @ N @ M5 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList3: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M5 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
=> ( ( M5 = N )
=> ( ( Deg2
= ( plus_plus_nat @ N @ M5 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi = Ma2 )
=> ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
=> ( ( ord_less_eq_nat @ Mi @ Ma2 )
=> ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ~ ( ( Mi != Ma2 )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
=> ( ( ord_less_nat @ Mi @ X4 )
& ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList3: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M5 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
=> ( ( M5
= ( suc @ N ) )
=> ( ( Deg2
= ( plus_plus_nat @ N @ M5 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi = Ma2 )
=> ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
=> ( ( ord_less_eq_nat @ Mi @ Ma2 )
=> ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ~ ( ( Mi != Ma2 )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
=> ( ( ord_less_nat @ Mi @ X4 )
& ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_3735_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A12: vEBT_VEBT,A23: nat] :
( ( ? [A3: $o,B2: $o] :
( A12
= ( vEBT_Leaf @ A3 @ B2 ) )
& ( A23
= ( suc @ zero_zero_nat ) ) )
| ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList2 @ Summary3 ) )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N4 ) )
& ( vEBT_invar_vebt @ Summary3 @ N4 )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
& ( A23
= ( plus_plus_nat @ N4 @ N4 ) )
& ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
| ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList2 @ Summary3 ) )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N4 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
& ( A23
= ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) )
& ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
| ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi2: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma3 ) ) @ A23 @ TreeList2 @ Summary3 ) )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N4 ) )
& ( vEBT_invar_vebt @ Summary3 @ N4 )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
& ( A23
= ( plus_plus_nat @ N4 @ N4 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
& ( ord_less_eq_nat @ Mi2 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
& ( ( Mi2 != Ma3 )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N4 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N4 ) ) )
=> ( ( ord_less_nat @ Mi2 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) )
| ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi2: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma3 ) ) @ A23 @ TreeList2 @ Summary3 ) )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N4 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
& ( A23
= ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
& ( ord_less_eq_nat @ Mi2 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
& ( ( Mi2 != Ma3 )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N4 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N4 ) ) )
=> ( ( ord_less_nat @ Mi2 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_3736_both__member__options__from__chilf__to__complete__tree,axiom,
! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi3: 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 @ TreeList ) )
=> ( ( ord_less_eq_nat @ one_one_nat @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_3737_sqrt__sum__squares__half__less,axiom,
! [X: real,U: real,Y: real] :
( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_3738_odd__mod__4__div__2,axiom,
! [N2: nat] :
( ( ( modulo_modulo_nat @ N2 @ ( 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 @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% odd_mod_4_div_2
thf(fact_3739_Bernoulli__inequality__even,axiom,
! [N2: nat,X: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N2 ) ) ) ).
% Bernoulli_inequality_even
thf(fact_3740_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_3741_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_3742_neg__equal__iff__equal,axiom,
! [A: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= ( uminus1482373934393186551omplex @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_3743_neg__equal__iff__equal,axiom,
! [A: code_integer,B: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= ( uminus1351360451143612070nteger @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_3744_neg__equal__iff__equal,axiom,
! [A: rat,B: rat] :
( ( ( uminus_uminus_rat @ A )
= ( uminus_uminus_rat @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_3745_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_3746_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_3747_add_Oinverse__inverse,axiom,
! [A: complex] :
( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_3748_add_Oinverse__inverse,axiom,
! [A: code_integer] :
( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_3749_add_Oinverse__inverse,axiom,
! [A: rat] :
( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_3750_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_3751_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_3752_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_3753_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri8010041392384452111omplex @ M )
= ( semiri8010041392384452111omplex @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_3754_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri681578069525770553at_rat @ M )
= ( semiri681578069525770553at_rat @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_3755_semiring__norm_I90_J,axiom,
! [M: num,N2: num] :
( ( ( bit1 @ M )
= ( bit1 @ N2 ) )
= ( M = N2 ) ) ).
% semiring_norm(90)
thf(fact_3756_real__sqrt__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ X )
= ( sqrt @ Y ) )
= ( X = Y ) ) ).
% real_sqrt_eq_iff
thf(fact_3757_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_3758_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_3759_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_3760_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_3761_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_3762_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_3763_neg__equal__zero,axiom,
! [A: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= A )
= ( A = zero_z3403309356797280102nteger ) ) ).
% neg_equal_zero
thf(fact_3764_neg__equal__zero,axiom,
! [A: rat] :
( ( ( uminus_uminus_rat @ A )
= A )
= ( A = zero_zero_rat ) ) ).
% neg_equal_zero
thf(fact_3765_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_3766_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_3767_equal__neg__zero,axiom,
! [A: code_integer] :
( ( A
= ( uminus1351360451143612070nteger @ A ) )
= ( A = zero_z3403309356797280102nteger ) ) ).
% equal_neg_zero
thf(fact_3768_equal__neg__zero,axiom,
! [A: rat] :
( ( A
= ( uminus_uminus_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% equal_neg_zero
thf(fact_3769_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_3770_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_3771_neg__equal__0__iff__equal,axiom,
! [A: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% neg_equal_0_iff_equal
thf(fact_3772_neg__equal__0__iff__equal,axiom,
! [A: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% neg_equal_0_iff_equal
thf(fact_3773_neg__equal__0__iff__equal,axiom,
! [A: rat] :
( ( ( uminus_uminus_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% neg_equal_0_iff_equal
thf(fact_3774_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_3775_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_3776_neg__0__equal__iff__equal,axiom,
! [A: complex] :
( ( zero_zero_complex
= ( uminus1482373934393186551omplex @ A ) )
= ( zero_zero_complex = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_3777_neg__0__equal__iff__equal,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger
= ( uminus1351360451143612070nteger @ A ) )
= ( zero_z3403309356797280102nteger = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_3778_neg__0__equal__iff__equal,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( uminus_uminus_rat @ A ) )
= ( zero_zero_rat = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_3779_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_3780_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_3781_add_Oinverse__neutral,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% add.inverse_neutral
thf(fact_3782_add_Oinverse__neutral,axiom,
( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% add.inverse_neutral
thf(fact_3783_add_Oinverse__neutral,axiom,
( ( uminus_uminus_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% add.inverse_neutral
thf(fact_3784_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_3785_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_3786_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_3787_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_3788_neg__numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( M = N2 ) ) ).
% neg_numeral_eq_iff
thf(fact_3789_neg__numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( M = N2 ) ) ).
% neg_numeral_eq_iff
thf(fact_3790_neg__numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
= ( M = N2 ) ) ).
% neg_numeral_eq_iff
thf(fact_3791_neg__numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
= ( M = N2 ) ) ).
% neg_numeral_eq_iff
thf(fact_3792_neg__numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
= ( M = N2 ) ) ).
% neg_numeral_eq_iff
thf(fact_3793_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_3794_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_3795_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_3796_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_3797_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_3798_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_3799_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_3800_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_3801_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_3802_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_3803_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_3804_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_3805_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_3806_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_3807_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_3808_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_3809_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_3810_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_3811_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_3812_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_3813_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_3814_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_3815_minus__add__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_3816_minus__add__cancel,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_3817_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_3818_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_3819_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_3820_add__minus__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_3821_add__minus__cancel,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_3822_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_3823_minus__diff__eq,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ).
% minus_diff_eq
thf(fact_3824_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_3825_minus__diff__eq,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
= ( minus_minus_complex @ B @ A ) ) ).
% minus_diff_eq
thf(fact_3826_minus__diff__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
= ( minus_8373710615458151222nteger @ B @ A ) ) ).
% minus_diff_eq
thf(fact_3827_minus__diff__eq,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
= ( minus_minus_rat @ B @ A ) ) ).
% minus_diff_eq
thf(fact_3828_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_3829_div__minus__minus,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
= ( divide6298287555418463151nteger @ A @ B ) ) ).
% div_minus_minus
thf(fact_3830_minus__dvd__iff,axiom,
! [X: real,Y: real] :
( ( dvd_dvd_real @ ( uminus_uminus_real @ X ) @ Y )
= ( dvd_dvd_real @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_3831_minus__dvd__iff,axiom,
! [X: int,Y: int] :
( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y )
= ( dvd_dvd_int @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_3832_minus__dvd__iff,axiom,
! [X: complex,Y: complex] :
( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X ) @ Y )
= ( dvd_dvd_complex @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_3833_minus__dvd__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ Y )
= ( dvd_dvd_Code_integer @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_3834_minus__dvd__iff,axiom,
! [X: rat,Y: rat] :
( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X ) @ Y )
= ( dvd_dvd_rat @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_3835_dvd__minus__iff,axiom,
! [X: real,Y: real] :
( ( dvd_dvd_real @ X @ ( uminus_uminus_real @ Y ) )
= ( dvd_dvd_real @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_3836_dvd__minus__iff,axiom,
! [X: int,Y: int] :
( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y ) )
= ( dvd_dvd_int @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_3837_dvd__minus__iff,axiom,
! [X: complex,Y: complex] :
( ( dvd_dvd_complex @ X @ ( uminus1482373934393186551omplex @ Y ) )
= ( dvd_dvd_complex @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_3838_dvd__minus__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( dvd_dvd_Code_integer @ X @ ( uminus1351360451143612070nteger @ Y ) )
= ( dvd_dvd_Code_integer @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_3839_dvd__minus__iff,axiom,
! [X: rat,Y: rat] :
( ( dvd_dvd_rat @ X @ ( uminus_uminus_rat @ Y ) )
= ( dvd_dvd_rat @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_3840_semiring__norm_I88_J,axiom,
! [M: num,N2: num] :
( ( bit0 @ M )
!= ( bit1 @ N2 ) ) ).
% semiring_norm(88)
thf(fact_3841_semiring__norm_I89_J,axiom,
! [M: num,N2: num] :
( ( bit1 @ M )
!= ( bit0 @ N2 ) ) ).
% semiring_norm(89)
thf(fact_3842_semiring__norm_I84_J,axiom,
! [N2: num] :
( one
!= ( bit1 @ N2 ) ) ).
% semiring_norm(84)
thf(fact_3843_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% semiring_norm(86)
thf(fact_3844_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_3845_mod__minus__minus,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% mod_minus_minus
thf(fact_3846_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_3847_real__sqrt__zero,axiom,
( ( sqrt @ zero_zero_real )
= zero_zero_real ) ).
% real_sqrt_zero
thf(fact_3848_real__sqrt__eq__zero__cancel__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% real_sqrt_eq_zero_cancel_iff
thf(fact_3849_real__sqrt__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% real_sqrt_less_iff
thf(fact_3850_real__sqrt__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% real_sqrt_le_iff
thf(fact_3851_not__Some__eq,axiom,
! [X: option4927543243414619207at_nat] :
( ( ! [Y4: product_prod_nat_nat] :
( X
!= ( some_P7363390416028606310at_nat @ Y4 ) ) )
= ( X = none_P5556105721700978146at_nat ) ) ).
% not_Some_eq
thf(fact_3852_not__Some__eq,axiom,
! [X: option_num] :
( ( ! [Y4: num] :
( X
!= ( some_num @ Y4 ) ) )
= ( X = none_num ) ) ).
% not_Some_eq
thf(fact_3853_not__None__eq,axiom,
! [X: option4927543243414619207at_nat] :
( ( X != none_P5556105721700978146at_nat )
= ( ? [Y4: product_prod_nat_nat] :
( X
= ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ).
% not_None_eq
thf(fact_3854_not__None__eq,axiom,
! [X: option_num] :
( ( X != none_num )
= ( ? [Y4: num] :
( X
= ( some_num @ Y4 ) ) ) ) ).
% not_None_eq
thf(fact_3855_real__sqrt__one,axiom,
( ( sqrt @ one_one_real )
= one_one_real ) ).
% real_sqrt_one
thf(fact_3856_real__sqrt__eq__1__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= one_one_real )
= ( X = one_one_real ) ) ).
% real_sqrt_eq_1_iff
thf(fact_3857_semiring__norm_I73_J,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% semiring_norm(73)
thf(fact_3858_semiring__norm_I80_J,axiom,
! [M: num,N2: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( ord_less_num @ M @ N2 ) ) ).
% semiring_norm(80)
thf(fact_3859_mi__ma__2__deg,axiom,
! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N2 )
=> ( ( ord_less_eq_nat @ Mi3 @ Ma )
& ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% mi_ma_2_deg
thf(fact_3860_neg__less__eq__nonneg,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_3861_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_3862_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_3863_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_3864_less__eq__neg__nonpos,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% less_eq_neg_nonpos
thf(fact_3865_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_3866_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_3867_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_3868_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_3869_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_3870_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_3871_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_3872_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_3873_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_3874_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_3875_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_3876_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_3877_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_3878_less__neg__neg,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% less_neg_neg
thf(fact_3879_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_3880_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_3881_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_3882_neg__less__pos,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_less_pos
thf(fact_3883_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_3884_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_3885_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_3886_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_3887_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_3888_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_3889_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_3890_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_3891_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_3892_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_3893_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_3894_ab__left__minus,axiom,
! [A: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
= zero_zero_complex ) ).
% ab_left_minus
thf(fact_3895_ab__left__minus,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= zero_z3403309356797280102nteger ) ).
% ab_left_minus
thf(fact_3896_ab__left__minus,axiom,
! [A: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
= zero_zero_rat ) ).
% ab_left_minus
thf(fact_3897_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_3898_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_3899_add_Oright__inverse,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
= zero_zero_complex ) ).
% add.right_inverse
thf(fact_3900_add_Oright__inverse,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= zero_z3403309356797280102nteger ) ).
% add.right_inverse
thf(fact_3901_add_Oright__inverse,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
= zero_zero_rat ) ).
% add.right_inverse
thf(fact_3902_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_3903_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_3904_verit__minus__simplify_I3_J,axiom,
! [B: complex] :
( ( minus_minus_complex @ zero_zero_complex @ B )
= ( uminus1482373934393186551omplex @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_3905_verit__minus__simplify_I3_J,axiom,
! [B: code_integer] :
( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
= ( uminus1351360451143612070nteger @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_3906_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_3907_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_3908_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_3909_diff__0,axiom,
! [A: complex] :
( ( minus_minus_complex @ zero_zero_complex @ A )
= ( uminus1482373934393186551omplex @ A ) ) ).
% diff_0
thf(fact_3910_diff__0,axiom,
! [A: code_integer] :
( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
= ( uminus1351360451143612070nteger @ A ) ) ).
% diff_0
thf(fact_3911_diff__0,axiom,
! [A: rat] :
( ( minus_minus_rat @ zero_zero_rat @ A )
= ( uminus_uminus_rat @ A ) ) ).
% diff_0
thf(fact_3912_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N2: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_3913_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N2: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_3914_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N2: num] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
= ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_3915_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N2: num] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
= ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_3916_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N2: num] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
= ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_3917_mult__minus1__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1_right
thf(fact_3918_mult__minus1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1_right
thf(fact_3919_mult__minus1__right,axiom,
! [Z: complex] :
( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ Z ) ) ).
% mult_minus1_right
thf(fact_3920_mult__minus1__right,axiom,
! [Z: code_integer] :
( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ Z ) ) ).
% mult_minus1_right
thf(fact_3921_mult__minus1__right,axiom,
! [Z: rat] :
( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ Z ) ) ).
% mult_minus1_right
thf(fact_3922_mult__minus1,axiom,
! [Z: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1
thf(fact_3923_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_3924_mult__minus1,axiom,
! [Z: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
= ( uminus1482373934393186551omplex @ Z ) ) ).
% mult_minus1
thf(fact_3925_mult__minus1,axiom,
! [Z: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
= ( uminus1351360451143612070nteger @ Z ) ) ).
% mult_minus1
thf(fact_3926_mult__minus1,axiom,
! [Z: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
= ( uminus_uminus_rat @ Z ) ) ).
% mult_minus1
thf(fact_3927_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_3928_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_3929_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_3930_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_3931_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_3932_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_3933_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_3934_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_3935_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_3936_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_3937_divide__minus1,axiom,
! [X: real] :
( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ X ) ) ).
% divide_minus1
thf(fact_3938_divide__minus1,axiom,
! [X: complex] :
( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ X ) ) ).
% divide_minus1
thf(fact_3939_divide__minus1,axiom,
! [X: rat] :
( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ X ) ) ).
% divide_minus1
thf(fact_3940_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_3941_div__minus1__right,axiom,
! [A: code_integer] :
( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ A ) ) ).
% div_minus1_right
thf(fact_3942_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_3943_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_3944_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_3945_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri8010041392384452111omplex @ M )
= zero_zero_complex )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_3946_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri681578069525770553at_rat @ M )
= zero_zero_rat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_3947_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_3948_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_3949_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_3950_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_complex
= ( semiri8010041392384452111omplex @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_3951_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_rat
= ( semiri681578069525770553at_rat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_3952_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_3953_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_3954_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_3955_of__nat__0,axiom,
( ( semiri8010041392384452111omplex @ zero_zero_nat )
= zero_zero_complex ) ).
% of_nat_0
thf(fact_3956_of__nat__0,axiom,
( ( semiri681578069525770553at_rat @ zero_zero_nat )
= zero_zero_rat ) ).
% of_nat_0
thf(fact_3957_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_3958_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_3959_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_3960_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_3961_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_3962_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_3963_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_3964_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_3965_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_real @ N2 ) ) ).
% of_nat_numeral
thf(fact_3966_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_int @ N2 ) ) ).
% of_nat_numeral
thf(fact_3967_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ N2 ) ) ).
% of_nat_numeral
thf(fact_3968_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N2 ) )
= ( numera6690914467698888265omplex @ N2 ) ) ).
% of_nat_numeral
thf(fact_3969_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_rat @ N2 ) ) ).
% of_nat_numeral
thf(fact_3970_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_3971_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_3972_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% of_nat_add
thf(fact_3973_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_add
thf(fact_3974_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_add
thf(fact_3975_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).
% of_nat_add
thf(fact_3976_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% of_nat_add
thf(fact_3977_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% of_nat_mult
thf(fact_3978_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_mult
thf(fact_3979_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N2 ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_mult
thf(fact_3980_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N2 ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).
% of_nat_mult
thf(fact_3981_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N2 ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% of_nat_mult
thf(fact_3982_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_3983_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_3984_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_3985_of__nat__1,axiom,
( ( semiri8010041392384452111omplex @ one_one_nat )
= one_one_complex ) ).
% of_nat_1
thf(fact_3986_of__nat__1,axiom,
( ( semiri681578069525770553at_rat @ one_one_nat )
= one_one_rat ) ).
% of_nat_1
thf(fact_3987_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_3988_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_3989_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_3990_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_complex
= ( semiri8010041392384452111omplex @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_3991_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_rat
= ( semiri681578069525770553at_rat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_3992_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri5074537144036343181t_real @ N2 )
= one_one_real )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_3993_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1314217659103216013at_int @ N2 )
= one_one_int )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_3994_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1316708129612266289at_nat @ N2 )
= one_one_nat )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_3995_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri8010041392384452111omplex @ N2 )
= one_one_complex )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_3996_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri681578069525770553at_rat @ N2 )
= one_one_rat )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_3997_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri5074537144036343181t_real @ X )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3998_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3999_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_4000_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri8010041392384452111omplex @ X )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_4001_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri681578069525770553at_rat @ X )
= ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_4002_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
= ( semiri5074537144036343181t_real @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_4003_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_4004_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_4005_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
= ( semiri8010041392384452111omplex @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_4006_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W )
= ( semiri681578069525770553at_rat @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_4007_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N2 ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_4008_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N2 ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_4009_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N2 ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_4010_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N2 ) )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_4011_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N2 ) )
= ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_4012_real__sqrt__gt__0__iff,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
= ( ord_less_real @ zero_zero_real @ Y ) ) ).
% real_sqrt_gt_0_iff
thf(fact_4013_real__sqrt__lt__0__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% real_sqrt_lt_0_iff
thf(fact_4014_real__sqrt__le__0__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% real_sqrt_le_0_iff
thf(fact_4015_real__sqrt__ge__0__iff,axiom,
! [Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
= ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).
% real_sqrt_ge_0_iff
thf(fact_4016_signed__take__bit__of__minus__1,axiom,
! [N2: nat] :
( ( bit_ri6519982836138164636nteger @ N2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% signed_take_bit_of_minus_1
thf(fact_4017_signed__take__bit__of__minus__1,axiom,
! [N2: nat] :
( ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% signed_take_bit_of_minus_1
thf(fact_4018_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_4019_real__sqrt__gt__1__iff,axiom,
! [Y: real] :
( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
= ( ord_less_real @ one_one_real @ Y ) ) ).
% real_sqrt_gt_1_iff
thf(fact_4020_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_4021_real__sqrt__ge__1__iff,axiom,
! [Y: real] :
( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
= ( ord_less_eq_real @ one_one_real @ Y ) ) ).
% real_sqrt_ge_1_iff
thf(fact_4022_semiring__norm_I9_J,axiom,
! [M: num,N2: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( bit1 @ ( plus_plus_num @ M @ N2 ) ) ) ).
% semiring_norm(9)
thf(fact_4023_semiring__norm_I7_J,axiom,
! [M: num,N2: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( bit1 @ ( plus_plus_num @ M @ N2 ) ) ) ).
% semiring_norm(7)
thf(fact_4024_semiring__norm_I15_J,axiom,
! [M: num,N2: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N2 ) ) ) ).
% semiring_norm(15)
thf(fact_4025_semiring__norm_I14_J,axiom,
! [M: num,N2: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( bit0 @ ( times_times_num @ M @ ( bit1 @ N2 ) ) ) ) ).
% semiring_norm(14)
thf(fact_4026_semiring__norm_I72_J,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% semiring_norm(72)
thf(fact_4027_semiring__norm_I81_J,axiom,
! [M: num,N2: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less_num @ M @ N2 ) ) ).
% semiring_norm(81)
thf(fact_4028_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_4029_semiring__norm_I77_J,axiom,
! [N2: num] : ( ord_less_num @ one @ ( bit1 @ N2 ) ) ).
% semiring_norm(77)
thf(fact_4030_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_4031_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_4032_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
= ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_4033_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_4034_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_4035_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n3304061248610475627l_real @ P ) ) ).
% of_nat_of_bool
thf(fact_4036_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n1201886186963655149omplex @ P ) ) ).
% of_nat_of_bool
thf(fact_4037_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n2052037380579107095ol_rat @ P ) ) ).
% of_nat_of_bool
thf(fact_4038_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n2687167440665602831ol_nat @ P ) ) ).
% of_nat_of_bool
thf(fact_4039_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n2684676970156552555ol_int @ P ) ) ).
% of_nat_of_bool
thf(fact_4040_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n356916108424825756nteger @ P ) ) ).
% of_nat_of_bool
thf(fact_4041_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi3: nat,Ma: nat,TreeList: 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 @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( 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 = Mi3 )
| ( X = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_4042_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_4043_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_4044_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_4045_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_4046_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_4047_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_4048_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_4049_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_4050_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_4051_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_4052_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_4053_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_4054_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_4055_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_4056_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_4057_numeral__eq__neg__one__iff,axiom,
! [N2: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) )
= ( uminus_uminus_real @ one_one_real ) )
= ( N2 = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_4058_numeral__eq__neg__one__iff,axiom,
! [N2: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) )
= ( uminus_uminus_int @ one_one_int ) )
= ( N2 = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_4059_numeral__eq__neg__one__iff,axiom,
! [N2: num] :
( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( N2 = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_4060_numeral__eq__neg__one__iff,axiom,
! [N2: num] :
( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( N2 = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_4061_numeral__eq__neg__one__iff,axiom,
! [N2: num] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) )
= ( uminus_uminus_rat @ one_one_rat ) )
= ( N2 = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_4062_neg__one__eq__numeral__iff,axiom,
! [N2: num] :
( ( ( uminus_uminus_real @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( N2 = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_4063_neg__one__eq__numeral__iff,axiom,
! [N2: num] :
( ( ( uminus_uminus_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( N2 = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_4064_neg__one__eq__numeral__iff,axiom,
! [N2: num] :
( ( ( uminus1482373934393186551omplex @ one_one_complex )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
= ( N2 = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_4065_neg__one__eq__numeral__iff,axiom,
! [N2: num] :
( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
= ( N2 = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_4066_neg__one__eq__numeral__iff,axiom,
! [N2: num] :
( ( ( uminus_uminus_rat @ one_one_rat )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
= ( N2 = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_4067_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_4068_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_4069_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_4070_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_4071_minus__one__mult__self,axiom,
! [N2: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) )
= one_one_real ) ).
% minus_one_mult_self
thf(fact_4072_minus__one__mult__self,axiom,
! [N2: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) )
= one_one_int ) ).
% minus_one_mult_self
thf(fact_4073_minus__one__mult__self,axiom,
! [N2: nat] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) )
= one_one_complex ) ).
% minus_one_mult_self
thf(fact_4074_minus__one__mult__self,axiom,
! [N2: nat] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) )
= one_one_Code_integer ) ).
% minus_one_mult_self
thf(fact_4075_minus__one__mult__self,axiom,
! [N2: nat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) )
= one_one_rat ) ).
% minus_one_mult_self
thf(fact_4076_left__minus__one__mult__self,axiom,
! [N2: nat,A: real] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_4077_left__minus__one__mult__self,axiom,
! [N2: nat,A: int] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_4078_left__minus__one__mult__self,axiom,
! [N2: nat,A: complex] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_4079_left__minus__one__mult__self,axiom,
! [N2: nat,A: code_integer] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_4080_left__minus__one__mult__self,axiom,
! [N2: nat,A: rat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_4081_mod__minus1__right,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% mod_minus1_right
thf(fact_4082_mod__minus1__right,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% mod_minus1_right
thf(fact_4083_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ M ) )
= ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% of_nat_Suc
thf(fact_4084_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_4085_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_4086_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ M ) )
= ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% of_nat_Suc
thf(fact_4087_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri681578069525770553at_rat @ ( suc @ M ) )
= ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% of_nat_Suc
thf(fact_4088_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
= ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_4089_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_4090_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_4091_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_4092_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_4093_real__sqrt__four,axiom,
( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% real_sqrt_four
thf(fact_4094_diff__numeral__simps_I2_J,axiom,
! [M: num,N2: num] :
( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).
% diff_numeral_simps(2)
thf(fact_4095_diff__numeral__simps_I2_J,axiom,
! [M: num,N2: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).
% diff_numeral_simps(2)
thf(fact_4096_diff__numeral__simps_I2_J,axiom,
! [M: num,N2: num] :
( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ).
% diff_numeral_simps(2)
thf(fact_4097_diff__numeral__simps_I2_J,axiom,
! [M: num,N2: num] :
( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
= ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N2 ) ) ) ).
% diff_numeral_simps(2)
thf(fact_4098_diff__numeral__simps_I2_J,axiom,
! [M: num,N2: num] :
( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% diff_numeral_simps(2)
thf(fact_4099_diff__numeral__simps_I3_J,axiom,
! [M: num,N2: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_4100_diff__numeral__simps_I3_J,axiom,
! [M: num,N2: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_4101_diff__numeral__simps_I3_J,axiom,
! [M: num,N2: num] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N2 ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_4102_diff__numeral__simps_I3_J,axiom,
! [M: num,N2: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_4103_diff__numeral__simps_I3_J,axiom,
! [M: num,N2: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_4104_zdiv__numeral__Bit1,axiom,
! [V: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit1
thf(fact_4105_semiring__norm_I3_J,axiom,
! [N2: num] :
( ( plus_plus_num @ one @ ( bit0 @ N2 ) )
= ( bit1 @ N2 ) ) ).
% semiring_norm(3)
thf(fact_4106_semiring__norm_I4_J,axiom,
! [N2: num] :
( ( plus_plus_num @ one @ ( bit1 @ N2 ) )
= ( bit0 @ ( plus_plus_num @ N2 @ one ) ) ) ).
% semiring_norm(4)
thf(fact_4107_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ one )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_4108_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ one )
= ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% semiring_norm(8)
thf(fact_4109_semiring__norm_I10_J,axiom,
! [M: num,N2: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N2 ) @ one ) ) ) ).
% semiring_norm(10)
thf(fact_4110_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% semiring_norm(172)
thf(fact_4111_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% semiring_norm(172)
thf(fact_4112_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
= ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% semiring_norm(172)
thf(fact_4113_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% semiring_norm(172)
thf(fact_4114_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
= ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% semiring_norm(172)
thf(fact_4115_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_4116_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_4117_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
= ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_4118_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
= ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_4119_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_4120_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y ) )
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_4121_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_4122_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y ) )
= ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_4123_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
= ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_4124_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_4125_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N2: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_4126_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N2: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_4127_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N2: num] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_4128_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N2: num] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_4129_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N2: num] :
( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_4130_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N2: num] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_4131_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N2: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_4132_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N2: num] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N2 ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_4133_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N2: num] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_4134_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N2: num] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_4135_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N2: num] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_4136_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N2: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_4137_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N2: num] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
= ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_4138_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N2: num] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
= ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_4139_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N2: num] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
= ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_4140_neg__numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
= ( ord_less_eq_num @ N2 @ M ) ) ).
% neg_numeral_le_iff
thf(fact_4141_neg__numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
= ( ord_less_eq_num @ N2 @ M ) ) ).
% neg_numeral_le_iff
thf(fact_4142_neg__numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( ord_less_eq_num @ N2 @ M ) ) ).
% neg_numeral_le_iff
thf(fact_4143_neg__numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( ord_less_eq_num @ N2 @ M ) ) ).
% neg_numeral_le_iff
thf(fact_4144_neg__numeral__less__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( ord_less_num @ N2 @ M ) ) ).
% neg_numeral_less_iff
thf(fact_4145_neg__numeral__less__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( ord_less_num @ N2 @ M ) ) ).
% neg_numeral_less_iff
thf(fact_4146_neg__numeral__less__iff,axiom,
! [M: num,N2: num] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
= ( ord_less_num @ N2 @ M ) ) ).
% neg_numeral_less_iff
thf(fact_4147_neg__numeral__less__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
= ( ord_less_num @ N2 @ M ) ) ).
% neg_numeral_less_iff
thf(fact_4148_semiring__norm_I16_J,axiom,
! [M: num,N2: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N2 ) @ ( bit0 @ ( times_times_num @ M @ N2 ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_4149_semiring__norm_I74_J,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less_num @ M @ N2 ) ) ).
% semiring_norm(74)
thf(fact_4150_semiring__norm_I79_J,axiom,
! [M: num,N2: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% semiring_norm(79)
thf(fact_4151_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_4152_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_4153_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_4154_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_4155_le__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_4156_le__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_4157_divide__le__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_4158_divide__le__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_4159_eq__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( ( A
= ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= B ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_4160_eq__divide__eq__numeral1_I2_J,axiom,
! [A: complex,B: complex,W: num] :
( ( A
= ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
= ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
!= zero_zero_complex )
=> ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
= B ) )
& ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_4161_eq__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( ( A
= ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
= ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
!= zero_zero_rat )
=> ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
= B ) )
& ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_4162_divide__eq__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= A )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_4163_divide__eq__eq__numeral1_I2_J,axiom,
! [B: complex,W: num,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
= A )
= ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
!= zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
& ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_4164_divide__eq__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
= A )
= ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
!= zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
& ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_4165_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_4166_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_4167_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_4168_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_4169_less__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_4170_less__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_4171_divide__less__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_4172_divide__less__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
= ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_4173_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_4174_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_4175_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_4176_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_4177_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_4178_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_4179_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_4180_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_4181_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_4182_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_4183_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_4184_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_4185_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_4186_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_4187_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_4188_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_4189_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_4190_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_4191_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_4192_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_4193_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_4194_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_4195_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_4196_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_4197_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_4198_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N2: nat] :
( ( ( semiri5074537144036343181t_real @ Y )
= ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_4199_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N2: nat] :
( ( ( semiri1314217659103216013at_int @ Y )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_4200_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N2: nat] :
( ( ( semiri1316708129612266289at_nat @ Y )
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_4201_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N2: nat] :
( ( ( semiri8010041392384452111omplex @ Y )
= ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_4202_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N2: nat] :
( ( ( semiri681578069525770553at_rat @ Y )
= ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_4203_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: nat] :
( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 )
= ( semiri5074537144036343181t_real @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_4204_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
= ( semiri1314217659103216013at_int @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_4205_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= ( semiri1316708129612266289at_nat @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_4206_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: nat] :
( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 )
= ( semiri8010041392384452111omplex @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_4207_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: nat] :
( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 )
= ( semiri681578069525770553at_rat @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_4208_real__of__nat__less__numeral__iff,axiom,
! [N2: nat,W: num] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( numeral_numeral_real @ W ) )
= ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ W ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_4209_numeral__less__real__of__nat__iff,axiom,
! [W: num,N2: nat] :
( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N2 ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_4210_numeral__le__real__of__nat__iff,axiom,
! [N2: num,M: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( semiri5074537144036343181t_real @ M ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_4211_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_4212_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_4213_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_4214_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_4215_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_4216_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_4217_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_4218_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_4219_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_4220_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_4221_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_4222_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_4223_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_4224_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_4225_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_4226_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_4227_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_4228_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_4229_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_4230_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_4231_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_4232_of__nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_4233_of__nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_4234_of__nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_4235_of__nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_4236_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_4237_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: int,N2: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_4238_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: complex,N2: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_4239_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: code_integer,N2: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_4240_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: rat,N2: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_4241_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N2: nat,A: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
= ( power_power_real @ A @ N2 ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_4242_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N2: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
= ( power_power_int @ A @ N2 ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_4243_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N2: nat,A: complex] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
= ( power_power_complex @ A @ N2 ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_4244_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N2: nat,A: code_integer] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
= ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_4245_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N2: nat,A: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
= ( power_power_rat @ A @ N2 ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_4246_power__minus__odd,axiom,
! [N2: nat,A: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
= ( uminus_uminus_real @ ( power_power_real @ A @ N2 ) ) ) ) ).
% power_minus_odd
thf(fact_4247_power__minus__odd,axiom,
! [N2: nat,A: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
= ( uminus_uminus_int @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_minus_odd
thf(fact_4248_power__minus__odd,axiom,
! [N2: nat,A: complex] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
= ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N2 ) ) ) ) ).
% power_minus_odd
thf(fact_4249_power__minus__odd,axiom,
! [N2: nat,A: code_integer] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
= ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ).
% power_minus_odd
thf(fact_4250_power__minus__odd,axiom,
! [N2: nat,A: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
= ( uminus_uminus_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% power_minus_odd
thf(fact_4251_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_4252_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_4253_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_4254_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_4255_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_4256_diff__numeral__special_I3_J,axiom,
! [N2: num] :
( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).
% diff_numeral_special(3)
thf(fact_4257_diff__numeral__special_I3_J,axiom,
! [N2: num] :
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).
% diff_numeral_special(3)
thf(fact_4258_diff__numeral__special_I3_J,axiom,
! [N2: num] :
( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N2 ) ) ) ).
% diff_numeral_special(3)
thf(fact_4259_diff__numeral__special_I3_J,axiom,
! [N2: num] :
( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
= ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N2 ) ) ) ).
% diff_numeral_special(3)
thf(fact_4260_diff__numeral__special_I3_J,axiom,
! [N2: num] :
( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% diff_numeral_special(3)
thf(fact_4261_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_4262_div__Suc__eq__div__add3,axiom,
! [M: nat,N2: nat] :
( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
= ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_4263_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_4264_mod__Suc__eq__mod__add3,axiom,
! [M: nat,N2: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
= ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).
% mod_Suc_eq_mod_add3
thf(fact_4265_signed__take__bit__Suc__minus__bit0,axiom,
! [N2: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_4266_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_4267_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_4268_dbl__simps_I4_J,axiom,
( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_4269_dbl__simps_I4_J,axiom,
( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_4270_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_4271_power__minus1__even,axiom,
! [N2: nat] :
( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= one_one_real ) ).
% power_minus1_even
thf(fact_4272_power__minus1__even,axiom,
! [N2: nat] :
( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= one_one_int ) ).
% power_minus1_even
thf(fact_4273_power__minus1__even,axiom,
! [N2: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= one_one_complex ) ).
% power_minus1_even
thf(fact_4274_power__minus1__even,axiom,
! [N2: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= one_one_Code_integer ) ).
% power_minus1_even
thf(fact_4275_power__minus1__even,axiom,
! [N2: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= one_one_rat ) ).
% power_minus1_even
thf(fact_4276_neg__one__even__power,axiom,
! [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
= one_one_real ) ) ).
% neg_one_even_power
thf(fact_4277_neg__one__even__power,axiom,
! [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
= one_one_int ) ) ).
% neg_one_even_power
thf(fact_4278_neg__one__even__power,axiom,
! [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
= one_one_complex ) ) ).
% neg_one_even_power
thf(fact_4279_neg__one__even__power,axiom,
! [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
= one_one_Code_integer ) ) ).
% neg_one_even_power
thf(fact_4280_neg__one__even__power,axiom,
! [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
= one_one_rat ) ) ).
% neg_one_even_power
thf(fact_4281_neg__one__odd__power,axiom,
! [N2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
= ( uminus_uminus_real @ one_one_real ) ) ) ).
% neg_one_odd_power
thf(fact_4282_neg__one__odd__power,axiom,
! [N2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% neg_one_odd_power
thf(fact_4283_neg__one__odd__power,axiom,
! [N2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% neg_one_odd_power
thf(fact_4284_neg__one__odd__power,axiom,
! [N2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% neg_one_odd_power
thf(fact_4285_neg__one__odd__power,axiom,
! [N2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
= ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% neg_one_odd_power
thf(fact_4286_even__of__nat,axiom,
! [N2: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N2 ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% even_of_nat
thf(fact_4287_even__of__nat,axiom,
! [N2: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% even_of_nat
thf(fact_4288_even__of__nat,axiom,
! [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% even_of_nat
thf(fact_4289_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_4290_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_4291_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_4292_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N2: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_4293_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N2: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_4294_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N2: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_4295_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N2: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_4296_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N2: nat,X: nat] :
( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_4297_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N2: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_4298_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_4299_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_4300_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_4301_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N2: nat,X: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_4302_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N2: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_4303_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N2: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_4304_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N2: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_4305_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_4306_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_4307_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_4308_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_4309_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [X: real,Y: real,Xa: 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 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( 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 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_squared_eq
thf(fact_4310_zmod__numeral__Bit1,axiom,
! [V: num,W: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).
% zmod_numeral_Bit1
thf(fact_4311_signed__take__bit__Suc__bit1,axiom,
! [N2: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_4312_signed__take__bit__Suc__minus__bit1,axiom,
! [N2: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( 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_4313_take__bit__minus,axiom,
! [N2: nat,K: int] :
( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) )
= ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ K ) ) ) ).
% take_bit_minus
thf(fact_4314_signed__take__bit__minus,axiom,
! [N2: nat,K: int] :
( ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) )
= ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ K ) ) ) ).
% signed_take_bit_minus
thf(fact_4315_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_4316_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_4317_minus__equation__iff,axiom,
! [A: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= B )
= ( ( uminus1482373934393186551omplex @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_4318_minus__equation__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= B )
= ( ( uminus1351360451143612070nteger @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_4319_minus__equation__iff,axiom,
! [A: rat,B: rat] :
( ( ( uminus_uminus_rat @ A )
= B )
= ( ( uminus_uminus_rat @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_4320_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_4321_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_4322_equation__minus__iff,axiom,
! [A: complex,B: complex] :
( ( A
= ( uminus1482373934393186551omplex @ B ) )
= ( B
= ( uminus1482373934393186551omplex @ A ) ) ) ).
% equation_minus_iff
thf(fact_4323_equation__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( A
= ( uminus1351360451143612070nteger @ B ) )
= ( B
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% equation_minus_iff
thf(fact_4324_equation__minus__iff,axiom,
! [A: rat,B: rat] :
( ( A
= ( uminus_uminus_rat @ B ) )
= ( B
= ( uminus_uminus_rat @ A ) ) ) ).
% equation_minus_iff
thf(fact_4325_real__sqrt__minus,axiom,
! [X: real] :
( ( sqrt @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).
% real_sqrt_minus
thf(fact_4326_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_4327_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_4328_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_4329_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_4330_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_4331_real__sqrt__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_less_mono
thf(fact_4332_real__sqrt__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_le_mono
thf(fact_4333_real__sqrt__divide,axiom,
! [X: real,Y: real] :
( ( sqrt @ ( divide_divide_real @ X @ Y ) )
= ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_divide
thf(fact_4334_real__sqrt__mult,axiom,
! [X: real,Y: real] :
( ( sqrt @ ( times_times_real @ X @ Y ) )
= ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_mult
thf(fact_4335_mult__of__nat__commute,axiom,
! [X: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_4336_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_4337_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_4338_mult__of__nat__commute,axiom,
! [X: nat,Y: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
= ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_4339_mult__of__nat__commute,axiom,
! [X: nat,Y: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
= ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_4340_real__sqrt__power,axiom,
! [X: real,K: nat] :
( ( sqrt @ ( power_power_real @ X @ K ) )
= ( power_power_real @ ( sqrt @ X ) @ K ) ) ).
% real_sqrt_power
thf(fact_4341_combine__options__cases,axiom,
! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
( ( ( X = none_P5556105721700978146at_nat )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_P5556105721700978146at_nat )
=> ( P @ X @ Y ) )
=> ( ! [A5: product_prod_nat_nat,B5: product_prod_nat_nat] :
( ( X
= ( some_P7363390416028606310at_nat @ A5 ) )
=> ( ( Y
= ( some_P7363390416028606310at_nat @ B5 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_4342_combine__options__cases,axiom,
! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
( ( ( X = none_P5556105721700978146at_nat )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_num )
=> ( P @ X @ Y ) )
=> ( ! [A5: product_prod_nat_nat,B5: num] :
( ( X
= ( some_P7363390416028606310at_nat @ A5 ) )
=> ( ( Y
= ( some_num @ B5 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_4343_combine__options__cases,axiom,
! [X: option_num,P: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
( ( ( X = none_num )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_P5556105721700978146at_nat )
=> ( P @ X @ Y ) )
=> ( ! [A5: num,B5: product_prod_nat_nat] :
( ( X
= ( some_num @ A5 ) )
=> ( ( Y
= ( some_P7363390416028606310at_nat @ B5 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_4344_combine__options__cases,axiom,
! [X: option_num,P: option_num > option_num > $o,Y: option_num] :
( ( ( X = none_num )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_num )
=> ( P @ X @ Y ) )
=> ( ! [A5: num,B5: num] :
( ( X
= ( some_num @ A5 ) )
=> ( ( Y
= ( some_num @ B5 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_4345_split__option__all,axiom,
( ( ^ [P5: option4927543243414619207at_nat > $o] :
! [X7: option4927543243414619207at_nat] : ( P5 @ X7 ) )
= ( ^ [P6: option4927543243414619207at_nat > $o] :
( ( P6 @ none_P5556105721700978146at_nat )
& ! [X3: product_prod_nat_nat] : ( P6 @ ( some_P7363390416028606310at_nat @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_4346_split__option__all,axiom,
( ( ^ [P5: option_num > $o] :
! [X7: option_num] : ( P5 @ X7 ) )
= ( ^ [P6: option_num > $o] :
( ( P6 @ none_num )
& ! [X3: num] : ( P6 @ ( some_num @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_4347_split__option__ex,axiom,
( ( ^ [P5: option4927543243414619207at_nat > $o] :
? [X7: option4927543243414619207at_nat] : ( P5 @ X7 ) )
= ( ^ [P6: option4927543243414619207at_nat > $o] :
( ( P6 @ none_P5556105721700978146at_nat )
| ? [X3: product_prod_nat_nat] : ( P6 @ ( some_P7363390416028606310at_nat @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_4348_split__option__ex,axiom,
( ( ^ [P5: option_num > $o] :
? [X7: option_num] : ( P5 @ X7 ) )
= ( ^ [P6: option_num > $o] :
( ( P6 @ none_num )
| ? [X3: num] : ( P6 @ ( some_num @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_4349_option_Oexhaust,axiom,
! [Y: option4927543243414619207at_nat] :
( ( Y != none_P5556105721700978146at_nat )
=> ~ ! [X23: product_prod_nat_nat] :
( Y
!= ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% option.exhaust
thf(fact_4350_option_Oexhaust,axiom,
! [Y: option_num] :
( ( Y != none_num )
=> ~ ! [X23: num] :
( Y
!= ( some_num @ X23 ) ) ) ).
% option.exhaust
thf(fact_4351_option_OdiscI,axiom,
! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
( ( Option
= ( some_P7363390416028606310at_nat @ X22 ) )
=> ( Option != none_P5556105721700978146at_nat ) ) ).
% option.discI
thf(fact_4352_option_OdiscI,axiom,
! [Option: option_num,X22: num] :
( ( Option
= ( some_num @ X22 ) )
=> ( Option != none_num ) ) ).
% option.discI
thf(fact_4353_option_Odistinct_I1_J,axiom,
! [X22: product_prod_nat_nat] :
( none_P5556105721700978146at_nat
!= ( some_P7363390416028606310at_nat @ X22 ) ) ).
% option.distinct(1)
thf(fact_4354_option_Odistinct_I1_J,axiom,
! [X22: num] :
( none_num
!= ( some_num @ X22 ) ) ).
% option.distinct(1)
thf(fact_4355_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_4356_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_4357_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_4358_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_4359_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_4360_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_4361_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_4362_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_4363_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_4364_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_4365_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_4366_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_4367_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_4368_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_4369_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_4370_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_4371_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_4372_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_4373_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_4374_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_4375_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_4376_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_4377_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_4378_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_4379_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_4380_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_4381_numeral__neq__neg__numeral,axiom,
! [M: num,N2: num] :
( ( numeral_numeral_real @ M )
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_4382_numeral__neq__neg__numeral,axiom,
! [M: num,N2: num] :
( ( numeral_numeral_int @ M )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_4383_numeral__neq__neg__numeral,axiom,
! [M: num,N2: num] :
( ( numera6690914467698888265omplex @ M )
!= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_4384_numeral__neq__neg__numeral,axiom,
! [M: num,N2: num] :
( ( numera6620942414471956472nteger @ M )
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_4385_numeral__neq__neg__numeral,axiom,
! [M: num,N2: num] :
( ( numeral_numeral_rat @ M )
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_4386_neg__numeral__neq__numeral,axiom,
! [M: num,N2: num] :
( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
!= ( numeral_numeral_real @ N2 ) ) ).
% neg_numeral_neq_numeral
thf(fact_4387_neg__numeral__neq__numeral,axiom,
! [M: num,N2: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
!= ( numeral_numeral_int @ N2 ) ) ).
% neg_numeral_neq_numeral
thf(fact_4388_neg__numeral__neq__numeral,axiom,
! [M: num,N2: num] :
( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
!= ( numera6690914467698888265omplex @ N2 ) ) ).
% neg_numeral_neq_numeral
thf(fact_4389_neg__numeral__neq__numeral,axiom,
! [M: num,N2: num] :
( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
!= ( numera6620942414471956472nteger @ N2 ) ) ).
% neg_numeral_neq_numeral
thf(fact_4390_neg__numeral__neq__numeral,axiom,
! [M: num,N2: num] :
( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
!= ( numeral_numeral_rat @ N2 ) ) ).
% neg_numeral_neq_numeral
thf(fact_4391_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_4392_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_4393_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_4394_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_4395_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_4396_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_4397_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_4398_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_4399_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_4400_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_4401_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_4402_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_4403_one__neq__neg__one,axiom,
( one_one_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% one_neq_neg_one
thf(fact_4404_one__neq__neg__one,axiom,
( one_one_Code_integer
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% one_neq_neg_one
thf(fact_4405_one__neq__neg__one,axiom,
( one_one_rat
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% one_neq_neg_one
thf(fact_4406_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_4407_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_4408_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_4409_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_4410_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_4411_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_4412_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_4413_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_4414_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_4415_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_4416_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_4417_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_4418_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_4419_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_4420_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_4421_minus__diff__commute,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
= ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_4422_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_4423_minus__diff__commute,axiom,
! [B: complex,A: complex] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
= ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_4424_minus__diff__commute,axiom,
! [B: code_integer,A: code_integer] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
= ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_4425_minus__diff__commute,axiom,
! [B: rat,A: rat] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
= ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_4426_minus__diff__minus,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_4427_minus__diff__minus,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_4428_minus__diff__minus,axiom,
! [A: complex,B: complex] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
= ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_4429_minus__diff__minus,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_4430_minus__diff__minus,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
= ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_4431_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_4432_minus__divide__right,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% minus_divide_right
thf(fact_4433_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_4434_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_4435_minus__divide__divide,axiom,
! [A: complex,B: complex] :
( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ).
% minus_divide_divide
thf(fact_4436_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_4437_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_4438_minus__divide__left,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_4439_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_4440_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_4441_div__minus__right,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% div_minus_right
thf(fact_4442_take__bit__of__nat,axiom,
! [N2: nat,M: nat] :
( ( bit_se2923211474154528505it_int @ N2 @ ( semiri1314217659103216013at_int @ M ) )
= ( semiri1314217659103216013at_int @ ( bit_se2925701944663578781it_nat @ N2 @ M ) ) ) ).
% take_bit_of_nat
thf(fact_4443_take__bit__of__nat,axiom,
! [N2: nat,M: nat] :
( ( bit_se2925701944663578781it_nat @ N2 @ ( semiri1316708129612266289at_nat @ M ) )
= ( semiri1316708129612266289at_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) ) ) ).
% take_bit_of_nat
thf(fact_4444_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_4445_mod__minus__right,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% mod_minus_right
thf(fact_4446_mod__minus__cong,axiom,
! [A: int,B: int,A4: int] :
( ( ( modulo_modulo_int @ A @ B )
= ( modulo_modulo_int @ A4 @ B ) )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A4 ) @ B ) ) ) ).
% mod_minus_cong
thf(fact_4447_mod__minus__cong,axiom,
! [A: code_integer,B: code_integer,A4: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ B )
= ( modulo364778990260209775nteger @ A4 @ B ) )
=> ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A4 ) @ B ) ) ) ).
% mod_minus_cong
thf(fact_4448_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_4449_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_4450_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_4451_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi3: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma ) ) @ zero_zero_nat @ Va2 @ Vb ) @ X )
= ( ( X = Mi3 )
| ( X = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_4452_real__sqrt__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% real_sqrt_gt_zero
thf(fact_4453_real__sqrt__eq__zero__cancel,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( sqrt @ X )
= zero_zero_real )
=> ( X = zero_zero_real ) ) ) ).
% real_sqrt_eq_zero_cancel
thf(fact_4454_real__sqrt__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_zero
thf(fact_4455_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_4456_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_4457_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_4458_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_4459_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_4460_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_4461_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_4462_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_4463_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% of_nat_less_0_iff
thf(fact_4464_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_4465_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_4466_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_4467_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ N2 ) )
!= zero_zero_complex ) ).
% of_nat_neq_0
thf(fact_4468_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri681578069525770553at_rat @ ( suc @ N2 ) )
!= zero_zero_rat ) ).
% of_nat_neq_0
thf(fact_4469_div__mult2__eq_H,axiom,
! [A: int,M: nat,N2: nat] :
( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% div_mult2_eq'
thf(fact_4470_div__mult2__eq_H,axiom,
! [A: nat,M: nat,N2: nat] :
( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% div_mult2_eq'
thf(fact_4471_of__nat__less__imp__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_4472_of__nat__less__imp__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_4473_of__nat__less__imp__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_4474_of__nat__less__imp__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_4475_less__imp__of__nat__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_4476_less__imp__of__nat__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_4477_less__imp__of__nat__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_4478_less__imp__of__nat__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_4479_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% of_nat_mono
thf(fact_4480_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_4481_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_4482_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_4483_not__numeral__le__neg__numeral,axiom,
! [M: num,N2: num] :
~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_4484_not__numeral__le__neg__numeral,axiom,
! [M: num,N2: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_4485_not__numeral__le__neg__numeral,axiom,
! [M: num,N2: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_4486_not__numeral__le__neg__numeral,axiom,
! [M: num,N2: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_4487_neg__numeral__le__numeral,axiom,
! [M: num,N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).
% neg_numeral_le_numeral
thf(fact_4488_neg__numeral__le__numeral,axiom,
! [M: num,N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).
% neg_numeral_le_numeral
thf(fact_4489_neg__numeral__le__numeral,axiom,
! [M: num,N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).
% neg_numeral_le_numeral
thf(fact_4490_neg__numeral__le__numeral,axiom,
! [M: num,N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).
% neg_numeral_le_numeral
thf(fact_4491_zero__neq__neg__numeral,axiom,
! [N2: num] :
( zero_zero_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% zero_neq_neg_numeral
thf(fact_4492_zero__neq__neg__numeral,axiom,
! [N2: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% zero_neq_neg_numeral
thf(fact_4493_zero__neq__neg__numeral,axiom,
! [N2: num] :
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% zero_neq_neg_numeral
thf(fact_4494_zero__neq__neg__numeral,axiom,
! [N2: num] :
( zero_z3403309356797280102nteger
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% zero_neq_neg_numeral
thf(fact_4495_zero__neq__neg__numeral,axiom,
! [N2: num] :
( zero_zero_rat
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% zero_neq_neg_numeral
thf(fact_4496_neg__numeral__less__numeral,axiom,
! [M: num,N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).
% neg_numeral_less_numeral
thf(fact_4497_neg__numeral__less__numeral,axiom,
! [M: num,N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).
% neg_numeral_less_numeral
thf(fact_4498_neg__numeral__less__numeral,axiom,
! [M: num,N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).
% neg_numeral_less_numeral
thf(fact_4499_neg__numeral__less__numeral,axiom,
! [M: num,N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).
% neg_numeral_less_numeral
thf(fact_4500_not__numeral__less__neg__numeral,axiom,
! [M: num,N2: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_4501_not__numeral__less__neg__numeral,axiom,
! [M: num,N2: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_4502_not__numeral__less__neg__numeral,axiom,
! [M: num,N2: num] :
~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_4503_not__numeral__less__neg__numeral,axiom,
! [M: num,N2: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_4504_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_4505_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_4506_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_4507_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_4508_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_4509_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_4510_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_4511_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_4512_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_4513_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_4514_zero__neq__neg__one,axiom,
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% zero_neq_neg_one
thf(fact_4515_zero__neq__neg__one,axiom,
( zero_z3403309356797280102nteger
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% zero_neq_neg_one
thf(fact_4516_zero__neq__neg__one,axiom,
( zero_zero_rat
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% zero_neq_neg_one
thf(fact_4517_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_4518_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_4519_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_4520_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_4521_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_4522_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_4523_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_4524_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_4525_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_4526_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_4527_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_4528_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_4529_add_Oinverse__unique,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_4530_add_Oinverse__unique,axiom,
! [A: code_integer,B: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger )
=> ( ( uminus1351360451143612070nteger @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_4531_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_4532_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_4533_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_4534_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_4535_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_4536_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_4537_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_4538_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_4539_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_4540_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_4541_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_4542_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X23: num] :
( Y
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_4543_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_4544_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_4545_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_4546_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_4547_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_4548_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_4549_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_4550_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_4551_numeral__times__minus__swap,axiom,
! [W: num,X: real] :
( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
= ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_4552_numeral__times__minus__swap,axiom,
! [W: num,X: int] :
( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
= ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_4553_numeral__times__minus__swap,axiom,
! [W: num,X: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X ) )
= ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_4554_numeral__times__minus__swap,axiom,
! [W: num,X: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
= ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_4555_numeral__times__minus__swap,axiom,
! [W: num,X: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
= ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_4556_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_4557_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_4558_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_4559_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_4560_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_4561_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_4562_numeral__neq__neg__one,axiom,
! [N2: num] :
( ( numeral_numeral_real @ N2 )
!= ( uminus_uminus_real @ one_one_real ) ) ).
% numeral_neq_neg_one
thf(fact_4563_numeral__neq__neg__one,axiom,
! [N2: num] :
( ( numeral_numeral_int @ N2 )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_4564_numeral__neq__neg__one,axiom,
! [N2: num] :
( ( numera6690914467698888265omplex @ N2 )
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% numeral_neq_neg_one
thf(fact_4565_numeral__neq__neg__one,axiom,
! [N2: num] :
( ( numera6620942414471956472nteger @ N2 )
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% numeral_neq_neg_one
thf(fact_4566_numeral__neq__neg__one,axiom,
! [N2: num] :
( ( numeral_numeral_rat @ N2 )
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% numeral_neq_neg_one
thf(fact_4567_one__neq__neg__numeral,axiom,
! [N2: num] :
( one_one_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% one_neq_neg_numeral
thf(fact_4568_one__neq__neg__numeral,axiom,
! [N2: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% one_neq_neg_numeral
thf(fact_4569_one__neq__neg__numeral,axiom,
! [N2: num] :
( one_one_complex
!= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% one_neq_neg_numeral
thf(fact_4570_one__neq__neg__numeral,axiom,
! [N2: num] :
( one_one_Code_integer
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% one_neq_neg_numeral
thf(fact_4571_one__neq__neg__numeral,axiom,
! [N2: num] :
( one_one_rat
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% one_neq_neg_numeral
thf(fact_4572_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_4573_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_4574_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_4575_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_4576_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_4577_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_4578_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N2 ) )
= ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_4579_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A3: real,B2: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4580_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4581_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_complex
= ( ^ [A3: complex,B2: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4582_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_8373710615458151222nteger
= ( ^ [A3: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4583_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4584_diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A3: real,B2: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_4585_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_4586_diff__conv__add__uminus,axiom,
( minus_minus_complex
= ( ^ [A3: complex,B2: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_4587_diff__conv__add__uminus,axiom,
( minus_8373710615458151222nteger
= ( ^ [A3: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_4588_diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_4589_group__cancel_Osub2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( minus_minus_real @ A @ B3 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_4590_group__cancel_Osub2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_4591_group__cancel_Osub2,axiom,
! [B3: complex,K: complex,B: complex,A: complex] :
( ( B3
= ( plus_plus_complex @ K @ B ) )
=> ( ( minus_minus_complex @ A @ B3 )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_4592_group__cancel_Osub2,axiom,
! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
( ( B3
= ( plus_p5714425477246183910nteger @ K @ B ) )
=> ( ( minus_8373710615458151222nteger @ A @ B3 )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_4593_group__cancel_Osub2,axiom,
! [B3: rat,K: rat,B: rat,A: rat] :
( ( B3
= ( plus_plus_rat @ K @ B ) )
=> ( ( minus_minus_rat @ A @ B3 )
= ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_4594_of__nat__dvd__iff,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) )
= ( dvd_dvd_nat @ M @ N2 ) ) ).
% of_nat_dvd_iff
thf(fact_4595_of__nat__dvd__iff,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( dvd_dvd_nat @ M @ N2 ) ) ).
% of_nat_dvd_iff
thf(fact_4596_of__nat__dvd__iff,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( dvd_dvd_nat @ M @ N2 ) ) ).
% of_nat_dvd_iff
thf(fact_4597_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_4598_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_4599_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_4600_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_4601_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_4602_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_4603_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_4604_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_4605_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_4606_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_4607_of__nat__mod,axiom,
! [M: nat,N2: nat] :
( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N2 ) )
= ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).
% of_nat_mod
thf(fact_4608_of__nat__mod,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) )
= ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_mod
thf(fact_4609_of__nat__mod,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N2 ) )
= ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_mod
thf(fact_4610_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_4611_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N2: int] :
( ( ( times_times_int @ M @ N2 )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_4612_zmult__eq__1__iff,axiom,
! [M: int,N2: int] :
( ( ( times_times_int @ M @ N2 )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N2 = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N2
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_4613_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_4614_zmod__zminus1__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ K @ L )
!= zero_zero_int ) ) ).
% zmod_zminus1_not_zero
thf(fact_4615_zmod__zminus2__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ K @ L )
!= zero_zero_int ) ) ).
% zmod_zminus2_not_zero
thf(fact_4616_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X3: real,Y4: real] : ( plus_plus_real @ X3 @ ( uminus_uminus_real @ Y4 ) ) ) ) ).
% minus_real_def
thf(fact_4617_Bernoulli__inequality,axiom,
! [X: real,N2: 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 @ N2 ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N2 ) ) ) ).
% Bernoulli_inequality
thf(fact_4618_real__div__sqrt,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( divide_divide_real @ X @ ( sqrt @ X ) )
= ( sqrt @ X ) ) ) ).
% real_div_sqrt
thf(fact_4619_sqrt__add__le__add__sqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).
% sqrt_add_le_add_sqrt
thf(fact_4620_le__real__sqrt__sumsq,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).
% le_real_sqrt_sumsq
thf(fact_4621_not__zero__le__neg__numeral,axiom,
! [N2: num] :
~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_4622_not__zero__le__neg__numeral,axiom,
! [N2: num] :
~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_4623_not__zero__le__neg__numeral,axiom,
! [N2: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_4624_not__zero__le__neg__numeral,axiom,
! [N2: num] :
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_4625_neg__numeral__le__zero,axiom,
! [N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).
% neg_numeral_le_zero
thf(fact_4626_neg__numeral__le__zero,axiom,
! [N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).
% neg_numeral_le_zero
thf(fact_4627_neg__numeral__le__zero,axiom,
! [N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).
% neg_numeral_le_zero
thf(fact_4628_neg__numeral__le__zero,axiom,
! [N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).
% neg_numeral_le_zero
thf(fact_4629_not__zero__less__neg__numeral,axiom,
! [N2: num] :
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_4630_not__zero__less__neg__numeral,axiom,
! [N2: num] :
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_4631_not__zero__less__neg__numeral,axiom,
! [N2: num] :
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_4632_not__zero__less__neg__numeral,axiom,
! [N2: num] :
~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_4633_neg__numeral__less__zero,axiom,
! [N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).
% neg_numeral_less_zero
thf(fact_4634_neg__numeral__less__zero,axiom,
! [N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).
% neg_numeral_less_zero
thf(fact_4635_neg__numeral__less__zero,axiom,
! [N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).
% neg_numeral_less_zero
thf(fact_4636_neg__numeral__less__zero,axiom,
! [N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).
% neg_numeral_less_zero
thf(fact_4637_le__minus__one__simps_I1_J,axiom,
ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% le_minus_one_simps(1)
thf(fact_4638_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_4639_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_4640_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_4641_le__minus__one__simps_I3_J,axiom,
~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% le_minus_one_simps(3)
thf(fact_4642_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_4643_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_4644_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_4645_numeral__Bit1,axiom,
! [N2: num] :
( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).
% numeral_Bit1
thf(fact_4646_numeral__Bit1,axiom,
! [N2: num] :
( ( numeral_numeral_real @ ( bit1 @ N2 ) )
= ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).
% numeral_Bit1
thf(fact_4647_numeral__Bit1,axiom,
! [N2: num] :
( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).
% numeral_Bit1
thf(fact_4648_numeral__Bit1,axiom,
! [N2: num] :
( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).
% numeral_Bit1
thf(fact_4649_numeral__Bit1,axiom,
! [N2: num] :
( ( numeral_numeral_int @ ( bit1 @ N2 ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).
% numeral_Bit1
thf(fact_4650_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_4651_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_4652_less__minus__one__simps_I3_J,axiom,
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% less_minus_one_simps(3)
thf(fact_4653_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_4654_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_4655_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_4656_less__minus__one__simps_I1_J,axiom,
ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% less_minus_one_simps(1)
thf(fact_4657_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_4658_of__nat__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N2 ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% of_nat_diff
thf(fact_4659_of__nat__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N2 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% of_nat_diff
thf(fact_4660_of__nat__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N2 ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).
% of_nat_diff
thf(fact_4661_of__nat__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N2 ) )
= ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ) ).
% of_nat_diff
thf(fact_4662_of__nat__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N2 ) )
= ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).
% of_nat_diff
thf(fact_4663_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_4664_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_4665_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_4666_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_4667_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% not_numeral_le_neg_one
thf(fact_4668_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_4669_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_4670_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_4671_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_4672_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_4673_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_4674_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_4675_neg__one__le__numeral,axiom,
! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% neg_one_le_numeral
thf(fact_4676_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_4677_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_4678_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_4679_neg__numeral__le__one,axiom,
! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% neg_numeral_le_one
thf(fact_4680_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_4681_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_4682_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_4683_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_4684_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_4685_neg__numeral__less__one,axiom,
! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% neg_numeral_less_one
thf(fact_4686_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_4687_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_4688_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_4689_neg__one__less__numeral,axiom,
! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% neg_one_less_numeral
thf(fact_4690_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_4691_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_4692_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_4693_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% not_numeral_less_neg_one
thf(fact_4694_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_4695_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_4696_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_4697_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_4698_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_4699_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_4700_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_4701_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_4702_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_4703_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_4704_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_4705_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_4706_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_4707_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_4708_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_4709_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_4710_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_4711_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_4712_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_4713_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_4714_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_4715_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_4716_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_4717_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_4718_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_4719_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_4720_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_4721_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_4722_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_4723_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_4724_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_4725_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_4726_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_4727_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_4728_uminus__numeral__One,axiom,
( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% uminus_numeral_One
thf(fact_4729_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_4730_uminus__numeral__One,axiom,
( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% uminus_numeral_One
thf(fact_4731_uminus__numeral__One,axiom,
( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% uminus_numeral_One
thf(fact_4732_uminus__numeral__One,axiom,
( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% uminus_numeral_One
thf(fact_4733_eval__nat__numeral_I3_J,axiom,
! [N2: num] :
( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
= ( suc @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_4734_power__minus,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( power_power_real @ A @ N2 ) ) ) ).
% power_minus
thf(fact_4735_power__minus,axiom,
! [A: int,N2: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ A @ N2 ) ) ) ).
% power_minus
thf(fact_4736_power__minus,axiom,
! [A: complex,N2: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( power_power_complex @ A @ N2 ) ) ) ).
% power_minus
thf(fact_4737_power__minus,axiom,
! [A: code_integer,N2: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).
% power_minus
thf(fact_4738_power__minus,axiom,
! [A: rat,N2: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ A @ N2 ) ) ) ).
% power_minus
thf(fact_4739_cong__exp__iff__simps_I10_J,axiom,
! [M: num,Q3: num,N2: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_4740_cong__exp__iff__simps_I10_J,axiom,
! [M: num,Q3: num,N2: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_4741_cong__exp__iff__simps_I10_J,axiom,
! [M: num,Q3: num,N2: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
!= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_4742_cong__exp__iff__simps_I12_J,axiom,
! [M: num,Q3: num,N2: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_4743_cong__exp__iff__simps_I12_J,axiom,
! [M: num,Q3: num,N2: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_4744_cong__exp__iff__simps_I12_J,axiom,
! [M: num,Q3: num,N2: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
!= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_4745_cong__exp__iff__simps_I13_J,axiom,
! [M: num,Q3: num,N2: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_4746_cong__exp__iff__simps_I13_J,axiom,
! [M: num,Q3: num,N2: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_4747_cong__exp__iff__simps_I13_J,axiom,
! [M: num,Q3: num,N2: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_4748_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_4749_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_4750_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_4751_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_4752_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_4753_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y3: real] :
? [N: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_4754_real__of__nat__div4,axiom,
! [N2: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_4755_real__of__nat__div,axiom,
! [D: nat,N2: nat] :
( ( dvd_dvd_nat @ D @ N2 )
=> ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ D ) )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% real_of_nat_div
thf(fact_4756_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_4757_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_4758_real__add__le__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_4759_real__0__le__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_4760_zmod__zminus1__eq__if,axiom,
! [A: int,B: int] :
( ( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
= zero_zero_int ) )
& ( ( ( modulo_modulo_int @ A @ B )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% zmod_zminus1_eq_if
thf(fact_4761_zmod__zminus2__eq__if,axiom,
! [A: int,B: int] :
( ( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
=> ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
= zero_zero_int ) )
& ( ( ( modulo_modulo_int @ A @ B )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
= ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).
% zmod_zminus2_eq_if
thf(fact_4762_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_4763_option_Osize_I4_J,axiom,
! [X22: num] :
( ( size_size_option_num @ ( some_num @ X22 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_4764_sqrt2__less__2,axiom,
ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% sqrt2_less_2
thf(fact_4765_mod__mult2__eq_H,axiom,
! [A: code_integer,M: nat,N2: nat] :
( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_4766_mod__mult2__eq_H,axiom,
! [A: int,M: nat,N2: nat] :
( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
= ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_4767_mod__mult2__eq_H,axiom,
! [A: nat,M: nat,N2: nat] :
( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_4768_numeral__Bit1__div__2,axiom,
! [N2: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N2 ) ) ).
% numeral_Bit1_div_2
thf(fact_4769_numeral__Bit1__div__2,axiom,
! [N2: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N2 ) ) ).
% numeral_Bit1_div_2
thf(fact_4770_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_4771_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_4772_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_4773_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_4774_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_4775_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_4776_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_4777_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_4778_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_4779_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_4780_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_4781_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_4782_eq__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_4783_eq__divide__eq__numeral_I2_J,axiom,
! [W: num,B: complex,C: complex] :
( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
= B ) )
& ( ( C = zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
= zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_4784_eq__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
= ( divide_divide_rat @ B @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
= B ) )
& ( ( C = zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
= zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_4785_divide__eq__eq__numeral_I2_J,axiom,
! [B: real,C: real,W: num] :
( ( ( divide_divide_real @ B @ C )
= ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_4786_divide__eq__eq__numeral_I2_J,axiom,
! [B: complex,C: complex,W: num] :
( ( ( divide1717551699836669952omplex @ B @ C )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
= ( ( ( C != zero_zero_complex )
=> ( B
= ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
= zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_4787_divide__eq__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ( divide_divide_rat @ B @ C )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
= ( ( ( C != zero_zero_rat )
=> ( B
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
= zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_4788_odd__numeral,axiom,
! [N2: num] :
~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) ) ).
% odd_numeral
thf(fact_4789_odd__numeral,axiom,
! [N2: num] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) ) ).
% odd_numeral
thf(fact_4790_odd__numeral,axiom,
! [N2: num] :
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ).
% odd_numeral
thf(fact_4791_add__divide__eq__if__simps_I3_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= B ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_4792_add__divide__eq__if__simps_I3_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
= B ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_4793_add__divide__eq__if__simps_I3_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
= B ) )
& ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
= ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_4794_minus__divide__add__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_4795_minus__divide__add__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_4796_minus__divide__add__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
= ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_4797_cong__exp__iff__simps_I3_J,axiom,
! [N2: num,Q3: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
!= zero_zero_nat ) ).
% cong_exp_iff_simps(3)
thf(fact_4798_cong__exp__iff__simps_I3_J,axiom,
! [N2: num,Q3: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
!= zero_zero_int ) ).
% cong_exp_iff_simps(3)
thf(fact_4799_cong__exp__iff__simps_I3_J,axiom,
! [N2: num,Q3: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
!= zero_z3403309356797280102nteger ) ).
% cong_exp_iff_simps(3)
thf(fact_4800_add__divide__eq__if__simps_I6_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_4801_add__divide__eq__if__simps_I6_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
= ( uminus1482373934393186551omplex @ B ) ) )
& ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_4802_add__divide__eq__if__simps_I6_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
= ( uminus_uminus_rat @ B ) ) )
& ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
= ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_4803_add__divide__eq__if__simps_I5_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_4804_add__divide__eq__if__simps_I5_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
= ( uminus1482373934393186551omplex @ B ) ) )
& ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_4805_add__divide__eq__if__simps_I5_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= ( uminus_uminus_rat @ B ) ) )
& ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_4806_minus__divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_4807_minus__divide__diff__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_4808_minus__divide__diff__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
= ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_4809_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_4810_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_4811_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_4812_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_4813_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_4814_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_4815_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_4816_numeral__3__eq__3,axiom,
( ( numeral_numeral_nat @ ( bit1 @ one ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% numeral_3_eq_3
thf(fact_4817_power2__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_real @ Y ) ) ) ) ).
% power2_eq_iff
thf(fact_4818_power2__eq__iff,axiom,
! [X: int,Y: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_int @ Y ) ) ) ) ).
% power2_eq_iff
thf(fact_4819_power2__eq__iff,axiom,
! [X: complex,Y: complex] :
( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y )
| ( X
= ( uminus1482373934393186551omplex @ Y ) ) ) ) ).
% power2_eq_iff
thf(fact_4820_power2__eq__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y )
| ( X
= ( uminus1351360451143612070nteger @ Y ) ) ) ) ).
% power2_eq_iff
thf(fact_4821_power2__eq__iff,axiom,
! [X: rat,Y: rat] :
( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_rat @ Y ) ) ) ) ).
% power2_eq_iff
thf(fact_4822_Suc3__eq__add__3,axiom,
! [N2: nat] :
( ( suc @ ( suc @ ( suc @ N2 ) ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ).
% Suc3_eq_add_3
thf(fact_4823_field__char__0__class_Oof__nat__div,axiom,
! [M: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_4824_field__char__0__class_Oof__nat__div,axiom,
! [M: nat,N2: nat] :
( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_4825_field__char__0__class_Oof__nat__div,axiom,
! [M: nat,N2: nat] :
( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N2 ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_4826_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N4: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N4 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% nat_less_real_le
thf(fact_4827_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N4: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_4828_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_4829_verit__less__mono__div__int2,axiom,
! [A2: int,B3: int,N2: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B3 @ N2 ) @ ( divide_divide_int @ A2 @ N2 ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_4830_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_4831_real__less__rsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
=> ( ord_less_real @ X @ ( sqrt @ Y ) ) ) ).
% real_less_rsqrt
thf(fact_4832_real__le__rsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
=> ( ord_less_eq_real @ X @ ( sqrt @ Y ) ) ) ).
% real_le_rsqrt
thf(fact_4833_sqrt__le__D,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( sqrt @ X ) @ Y )
=> ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sqrt_le_D
thf(fact_4834_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_4835_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_4836_of__nat__less__two__power,axiom,
! [N2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).
% of_nat_less_two_power
thf(fact_4837_of__nat__less__two__power,axiom,
! [N2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).
% of_nat_less_two_power
thf(fact_4838_of__nat__less__two__power,axiom,
! [N2: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ).
% of_nat_less_two_power
thf(fact_4839_inverse__of__nat__le,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( N2 != zero_zero_nat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_4840_inverse__of__nat__le,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( N2 != zero_zero_nat )
=> ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_4841_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_4842_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_4843_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_4844_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_4845_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_4846_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_4847_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_4848_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_4849_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_4850_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_4851_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_4852_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_4853_divide__less__eq__numeral_I2_J,axiom,
! [B: real,C: real,W: num] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ 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 @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_4854_divide__less__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ 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 @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_4855_less__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ 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 @ W ) ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_4856_less__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ 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 @ W ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_4857_cong__exp__iff__simps_I7_J,axiom,
! [Q3: num,N2: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q3 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(7)
thf(fact_4858_cong__exp__iff__simps_I7_J,axiom,
! [Q3: num,N2: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q3 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(7)
thf(fact_4859_cong__exp__iff__simps_I7_J,axiom,
! [Q3: num,N2: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q3 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(7)
thf(fact_4860_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q3: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(11)
thf(fact_4861_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q3: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(11)
thf(fact_4862_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q3: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(11)
thf(fact_4863_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_4864_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_4865_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_4866_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_4867_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_4868_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 )
=> ( ! [M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_4869_Suc__div__eq__add3__div,axiom,
! [M: nat,N2: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
= ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).
% Suc_div_eq_add3_div
thf(fact_4870_uminus__power__if,axiom,
! [N2: nat,A: real] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
= ( power_power_real @ A @ N2 ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
= ( uminus_uminus_real @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% uminus_power_if
thf(fact_4871_uminus__power__if,axiom,
! [N2: nat,A: int] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
= ( power_power_int @ A @ N2 ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
= ( uminus_uminus_int @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% uminus_power_if
thf(fact_4872_uminus__power__if,axiom,
! [N2: nat,A: complex] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
= ( power_power_complex @ A @ N2 ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
= ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N2 ) ) ) ) ) ).
% uminus_power_if
thf(fact_4873_uminus__power__if,axiom,
! [N2: nat,A: code_integer] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
= ( power_8256067586552552935nteger @ A @ N2 ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
= ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ) ).
% uminus_power_if
thf(fact_4874_uminus__power__if,axiom,
! [N2: nat,A: rat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
= ( power_power_rat @ A @ N2 ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
= ( uminus_uminus_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% uminus_power_if
thf(fact_4875_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N2 @ K ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4876_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N2 @ K ) )
= ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4877_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N2 @ K ) )
= ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4878_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N2 @ K ) )
= ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4879_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N2 @ K ) )
= ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4880_Suc__mod__eq__add3__mod,axiom,
! [M: nat,N2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
= ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).
% Suc_mod_eq_add3_mod
thf(fact_4881_realpow__square__minus__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% realpow_square_minus_le
thf(fact_4882_take__bit__Suc__minus__bit0,axiom,
! [N2: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_4883_real__of__nat__div2,axiom,
! [N2: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_4884_real__of__nat__div3,axiom,
! [N2: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) ) @ one_one_real ) ).
% real_of_nat_div3
thf(fact_4885_signed__take__bit__int__less__eq__self__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ K )
= ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_4886_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N2: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_4887_signed__take__bit__int__greater__self__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
= ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_4888_minus__mod__int__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
= ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).
% minus_mod_int_eq
thf(fact_4889_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_4890_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_4891_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_4892_real__sqrt__unique,axiom,
! [Y: real,X: real] :
( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( sqrt @ X )
= Y ) ) ) ).
% real_sqrt_unique
thf(fact_4893_real__le__lsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% real_le_lsqrt
thf(fact_4894_lemma__real__divide__sqrt__less,axiom,
! [U: real] :
( ( ord_less_real @ zero_zero_real @ U )
=> ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% lemma_real_divide_sqrt_less
thf(fact_4895_real__sqrt__sum__squares__eq__cancel2,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= Y )
=> ( X = zero_zero_real ) ) ).
% real_sqrt_sum_squares_eq_cancel2
thf(fact_4896_real__sqrt__sum__squares__eq__cancel,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= X )
=> ( Y = zero_zero_real ) ) ).
% real_sqrt_sum_squares_eq_cancel
thf(fact_4897_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_4898_real__sqrt__sum__squares__ge2,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge2
thf(fact_4899_real__sqrt__sum__squares__ge1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge1
thf(fact_4900_le__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( 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 @ W ) ) @ 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 @ W ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_4901_le__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( 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 @ W ) ) @ 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 @ W ) ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_4902_divide__le__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ 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 @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_4903_divide__le__eq__numeral_I2_J,axiom,
! [B: real,C: real,W: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ 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 @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_4904_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_4905_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_4906_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_4907_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_4908_minus__power__mult__self,axiom,
! [A: real,N2: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 ) )
= ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% minus_power_mult_self
thf(fact_4909_minus__power__mult__self,axiom,
! [A: int,N2: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) )
= ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% minus_power_mult_self
thf(fact_4910_minus__power__mult__self,axiom,
! [A: complex,N2: nat] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 ) )
= ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% minus_power_mult_self
thf(fact_4911_minus__power__mult__self,axiom,
! [A: code_integer,N2: nat] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 ) )
= ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% minus_power_mult_self
thf(fact_4912_minus__power__mult__self,axiom,
! [A: rat,N2: nat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 ) )
= ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% minus_power_mult_self
thf(fact_4913_minus__one__power__iff,axiom,
! [N2: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
= one_one_real ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% minus_one_power_iff
thf(fact_4914_minus__one__power__iff,axiom,
! [N2: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
= one_one_int ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% minus_one_power_iff
thf(fact_4915_minus__one__power__iff,axiom,
! [N2: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
= one_one_complex ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% minus_one_power_iff
thf(fact_4916_minus__one__power__iff,axiom,
! [N2: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
= one_one_Code_integer ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% minus_one_power_iff
thf(fact_4917_minus__one__power__iff,axiom,
! [N2: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
= one_one_rat ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
= ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% minus_one_power_iff
thf(fact_4918_linear__plus__1__le__power,axiom,
! [X: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N2 ) ) ) ).
% linear_plus_1_le_power
thf(fact_4919_signed__take__bit__int__eq__self__iff,axiom,
! [N2: nat,K: int] :
( ( ( bit_ri631733984087533419it_int @ N2 @ K )
= K )
= ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
& ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_4920_signed__take__bit__int__eq__self,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
=> ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
=> ( ( bit_ri631733984087533419it_int @ N2 @ K )
= K ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_4921_minus__1__div__exp__eq__int,axiom,
! [N2: nat] :
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% minus_1_div_exp_eq_int
thf(fact_4922_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( divide_divide_int @ K @ L )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_4923_real__less__lsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% real_less_lsqrt
thf(fact_4924_sqrt__sum__squares__le__sum,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y ) ) ) ) ).
% sqrt_sum_squares_le_sum
thf(fact_4925_sqrt__even__pow2,axiom,
! [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% sqrt_even_pow2
thf(fact_4926_power__minus1__odd,axiom,
! [N2: nat] :
( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% power_minus1_odd
thf(fact_4927_power__minus1__odd,axiom,
! [N2: nat] :
( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% power_minus1_odd
thf(fact_4928_power__minus1__odd,axiom,
! [N2: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% power_minus1_odd
thf(fact_4929_power__minus1__odd,axiom,
! [N2: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% power_minus1_odd
thf(fact_4930_power__minus1__odd,axiom,
! [N2: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% power_minus1_odd
thf(fact_4931_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ zero_zero_int )
=> ( ( P @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K3: int] :
( ( P @ K3 )
=> ( ( K3 != zero_zero_int )
=> ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K3: int] :
( ( P @ K3 )
=> ( ( K3
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_4932_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X: real,Y: real,Xa: 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 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_ge_zero
thf(fact_4933_real__sqrt__power__even,axiom,
! [N2: nat,X: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( sqrt @ X ) @ N2 )
= ( power_power_real @ X @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% real_sqrt_power_even
thf(fact_4934_arith__geo__mean__sqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% arith_geo_mean_sqrt
thf(fact_4935_invar__vebt_Ointros_I4_J,axiom,
! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi3: nat,Ma: nat] :
( ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M = N2 )
=> ( ( Deg
= ( plus_plus_nat @ N2 @ M ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi3 = Ma )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) )
=> ( ( ord_less_eq_nat @ Mi3 @ Ma )
=> ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( ( Mi3 != Ma )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_4936_take__bit__Suc__bit1,axiom,
! [N2: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_Suc_bit1
thf(fact_4937_take__bit__Suc__bit1,axiom,
! [N2: nat,K: num] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% take_bit_Suc_bit1
thf(fact_4938_take__bit__Suc__minus__1__eq,axiom,
! [N2: nat] :
( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ one_one_Code_integer ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_4939_take__bit__Suc__minus__1__eq,axiom,
! [N2: nat] :
( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ one_one_int ) )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ one_one_int ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_4940_take__bit__numeral__minus__1__eq,axiom,
! [K: num] :
( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).
% take_bit_numeral_minus_1_eq
thf(fact_4941_take__bit__numeral__minus__1__eq,axiom,
! [K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).
% take_bit_numeral_minus_1_eq
thf(fact_4942_invar__vebt_Ointros_I5_J,axiom,
! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi3: nat,Ma: nat] :
( ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M
= ( suc @ N2 ) )
=> ( ( Deg
= ( plus_plus_nat @ N2 @ M ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi3 = Ma )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) )
=> ( ( ord_less_eq_nat @ Mi3 @ Ma )
=> ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( ( Mi3 != Ma )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_4943_signed__take__bit__int__greater__eq,axiom,
! [K: int,N2: nat] :
( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_4944_take__bit__minus__small__eq,axiom,
! [K: int,N2: nat] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
=> ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ K ) )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_4945_member__inv,axiom,
! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
& ( ( X = Mi3 )
| ( X = Ma )
| ( ( ord_less_nat @ X @ Ma )
& ( ord_less_nat @ Mi3 @ X )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
& ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( 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_4946_divmod__step__eq,axiom,
! [L: num,R4: nat,Q3: nat] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R4 )
=> ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R4 ) )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R4 @ ( numeral_numeral_nat @ L ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R4 )
=> ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R4 ) )
= ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R4 ) ) ) ) ).
% divmod_step_eq
thf(fact_4947_divmod__step__eq,axiom,
! [L: num,R4: int,Q3: int] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R4 )
=> ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R4 ) )
= ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R4 @ ( numeral_numeral_int @ L ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R4 )
=> ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R4 ) )
= ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R4 ) ) ) ) ).
% divmod_step_eq
thf(fact_4948_divmod__step__eq,axiom,
! [L: num,R4: code_integer,Q3: code_integer] :
( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R4 )
=> ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R4 ) )
= ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R4 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R4 )
=> ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R4 ) )
= ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R4 ) ) ) ) ).
% divmod_step_eq
thf(fact_4949_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_4950_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_4951_nat__approx__posE,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ~ ! [N: nat] :
~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ E2 ) ) ).
% nat_approx_posE
thf(fact_4952_nat__approx__posE,axiom,
! [E2: rat] :
( ( ord_less_rat @ zero_zero_rat @ E2 )
=> ~ ! [N: nat] :
~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) ) @ E2 ) ) ).
% nat_approx_posE
thf(fact_4953_compl__le__compl__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
= ( ord_less_eq_set_nat @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_4954_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_4955_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_4956_dbl__dec__simps_I4_J,axiom,
( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_4957_dbl__dec__simps_I4_J,axiom,
( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_4958_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_4959_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_4960_negative__eq__positive,axiom,
! [N2: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N2 = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_4961_negative__zle,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_4962_int__dvd__int__iff,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( dvd_dvd_nat @ M @ N2 ) ) ).
% int_dvd_int_iff
thf(fact_4963_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6511756317524482435omplex @ one_one_complex )
= one_one_complex ) ).
% dbl_dec_simps(3)
thf(fact_4964_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6075765906172075777c_real @ one_one_real )
= one_one_real ) ).
% dbl_dec_simps(3)
thf(fact_4965_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
= one_one_rat ) ).
% dbl_dec_simps(3)
thf(fact_4966_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_4967_negative__zless,axiom,
! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_4968_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_4969_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_4970_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% dbl_dec_simps(2)
thf(fact_4971_dbl__dec__simps_I2_J,axiom,
( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% dbl_dec_simps(2)
thf(fact_4972_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% dbl_dec_simps(2)
thf(fact_4973_int__cases2,axiom,
! [Z: int] :
( ! [N: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N ) )
=> ~ ! [N: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% int_cases2
thf(fact_4974_int__cases,axiom,
! [Z: int] :
( ! [N: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N ) )
=> ~ ! [N: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ).
% int_cases
thf(fact_4975_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N: nat] : ( P @ ( semiri1314217659103216013at_int @ N ) )
=> ( ! [N: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_4976_not__int__zless__negative,axiom,
! [N2: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_4977_int__diff__cases,axiom,
! [Z: int] :
~ ! [M5: nat,N: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% int_diff_cases
thf(fact_4978_int__cases4,axiom,
! [M: int] :
( ! [N: nat] :
( M
!= ( semiri1314217659103216013at_int @ N ) )
=> ~ ! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% int_cases4
thf(fact_4979_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_4980_int__ops_I3_J,axiom,
! [N2: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_int @ N2 ) ) ).
% int_ops(3)
thf(fact_4981_int__zle__neg,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N2 = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_4982_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_4983_zle__int,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% zle_int
thf(fact_4984_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_4985_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N: nat] :
( K
= ( semiri1314217659103216013at_int @ N ) ) ) ).
% zero_le_imp_eq_int
thf(fact_4986_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N: nat] :
( K
!= ( semiri1314217659103216013at_int @ N ) ) ) ).
% nonneg_int_cases
thf(fact_4987_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% nonpos_int_cases
thf(fact_4988_negative__zle__0,axiom,
! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_4989_zadd__int__left,axiom,
! [M: nat,N2: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_4990_int__plus,axiom,
! [N2: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_4991_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_4992_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_4993_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z5: int] :
? [N4: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_4994_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_4995_zmod__int,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
= ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% zmod_int
thf(fact_4996_VEBT__internal_Ovalid_H_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [Uu2: $o,Uv2: $o,D4: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D4 ) )
=> ~ ! [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_4997_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N ) )
=> ~ ! [N: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% int_cases3
thf(fact_4998_xor__num_Ocases,axiom,
! [X: product_prod_num_num] :
( ( X
!= ( product_Pair_num_num @ one @ one ) )
=> ( ! [N: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit0 @ N ) ) )
=> ( ! [N: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit1 @ N ) ) )
=> ( ! [M5: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) )
=> ( ! [M5: num,N: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N ) ) )
=> ( ! [M5: num,N: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N ) ) )
=> ( ! [M5: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) )
=> ( ! [M5: num,N: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N ) ) )
=> ~ ! [M5: num,N: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_4999_not__zle__0__negative,axiom,
! [N2: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ).
% not_zle_0_negative
thf(fact_5000_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ).
% negD
thf(fact_5001_negative__zless__0,axiom,
! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_5002_int__Suc,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).
% int_Suc
thf(fact_5003_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_5004_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z5: int] :
? [N4: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_5005_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_5006_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% pos_int_cases
thf(fact_5007_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
& ( K
= ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_5008_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% neg_int_cases
thf(fact_5009_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_5010_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_5011_VEBT__internal_Onaive__member_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A5: $o,B5: $o,X2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X2 ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT,X2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ X2 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_5012_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_5013_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_5014_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_5015_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_5016_dbl__dec__def,axiom,
( neg_nu6511756317524482435omplex
= ( ^ [X3: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).
% dbl_dec_def
thf(fact_5017_dbl__dec__def,axiom,
( neg_nu6075765906172075777c_real
= ( ^ [X3: real] : ( minus_minus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).
% dbl_dec_def
thf(fact_5018_dbl__dec__def,axiom,
( neg_nu3179335615603231917ec_rat
= ( ^ [X3: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).
% dbl_dec_def
thf(fact_5019_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_5020_compl__mono,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).
% compl_mono
thf(fact_5021_compl__le__swap1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
=> ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% compl_le_swap1
thf(fact_5022_compl__le__swap2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_5023_real__arch__simple,axiom,
! [X: rat] :
? [N: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N ) ) ).
% real_arch_simple
thf(fact_5024_real__arch__simple,axiom,
! [X: real] :
? [N: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ).
% real_arch_simple
thf(fact_5025_reals__Archimedean2,axiom,
! [X: real] :
? [N: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ).
% reals_Archimedean2
thf(fact_5026_reals__Archimedean2,axiom,
! [X: rat] :
? [N: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N ) ) ).
% reals_Archimedean2
thf(fact_5027_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N: nat] :
( ~ ( P @ N )
& ( P @ ( suc @ N ) ) ) ) ) ).
% exists_least_lemma
thf(fact_5028_eq__diff__eq_H,axiom,
! [X: real,Y: real,Z: real] :
( ( X
= ( minus_minus_real @ Y @ Z ) )
= ( Y
= ( plus_plus_real @ X @ Z ) ) ) ).
% eq_diff_eq'
thf(fact_5029_VEBT__internal_Omembermima_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ 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 ) )
=> ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X2 ) )
=> ( ! [Mi: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X2 ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT,X2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ X2 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_5030_vebt__member_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A5: $o,B5: $o,X2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X2 ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X2 ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X2 ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X2 ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X2 ) ) ) ) ) ) ).
% vebt_member.cases
thf(fact_5031_ex__less__of__nat__mult,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_5032_ex__less__of__nat__mult,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ? [N: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_5033_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_5034_lemma__termdiff3,axiom,
! [H2: real,Z: real,K5: real,N2: nat] :
( ( H2 != zero_zero_real )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N2 ) @ ( power_power_real @ Z @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_5035_lemma__termdiff3,axiom,
! [H2: complex,Z: complex,K5: real,N2: nat] :
( ( H2 != zero_zero_complex )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_5036_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( 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_5037_divmod__algorithm__code_I8_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_num @ M @ N2 )
=> ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less_num @ M @ N2 )
=> ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_5038_divmod__algorithm__code_I8_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_num @ M @ N2 )
=> ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less_num @ M @ N2 )
=> ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_5039_divmod__algorithm__code_I8_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_num @ M @ N2 )
=> ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less_num @ M @ N2 )
=> ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_5040_divmod__algorithm__code_I7_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_eq_num @ M @ N2 )
=> ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M @ N2 )
=> ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_5041_divmod__algorithm__code_I7_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_eq_num @ M @ N2 )
=> ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M @ N2 )
=> ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_5042_divmod__algorithm__code_I7_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_eq_num @ M @ N2 )
=> ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M @ N2 )
=> ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_5043_divides__aux__eq,axiom,
! [Q3: nat,R4: nat] :
( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q3 @ R4 ) )
= ( R4 = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_5044_divides__aux__eq,axiom,
! [Q3: int,R4: int] :
( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q3 @ R4 ) )
= ( R4 = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_5045_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_5046_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one )
= zero_zero_nat ) ).
% pred_numeral_simps(1)
thf(fact_5047_Suc__eq__numeral,axiom,
! [N2: nat,K: num] :
( ( ( suc @ N2 )
= ( numeral_numeral_nat @ K ) )
= ( N2
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_5048_eq__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( ( numeral_numeral_nat @ K )
= ( suc @ N2 ) )
= ( ( pred_numeral @ K )
= N2 ) ) ).
% eq_numeral_Suc
thf(fact_5049_pred__numeral__simps_I3_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit1 @ K ) )
= ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% pred_numeral_simps(3)
thf(fact_5050_less__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
= ( ord_less_nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% less_numeral_Suc
thf(fact_5051_less__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( ord_less_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
= ( ord_less_nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% less_Suc_numeral
thf(fact_5052_le__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% le_numeral_Suc
thf(fact_5053_le__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
= ( ord_less_eq_nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% le_Suc_numeral
thf(fact_5054_diff__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% diff_numeral_Suc
thf(fact_5055_diff__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( minus_minus_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
= ( minus_minus_nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% diff_Suc_numeral
thf(fact_5056_dvd__numeral__simp,axiom,
! [M: num,N2: num] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
= ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N2 @ M ) ) ) ).
% dvd_numeral_simp
thf(fact_5057_dvd__numeral__simp,axiom,
! [M: num,N2: num] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
= ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N2 @ M ) ) ) ).
% dvd_numeral_simp
thf(fact_5058_dvd__numeral__simp,axiom,
! [M: num,N2: num] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) )
= ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N2 @ M ) ) ) ).
% dvd_numeral_simp
thf(fact_5059_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_5060_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_5061_divmod__algorithm__code_I2_J,axiom,
! [M: num] :
( ( unique3479559517661332726nteger @ M @ one )
= ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% divmod_algorithm_code(2)
thf(fact_5062_divmod__algorithm__code_I3_J,axiom,
! [N2: num] :
( ( unique5052692396658037445od_int @ one @ ( bit0 @ N2 ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_5063_divmod__algorithm__code_I3_J,axiom,
! [N2: num] :
( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N2 ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_5064_divmod__algorithm__code_I3_J,axiom,
! [N2: num] :
( ( unique3479559517661332726nteger @ one @ ( bit0 @ N2 ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_5065_divmod__algorithm__code_I4_J,axiom,
! [N2: num] :
( ( unique5052692396658037445od_int @ one @ ( bit1 @ N2 ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_5066_divmod__algorithm__code_I4_J,axiom,
! [N2: num] :
( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N2 ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_5067_divmod__algorithm__code_I4_J,axiom,
! [N2: num] :
( ( unique3479559517661332726nteger @ one @ ( bit1 @ N2 ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_5068_signed__take__bit__numeral__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_numeral_bit0
thf(fact_5069_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_5070_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_5071_int__int__eq,axiom,
! [M: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M = N2 ) ) ).
% int_int_eq
thf(fact_5072_numeral__eq__Suc,axiom,
( numeral_numeral_nat
= ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_5073_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K2: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ one_one_nat ) ) ) ).
% pred_numeral_def
thf(fact_5074_divmod__int__def,axiom,
( unique5052692396658037445od_int
= ( ^ [M2: num,N4: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N4 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N4 ) ) ) ) ) ).
% divmod_int_def
thf(fact_5075_divmod__def,axiom,
( unique5052692396658037445od_int
= ( ^ [M2: num,N4: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N4 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N4 ) ) ) ) ) ).
% divmod_def
thf(fact_5076_divmod__def,axiom,
( unique5055182867167087721od_nat
= ( ^ [M2: num,N4: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N4 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N4 ) ) ) ) ) ).
% divmod_def
thf(fact_5077_divmod__def,axiom,
( unique3479559517661332726nteger
= ( ^ [M2: num,N4: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N4 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N4 ) ) ) ) ) ).
% divmod_def
thf(fact_5078_divmod_H__nat__def,axiom,
( unique5055182867167087721od_nat
= ( ^ [M2: num,N4: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N4 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N4 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_5079_take__bit__numeral__bit0,axiom,
! [L: num,K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_5080_take__bit__numeral__bit0,axiom,
! [L: num,K: num] :
( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_5081_take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_5082_divmod__divmod__step,axiom,
( unique5055182867167087721od_nat
= ( ^ [M2: num,N4: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M2 @ N4 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M2 ) ) @ ( unique5026877609467782581ep_nat @ N4 @ ( unique5055182867167087721od_nat @ M2 @ ( bit0 @ N4 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_5083_divmod__divmod__step,axiom,
( unique5052692396658037445od_int
= ( ^ [M2: num,N4: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M2 @ N4 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M2 ) ) @ ( unique5024387138958732305ep_int @ N4 @ ( unique5052692396658037445od_int @ M2 @ ( bit0 @ N4 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_5084_divmod__divmod__step,axiom,
( unique3479559517661332726nteger
= ( ^ [M2: num,N4: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M2 @ N4 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( unique4921790084139445826nteger @ N4 @ ( unique3479559517661332726nteger @ M2 @ ( bit0 @ N4 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_5085_take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_numeral_bit1
thf(fact_5086_take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% take_bit_numeral_bit1
thf(fact_5087_norm__divide__numeral,axiom,
! [A: real,W: num] :
( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
= ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% norm_divide_numeral
thf(fact_5088_norm__divide__numeral,axiom,
! [A: complex,W: num] :
( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
= ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% norm_divide_numeral
thf(fact_5089_norm__mult__numeral1,axiom,
! [W: num,A: real] :
( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
= ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% norm_mult_numeral1
thf(fact_5090_norm__mult__numeral1,axiom,
! [W: num,A: complex] :
( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
= ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% norm_mult_numeral1
thf(fact_5091_norm__mult__numeral2,axiom,
! [A: real,W: num] :
( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
= ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% norm_mult_numeral2
thf(fact_5092_norm__mult__numeral2,axiom,
! [A: complex,W: num] :
( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
= ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% norm_mult_numeral2
thf(fact_5093_norm__neg__numeral,axiom,
! [W: num] :
( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( numeral_numeral_real @ W ) ) ).
% norm_neg_numeral
thf(fact_5094_norm__neg__numeral,axiom,
! [W: num] :
( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
= ( numeral_numeral_real @ W ) ) ).
% norm_neg_numeral
thf(fact_5095_norm__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_le_zero_iff
thf(fact_5096_norm__le__zero__iff,axiom,
! [X: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_le_zero_iff
thf(fact_5097_zero__less__norm__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
= ( X != zero_zero_real ) ) ).
% zero_less_norm_iff
thf(fact_5098_zero__less__norm__iff,axiom,
! [X: complex] :
( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
= ( X != zero_zero_complex ) ) ).
% zero_less_norm_iff
thf(fact_5099_norm__numeral,axiom,
! [W: num] :
( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
= ( numeral_numeral_real @ W ) ) ).
% norm_numeral
thf(fact_5100_norm__numeral,axiom,
! [W: num] :
( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
= ( numeral_numeral_real @ W ) ) ).
% norm_numeral
thf(fact_5101_norm__zero,axiom,
( ( real_V7735802525324610683m_real @ zero_zero_real )
= zero_zero_real ) ).
% norm_zero
thf(fact_5102_norm__zero,axiom,
( ( real_V1022390504157884413omplex @ zero_zero_complex )
= zero_zero_real ) ).
% norm_zero
thf(fact_5103_norm__eq__zero,axiom,
! [X: real] :
( ( ( real_V7735802525324610683m_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_eq_zero
thf(fact_5104_norm__eq__zero,axiom,
! [X: complex] :
( ( ( real_V1022390504157884413omplex @ X )
= zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_eq_zero
thf(fact_5105_norm__one,axiom,
( ( real_V7735802525324610683m_real @ one_one_real )
= one_one_real ) ).
% norm_one
thf(fact_5106_norm__one,axiom,
( ( real_V1022390504157884413omplex @ one_one_complex )
= one_one_real ) ).
% norm_one
thf(fact_5107_norm__not__less__zero,axiom,
! [X: complex] :
~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).
% norm_not_less_zero
thf(fact_5108_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_5109_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_5110_norm__power,axiom,
! [X: real,N2: nat] :
( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N2 ) )
= ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N2 ) ) ).
% norm_power
thf(fact_5111_norm__power,axiom,
! [X: complex,N2: nat] :
( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N2 ) )
= ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N2 ) ) ).
% norm_power
thf(fact_5112_norm__uminus__minus,axiom,
! [X: real,Y: real] :
( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y ) )
= ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% norm_uminus_minus
thf(fact_5113_norm__uminus__minus,axiom,
! [X: complex,Y: complex] :
( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y ) )
= ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% norm_uminus_minus
thf(fact_5114_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_5115_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_5116_power__eq__imp__eq__norm,axiom,
! [W: real,N2: nat,Z: real] :
( ( ( power_power_real @ W @ N2 )
= ( power_power_real @ Z @ N2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( real_V7735802525324610683m_real @ W )
= ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_5117_power__eq__imp__eq__norm,axiom,
! [W: complex,N2: nat,Z: complex] :
( ( ( power_power_complex @ W @ N2 )
= ( power_power_complex @ Z @ N2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( real_V1022390504157884413omplex @ W )
= ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_5118_norm__mult__less,axiom,
! [X: real,R4: real,Y: real,S: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R4 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R4 @ S ) ) ) ) ).
% norm_mult_less
thf(fact_5119_norm__mult__less,axiom,
! [X: complex,R4: real,Y: complex,S: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R4 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R4 @ S ) ) ) ) ).
% norm_mult_less
thf(fact_5120_norm__triangle__lt,axiom,
! [X: real,Y: real,E2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E2 ) ) ).
% norm_triangle_lt
thf(fact_5121_norm__triangle__lt,axiom,
! [X: complex,Y: complex,E2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E2 ) ) ).
% norm_triangle_lt
thf(fact_5122_norm__add__less,axiom,
! [X: real,R4: real,Y: real,S: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R4 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R4 @ S ) ) ) ) ).
% norm_add_less
thf(fact_5123_norm__add__less,axiom,
! [X: complex,R4: real,Y: complex,S: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R4 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R4 @ S ) ) ) ) ).
% norm_add_less
thf(fact_5124_norm__triangle__mono,axiom,
! [A: real,R4: real,B: real,S: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R4 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R4 @ S ) ) ) ) ).
% norm_triangle_mono
thf(fact_5125_norm__triangle__mono,axiom,
! [A: complex,R4: real,B: complex,S: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R4 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R4 @ S ) ) ) ) ).
% norm_triangle_mono
thf(fact_5126_norm__triangle__ineq,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% norm_triangle_ineq
thf(fact_5127_norm__triangle__ineq,axiom,
! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% norm_triangle_ineq
thf(fact_5128_norm__triangle__le,axiom,
! [X: real,Y: real,E2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E2 ) ) ).
% norm_triangle_le
thf(fact_5129_norm__triangle__le,axiom,
! [X: complex,Y: complex,E2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E2 ) ) ).
% norm_triangle_le
thf(fact_5130_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_5131_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_5132_norm__power__ineq,axiom,
! [X: real,N2: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N2 ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N2 ) ) ).
% norm_power_ineq
thf(fact_5133_norm__power__ineq,axiom,
! [X: complex,N2: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N2 ) ) ).
% norm_power_ineq
thf(fact_5134_norm__diff__triangle__less,axiom,
! [X: real,Y: real,E1: real,Z: real,E22: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_5135_norm__diff__triangle__less,axiom,
! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_5136_norm__triangle__le__diff,axiom,
! [X: real,Y: real,E2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E2 ) ) ).
% norm_triangle_le_diff
thf(fact_5137_norm__triangle__le__diff,axiom,
! [X: complex,Y: complex,E2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E2 ) ) ).
% norm_triangle_le_diff
thf(fact_5138_norm__diff__triangle__le,axiom,
! [X: real,Y: real,E1: real,Z: real,E22: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_5139_norm__diff__triangle__le,axiom,
! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_5140_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_5141_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_5142_norm__triangle__sub,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).
% norm_triangle_sub
thf(fact_5143_norm__triangle__sub,axiom,
! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).
% norm_triangle_sub
thf(fact_5144_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_5145_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_5146_power__eq__1__iff,axiom,
! [W: real,N2: nat] :
( ( ( power_power_real @ W @ N2 )
= one_one_real )
=> ( ( ( real_V7735802525324610683m_real @ W )
= one_one_real )
| ( N2 = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_5147_power__eq__1__iff,axiom,
! [W: complex,N2: nat] :
( ( ( power_power_complex @ W @ N2 )
= one_one_complex )
=> ( ( ( real_V1022390504157884413omplex @ W )
= one_one_real )
| ( N2 = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_5148_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_5149_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_5150_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_5151_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_5152_norm__power__diff,axiom,
! [Z: real,W: real,M: nat] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z @ M ) @ ( power_power_real @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_5153_norm__power__diff,axiom,
! [Z: complex,W: complex,M: nat] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z @ M ) @ ( power_power_complex @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_5154_minus__one__div__numeral,axiom,
! [N2: num] :
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).
% minus_one_div_numeral
thf(fact_5155_one__div__minus__numeral,axiom,
! [N2: num] :
( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).
% one_div_minus_numeral
thf(fact_5156_minus__numeral__div__numeral,axiom,
! [M: num,N2: num] :
( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).
% minus_numeral_div_numeral
thf(fact_5157_numeral__div__minus__numeral,axiom,
! [M: num,N2: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).
% numeral_div_minus_numeral
thf(fact_5158_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_5159_neg__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q3: int,R4: int] :
( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q3 @ R4 ) )
=> ( 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 @ Q3 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R4 ) @ one_one_int ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_5160_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_5161_pred__numeral__inc,axiom,
! [K: num] :
( ( pred_numeral @ ( inc @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% pred_numeral_inc
thf(fact_5162_Divides_Oadjust__div__eq,axiom,
! [Q3: int,R4: int] :
( ( adjust_div @ ( product_Pair_int_int @ Q3 @ R4 ) )
= ( plus_plus_int @ Q3 @ ( zero_n2684676970156552555ol_int @ ( R4 != zero_zero_int ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_5163_add__neg__numeral__special_I5_J,axiom,
! [N2: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N2 ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_5164_add__neg__numeral__special_I5_J,axiom,
! [N2: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N2 ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_5165_add__neg__numeral__special_I5_J,axiom,
! [N2: num] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N2 ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_5166_add__neg__numeral__special_I5_J,axiom,
! [N2: num] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N2 ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_5167_add__neg__numeral__special_I5_J,axiom,
! [N2: num] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N2 ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_5168_add__neg__numeral__special_I6_J,axiom,
! [M: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_5169_add__neg__numeral__special_I6_J,axiom,
! [M: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_5170_add__neg__numeral__special_I6_J,axiom,
! [M: num] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_5171_add__neg__numeral__special_I6_J,axiom,
! [M: num] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_5172_add__neg__numeral__special_I6_J,axiom,
! [M: num] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_5173_diff__numeral__special_I6_J,axiom,
! [M: num] :
( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
= ( numeral_numeral_real @ ( inc @ M ) ) ) ).
% diff_numeral_special(6)
thf(fact_5174_diff__numeral__special_I6_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
= ( numeral_numeral_int @ ( inc @ M ) ) ) ).
% diff_numeral_special(6)
thf(fact_5175_diff__numeral__special_I6_J,axiom,
! [M: num] :
( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ).
% diff_numeral_special(6)
thf(fact_5176_diff__numeral__special_I6_J,axiom,
! [M: num] :
( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ).
% diff_numeral_special(6)
thf(fact_5177_diff__numeral__special_I6_J,axiom,
! [M: num] :
( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( numeral_numeral_rat @ ( inc @ M ) ) ) ).
% diff_numeral_special(6)
thf(fact_5178_diff__numeral__special_I5_J,axiom,
! [N2: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N2 ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N2 ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_5179_diff__numeral__special_I5_J,axiom,
! [N2: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N2 ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_5180_diff__numeral__special_I5_J,axiom,
! [N2: num] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N2 ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N2 ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_5181_diff__numeral__special_I5_J,axiom,
! [N2: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N2 ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N2 ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_5182_diff__numeral__special_I5_J,axiom,
! [N2: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N2 ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N2 ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_5183_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one )
=> ( ! [X2: num] :
( ( P @ X2 )
=> ( P @ ( inc @ X2 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_5184_add__inc,axiom,
! [X: num,Y: num] :
( ( plus_plus_num @ X @ ( inc @ Y ) )
= ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).
% add_inc
thf(fact_5185_unique__quotient,axiom,
! [A: int,B: int,Q3: int,R4: int,Q6: int,R3: int] :
( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R4 ) )
=> ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R3 ) )
=> ( Q3 = Q6 ) ) ) ).
% unique_quotient
thf(fact_5186_unique__remainder,axiom,
! [A: int,B: int,Q3: int,R4: int,Q6: int,R3: int] :
( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R4 ) )
=> ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R3 ) )
=> ( R4 = R3 ) ) ) ).
% unique_remainder
thf(fact_5187_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_5188_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_5189_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_5190_add__One,axiom,
! [X: num] :
( ( plus_plus_num @ X @ one )
= ( inc @ X ) ) ).
% add_One
thf(fact_5191_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_5192_mult__inc,axiom,
! [X: num,Y: num] :
( ( times_times_num @ X @ ( inc @ Y ) )
= ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).
% mult_inc
thf(fact_5193_div__int__unique,axiom,
! [K: int,L: int,Q3: int,R4: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R4 ) )
=> ( ( divide_divide_int @ K @ L )
= Q3 ) ) ).
% div_int_unique
thf(fact_5194_mod__int__unique,axiom,
! [K: int,L: int,Q3: int,R4: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R4 ) )
=> ( ( modulo_modulo_int @ K @ L )
= R4 ) ) ).
% mod_int_unique
thf(fact_5195_numeral__inc,axiom,
! [X: num] :
( ( numera6690914467698888265omplex @ ( inc @ X ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% numeral_inc
thf(fact_5196_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_real @ ( inc @ X ) )
= ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% numeral_inc
thf(fact_5197_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_rat @ ( inc @ X ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% numeral_inc
thf(fact_5198_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_nat @ ( inc @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% numeral_inc
thf(fact_5199_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_int @ ( inc @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% numeral_inc
thf(fact_5200_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q3: int] :
( ( L != zero_zero_int )
=> ( ( K
= ( times_times_int @ Q3 @ L ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ zero_zero_int ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_5201_eucl__rel__int,axiom,
! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).
% eucl_rel_int
thf(fact_5202_zminus1__lemma,axiom,
! [A: int,B: int,Q3: int,R4: int] :
( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R4 ) )
=> ( ( B != zero_zero_int )
=> ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R4 = zero_zero_int ) @ ( uminus_uminus_int @ Q3 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q3 ) @ one_one_int ) ) @ ( if_int @ ( R4 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R4 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_5203_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q3: int,R4: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R4 ) )
= ( ( K
= ( plus_plus_int @ ( times_times_int @ L @ Q3 ) @ R4 ) )
& ( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ zero_zero_int @ R4 )
& ( ord_less_int @ R4 @ L ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ L )
=> ( ( ( ord_less_int @ L @ zero_zero_int )
=> ( ( ord_less_int @ L @ R4 )
& ( ord_less_eq_int @ R4 @ zero_zero_int ) ) )
& ( ~ ( ord_less_int @ L @ zero_zero_int )
=> ( Q3 = zero_zero_int ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_5204_pos__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q3: int,R4: int] :
( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R4 ) )
=> ( 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 @ Q3 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R4 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_5205_take__bit__Suc__minus__bit1,axiom,
! [N2: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_5206_artanh__def,axiom,
( artanh_real
= ( ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% artanh_def
thf(fact_5207_product__nth,axiom,
! [N2: nat,Xs: list_num,Ys: list_num] :
( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_num @ Xs ) @ ( size_size_list_num @ Ys ) ) )
=> ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs @ Ys ) @ N2 )
= ( product_Pair_num_num @ ( nth_num @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) @ ( nth_num @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_5208_product__nth,axiom,
! [N2: nat,Xs: list_Code_integer,Ys: list_o] :
( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s3445333598471063425nteger @ Xs ) @ ( size_size_list_o @ Ys ) ) )
=> ( ( nth_Pr8522763379788166057eger_o @ ( produc3607205314601156340eger_o @ Xs @ Ys ) @ N2 )
= ( produc6677183202524767010eger_o @ ( nth_Code_integer @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_5209_product__nth,axiom,
! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
=> ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) @ N2 )
= ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_5210_product__nth,axiom,
! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) )
=> ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) @ N2 )
= ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_5211_product__nth,axiom,
! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
=> ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) @ N2 )
= ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_5212_product__nth,axiom,
! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) )
=> ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) @ N2 )
= ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_5213_product__nth,axiom,
! [N2: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
=> ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) @ N2 )
= ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_5214_product__nth,axiom,
! [N2: nat,Xs: list_o,Ys: list_o] :
( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) )
=> ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs @ Ys ) @ N2 )
= ( product_Pair_o_o @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_5215_product__nth,axiom,
! [N2: nat,Xs: list_o,Ys: list_nat] :
( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
=> ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs @ Ys ) @ N2 )
= ( product_Pair_o_nat @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_5216_product__nth,axiom,
! [N2: nat,Xs: list_o,Ys: list_int] :
( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) )
=> ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs @ Ys ) @ N2 )
= ( product_Pair_o_int @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_5217_dbl__inc__simps_I3_J,axiom,
( ( neg_nu8557863876264182079omplex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_5218_dbl__inc__simps_I3_J,axiom,
( ( neg_nu8295874005876285629c_real @ one_one_real )
= ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_5219_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_5220_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5851722552734809277nc_int @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_5221_pochhammer__double,axiom,
! [Z: real,N2: nat] :
( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s7457072308508201937r_real @ Z @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).
% pochhammer_double
thf(fact_5222_pochhammer__double,axiom,
! [Z: complex,N2: nat] :
( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s2602460028002588243omplex @ Z @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).
% pochhammer_double
thf(fact_5223_pochhammer__double,axiom,
! [Z: rat,N2: nat] :
( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).
% pochhammer_double
thf(fact_5224_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_5225_central__binomial__lower__bound,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ) ) ).
% central_binomial_lower_bound
thf(fact_5226_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_5227_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_5228_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_5229_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X )
= ( ln_ln_real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_5230_pochhammer__0,axiom,
! [A: complex] :
( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
= one_one_complex ) ).
% pochhammer_0
thf(fact_5231_pochhammer__0,axiom,
! [A: real] :
( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
= one_one_real ) ).
% pochhammer_0
thf(fact_5232_pochhammer__0,axiom,
! [A: rat] :
( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% pochhammer_0
thf(fact_5233_pochhammer__0,axiom,
! [A: nat] :
( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% pochhammer_0
thf(fact_5234_pochhammer__0,axiom,
! [A: int] :
( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
= one_one_int ) ).
% pochhammer_0
thf(fact_5235_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
= one_one_complex ) ).
% dbl_inc_simps(2)
thf(fact_5236_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_5237_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
= one_one_rat ) ).
% dbl_inc_simps(2)
thf(fact_5238_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_5239_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_5240_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_5241_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% dbl_inc_simps(4)
thf(fact_5242_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% dbl_inc_simps(4)
thf(fact_5243_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_5244_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
= ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_5245_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_5246_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_5247_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_5248_length__product,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% length_product
thf(fact_5249_length__product,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_o] :
( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% length_product
thf(fact_5250_length__product,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_nat] :
( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_5251_length__product,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_int] :
( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% length_product
thf(fact_5252_length__product,axiom,
! [Xs: list_o,Ys: list_VEBT_VEBT] :
( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% length_product
thf(fact_5253_length__product,axiom,
! [Xs: list_o,Ys: list_o] :
( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% length_product
thf(fact_5254_length__product,axiom,
! [Xs: list_o,Ys: list_nat] :
( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_5255_length__product,axiom,
! [Xs: list_o,Ys: list_int] :
( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% length_product
thf(fact_5256_length__product,axiom,
! [Xs: list_nat,Ys: list_VEBT_VEBT] :
( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% length_product
thf(fact_5257_length__product,axiom,
! [Xs: list_nat,Ys: list_o] :
( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% length_product
thf(fact_5258_dbl__dec__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
= ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).
% dbl_dec_simps(5)
thf(fact_5259_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_5260_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_5261_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_5262_ln__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_5263_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_5264_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_5265_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_5266_pred__numeral__simps_I2_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit0 @ K ) )
= ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% pred_numeral_simps(2)
thf(fact_5267_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_5268_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_5269_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
= ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_5270_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_5271_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_5272_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_5273_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_5274_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
= ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_5275_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_5276_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_5277_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_5278_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_5279_semiring__norm_I26_J,axiom,
( ( bitM @ one )
= one ) ).
% semiring_norm(26)
thf(fact_5280_ln__less__self,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_5281_pochhammer__pos,axiom,
! [X: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N2 ) ) ) ).
% pochhammer_pos
thf(fact_5282_pochhammer__pos,axiom,
! [X: rat,N2: nat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N2 ) ) ) ).
% pochhammer_pos
thf(fact_5283_pochhammer__pos,axiom,
! [X: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N2 ) ) ) ).
% pochhammer_pos
thf(fact_5284_pochhammer__pos,axiom,
! [X: int,N2: nat] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N2 ) ) ) ).
% pochhammer_pos
thf(fact_5285_pochhammer__eq__0__mono,axiom,
! [A: complex,N2: nat,M: nat] :
( ( ( comm_s2602460028002588243omplex @ A @ N2 )
= zero_zero_complex )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( comm_s2602460028002588243omplex @ A @ M )
= zero_zero_complex ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_5286_pochhammer__eq__0__mono,axiom,
! [A: real,N2: nat,M: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ N2 )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( comm_s7457072308508201937r_real @ A @ M )
= zero_zero_real ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_5287_pochhammer__eq__0__mono,axiom,
! [A: rat,N2: nat,M: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
= zero_zero_rat )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( comm_s4028243227959126397er_rat @ A @ M )
= zero_zero_rat ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_5288_pochhammer__neq__0__mono,axiom,
! [A: complex,M: nat,N2: nat] :
( ( ( comm_s2602460028002588243omplex @ A @ M )
!= zero_zero_complex )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( comm_s2602460028002588243omplex @ A @ N2 )
!= zero_zero_complex ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_5289_pochhammer__neq__0__mono,axiom,
! [A: real,M: nat,N2: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ M )
!= zero_zero_real )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( comm_s7457072308508201937r_real @ A @ N2 )
!= zero_zero_real ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_5290_pochhammer__neq__0__mono,axiom,
! [A: rat,M: nat,N2: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ M )
!= zero_zero_rat )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( comm_s4028243227959126397er_rat @ A @ N2 )
!= zero_zero_rat ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_5291_semiring__norm_I28_J,axiom,
! [N2: num] :
( ( bitM @ ( bit1 @ N2 ) )
= ( bit1 @ ( bit0 @ N2 ) ) ) ).
% semiring_norm(28)
thf(fact_5292_semiring__norm_I27_J,axiom,
! [N2: num] :
( ( bitM @ ( bit0 @ N2 ) )
= ( bit1 @ ( bitM @ N2 ) ) ) ).
% semiring_norm(27)
thf(fact_5293_inc__BitM__eq,axiom,
! [N2: num] :
( ( inc @ ( bitM @ N2 ) )
= ( bit0 @ N2 ) ) ).
% inc_BitM_eq
thf(fact_5294_BitM__inc__eq,axiom,
! [N2: num] :
( ( bitM @ ( inc @ N2 ) )
= ( bit1 @ N2 ) ) ).
% BitM_inc_eq
thf(fact_5295_ln__bound,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_bound
thf(fact_5296_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_5297_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_5298_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_5299_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_5300_pochhammer__nonneg,axiom,
! [X: rat,N2: nat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N2 ) ) ) ).
% pochhammer_nonneg
thf(fact_5301_pochhammer__nonneg,axiom,
! [X: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N2 ) ) ) ).
% pochhammer_nonneg
thf(fact_5302_pochhammer__nonneg,axiom,
! [X: int,N2: nat] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N2 ) ) ) ).
% pochhammer_nonneg
thf(fact_5303_pochhammer__nonneg,axiom,
! [X: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N2 ) ) ) ).
% pochhammer_nonneg
thf(fact_5304_pochhammer__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
= one_one_complex ) )
& ( ( N2 != zero_zero_nat )
=> ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
= zero_zero_complex ) ) ) ).
% pochhammer_0_left
thf(fact_5305_pochhammer__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
= one_one_real ) )
& ( ( N2 != zero_zero_nat )
=> ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
= zero_zero_real ) ) ) ).
% pochhammer_0_left
thf(fact_5306_pochhammer__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
= one_one_rat ) )
& ( ( N2 != zero_zero_nat )
=> ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
= zero_zero_rat ) ) ) ).
% pochhammer_0_left
thf(fact_5307_pochhammer__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
= one_one_nat ) )
& ( ( N2 != zero_zero_nat )
=> ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ) ) ).
% pochhammer_0_left
thf(fact_5308_pochhammer__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
= one_one_int ) )
& ( ( N2 != zero_zero_nat )
=> ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
= zero_zero_int ) ) ) ).
% pochhammer_0_left
thf(fact_5309_eval__nat__numeral_I2_J,axiom,
! [N2: num] :
( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
= ( suc @ ( numeral_numeral_nat @ ( bitM @ N2 ) ) ) ) ).
% eval_nat_numeral(2)
thf(fact_5310_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_5311_BitM__plus__one,axiom,
! [N2: num] :
( ( plus_plus_num @ ( bitM @ N2 ) @ one )
= ( bit0 @ N2 ) ) ).
% BitM_plus_one
thf(fact_5312_one__plus__BitM,axiom,
! [N2: num] :
( ( plus_plus_num @ one @ ( bitM @ N2 ) )
= ( bit0 @ N2 ) ) ).
% one_plus_BitM
thf(fact_5313_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_5314_ln__mult,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
= ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_mult
thf(fact_5315_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_5316_ln__div,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
= ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_div
thf(fact_5317_pochhammer__rec,axiom,
! [A: complex,N2: nat] :
( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
= ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N2 ) ) ) ).
% pochhammer_rec
thf(fact_5318_pochhammer__rec,axiom,
! [A: real,N2: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
= ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N2 ) ) ) ).
% pochhammer_rec
thf(fact_5319_pochhammer__rec,axiom,
! [A: rat,N2: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
= ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N2 ) ) ) ).
% pochhammer_rec
thf(fact_5320_pochhammer__rec,axiom,
! [A: nat,N2: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
= ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N2 ) ) ) ).
% pochhammer_rec
thf(fact_5321_pochhammer__rec,axiom,
! [A: int,N2: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
= ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N2 ) ) ) ).
% pochhammer_rec
thf(fact_5322_dbl__inc__def,axiom,
( neg_nu8557863876264182079omplex
= ( ^ [X3: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).
% dbl_inc_def
thf(fact_5323_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X3: real] : ( plus_plus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_5324_dbl__inc__def,axiom,
( neg_nu5219082963157363817nc_rat
= ( ^ [X3: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).
% dbl_inc_def
thf(fact_5325_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_5326_pochhammer__Suc,axiom,
! [A: real,N2: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N2 ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% pochhammer_Suc
thf(fact_5327_pochhammer__Suc,axiom,
! [A: int,N2: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N2 ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% pochhammer_Suc
thf(fact_5328_pochhammer__Suc,axiom,
! [A: nat,N2: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N2 ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).
% pochhammer_Suc
thf(fact_5329_pochhammer__Suc,axiom,
! [A: complex,N2: nat] :
( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
= ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N2 ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N2 ) ) ) ) ).
% pochhammer_Suc
thf(fact_5330_pochhammer__Suc,axiom,
! [A: rat,N2: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N2 ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).
% pochhammer_Suc
thf(fact_5331_pochhammer__rec_H,axiom,
! [Z: real,N2: nat] :
( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N2 ) )
= ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( comm_s7457072308508201937r_real @ Z @ N2 ) ) ) ).
% pochhammer_rec'
thf(fact_5332_pochhammer__rec_H,axiom,
! [Z: int,N2: nat] :
( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N2 ) )
= ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( comm_s4660882817536571857er_int @ Z @ N2 ) ) ) ).
% pochhammer_rec'
thf(fact_5333_pochhammer__rec_H,axiom,
! [Z: nat,N2: nat] :
( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N2 ) )
= ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) ) ) ).
% pochhammer_rec'
thf(fact_5334_pochhammer__rec_H,axiom,
! [Z: complex,N2: nat] :
( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N2 ) )
= ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) @ ( comm_s2602460028002588243omplex @ Z @ N2 ) ) ) ).
% pochhammer_rec'
thf(fact_5335_pochhammer__rec_H,axiom,
! [Z: rat,N2: nat] :
( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N2 ) )
= ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) ) ) ).
% pochhammer_rec'
thf(fact_5336_pochhammer__of__nat__eq__0__lemma,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ N2 @ K )
=> ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
= zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_5337_pochhammer__of__nat__eq__0__lemma,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ N2 @ K )
=> ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
= zero_zero_real ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_5338_pochhammer__of__nat__eq__0__lemma,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ N2 @ K )
=> ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
= zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_5339_pochhammer__of__nat__eq__0__lemma,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ N2 @ K )
=> ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
= zero_zero_complex ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_5340_pochhammer__of__nat__eq__0__lemma,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ N2 @ K )
=> ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
= zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_5341_pochhammer__of__nat__eq__0__iff,axiom,
! [N2: nat,K: nat] :
( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
= zero_z3403309356797280102nteger )
= ( ord_less_nat @ N2 @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_5342_pochhammer__of__nat__eq__0__iff,axiom,
! [N2: nat,K: nat] :
( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
= zero_zero_real )
= ( ord_less_nat @ N2 @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_5343_pochhammer__of__nat__eq__0__iff,axiom,
! [N2: nat,K: nat] :
( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
= zero_zero_int )
= ( ord_less_nat @ N2 @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_5344_pochhammer__of__nat__eq__0__iff,axiom,
! [N2: nat,K: nat] :
( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
= zero_zero_complex )
= ( ord_less_nat @ N2 @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_5345_pochhammer__of__nat__eq__0__iff,axiom,
! [N2: nat,K: nat] :
( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
= zero_zero_rat )
= ( ord_less_nat @ N2 @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_5346_pochhammer__eq__0__iff,axiom,
! [A: real,N2: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ N2 )
= zero_zero_real )
= ( ? [K2: nat] :
( ( ord_less_nat @ K2 @ N2 )
& ( A
= ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_5347_pochhammer__eq__0__iff,axiom,
! [A: complex,N2: nat] :
( ( ( comm_s2602460028002588243omplex @ A @ N2 )
= zero_zero_complex )
= ( ? [K2: nat] :
( ( ord_less_nat @ K2 @ N2 )
& ( A
= ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_5348_pochhammer__eq__0__iff,axiom,
! [A: rat,N2: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
= zero_zero_rat )
= ( ? [K2: nat] :
( ( ord_less_nat @ K2 @ N2 )
& ( A
= ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K2 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_5349_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
!= zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_5350_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
!= zero_zero_real ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_5351_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
!= zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_5352_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
!= zero_zero_complex ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_5353_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
!= zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_5354_pochhammer__product_H,axiom,
! [Z: real,N2: nat,M: nat] :
( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N2 @ M ) )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N2 ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_5355_pochhammer__product_H,axiom,
! [Z: int,N2: nat,M: nat] :
( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N2 @ M ) )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_5356_pochhammer__product_H,axiom,
! [Z: nat,N2: nat,M: nat] :
( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N2 @ M ) )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_5357_pochhammer__product_H,axiom,
! [Z: complex,N2: nat,M: nat] :
( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N2 @ M ) )
= ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_5358_pochhammer__product_H,axiom,
! [Z: rat,N2: nat,M: nat] :
( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N2 @ M ) )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N2 ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_5359_binomial__mono,axiom,
! [K: nat,K6: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ K6 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
=> ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).
% binomial_mono
thf(fact_5360_binomial__maximum_H,axiom,
! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).
% binomial_maximum'
thf(fact_5361_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_5362_binomial__maximum,axiom,
! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% binomial_maximum
thf(fact_5363_binomial__antimono,axiom,
! [K: nat,K6: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ K6 )
=> ( ( ord_less_eq_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
=> ( ( ord_less_eq_nat @ K6 @ N2 )
=> ( ord_less_eq_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).
% binomial_antimono
thf(fact_5364_numeral__BitM,axiom,
! [N2: num] :
( ( numera6690914467698888265omplex @ ( bitM @ N2 ) )
= ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N2 ) ) @ one_one_complex ) ) ).
% numeral_BitM
thf(fact_5365_numeral__BitM,axiom,
! [N2: num] :
( ( numeral_numeral_real @ ( bitM @ N2 ) )
= ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N2 ) ) @ one_one_real ) ) ).
% numeral_BitM
thf(fact_5366_numeral__BitM,axiom,
! [N2: num] :
( ( numeral_numeral_rat @ ( bitM @ N2 ) )
= ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N2 ) ) @ one_one_rat ) ) ).
% numeral_BitM
thf(fact_5367_numeral__BitM,axiom,
! [N2: num] :
( ( numeral_numeral_int @ ( bitM @ N2 ) )
= ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ one_one_int ) ) ).
% numeral_BitM
thf(fact_5368_odd__numeral__BitM,axiom,
! [W: num] :
~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).
% odd_numeral_BitM
thf(fact_5369_odd__numeral__BitM,axiom,
! [W: num] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).
% odd_numeral_BitM
thf(fact_5370_odd__numeral__BitM,axiom,
! [W: num] :
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).
% odd_numeral_BitM
thf(fact_5371_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_5372_ln__diff__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).
% ln_diff_le
thf(fact_5373_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_5374_ln__realpow,axiom,
! [X: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( power_power_real @ X @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ln_ln_real @ X ) ) ) ) ).
% ln_realpow
thf(fact_5375_pochhammer__product,axiom,
! [M: nat,N2: nat,Z: real] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( comm_s7457072308508201937r_real @ Z @ N2 )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_5376_pochhammer__product,axiom,
! [M: nat,N2: nat,Z: int] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( comm_s4660882817536571857er_int @ Z @ N2 )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_5377_pochhammer__product,axiom,
! [M: nat,N2: nat,Z: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( comm_s4663373288045622133er_nat @ Z @ N2 )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_5378_pochhammer__product,axiom,
! [M: nat,N2: nat,Z: complex] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( comm_s2602460028002588243omplex @ Z @ N2 )
= ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_5379_pochhammer__product,axiom,
! [M: nat,N2: nat,Z: rat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( comm_s4028243227959126397er_rat @ Z @ N2 )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_5380_binomial__strict__mono,axiom,
! [K: nat,K6: nat,N2: nat] :
( ( ord_less_nat @ K @ K6 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
=> ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).
% binomial_strict_mono
thf(fact_5381_binomial__strict__antimono,axiom,
! [K: nat,K6: nat,N2: nat] :
( ( ord_less_nat @ K @ K6 )
=> ( ( ord_less_eq_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
=> ( ( ord_less_eq_nat @ K6 @ N2 )
=> ( ord_less_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_5382_binomial__less__binomial__Suc,axiom,
! [K: nat,N2: nat] :
( ( ord_less_nat @ K @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).
% binomial_less_binomial_Suc
thf(fact_5383_central__binomial__odd,axiom,
! [N2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( binomial @ N2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% central_binomial_odd
thf(fact_5384_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_5385_pochhammer__absorb__comp,axiom,
! [R4: code_integer,K: nat] :
( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R4 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R4 ) @ K ) )
= ( times_3573771949741848930nteger @ R4 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R4 ) @ one_one_Code_integer ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_5386_pochhammer__absorb__comp,axiom,
! [R4: real,K: nat] :
( ( times_times_real @ ( minus_minus_real @ R4 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R4 ) @ K ) )
= ( times_times_real @ R4 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R4 ) @ one_one_real ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_5387_pochhammer__absorb__comp,axiom,
! [R4: int,K: nat] :
( ( times_times_int @ ( minus_minus_int @ R4 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R4 ) @ K ) )
= ( times_times_int @ R4 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R4 ) @ one_one_int ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_5388_pochhammer__absorb__comp,axiom,
! [R4: complex,K: nat] :
( ( times_times_complex @ ( minus_minus_complex @ R4 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R4 ) @ K ) )
= ( times_times_complex @ R4 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R4 ) @ one_one_complex ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_5389_pochhammer__absorb__comp,axiom,
! [R4: rat,K: nat] :
( ( times_times_rat @ ( minus_minus_rat @ R4 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R4 ) @ K ) )
= ( times_times_rat @ R4 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R4 ) @ one_one_rat ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_5390_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_5391_arsinh__real__def,axiom,
( arsinh_real
= ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% arsinh_real_def
thf(fact_5392_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_5393_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_5394_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_5395_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_5396_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_5397_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_5398_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_5399_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_5400_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_5401_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_5402_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_5403_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_5404_zero__less__binomial__iff,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) )
= ( ord_less_eq_nat @ K @ N2 ) ) ).
% zero_less_binomial_iff
thf(fact_5405_choose__two,axiom,
! [N2: nat] :
( ( binomial @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% choose_two
thf(fact_5406_binomial__n__0,axiom,
! [N2: nat] :
( ( binomial @ N2 @ zero_zero_nat )
= one_one_nat ) ).
% binomial_n_0
thf(fact_5407_binomial__Suc__Suc,axiom,
! [N2: nat,K: nat] :
( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_5408_binomial__eq__0__iff,axiom,
! [N2: nat,K: nat] :
( ( ( binomial @ N2 @ K )
= zero_zero_nat )
= ( ord_less_nat @ N2 @ K ) ) ).
% binomial_eq_0_iff
thf(fact_5409_binomial__1,axiom,
! [N2: nat] :
( ( binomial @ N2 @ ( suc @ zero_zero_nat ) )
= N2 ) ).
% binomial_1
thf(fact_5410_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% binomial_0_Suc
thf(fact_5411_binomial__Suc__n,axiom,
! [N2: nat] :
( ( binomial @ ( suc @ N2 ) @ N2 )
= ( suc @ N2 ) ) ).
% binomial_Suc_n
thf(fact_5412_binomial__n__n,axiom,
! [N2: nat] :
( ( binomial @ N2 @ N2 )
= one_one_nat ) ).
% binomial_n_n
thf(fact_5413_choose__one,axiom,
! [N2: nat] :
( ( binomial @ N2 @ one_one_nat )
= N2 ) ).
% choose_one
thf(fact_5414_binomial__eq__0,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ N2 @ K )
=> ( ( binomial @ N2 @ K )
= zero_zero_nat ) ) ).
% binomial_eq_0
thf(fact_5415_Suc__times__binomial__eq,axiom,
! [N2: nat,K: nat] :
( ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) )
= ( times_times_nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% Suc_times_binomial_eq
thf(fact_5416_Suc__times__binomial,axiom,
! [K: nat,N2: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) )
= ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) ) ).
% Suc_times_binomial
thf(fact_5417_binomial__symmetric,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( binomial @ N2 @ K )
= ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_5418_choose__mult__lemma,axiom,
! [M: nat,R4: nat,K: nat] :
( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R4 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
= ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R4 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R4 ) @ M ) ) ) ).
% choose_mult_lemma
thf(fact_5419_binomial__le__pow,axiom,
! [R4: nat,N2: nat] :
( ( ord_less_eq_nat @ R4 @ N2 )
=> ( ord_less_eq_nat @ ( binomial @ N2 @ R4 ) @ ( power_power_nat @ N2 @ R4 ) ) ) ).
% binomial_le_pow
thf(fact_5420_zero__less__binomial,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) ) ) ).
% zero_less_binomial
thf(fact_5421_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_5422_choose__mult,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( ( times_times_nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) )
= ( times_times_nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus_nat @ N2 @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_5423_binomial__Suc__Suc__eq__times,axiom,
! [N2: nat,K: nat] :
( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
= ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) @ ( suc @ K ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_5424_binomial__absorb__comp,axiom,
! [N2: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ N2 @ K ) @ ( binomial @ N2 @ K ) )
= ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_5425_binomial__absorption,axiom,
! [K: nat,N2: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) )
= ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).
% binomial_absorption
thf(fact_5426_binomial__ge__n__over__k__pow__k,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_5427_binomial__ge__n__over__k__pow__k,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_5428_binomial__le__pow2,axiom,
! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% binomial_le_pow2
thf(fact_5429_choose__reduce__nat,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( binomial @ N2 @ K )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_5430_times__binomial__minus1__eq,axiom,
! [K: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( times_times_nat @ K @ ( binomial @ N2 @ K ) )
= ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_5431_binomial__addition__formula,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( binomial @ N2 @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_5432_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_5433_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_5434_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_5435_arctan__half,axiom,
( arctan
= ( ^ [X3: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X3 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% arctan_half
thf(fact_5436_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_5437_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_5438_abs__abs,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_abs
thf(fact_5439_abs__abs,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_abs
thf(fact_5440_abs__abs,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
= ( abs_abs_Code_integer @ A ) ) ).
% abs_abs
thf(fact_5441_abs__abs,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
= ( abs_abs_rat @ A ) ) ).
% abs_abs
thf(fact_5442_abs__idempotent,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_idempotent
thf(fact_5443_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_5444_abs__idempotent,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
= ( abs_abs_Code_integer @ A ) ) ).
% abs_idempotent
thf(fact_5445_abs__idempotent,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
= ( abs_abs_rat @ A ) ) ).
% abs_idempotent
thf(fact_5446_abs__0__eq,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger
= ( abs_abs_Code_integer @ A ) )
= ( A = zero_z3403309356797280102nteger ) ) ).
% abs_0_eq
thf(fact_5447_abs__0__eq,axiom,
! [A: real] :
( ( zero_zero_real
= ( abs_abs_real @ A ) )
= ( A = zero_zero_real ) ) ).
% abs_0_eq
thf(fact_5448_abs__0__eq,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( abs_abs_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% abs_0_eq
thf(fact_5449_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_5450_abs__eq__0,axiom,
! [A: code_integer] :
( ( ( abs_abs_Code_integer @ A )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% abs_eq_0
thf(fact_5451_abs__eq__0,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0
thf(fact_5452_abs__eq__0,axiom,
! [A: rat] :
( ( ( abs_abs_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% abs_eq_0
thf(fact_5453_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_5454_abs__zero,axiom,
( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% abs_zero
thf(fact_5455_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_zero
thf(fact_5456_abs__zero,axiom,
( ( abs_abs_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% abs_zero
thf(fact_5457_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_5458_abs__0,axiom,
( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% abs_0
thf(fact_5459_abs__0,axiom,
( ( abs_abs_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% abs_0
thf(fact_5460_abs__0,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_0
thf(fact_5461_abs__0,axiom,
( ( abs_abs_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% abs_0
thf(fact_5462_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_5463_abs__numeral,axiom,
! [N2: num] :
( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
= ( numera6620942414471956472nteger @ N2 ) ) ).
% abs_numeral
thf(fact_5464_abs__numeral,axiom,
! [N2: num] :
( ( abs_abs_real @ ( numeral_numeral_real @ N2 ) )
= ( numeral_numeral_real @ N2 ) ) ).
% abs_numeral
thf(fact_5465_abs__numeral,axiom,
! [N2: num] :
( ( abs_abs_rat @ ( numeral_numeral_rat @ N2 ) )
= ( numeral_numeral_rat @ N2 ) ) ).
% abs_numeral
thf(fact_5466_abs__numeral,axiom,
! [N2: num] :
( ( abs_abs_int @ ( numeral_numeral_int @ N2 ) )
= ( numeral_numeral_int @ N2 ) ) ).
% abs_numeral
thf(fact_5467_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_5468_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_5469_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_5470_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_5471_abs__1,axiom,
( ( abs_abs_Code_integer @ one_one_Code_integer )
= one_one_Code_integer ) ).
% abs_1
thf(fact_5472_abs__1,axiom,
( ( abs_abs_complex @ one_one_complex )
= one_one_complex ) ).
% abs_1
thf(fact_5473_abs__1,axiom,
( ( abs_abs_real @ one_one_real )
= one_one_real ) ).
% abs_1
thf(fact_5474_abs__1,axiom,
( ( abs_abs_rat @ one_one_rat )
= one_one_rat ) ).
% abs_1
thf(fact_5475_abs__1,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_1
thf(fact_5476_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_5477_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_5478_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_5479_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_5480_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_5481_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_5482_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_5483_abs__minus,axiom,
! [A: real] :
( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_minus
thf(fact_5484_abs__minus,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus
thf(fact_5485_abs__minus,axiom,
! [A: complex] :
( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
= ( abs_abs_complex @ A ) ) ).
% abs_minus
thf(fact_5486_abs__minus,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
= ( abs_abs_Code_integer @ A ) ) ).
% abs_minus
thf(fact_5487_abs__minus,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
= ( abs_abs_rat @ A ) ) ).
% abs_minus
thf(fact_5488_abs__minus__cancel,axiom,
! [A: real] :
( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_minus_cancel
thf(fact_5489_abs__minus__cancel,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus_cancel
thf(fact_5490_abs__minus__cancel,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
= ( abs_abs_Code_integer @ A ) ) ).
% abs_minus_cancel
thf(fact_5491_abs__minus__cancel,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
= ( abs_abs_rat @ A ) ) ).
% abs_minus_cancel
thf(fact_5492_abs__dvd__iff,axiom,
! [M: real,K: real] :
( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
= ( dvd_dvd_real @ M @ K ) ) ).
% abs_dvd_iff
thf(fact_5493_abs__dvd__iff,axiom,
! [M: int,K: int] :
( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
= ( dvd_dvd_int @ M @ K ) ) ).
% abs_dvd_iff
thf(fact_5494_abs__dvd__iff,axiom,
! [M: code_integer,K: code_integer] :
( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
= ( dvd_dvd_Code_integer @ M @ K ) ) ).
% abs_dvd_iff
thf(fact_5495_abs__dvd__iff,axiom,
! [M: rat,K: rat] :
( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
= ( dvd_dvd_rat @ M @ K ) ) ).
% abs_dvd_iff
thf(fact_5496_dvd__abs__iff,axiom,
! [M: real,K: real] :
( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
= ( dvd_dvd_real @ M @ K ) ) ).
% dvd_abs_iff
thf(fact_5497_dvd__abs__iff,axiom,
! [M: int,K: int] :
( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
= ( dvd_dvd_int @ M @ K ) ) ).
% dvd_abs_iff
thf(fact_5498_dvd__abs__iff,axiom,
! [M: code_integer,K: code_integer] :
( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
= ( dvd_dvd_Code_integer @ M @ K ) ) ).
% dvd_abs_iff
thf(fact_5499_dvd__abs__iff,axiom,
! [M: rat,K: rat] :
( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
= ( dvd_dvd_rat @ M @ K ) ) ).
% dvd_abs_iff
thf(fact_5500_abs__of__nat,axiom,
! [N2: nat] :
( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N2 ) )
= ( semiri4939895301339042750nteger @ N2 ) ) ).
% abs_of_nat
thf(fact_5501_abs__of__nat,axiom,
! [N2: nat] :
( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( semiri5074537144036343181t_real @ N2 ) ) ).
% abs_of_nat
thf(fact_5502_abs__of__nat,axiom,
! [N2: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri1314217659103216013at_int @ N2 ) ) ).
% abs_of_nat
thf(fact_5503_abs__of__nat,axiom,
! [N2: nat] :
( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N2 ) )
= ( semiri681578069525770553at_rat @ N2 ) ) ).
% abs_of_nat
thf(fact_5504_abs__bool__eq,axiom,
! [P: $o] :
( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
= ( zero_n3304061248610475627l_real @ P ) ) ).
% abs_bool_eq
thf(fact_5505_abs__bool__eq,axiom,
! [P: $o] :
( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
= ( zero_n2052037380579107095ol_rat @ P ) ) ).
% abs_bool_eq
thf(fact_5506_abs__bool__eq,axiom,
! [P: $o] :
( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
= ( zero_n2684676970156552555ol_int @ P ) ) ).
% abs_bool_eq
thf(fact_5507_abs__bool__eq,axiom,
! [P: $o] :
( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
= ( zero_n356916108424825756nteger @ P ) ) ).
% abs_bool_eq
thf(fact_5508_tanh__0,axiom,
( ( tanh_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% tanh_0
thf(fact_5509_tanh__0,axiom,
( ( tanh_real @ zero_zero_real )
= zero_zero_real ) ).
% tanh_0
thf(fact_5510_tanh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% tanh_real_less_iff
thf(fact_5511_abs__of__nonneg,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( abs_abs_Code_integer @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_5512_abs__of__nonneg,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( abs_abs_rat @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_5513_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_5514_abs__of__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_5515_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_5516_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_5517_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_5518_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_5519_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_5520_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_5521_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_5522_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_5523_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_5524_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_5525_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_5526_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_5527_abs__neg__numeral,axiom,
! [N2: num] :
( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( numeral_numeral_real @ N2 ) ) ).
% abs_neg_numeral
thf(fact_5528_abs__neg__numeral,axiom,
! [N2: num] :
( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( numeral_numeral_int @ N2 ) ) ).
% abs_neg_numeral
thf(fact_5529_abs__neg__numeral,axiom,
! [N2: num] :
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
= ( numera6620942414471956472nteger @ N2 ) ) ).
% abs_neg_numeral
thf(fact_5530_abs__neg__numeral,axiom,
! [N2: num] :
( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
= ( numeral_numeral_rat @ N2 ) ) ).
% abs_neg_numeral
thf(fact_5531_abs__neg__one,axiom,
( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
= one_one_real ) ).
% abs_neg_one
thf(fact_5532_abs__neg__one,axiom,
( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
= one_one_int ) ).
% abs_neg_one
thf(fact_5533_abs__neg__one,axiom,
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= one_one_Code_integer ) ).
% abs_neg_one
thf(fact_5534_abs__neg__one,axiom,
( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= one_one_rat ) ).
% abs_neg_one
thf(fact_5535_abs__power__minus,axiom,
! [A: real,N2: nat] :
( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 ) )
= ( abs_abs_real @ ( power_power_real @ A @ N2 ) ) ) ).
% abs_power_minus
thf(fact_5536_abs__power__minus,axiom,
! [A: int,N2: nat] :
( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) )
= ( abs_abs_int @ ( power_power_int @ A @ N2 ) ) ) ).
% abs_power_minus
thf(fact_5537_abs__power__minus,axiom,
! [A: code_integer,N2: nat] :
( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 ) )
= ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).
% abs_power_minus
thf(fact_5538_abs__power__minus,axiom,
! [A: rat,N2: nat] :
( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 ) )
= ( abs_abs_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% abs_power_minus
thf(fact_5539_zero__less__arctan__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% zero_less_arctan_iff
thf(fact_5540_arctan__less__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% arctan_less_zero_iff
thf(fact_5541_real__sqrt__mult__self,axiom,
! [A: real] :
( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
= ( abs_abs_real @ A ) ) ).
% real_sqrt_mult_self
thf(fact_5542_real__sqrt__abs2,axiom,
! [X: real] :
( ( sqrt @ ( times_times_real @ X @ X ) )
= ( abs_abs_real @ X ) ) ).
% real_sqrt_abs2
thf(fact_5543_tanh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% tanh_real_pos_iff
thf(fact_5544_tanh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% tanh_real_neg_iff
thf(fact_5545_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_5546_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_5547_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_5548_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_5549_abs__of__nonpos,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ A )
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% abs_of_nonpos
thf(fact_5550_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_5551_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_5552_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_5553_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_5554_zero__less__power__abs__iff,axiom,
! [A: code_integer,N2: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) )
= ( ( A != zero_z3403309356797280102nteger )
| ( N2 = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_5555_zero__less__power__abs__iff,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
= ( ( A != zero_zero_real )
| ( N2 = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_5556_zero__less__power__abs__iff,axiom,
! [A: rat,N2: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) )
= ( ( A != zero_zero_rat )
| ( N2 = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_5557_zero__less__power__abs__iff,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) )
= ( ( A != zero_zero_int )
| ( N2 = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_5558_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_5559_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_5560_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_5561_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_5562_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_5563_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_5564_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_5565_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_5566_power__even__abs__numeral,axiom,
! [W: num,A: code_integer] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
=> ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
= ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_even_abs_numeral
thf(fact_5567_power__even__abs__numeral,axiom,
! [W: num,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
=> ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_even_abs_numeral
thf(fact_5568_power__even__abs__numeral,axiom,
! [W: num,A: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
=> ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_even_abs_numeral
thf(fact_5569_power__even__abs__numeral,axiom,
! [W: num,A: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
=> ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_even_abs_numeral
thf(fact_5570_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_5571_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_5572_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_5573_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_5574_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_5575_abs__ge__self,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_self
thf(fact_5576_abs__ge__self,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% abs_ge_self
thf(fact_5577_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_5578_abs__ge__self,axiom,
! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% abs_ge_self
thf(fact_5579_abs__eq__0__iff,axiom,
! [A: code_integer] :
( ( ( abs_abs_Code_integer @ A )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% abs_eq_0_iff
thf(fact_5580_abs__eq__0__iff,axiom,
! [A: complex] :
( ( ( abs_abs_complex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% abs_eq_0_iff
thf(fact_5581_abs__eq__0__iff,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0_iff
thf(fact_5582_abs__eq__0__iff,axiom,
! [A: rat] :
( ( ( abs_abs_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% abs_eq_0_iff
thf(fact_5583_abs__eq__0__iff,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_5584_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_5585_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_5586_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_5587_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_5588_abs__one,axiom,
( ( abs_abs_Code_integer @ one_one_Code_integer )
= one_one_Code_integer ) ).
% abs_one
thf(fact_5589_abs__one,axiom,
( ( abs_abs_real @ one_one_real )
= one_one_real ) ).
% abs_one
thf(fact_5590_abs__one,axiom,
( ( abs_abs_rat @ one_one_rat )
= one_one_rat ) ).
% abs_one
thf(fact_5591_abs__one,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_one
thf(fact_5592_abs__minus__commute,axiom,
! [A: code_integer,B: code_integer] :
( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
= ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_5593_abs__minus__commute,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
= ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_5594_abs__minus__commute,axiom,
! [A: rat,B: rat] :
( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
= ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_5595_abs__minus__commute,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
= ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_5596_abs__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( abs_abs_real @ X )
= ( abs_abs_real @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_real @ Y ) ) ) ) ).
% abs_eq_iff
thf(fact_5597_abs__eq__iff,axiom,
! [X: int,Y: int] :
( ( ( abs_abs_int @ X )
= ( abs_abs_int @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_int @ Y ) ) ) ) ).
% abs_eq_iff
thf(fact_5598_abs__eq__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ( abs_abs_Code_integer @ X )
= ( abs_abs_Code_integer @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus1351360451143612070nteger @ Y ) ) ) ) ).
% abs_eq_iff
thf(fact_5599_abs__eq__iff,axiom,
! [X: rat,Y: rat] :
( ( ( abs_abs_rat @ X )
= ( abs_abs_rat @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_rat @ Y ) ) ) ) ).
% abs_eq_iff
thf(fact_5600_power__abs,axiom,
! [A: code_integer,N2: nat] :
( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) )
= ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) ) ).
% power_abs
thf(fact_5601_power__abs,axiom,
! [A: int,N2: nat] :
( ( abs_abs_int @ ( power_power_int @ A @ N2 ) )
= ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) ) ).
% power_abs
thf(fact_5602_power__abs,axiom,
! [A: real,N2: nat] :
( ( abs_abs_real @ ( power_power_real @ A @ N2 ) )
= ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) ) ).
% power_abs
thf(fact_5603_power__abs,axiom,
! [A: rat,N2: nat] :
( ( abs_abs_rat @ ( power_power_rat @ A @ N2 ) )
= ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) ) ).
% power_abs
thf(fact_5604_dvd__if__abs__eq,axiom,
! [L: real,K: real] :
( ( ( abs_abs_real @ L )
= ( abs_abs_real @ K ) )
=> ( dvd_dvd_real @ L @ K ) ) ).
% dvd_if_abs_eq
thf(fact_5605_dvd__if__abs__eq,axiom,
! [L: int,K: int] :
( ( ( abs_abs_int @ L )
= ( abs_abs_int @ K ) )
=> ( dvd_dvd_int @ L @ K ) ) ).
% dvd_if_abs_eq
thf(fact_5606_dvd__if__abs__eq,axiom,
! [L: code_integer,K: code_integer] :
( ( ( abs_abs_Code_integer @ L )
= ( abs_abs_Code_integer @ K ) )
=> ( dvd_dvd_Code_integer @ L @ K ) ) ).
% dvd_if_abs_eq
thf(fact_5607_dvd__if__abs__eq,axiom,
! [L: rat,K: rat] :
( ( ( abs_abs_rat @ L )
= ( abs_abs_rat @ K ) )
=> ( dvd_dvd_rat @ L @ K ) ) ).
% dvd_if_abs_eq
thf(fact_5608_arctan__monotone,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% arctan_monotone
thf(fact_5609_arctan__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% arctan_less_iff
thf(fact_5610_abs__ge__zero,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_zero
thf(fact_5611_abs__ge__zero,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% abs_ge_zero
thf(fact_5612_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_5613_abs__ge__zero,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% abs_ge_zero
thf(fact_5614_abs__not__less__zero,axiom,
! [A: code_integer] :
~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% abs_not_less_zero
thf(fact_5615_abs__not__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% abs_not_less_zero
thf(fact_5616_abs__not__less__zero,axiom,
! [A: rat] :
~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% abs_not_less_zero
thf(fact_5617_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_5618_abs__of__pos,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( abs_abs_Code_integer @ A )
= A ) ) ).
% abs_of_pos
thf(fact_5619_abs__of__pos,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_pos
thf(fact_5620_abs__of__pos,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( abs_abs_rat @ A )
= A ) ) ).
% abs_of_pos
thf(fact_5621_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_5622_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_5623_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_5624_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_5625_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_5626_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_5627_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_5628_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_5629_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_5630_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_5631_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_5632_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_5633_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_5634_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_5635_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_5636_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_5637_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_5638_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_5639_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_5640_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_5641_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_5642_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_5643_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_5644_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_5645_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_5646_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_5647_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_5648_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_5649_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_5650_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_5651_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_5652_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_5653_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_5654_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_5655_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_5656_abs__ge__minus__self,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_minus_self
thf(fact_5657_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_5658_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_5659_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_5660_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_5661_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_5662_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_5663_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_5664_tanh__real__lt__1,axiom,
! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).
% tanh_real_lt_1
thf(fact_5665_dense__eq0__I,axiom,
! [X: rat] :
( ! [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E ) )
=> ( X = zero_zero_rat ) ) ).
% dense_eq0_I
thf(fact_5666_dense__eq0__I,axiom,
! [X: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E ) )
=> ( X = zero_zero_real ) ) ).
% dense_eq0_I
thf(fact_5667_abs__mult__pos,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
=> ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
= ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_5668_abs__mult__pos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
= ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_5669_abs__mult__pos,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
= ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_5670_abs__mult__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
= ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_5671_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_5672_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_5673_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_5674_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_5675_abs__minus__le__zero,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% abs_minus_le_zero
thf(fact_5676_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_5677_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_5678_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_5679_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_5680_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_5681_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_5682_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_5683_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_5684_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_5685_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_5686_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_5687_zero__le__power__abs,axiom,
! [A: code_integer,N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) ) ).
% zero_le_power_abs
thf(fact_5688_zero__le__power__abs,axiom,
! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) ) ).
% zero_le_power_abs
thf(fact_5689_zero__le__power__abs,axiom,
! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) ) ).
% zero_le_power_abs
thf(fact_5690_zero__le__power__abs,axiom,
! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) ) ).
% zero_le_power_abs
thf(fact_5691_abs__div__pos,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
= ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% abs_div_pos
thf(fact_5692_abs__div__pos,axiom,
! [Y: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
= ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% abs_div_pos
thf(fact_5693_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_5694_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_5695_abs__of__neg,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ A )
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% abs_of_neg
thf(fact_5696_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_5697_abs__if,axiom,
( abs_abs_real
= ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% abs_if
thf(fact_5698_abs__if,axiom,
( abs_abs_int
= ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% abs_if
thf(fact_5699_abs__if,axiom,
( abs_abs_Code_integer
= ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% abs_if
thf(fact_5700_abs__if,axiom,
( abs_abs_rat
= ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% abs_if
thf(fact_5701_abs__if__raw,axiom,
( abs_abs_real
= ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% abs_if_raw
thf(fact_5702_abs__if__raw,axiom,
( abs_abs_int
= ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% abs_if_raw
thf(fact_5703_abs__if__raw,axiom,
( abs_abs_Code_integer
= ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% abs_if_raw
thf(fact_5704_abs__if__raw,axiom,
( abs_abs_rat
= ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% abs_if_raw
thf(fact_5705_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_5706_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_5707_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_5708_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_5709_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_5710_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_5711_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_5712_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_5713_abs__diff__le__iff,axiom,
! [X: code_integer,A: code_integer,R4: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R4 )
= ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R4 ) @ X )
& ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R4 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_5714_abs__diff__le__iff,axiom,
! [X: rat,A: rat,R4: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R4 )
= ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R4 ) @ X )
& ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R4 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_5715_abs__diff__le__iff,axiom,
! [X: int,A: int,R4: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R4 )
= ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R4 ) @ X )
& ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R4 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_5716_abs__diff__le__iff,axiom,
! [X: real,A: real,R4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R4 )
= ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R4 ) @ X )
& ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R4 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_5717_abs__diff__less__iff,axiom,
! [X: code_integer,A: code_integer,R4: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R4 )
= ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R4 ) @ X )
& ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R4 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_5718_abs__diff__less__iff,axiom,
! [X: real,A: real,R4: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R4 )
= ( ( ord_less_real @ ( minus_minus_real @ A @ R4 ) @ X )
& ( ord_less_real @ X @ ( plus_plus_real @ A @ R4 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_5719_abs__diff__less__iff,axiom,
! [X: rat,A: rat,R4: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R4 )
= ( ( ord_less_rat @ ( minus_minus_rat @ A @ R4 ) @ X )
& ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R4 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_5720_abs__diff__less__iff,axiom,
! [X: int,A: int,R4: int] :
( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R4 )
= ( ( ord_less_int @ ( minus_minus_int @ A @ R4 ) @ X )
& ( ord_less_int @ X @ ( plus_plus_int @ A @ R4 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_5721_abs__real__def,axiom,
( abs_abs_real
= ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% abs_real_def
thf(fact_5722_lemma__interval__lt,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y3: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D4 )
=> ( ( ord_less_real @ A @ Y3 )
& ( ord_less_real @ Y3 @ B ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_5723_sin__bound__lemma,axiom,
! [X: real,Y: real,U: real,V: real] :
( ( X = Y )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).
% sin_bound_lemma
thf(fact_5724_arctan__add,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).
% arctan_add
thf(fact_5725_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_5726_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_5727_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_5728_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_5729_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_5730_lemma__interval,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y3: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D4 )
=> ( ( ord_less_eq_real @ A @ Y3 )
& ( ord_less_eq_real @ Y3 @ B ) ) ) ) ) ) ).
% lemma_interval
thf(fact_5731_abs__le__square__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y ) )
= ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_5732_abs__le__square__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
= ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_5733_abs__le__square__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
= ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_5734_abs__le__square__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) )
= ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_5735_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_5736_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_5737_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_5738_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_5739_power__even__abs,axiom,
! [N2: nat,A: code_integer] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 )
= ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).
% power_even_abs
thf(fact_5740_power__even__abs,axiom,
! [N2: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_int @ ( abs_abs_int @ A ) @ N2 )
= ( power_power_int @ A @ N2 ) ) ) ).
% power_even_abs
thf(fact_5741_power__even__abs,axiom,
! [N2: nat,A: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_real @ ( abs_abs_real @ A ) @ N2 )
= ( power_power_real @ A @ N2 ) ) ) ).
% power_even_abs
thf(fact_5742_power__even__abs,axiom,
! [N2: nat,A: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 )
= ( power_power_rat @ A @ N2 ) ) ) ).
% power_even_abs
thf(fact_5743_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_5744_power2__le__iff__abs__le,axiom,
! [Y: code_integer,X: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
=> ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y ) ) ) ).
% power2_le_iff_abs_le
thf(fact_5745_power2__le__iff__abs__le,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y ) ) ) ).
% power2_le_iff_abs_le
thf(fact_5746_power2__le__iff__abs__le,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y ) ) ) ).
% power2_le_iff_abs_le
thf(fact_5747_power2__le__iff__abs__le,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y ) ) ) ).
% power2_le_iff_abs_le
thf(fact_5748_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_5749_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_5750_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_5751_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_5752_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_5753_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_5754_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_5755_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_5756_power__mono__even,axiom,
! [N2: nat,A: code_integer,B: code_integer] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
=> ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ) ).
% power_mono_even
thf(fact_5757_power__mono__even,axiom,
! [N2: nat,A: rat,B: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).
% power_mono_even
thf(fact_5758_power__mono__even,axiom,
! [N2: nat,A: int,B: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% power_mono_even
thf(fact_5759_power__mono__even,axiom,
! [N2: nat,A: real,B: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% power_mono_even
thf(fact_5760_sqrt__ge__absD,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y ) )
=> ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).
% sqrt_ge_absD
thf(fact_5761_sqrt__sum__squares__le__sum__abs,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) ) ) ).
% sqrt_sum_squares_le_sum_abs
thf(fact_5762_real__sqrt__ge__abs2,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_ge_abs2
thf(fact_5763_real__sqrt__ge__abs1,axiom,
! [X: real,Y: 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 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_ge_abs1
thf(fact_5764_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_5765_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_5766_cos__x__y__le__one,axiom,
! [X: real,Y: 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 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% cos_x_y_le_one
thf(fact_5767_real__sqrt__sum__squares__less,axiom,
! [X: real,U: real,Y: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( 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 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_5768_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_5769_abs__sqrt__wlog,axiom,
! [P: code_integer > code_integer > $o,X: code_integer] :
( ! [X2: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X2 )
=> ( P @ X2 @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_5770_abs__sqrt__wlog,axiom,
! [P: rat > rat > $o,X: rat] :
( ! [X2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
=> ( P @ X2 @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_5771_abs__sqrt__wlog,axiom,
! [P: int > int > $o,X: int] :
( ! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P @ X2 @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_5772_abs__sqrt__wlog,axiom,
! [P: real > real > $o,X: real] :
( ! [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( P @ X2 @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_5773_signed__take__bit__eq__take__bit__minus,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N4: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ K2 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N4 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_5774_fact__double,axiom,
! [N2: nat] :
( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).
% fact_double
thf(fact_5775_fact__double,axiom,
! [N2: nat] :
( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% fact_double
thf(fact_5776_fact__double,axiom,
! [N2: nat] :
( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( times_times_real @ ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% fact_double
thf(fact_5777_modulo__int__unfold,axiom,
! [L: int,K: int,N2: nat,M: nat] :
( ( ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N2 = zero_zero_nat ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
= ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
& ( ~ ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N2 = zero_zero_nat ) )
=> ( ( ( ( sgn_sgn_int @ K )
= ( sgn_sgn_int @ L ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
= ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) )
& ( ( ( sgn_sgn_int @ K )
!= ( sgn_sgn_int @ L ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
= ( times_times_int @ ( sgn_sgn_int @ L )
@ ( minus_minus_int
@ ( semiri1314217659103216013at_int
@ ( times_times_nat @ N2
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) )
@ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_5778_mask__numeral,axiom,
! [N2: num] :
( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N2 ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N2 ) ) ) ) ) ).
% mask_numeral
thf(fact_5779_mask__numeral,axiom,
! [N2: num] :
( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N2 ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N2 ) ) ) ) ) ).
% mask_numeral
thf(fact_5780_divide__int__unfold,axiom,
! [L: int,K: int,N2: nat,M: nat] :
( ( ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N2 = zero_zero_nat ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
= zero_zero_int ) )
& ( ~ ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N2 = zero_zero_nat ) )
=> ( ( ( ( sgn_sgn_int @ K )
= ( sgn_sgn_int @ L ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) )
& ( ( ( sgn_sgn_int @ K )
!= ( sgn_sgn_int @ L ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
= ( uminus_uminus_int
@ ( semiri1314217659103216013at_int
@ ( plus_plus_nat @ ( divide_divide_nat @ M @ N2 )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_5781_sgn__sgn,axiom,
! [A: int] :
( ( sgn_sgn_int @ ( sgn_sgn_int @ A ) )
= ( sgn_sgn_int @ A ) ) ).
% sgn_sgn
thf(fact_5782_sgn__sgn,axiom,
! [A: real] :
( ( sgn_sgn_real @ ( sgn_sgn_real @ A ) )
= ( sgn_sgn_real @ A ) ) ).
% sgn_sgn
thf(fact_5783_sgn__sgn,axiom,
! [A: complex] :
( ( sgn_sgn_complex @ ( sgn_sgn_complex @ A ) )
= ( sgn_sgn_complex @ A ) ) ).
% sgn_sgn
thf(fact_5784_sgn__sgn,axiom,
! [A: code_integer] :
( ( sgn_sgn_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
= ( sgn_sgn_Code_integer @ A ) ) ).
% sgn_sgn
thf(fact_5785_sgn__sgn,axiom,
! [A: rat] :
( ( sgn_sgn_rat @ ( sgn_sgn_rat @ A ) )
= ( sgn_sgn_rat @ A ) ) ).
% sgn_sgn
thf(fact_5786_mask__nat__positive__iff,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% mask_nat_positive_iff
thf(fact_5787_sgn__0,axiom,
( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% sgn_0
thf(fact_5788_sgn__0,axiom,
( ( sgn_sgn_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% sgn_0
thf(fact_5789_sgn__0,axiom,
( ( sgn_sgn_real @ zero_zero_real )
= zero_zero_real ) ).
% sgn_0
thf(fact_5790_sgn__0,axiom,
( ( sgn_sgn_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% sgn_0
thf(fact_5791_sgn__0,axiom,
( ( sgn_sgn_int @ zero_zero_int )
= zero_zero_int ) ).
% sgn_0
thf(fact_5792_sgn__zero,axiom,
( ( sgn_sgn_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% sgn_zero
thf(fact_5793_sgn__zero,axiom,
( ( sgn_sgn_real @ zero_zero_real )
= zero_zero_real ) ).
% sgn_zero
thf(fact_5794_sgn__1,axiom,
( ( sgn_sgn_int @ one_one_int )
= one_one_int ) ).
% sgn_1
thf(fact_5795_sgn__1,axiom,
( ( sgn_sgn_real @ one_one_real )
= one_one_real ) ).
% sgn_1
thf(fact_5796_sgn__1,axiom,
( ( sgn_sgn_complex @ one_one_complex )
= one_one_complex ) ).
% sgn_1
thf(fact_5797_sgn__1,axiom,
( ( sgn_sgn_Code_integer @ one_one_Code_integer )
= one_one_Code_integer ) ).
% sgn_1
thf(fact_5798_sgn__1,axiom,
( ( sgn_sgn_rat @ one_one_rat )
= one_one_rat ) ).
% sgn_1
thf(fact_5799_sgn__one,axiom,
( ( sgn_sgn_real @ one_one_real )
= one_one_real ) ).
% sgn_one
thf(fact_5800_sgn__one,axiom,
( ( sgn_sgn_complex @ one_one_complex )
= one_one_complex ) ).
% sgn_one
thf(fact_5801_sgn__divide,axiom,
! [A: complex,B: complex] :
( ( sgn_sgn_complex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B ) ) ) ).
% sgn_divide
thf(fact_5802_sgn__divide,axiom,
! [A: real,B: real] :
( ( sgn_sgn_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).
% sgn_divide
thf(fact_5803_sgn__divide,axiom,
! [A: rat,B: rat] :
( ( sgn_sgn_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).
% sgn_divide
thf(fact_5804_idom__abs__sgn__class_Osgn__minus,axiom,
! [A: real] :
( ( sgn_sgn_real @ ( uminus_uminus_real @ A ) )
= ( uminus_uminus_real @ ( sgn_sgn_real @ A ) ) ) ).
% idom_abs_sgn_class.sgn_minus
thf(fact_5805_idom__abs__sgn__class_Osgn__minus,axiom,
! [A: int] :
( ( sgn_sgn_int @ ( uminus_uminus_int @ A ) )
= ( uminus_uminus_int @ ( sgn_sgn_int @ A ) ) ) ).
% idom_abs_sgn_class.sgn_minus
thf(fact_5806_idom__abs__sgn__class_Osgn__minus,axiom,
! [A: complex] :
( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ A ) )
= ( uminus1482373934393186551omplex @ ( sgn_sgn_complex @ A ) ) ) ).
% idom_abs_sgn_class.sgn_minus
thf(fact_5807_idom__abs__sgn__class_Osgn__minus,axiom,
! [A: code_integer] :
( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
= ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ A ) ) ) ).
% idom_abs_sgn_class.sgn_minus
thf(fact_5808_idom__abs__sgn__class_Osgn__minus,axiom,
! [A: rat] :
( ( sgn_sgn_rat @ ( uminus_uminus_rat @ A ) )
= ( uminus_uminus_rat @ ( sgn_sgn_rat @ A ) ) ) ).
% idom_abs_sgn_class.sgn_minus
thf(fact_5809_power__sgn,axiom,
! [A: code_integer,N2: nat] :
( ( sgn_sgn_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) )
= ( power_8256067586552552935nteger @ ( sgn_sgn_Code_integer @ A ) @ N2 ) ) ).
% power_sgn
thf(fact_5810_power__sgn,axiom,
! [A: int,N2: nat] :
( ( sgn_sgn_int @ ( power_power_int @ A @ N2 ) )
= ( power_power_int @ ( sgn_sgn_int @ A ) @ N2 ) ) ).
% power_sgn
thf(fact_5811_power__sgn,axiom,
! [A: real,N2: nat] :
( ( sgn_sgn_real @ ( power_power_real @ A @ N2 ) )
= ( power_power_real @ ( sgn_sgn_real @ A ) @ N2 ) ) ).
% power_sgn
thf(fact_5812_power__sgn,axiom,
! [A: rat,N2: nat] :
( ( sgn_sgn_rat @ ( power_power_rat @ A @ N2 ) )
= ( power_power_rat @ ( sgn_sgn_rat @ A ) @ N2 ) ) ).
% power_sgn
thf(fact_5813_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
= ( ( abs_abs_int @ X )
= one_one_int ) ) ).
% zdvd1_eq
thf(fact_5814_sgn__greater,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% sgn_greater
thf(fact_5815_sgn__greater,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% sgn_greater
thf(fact_5816_sgn__greater,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% sgn_greater
thf(fact_5817_sgn__greater,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% sgn_greater
thf(fact_5818_sgn__less,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
= ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% sgn_less
thf(fact_5819_sgn__less,axiom,
! [A: real] :
( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% sgn_less
thf(fact_5820_sgn__less,axiom,
! [A: rat] :
( ( ord_less_rat @ ( sgn_sgn_rat @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% sgn_less
thf(fact_5821_sgn__less,axiom,
! [A: int] :
( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% sgn_less
thf(fact_5822_divide__sgn,axiom,
! [A: real,B: real] :
( ( divide_divide_real @ A @ ( sgn_sgn_real @ B ) )
= ( times_times_real @ A @ ( sgn_sgn_real @ B ) ) ) ).
% divide_sgn
thf(fact_5823_divide__sgn,axiom,
! [A: rat,B: rat] :
( ( divide_divide_rat @ A @ ( sgn_sgn_rat @ B ) )
= ( times_times_rat @ A @ ( sgn_sgn_rat @ B ) ) ) ).
% divide_sgn
thf(fact_5824_fact__0,axiom,
( ( semiri5044797733671781792omplex @ zero_zero_nat )
= one_one_complex ) ).
% fact_0
thf(fact_5825_fact__0,axiom,
( ( semiri773545260158071498ct_rat @ zero_zero_nat )
= one_one_rat ) ).
% fact_0
thf(fact_5826_fact__0,axiom,
( ( semiri1406184849735516958ct_int @ zero_zero_nat )
= one_one_int ) ).
% fact_0
thf(fact_5827_fact__0,axiom,
( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
= one_one_nat ) ).
% fact_0
thf(fact_5828_fact__0,axiom,
( ( semiri2265585572941072030t_real @ zero_zero_nat )
= one_one_real ) ).
% fact_0
thf(fact_5829_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_5830_mask__eq__0__iff,axiom,
! [N2: nat] :
( ( ( bit_se2002935070580805687sk_nat @ N2 )
= zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_5831_mask__eq__0__iff,axiom,
! [N2: nat] :
( ( ( bit_se2000444600071755411sk_int @ N2 )
= zero_zero_int )
= ( N2 = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_5832_mask__0,axiom,
( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% mask_0
thf(fact_5833_mask__0,axiom,
( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
= zero_zero_int ) ).
% mask_0
thf(fact_5834_fact__1,axiom,
( ( semiri5044797733671781792omplex @ one_one_nat )
= one_one_complex ) ).
% fact_1
thf(fact_5835_fact__1,axiom,
( ( semiri773545260158071498ct_rat @ one_one_nat )
= one_one_rat ) ).
% fact_1
thf(fact_5836_fact__1,axiom,
( ( semiri1406184849735516958ct_int @ one_one_nat )
= one_one_int ) ).
% fact_1
thf(fact_5837_fact__1,axiom,
( ( semiri1408675320244567234ct_nat @ one_one_nat )
= one_one_nat ) ).
% fact_1
thf(fact_5838_fact__1,axiom,
( ( semiri2265585572941072030t_real @ one_one_nat )
= one_one_real ) ).
% fact_1
thf(fact_5839_sgn__pos,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( sgn_sgn_Code_integer @ A )
= one_one_Code_integer ) ) ).
% sgn_pos
thf(fact_5840_sgn__pos,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( sgn_sgn_real @ A )
= one_one_real ) ) ).
% sgn_pos
thf(fact_5841_sgn__pos,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( sgn_sgn_rat @ A )
= one_one_rat ) ) ).
% sgn_pos
thf(fact_5842_sgn__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( sgn_sgn_int @ A )
= one_one_int ) ) ).
% sgn_pos
thf(fact_5843_bit__numeral__Bit0__Suc__iff,axiom,
! [M: num,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N2 ) )
= ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N2 ) ) ).
% bit_numeral_Bit0_Suc_iff
thf(fact_5844_bit__numeral__Bit0__Suc__iff,axiom,
! [M: num,N2: nat] :
( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N2 ) )
= ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).
% bit_numeral_Bit0_Suc_iff
thf(fact_5845_abs__sgn__eq__1,axiom,
! [A: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
= one_one_Code_integer ) ) ).
% abs_sgn_eq_1
thf(fact_5846_abs__sgn__eq__1,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
= one_one_real ) ) ).
% abs_sgn_eq_1
thf(fact_5847_abs__sgn__eq__1,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
= one_one_rat ) ) ).
% abs_sgn_eq_1
thf(fact_5848_abs__sgn__eq__1,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
= one_one_int ) ) ).
% abs_sgn_eq_1
thf(fact_5849_fact__Suc__0,axiom,
( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
= one_one_complex ) ).
% fact_Suc_0
thf(fact_5850_fact__Suc__0,axiom,
( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
= one_one_rat ) ).
% fact_Suc_0
thf(fact_5851_fact__Suc__0,axiom,
( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
% fact_Suc_0
thf(fact_5852_fact__Suc__0,axiom,
( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% fact_Suc_0
thf(fact_5853_fact__Suc__0,axiom,
( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
= one_one_real ) ).
% fact_Suc_0
thf(fact_5854_bit__numeral__Bit1__Suc__iff,axiom,
! [M: num,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N2 ) )
= ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N2 ) ) ).
% bit_numeral_Bit1_Suc_iff
thf(fact_5855_bit__numeral__Bit1__Suc__iff,axiom,
! [M: num,N2: nat] :
( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N2 ) )
= ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).
% bit_numeral_Bit1_Suc_iff
thf(fact_5856_fact__Suc,axiom,
! [N2: nat] :
( ( semiri1406184849735516958ct_int @ ( suc @ N2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% fact_Suc
thf(fact_5857_fact__Suc,axiom,
! [N2: nat] :
( ( semiri5044797733671781792omplex @ ( suc @ N2 ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).
% fact_Suc
thf(fact_5858_fact__Suc,axiom,
! [N2: nat] :
( ( semiri773545260158071498ct_rat @ ( suc @ N2 ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% fact_Suc
thf(fact_5859_fact__Suc,axiom,
! [N2: nat] :
( ( semiri1408675320244567234ct_nat @ ( suc @ N2 ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% fact_Suc
thf(fact_5860_fact__Suc,axiom,
! [N2: nat] :
( ( semiri2265585572941072030t_real @ ( suc @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% fact_Suc
thf(fact_5861_sgn__mult__self__eq,axiom,
! [A: real] :
( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ A ) )
= ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% sgn_mult_self_eq
thf(fact_5862_sgn__mult__self__eq,axiom,
! [A: rat] :
( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ A ) )
= ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% sgn_mult_self_eq
thf(fact_5863_sgn__mult__self__eq,axiom,
! [A: int] :
( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
= ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% sgn_mult_self_eq
thf(fact_5864_sgn__mult__self__eq,axiom,
! [A: code_integer] :
( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
= ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% sgn_mult_self_eq
thf(fact_5865_mask__Suc__0,axiom,
( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% mask_Suc_0
thf(fact_5866_mask__Suc__0,axiom,
( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
% mask_Suc_0
thf(fact_5867_sgn__abs,axiom,
! [A: complex] :
( ( abs_abs_complex @ ( sgn_sgn_complex @ A ) )
= ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% sgn_abs
thf(fact_5868_sgn__abs,axiom,
! [A: real] :
( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
= ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% sgn_abs
thf(fact_5869_sgn__abs,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
= ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% sgn_abs
thf(fact_5870_sgn__abs,axiom,
! [A: int] :
( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
= ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% sgn_abs
thf(fact_5871_sgn__abs,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
= ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% sgn_abs
thf(fact_5872_idom__abs__sgn__class_Oabs__sgn,axiom,
! [A: complex] :
( ( sgn_sgn_complex @ ( abs_abs_complex @ A ) )
= ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% idom_abs_sgn_class.abs_sgn
thf(fact_5873_idom__abs__sgn__class_Oabs__sgn,axiom,
! [A: real] :
( ( sgn_sgn_real @ ( abs_abs_real @ A ) )
= ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% idom_abs_sgn_class.abs_sgn
thf(fact_5874_idom__abs__sgn__class_Oabs__sgn,axiom,
! [A: rat] :
( ( sgn_sgn_rat @ ( abs_abs_rat @ A ) )
= ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% idom_abs_sgn_class.abs_sgn
thf(fact_5875_idom__abs__sgn__class_Oabs__sgn,axiom,
! [A: int] :
( ( sgn_sgn_int @ ( abs_abs_int @ A ) )
= ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% idom_abs_sgn_class.abs_sgn
thf(fact_5876_idom__abs__sgn__class_Oabs__sgn,axiom,
! [A: code_integer] :
( ( sgn_sgn_Code_integer @ ( abs_abs_Code_integer @ A ) )
= ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% idom_abs_sgn_class.abs_sgn
thf(fact_5877_take__bit__minus__one__eq__mask,axiom,
! [N2: nat] :
( ( bit_se1745604003318907178nteger @ N2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( bit_se2119862282449309892nteger @ N2 ) ) ).
% take_bit_minus_one_eq_mask
thf(fact_5878_take__bit__minus__one__eq__mask,axiom,
! [N2: nat] :
( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
= ( bit_se2000444600071755411sk_int @ N2 ) ) ).
% take_bit_minus_one_eq_mask
thf(fact_5879_sgn__mult__dvd__iff,axiom,
! [R4: int,L: int,K: int] :
( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R4 ) @ L ) @ K )
= ( ( dvd_dvd_int @ L @ K )
& ( ( R4 = zero_zero_int )
=> ( K = zero_zero_int ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_5880_mult__sgn__dvd__iff,axiom,
! [L: int,R4: int,K: int] :
( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R4 ) ) @ K )
= ( ( dvd_dvd_int @ L @ K )
& ( ( R4 = zero_zero_int )
=> ( K = zero_zero_int ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_5881_dvd__sgn__mult__iff,axiom,
! [L: int,R4: int,K: int] :
( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R4 ) @ K ) )
= ( ( dvd_dvd_int @ L @ K )
| ( R4 = zero_zero_int ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_5882_dvd__mult__sgn__iff,axiom,
! [L: int,K: int,R4: int] :
( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R4 ) ) )
= ( ( dvd_dvd_int @ L @ K )
| ( R4 = zero_zero_int ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_5883_signed__take__bit__nonnegative__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
= ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).
% signed_take_bit_nonnegative_iff
thf(fact_5884_signed__take__bit__negative__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ zero_zero_int )
= ( bit_se1146084159140164899it_int @ K @ N2 ) ) ).
% signed_take_bit_negative_iff
thf(fact_5885_sgn__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( sgn_sgn_real @ A )
= ( uminus_uminus_real @ one_one_real ) ) ) ).
% sgn_neg
thf(fact_5886_sgn__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( sgn_sgn_int @ A )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% sgn_neg
thf(fact_5887_sgn__neg,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( sgn_sgn_Code_integer @ A )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% sgn_neg
thf(fact_5888_sgn__neg,axiom,
! [A: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( sgn_sgn_rat @ A )
= ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% sgn_neg
thf(fact_5889_fact__2,axiom,
( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_5890_fact__2,axiom,
( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_5891_fact__2,axiom,
( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_5892_fact__2,axiom,
( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_5893_fact__2,axiom,
( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_5894_sgn__of__nat,axiom,
! [N2: nat] :
( ( sgn_sgn_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( zero_n3304061248610475627l_real @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% sgn_of_nat
thf(fact_5895_sgn__of__nat,axiom,
! [N2: nat] :
( ( sgn_sgn_rat @ ( semiri681578069525770553at_rat @ N2 ) )
= ( zero_n2052037380579107095ol_rat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% sgn_of_nat
thf(fact_5896_sgn__of__nat,axiom,
! [N2: nat] :
( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% sgn_of_nat
thf(fact_5897_sgn__of__nat,axiom,
! [N2: nat] :
( ( sgn_sgn_Code_integer @ ( semiri4939895301339042750nteger @ N2 ) )
= ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% sgn_of_nat
thf(fact_5898_bit__numeral__simps_I2_J,axiom,
! [W: num,N2: num] :
( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N2 ) ) ) ).
% bit_numeral_simps(2)
thf(fact_5899_bit__numeral__simps_I2_J,axiom,
! [W: num,N2: num] :
( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N2 ) ) ) ).
% bit_numeral_simps(2)
thf(fact_5900_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W: num,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N2 ) )
= ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N2 ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_5901_bit__numeral__simps_I3_J,axiom,
! [W: num,N2: num] :
( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N2 ) ) ) ).
% bit_numeral_simps(3)
thf(fact_5902_bit__numeral__simps_I3_J,axiom,
! [W: num,N2: num] :
( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N2 ) ) ) ).
% bit_numeral_simps(3)
thf(fact_5903_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W: num,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N2 ) )
= ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N2 ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_5904_bit__0,axiom,
! [A: code_integer] :
( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
= ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% bit_0
thf(fact_5905_bit__0,axiom,
! [A: int] :
( ( bit_se1146084159140164899it_int @ A @ zero_zero_nat )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% bit_0
thf(fact_5906_bit__0,axiom,
! [A: nat] :
( ( bit_se1148574629649215175it_nat @ A @ zero_zero_nat )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% bit_0
thf(fact_5907_bit__minus__numeral__int_I1_J,axiom,
! [W: num,N2: num] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N2 ) ) ) ).
% bit_minus_numeral_int(1)
thf(fact_5908_bit__minus__numeral__int_I2_J,axiom,
! [W: num,N2: num] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N2 ) ) ) ) ).
% bit_minus_numeral_int(2)
thf(fact_5909_bit__mod__2__iff,axiom,
! [A: code_integer,N2: nat] :
( ( bit_se9216721137139052372nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N2 )
= ( ( N2 = zero_zero_nat )
& ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% bit_mod_2_iff
thf(fact_5910_bit__mod__2__iff,axiom,
! [A: int,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N2 )
= ( ( N2 = zero_zero_nat )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% bit_mod_2_iff
thf(fact_5911_bit__mod__2__iff,axiom,
! [A: nat,N2: nat] :
( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
= ( ( N2 = zero_zero_nat )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% bit_mod_2_iff
thf(fact_5912_div__eq__sgn__abs,axiom,
! [K: int,L: int] :
( ( ( sgn_sgn_int @ K )
= ( sgn_sgn_int @ L ) )
=> ( ( divide_divide_int @ K @ L )
= ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% div_eq_sgn_abs
thf(fact_5913_of__nat__mask__eq,axiom,
! [N2: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) )
= ( bit_se2002935070580805687sk_nat @ N2 ) ) ).
% of_nat_mask_eq
thf(fact_5914_of__nat__mask__eq,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N2 ) )
= ( bit_se2000444600071755411sk_int @ N2 ) ) ).
% of_nat_mask_eq
thf(fact_5915_bit__of__nat__iff__bit,axiom,
! [M: nat,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( semiri1314217659103216013at_int @ M ) @ N2 )
= ( bit_se1148574629649215175it_nat @ M @ N2 ) ) ).
% bit_of_nat_iff_bit
thf(fact_5916_bit__of__nat__iff__bit,axiom,
! [M: nat,N2: nat] :
( ( bit_se1148574629649215175it_nat @ ( semiri1316708129612266289at_nat @ M ) @ N2 )
= ( bit_se1148574629649215175it_nat @ M @ N2 ) ) ).
% bit_of_nat_iff_bit
thf(fact_5917_bit__numeral__iff,axiom,
! [M: num,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N2 )
= ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).
% bit_numeral_iff
thf(fact_5918_bit__numeral__iff,axiom,
! [M: num,N2: nat] :
( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N2 )
= ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).
% bit_numeral_iff
thf(fact_5919_bit__disjunctive__add__iff,axiom,
! [A: int,B: int,N2: nat] :
( ! [N: nat] :
( ~ ( bit_se1146084159140164899it_int @ A @ N )
| ~ ( bit_se1146084159140164899it_int @ B @ N ) )
=> ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B ) @ N2 )
= ( ( bit_se1146084159140164899it_int @ A @ N2 )
| ( bit_se1146084159140164899it_int @ B @ N2 ) ) ) ) ).
% bit_disjunctive_add_iff
thf(fact_5920_bit__disjunctive__add__iff,axiom,
! [A: nat,B: nat,N2: nat] :
( ! [N: nat] :
( ~ ( bit_se1148574629649215175it_nat @ A @ N )
| ~ ( bit_se1148574629649215175it_nat @ B @ N ) )
=> ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
= ( ( bit_se1148574629649215175it_nat @ A @ N2 )
| ( bit_se1148574629649215175it_nat @ B @ N2 ) ) ) ) ).
% bit_disjunctive_add_iff
thf(fact_5921_fact__mono__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% fact_mono_nat
thf(fact_5922_fact__ge__self,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% fact_ge_self
thf(fact_5923_sgn__eq__0__iff,axiom,
! [A: code_integer] :
( ( ( sgn_sgn_Code_integer @ A )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% sgn_eq_0_iff
thf(fact_5924_sgn__eq__0__iff,axiom,
! [A: complex] :
( ( ( sgn_sgn_complex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% sgn_eq_0_iff
thf(fact_5925_sgn__eq__0__iff,axiom,
! [A: real] :
( ( ( sgn_sgn_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% sgn_eq_0_iff
thf(fact_5926_sgn__eq__0__iff,axiom,
! [A: rat] :
( ( ( sgn_sgn_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% sgn_eq_0_iff
thf(fact_5927_sgn__eq__0__iff,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% sgn_eq_0_iff
thf(fact_5928_sgn__0__0,axiom,
! [A: code_integer] :
( ( ( sgn_sgn_Code_integer @ A )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% sgn_0_0
thf(fact_5929_sgn__0__0,axiom,
! [A: real] :
( ( ( sgn_sgn_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% sgn_0_0
thf(fact_5930_sgn__0__0,axiom,
! [A: rat] :
( ( ( sgn_sgn_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% sgn_0_0
thf(fact_5931_sgn__0__0,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% sgn_0_0
thf(fact_5932_sgn__zero__iff,axiom,
! [X: complex] :
( ( ( sgn_sgn_complex @ X )
= zero_zero_complex )
= ( X = zero_zero_complex ) ) ).
% sgn_zero_iff
thf(fact_5933_sgn__zero__iff,axiom,
! [X: real] :
( ( ( sgn_sgn_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% sgn_zero_iff
thf(fact_5934_sgn__mult,axiom,
! [A: complex,B: complex] :
( ( sgn_sgn_complex @ ( times_times_complex @ A @ B ) )
= ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B ) ) ) ).
% sgn_mult
thf(fact_5935_sgn__mult,axiom,
! [A: code_integer,B: code_integer] :
( ( sgn_sgn_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
= ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ B ) ) ) ).
% sgn_mult
thf(fact_5936_sgn__mult,axiom,
! [A: real,B: real] :
( ( sgn_sgn_real @ ( times_times_real @ A @ B ) )
= ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).
% sgn_mult
thf(fact_5937_sgn__mult,axiom,
! [A: rat,B: rat] :
( ( sgn_sgn_rat @ ( times_times_rat @ A @ B ) )
= ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).
% sgn_mult
thf(fact_5938_sgn__mult,axiom,
! [A: int,B: int] :
( ( sgn_sgn_int @ ( times_times_int @ A @ B ) )
= ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ).
% sgn_mult
thf(fact_5939_same__sgn__sgn__add,axiom,
! [B: code_integer,A: code_integer] :
( ( ( sgn_sgn_Code_integer @ B )
= ( sgn_sgn_Code_integer @ A ) )
=> ( ( sgn_sgn_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( sgn_sgn_Code_integer @ A ) ) ) ).
% same_sgn_sgn_add
thf(fact_5940_same__sgn__sgn__add,axiom,
! [B: real,A: real] :
( ( ( sgn_sgn_real @ B )
= ( sgn_sgn_real @ A ) )
=> ( ( sgn_sgn_real @ ( plus_plus_real @ A @ B ) )
= ( sgn_sgn_real @ A ) ) ) ).
% same_sgn_sgn_add
thf(fact_5941_same__sgn__sgn__add,axiom,
! [B: rat,A: rat] :
( ( ( sgn_sgn_rat @ B )
= ( sgn_sgn_rat @ A ) )
=> ( ( sgn_sgn_rat @ ( plus_plus_rat @ A @ B ) )
= ( sgn_sgn_rat @ A ) ) ) ).
% same_sgn_sgn_add
thf(fact_5942_same__sgn__sgn__add,axiom,
! [B: int,A: int] :
( ( ( sgn_sgn_int @ B )
= ( sgn_sgn_int @ A ) )
=> ( ( sgn_sgn_int @ ( plus_plus_int @ A @ B ) )
= ( sgn_sgn_int @ A ) ) ) ).
% same_sgn_sgn_add
thf(fact_5943_fact__nonzero,axiom,
! [N2: nat] :
( ( semiri5044797733671781792omplex @ N2 )
!= zero_zero_complex ) ).
% fact_nonzero
thf(fact_5944_fact__nonzero,axiom,
! [N2: nat] :
( ( semiri773545260158071498ct_rat @ N2 )
!= zero_zero_rat ) ).
% fact_nonzero
thf(fact_5945_fact__nonzero,axiom,
! [N2: nat] :
( ( semiri1406184849735516958ct_int @ N2 )
!= zero_zero_int ) ).
% fact_nonzero
thf(fact_5946_fact__nonzero,axiom,
! [N2: nat] :
( ( semiri1408675320244567234ct_nat @ N2 )
!= zero_zero_nat ) ).
% fact_nonzero
thf(fact_5947_fact__nonzero,axiom,
! [N2: nat] :
( ( semiri2265585572941072030t_real @ N2 )
!= zero_zero_real ) ).
% fact_nonzero
thf(fact_5948_zdvd__antisym__abs,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( abs_abs_int @ A )
= ( abs_abs_int @ B ) ) ) ) ).
% zdvd_antisym_abs
thf(fact_5949_bit__unset__bit__iff,axiom,
! [M: nat,A: int,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_se4203085406695923979it_int @ M @ A ) @ N2 )
= ( ( bit_se1146084159140164899it_int @ A @ N2 )
& ( M != N2 ) ) ) ).
% bit_unset_bit_iff
thf(fact_5950_bit__unset__bit__iff,axiom,
! [M: nat,A: nat,N2: nat] :
( ( bit_se1148574629649215175it_nat @ ( bit_se4205575877204974255it_nat @ M @ A ) @ N2 )
= ( ( bit_se1148574629649215175it_nat @ A @ N2 )
& ( M != N2 ) ) ) ).
% bit_unset_bit_iff
thf(fact_5951_less__eq__mask,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ).
% less_eq_mask
thf(fact_5952_div__sgn__abs__cancel,axiom,
! [V: int,K: int,L: 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 @ L ) ) )
= ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% div_sgn_abs_cancel
thf(fact_5953_div__dvd__sgn__abs,axiom,
! [L: int,K: int] :
( ( dvd_dvd_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ).
% div_dvd_sgn_abs
thf(fact_5954_not__bit__1__Suc,axiom,
! [N2: nat] :
~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N2 ) ) ).
% not_bit_1_Suc
thf(fact_5955_not__bit__1__Suc,axiom,
! [N2: nat] :
~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N2 ) ) ).
% not_bit_1_Suc
thf(fact_5956_bit__1__iff,axiom,
! [N2: nat] :
( ( bit_se1146084159140164899it_int @ one_one_int @ N2 )
= ( N2 = zero_zero_nat ) ) ).
% bit_1_iff
thf(fact_5957_bit__1__iff,axiom,
! [N2: nat] :
( ( bit_se1148574629649215175it_nat @ one_one_nat @ N2 )
= ( N2 = zero_zero_nat ) ) ).
% bit_1_iff
thf(fact_5958_bit__numeral__simps_I1_J,axiom,
! [N2: num] :
~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N2 ) ) ).
% bit_numeral_simps(1)
thf(fact_5959_bit__numeral__simps_I1_J,axiom,
! [N2: num] :
~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% bit_numeral_simps(1)
thf(fact_5960_fact__less__mono__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).
% fact_less_mono_nat
thf(fact_5961_bit__take__bit__iff,axiom,
! [M: nat,A: int,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N2 )
= ( ( ord_less_nat @ N2 @ M )
& ( bit_se1146084159140164899it_int @ A @ N2 ) ) ) ).
% bit_take_bit_iff
thf(fact_5962_bit__take__bit__iff,axiom,
! [M: nat,A: nat,N2: nat] :
( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N2 )
= ( ( ord_less_nat @ N2 @ M )
& ( bit_se1148574629649215175it_nat @ A @ N2 ) ) ) ).
% bit_take_bit_iff
thf(fact_5963_sgn__not__eq__imp,axiom,
! [B: real,A: real] :
( ( ( sgn_sgn_real @ B )
!= ( sgn_sgn_real @ A ) )
=> ( ( ( sgn_sgn_real @ A )
!= zero_zero_real )
=> ( ( ( sgn_sgn_real @ B )
!= zero_zero_real )
=> ( ( sgn_sgn_real @ A )
= ( uminus_uminus_real @ ( sgn_sgn_real @ B ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_5964_sgn__not__eq__imp,axiom,
! [B: int,A: int] :
( ( ( sgn_sgn_int @ B )
!= ( sgn_sgn_int @ A ) )
=> ( ( ( sgn_sgn_int @ A )
!= zero_zero_int )
=> ( ( ( sgn_sgn_int @ B )
!= zero_zero_int )
=> ( ( sgn_sgn_int @ A )
= ( uminus_uminus_int @ ( sgn_sgn_int @ B ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_5965_sgn__not__eq__imp,axiom,
! [B: code_integer,A: code_integer] :
( ( ( sgn_sgn_Code_integer @ B )
!= ( sgn_sgn_Code_integer @ A ) )
=> ( ( ( sgn_sgn_Code_integer @ A )
!= zero_z3403309356797280102nteger )
=> ( ( ( sgn_sgn_Code_integer @ B )
!= zero_z3403309356797280102nteger )
=> ( ( sgn_sgn_Code_integer @ A )
= ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_5966_sgn__not__eq__imp,axiom,
! [B: rat,A: rat] :
( ( ( sgn_sgn_rat @ B )
!= ( sgn_sgn_rat @ A ) )
=> ( ( ( sgn_sgn_rat @ A )
!= zero_zero_rat )
=> ( ( ( sgn_sgn_rat @ B )
!= zero_zero_rat )
=> ( ( sgn_sgn_rat @ A )
= ( uminus_uminus_rat @ ( sgn_sgn_rat @ B ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_5967_bit__of__bool__iff,axiom,
! [B: $o,N2: nat] :
( ( bit_se9216721137139052372nteger @ ( zero_n356916108424825756nteger @ B ) @ N2 )
= ( B
& ( N2 = zero_zero_nat ) ) ) ).
% bit_of_bool_iff
thf(fact_5968_bit__of__bool__iff,axiom,
! [B: $o,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B ) @ N2 )
= ( B
& ( N2 = zero_zero_nat ) ) ) ).
% bit_of_bool_iff
thf(fact_5969_bit__of__bool__iff,axiom,
! [B: $o,N2: nat] :
( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ N2 )
= ( B
& ( N2 = zero_zero_nat ) ) ) ).
% bit_of_bool_iff
thf(fact_5970_sgn__minus__1,axiom,
( ( sgn_sgn_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% sgn_minus_1
thf(fact_5971_sgn__minus__1,axiom,
( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% sgn_minus_1
thf(fact_5972_sgn__minus__1,axiom,
( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% sgn_minus_1
thf(fact_5973_sgn__minus__1,axiom,
( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% sgn_minus_1
thf(fact_5974_sgn__minus__1,axiom,
( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% sgn_minus_1
thf(fact_5975_fact__ge__zero,axiom,
! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).
% fact_ge_zero
thf(fact_5976_fact__ge__zero,axiom,
! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).
% fact_ge_zero
thf(fact_5977_fact__ge__zero,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% fact_ge_zero
thf(fact_5978_fact__ge__zero,axiom,
! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).
% fact_ge_zero
thf(fact_5979_linordered__idom__class_Oabs__sgn,axiom,
( abs_abs_Code_integer
= ( ^ [K2: code_integer] : ( times_3573771949741848930nteger @ K2 @ ( sgn_sgn_Code_integer @ K2 ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_5980_linordered__idom__class_Oabs__sgn,axiom,
( abs_abs_real
= ( ^ [K2: real] : ( times_times_real @ K2 @ ( sgn_sgn_real @ K2 ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_5981_linordered__idom__class_Oabs__sgn,axiom,
( abs_abs_rat
= ( ^ [K2: rat] : ( times_times_rat @ K2 @ ( sgn_sgn_rat @ K2 ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_5982_linordered__idom__class_Oabs__sgn,axiom,
( abs_abs_int
= ( ^ [K2: int] : ( times_times_int @ K2 @ ( sgn_sgn_int @ K2 ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_5983_abs__mult__sgn,axiom,
! [A: complex] :
( ( times_times_complex @ ( abs_abs_complex @ A ) @ ( sgn_sgn_complex @ A ) )
= A ) ).
% abs_mult_sgn
thf(fact_5984_abs__mult__sgn,axiom,
! [A: code_integer] :
( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
= A ) ).
% abs_mult_sgn
thf(fact_5985_abs__mult__sgn,axiom,
! [A: real] :
( ( times_times_real @ ( abs_abs_real @ A ) @ ( sgn_sgn_real @ A ) )
= A ) ).
% abs_mult_sgn
thf(fact_5986_abs__mult__sgn,axiom,
! [A: rat] :
( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( sgn_sgn_rat @ A ) )
= A ) ).
% abs_mult_sgn
thf(fact_5987_abs__mult__sgn,axiom,
! [A: int] :
( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
= A ) ).
% abs_mult_sgn
thf(fact_5988_sgn__mult__abs,axiom,
! [A: complex] :
( ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( abs_abs_complex @ A ) )
= A ) ).
% sgn_mult_abs
thf(fact_5989_sgn__mult__abs,axiom,
! [A: code_integer] :
( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
= A ) ).
% sgn_mult_abs
thf(fact_5990_sgn__mult__abs,axiom,
! [A: real] :
( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( abs_abs_real @ A ) )
= A ) ).
% sgn_mult_abs
thf(fact_5991_sgn__mult__abs,axiom,
! [A: rat] :
( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( abs_abs_rat @ A ) )
= A ) ).
% sgn_mult_abs
thf(fact_5992_sgn__mult__abs,axiom,
! [A: int] :
( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
= A ) ).
% sgn_mult_abs
thf(fact_5993_mult__sgn__abs,axiom,
! [X: code_integer] :
( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ X ) @ ( abs_abs_Code_integer @ X ) )
= X ) ).
% mult_sgn_abs
thf(fact_5994_mult__sgn__abs,axiom,
! [X: real] :
( ( times_times_real @ ( sgn_sgn_real @ X ) @ ( abs_abs_real @ X ) )
= X ) ).
% mult_sgn_abs
thf(fact_5995_mult__sgn__abs,axiom,
! [X: rat] :
( ( times_times_rat @ ( sgn_sgn_rat @ X ) @ ( abs_abs_rat @ X ) )
= X ) ).
% mult_sgn_abs
thf(fact_5996_mult__sgn__abs,axiom,
! [X: int] :
( ( times_times_int @ ( sgn_sgn_int @ X ) @ ( abs_abs_int @ X ) )
= X ) ).
% mult_sgn_abs
thf(fact_5997_fact__not__neg,axiom,
! [N2: nat] :
~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ zero_zero_rat ) ).
% fact_not_neg
thf(fact_5998_fact__not__neg,axiom,
! [N2: nat] :
~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N2 ) @ zero_zero_int ) ).
% fact_not_neg
thf(fact_5999_fact__not__neg,axiom,
! [N2: nat] :
~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ zero_zero_nat ) ).
% fact_not_neg
thf(fact_6000_fact__not__neg,axiom,
! [N2: nat] :
~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N2 ) @ zero_zero_real ) ).
% fact_not_neg
thf(fact_6001_fact__gt__zero,axiom,
! [N2: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).
% fact_gt_zero
thf(fact_6002_fact__gt__zero,axiom,
! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).
% fact_gt_zero
thf(fact_6003_fact__gt__zero,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% fact_gt_zero
thf(fact_6004_fact__gt__zero,axiom,
! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).
% fact_gt_zero
thf(fact_6005_int__sgnE,axiom,
! [K: int] :
~ ! [N: nat,L3: int] :
( K
!= ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% int_sgnE
thf(fact_6006_same__sgn__abs__add,axiom,
! [B: code_integer,A: code_integer] :
( ( ( sgn_sgn_Code_integer @ B )
= ( sgn_sgn_Code_integer @ A ) )
=> ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% same_sgn_abs_add
thf(fact_6007_same__sgn__abs__add,axiom,
! [B: real,A: real] :
( ( ( sgn_sgn_real @ B )
= ( sgn_sgn_real @ A ) )
=> ( ( abs_abs_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% same_sgn_abs_add
thf(fact_6008_same__sgn__abs__add,axiom,
! [B: rat,A: rat] :
( ( ( sgn_sgn_rat @ B )
= ( sgn_sgn_rat @ A ) )
=> ( ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% same_sgn_abs_add
thf(fact_6009_same__sgn__abs__add,axiom,
! [B: int,A: int] :
( ( ( sgn_sgn_int @ B )
= ( sgn_sgn_int @ A ) )
=> ( ( abs_abs_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% same_sgn_abs_add
thf(fact_6010_fact__ge__1,axiom,
! [N2: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).
% fact_ge_1
thf(fact_6011_fact__ge__1,axiom,
! [N2: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).
% fact_ge_1
thf(fact_6012_fact__ge__1,axiom,
! [N2: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% fact_ge_1
thf(fact_6013_fact__ge__1,axiom,
! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).
% fact_ge_1
thf(fact_6014_fact__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% fact_mono
thf(fact_6015_fact__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% fact_mono
thf(fact_6016_fact__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% fact_mono
thf(fact_6017_fact__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% fact_mono
thf(fact_6018_signed__take__bit__eq__if__positive,axiom,
! [A: int,N2: nat] :
( ~ ( bit_se1146084159140164899it_int @ A @ N2 )
=> ( ( bit_ri631733984087533419it_int @ N2 @ A )
= ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ).
% signed_take_bit_eq_if_positive
thf(fact_6019_abs__zmult__eq__1,axiom,
! [M: int,N2: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N2 ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_6020_fact__dvd,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% fact_dvd
thf(fact_6021_fact__dvd,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% fact_dvd
thf(fact_6022_fact__dvd,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% fact_dvd
thf(fact_6023_fact__dvd,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% fact_dvd
thf(fact_6024_abs__div,axiom,
! [Y: int,X: int] :
( ( dvd_dvd_int @ Y @ X )
=> ( ( abs_abs_int @ ( divide_divide_int @ X @ Y ) )
= ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) ) ) ) ).
% abs_div
thf(fact_6025_pochhammer__fact,axiom,
( semiri5044797733671781792omplex
= ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).
% pochhammer_fact
thf(fact_6026_pochhammer__fact,axiom,
( semiri773545260158071498ct_rat
= ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).
% pochhammer_fact
thf(fact_6027_pochhammer__fact,axiom,
( semiri1406184849735516958ct_int
= ( comm_s4660882817536571857er_int @ one_one_int ) ) ).
% pochhammer_fact
thf(fact_6028_pochhammer__fact,axiom,
( semiri1408675320244567234ct_nat
= ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).
% pochhammer_fact
thf(fact_6029_pochhammer__fact,axiom,
( semiri2265585572941072030t_real
= ( comm_s7457072308508201937r_real @ one_one_real ) ) ).
% pochhammer_fact
thf(fact_6030_mask__nonnegative__int,axiom,
! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ).
% mask_nonnegative_int
thf(fact_6031_not__mask__negative__int,axiom,
! [N2: nat] :
~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ zero_zero_int ) ).
% not_mask_negative_int
thf(fact_6032_eucl__rel__int__remainderI,axiom,
! [R4: int,L: int,K: int,Q3: int] :
( ( ( sgn_sgn_int @ R4 )
= ( sgn_sgn_int @ L ) )
=> ( ( ord_less_int @ ( abs_abs_int @ R4 ) @ ( abs_abs_int @ L ) )
=> ( ( K
= ( plus_plus_int @ ( times_times_int @ Q3 @ L ) @ R4 ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R4 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_6033_fact__ge__Suc__0__nat,axiom,
! [N2: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% fact_ge_Suc_0_nat
thf(fact_6034_sgn__1__pos,axiom,
! [A: code_integer] :
( ( ( sgn_sgn_Code_integer @ A )
= one_one_Code_integer )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% sgn_1_pos
thf(fact_6035_sgn__1__pos,axiom,
! [A: real] :
( ( ( sgn_sgn_real @ A )
= one_one_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% sgn_1_pos
thf(fact_6036_sgn__1__pos,axiom,
! [A: rat] :
( ( ( sgn_sgn_rat @ A )
= one_one_rat )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% sgn_1_pos
thf(fact_6037_sgn__1__pos,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= one_one_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% sgn_1_pos
thf(fact_6038_abs__sgn__eq,axiom,
! [A: code_integer] :
( ( ( A = zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
= zero_z3403309356797280102nteger ) )
& ( ( A != zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
= one_one_Code_integer ) ) ) ).
% abs_sgn_eq
thf(fact_6039_abs__sgn__eq,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
= one_one_real ) ) ) ).
% abs_sgn_eq
thf(fact_6040_abs__sgn__eq,axiom,
! [A: rat] :
( ( ( A = zero_zero_rat )
=> ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
= zero_zero_rat ) )
& ( ( A != zero_zero_rat )
=> ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
= one_one_rat ) ) ) ).
% abs_sgn_eq
thf(fact_6041_abs__sgn__eq,axiom,
! [A: int] :
( ( ( A = zero_zero_int )
=> ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
= zero_zero_int ) )
& ( ( A != zero_zero_int )
=> ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
= one_one_int ) ) ) ).
% abs_sgn_eq
thf(fact_6042_dvd__fact,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).
% dvd_fact
thf(fact_6043_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A32: product_prod_int_int] :
( ( eucl_rel_int @ A1 @ A22 @ A32 )
=> ( ( ( A22 = zero_zero_int )
=> ( A32
!= ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
=> ( ! [Q2: int] :
( ( A32
= ( product_Pair_int_int @ Q2 @ zero_zero_int ) )
=> ( ( A22 != zero_zero_int )
=> ( A1
!= ( times_times_int @ Q2 @ A22 ) ) ) )
=> ~ ! [R2: int,Q2: int] :
( ( A32
= ( product_Pair_int_int @ Q2 @ R2 ) )
=> ( ( ( sgn_sgn_int @ R2 )
= ( sgn_sgn_int @ A22 ) )
=> ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ A22 ) )
=> ( A1
!= ( plus_plus_int @ ( times_times_int @ Q2 @ A22 ) @ R2 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_6044_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A12: int,A23: int,A33: product_prod_int_int] :
( ? [K2: int] :
( ( A12 = K2 )
& ( A23 = zero_zero_int )
& ( A33
= ( product_Pair_int_int @ zero_zero_int @ K2 ) ) )
| ? [L2: int,K2: int,Q4: int] :
( ( A12 = K2 )
& ( A23 = L2 )
& ( A33
= ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
& ( L2 != zero_zero_int )
& ( K2
= ( times_times_int @ Q4 @ L2 ) ) )
| ? [R5: int,L2: int,K2: int,Q4: int] :
( ( A12 = K2 )
& ( A23 = L2 )
& ( A33
= ( product_Pair_int_int @ Q4 @ R5 ) )
& ( ( sgn_sgn_int @ R5 )
= ( sgn_sgn_int @ L2 ) )
& ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
& ( K2
= ( plus_plus_int @ ( times_times_int @ Q4 @ L2 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_6045_div__noneq__sgn__abs,axiom,
! [L: int,K: int] :
( ( L != zero_zero_int )
=> ( ( ( sgn_sgn_int @ K )
!= ( sgn_sgn_int @ L ) )
=> ( ( divide_divide_int @ K @ L )
= ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) )
@ ( zero_n2684676970156552555ol_int
@ ~ ( dvd_dvd_int @ L @ K ) ) ) ) ) ) ).
% div_noneq_sgn_abs
thf(fact_6046_bit__not__int__iff_H,axiom,
! [K: int,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N2 )
= ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).
% bit_not_int_iff'
thf(fact_6047_fact__less__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N2 )
=> ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ) ).
% fact_less_mono
thf(fact_6048_fact__less__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N2 )
=> ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ) ).
% fact_less_mono
thf(fact_6049_fact__less__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).
% fact_less_mono
thf(fact_6050_fact__less__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N2 )
=> ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ).
% fact_less_mono
thf(fact_6051_sgn__mod,axiom,
! [L: int,K: int] :
( ( L != zero_zero_int )
=> ( ~ ( dvd_dvd_int @ L @ K )
=> ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
= ( sgn_sgn_int @ L ) ) ) ) ).
% sgn_mod
thf(fact_6052_fact__fact__dvd__fact,axiom,
! [K: nat,N2: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N2 ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% fact_fact_dvd_fact
thf(fact_6053_fact__fact__dvd__fact,axiom,
! [K: nat,N2: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N2 ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% fact_fact_dvd_fact
thf(fact_6054_fact__fact__dvd__fact,axiom,
! [K: nat,N2: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N2 ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% fact_fact_dvd_fact
thf(fact_6055_fact__fact__dvd__fact,axiom,
! [K: nat,N2: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% fact_fact_dvd_fact
thf(fact_6056_fact__fact__dvd__fact,axiom,
! [K: nat,N2: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% fact_fact_dvd_fact
thf(fact_6057_fact__mod,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) )
= zero_zero_int ) ) ).
% fact_mod
thf(fact_6058_fact__mod,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) )
= zero_z3403309356797280102nteger ) ) ).
% fact_mod
thf(fact_6059_fact__mod,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) )
= zero_zero_nat ) ) ).
% fact_mod
thf(fact_6060_zabs__def,axiom,
( abs_abs_int
= ( ^ [I3: int] : ( if_int @ ( ord_less_int @ I3 @ zero_zero_int ) @ ( uminus_uminus_int @ I3 ) @ I3 ) ) ) ).
% zabs_def
thf(fact_6061_dvd__imp__le__int,axiom,
! [I: int,D: int] :
( ( I != zero_zero_int )
=> ( ( dvd_dvd_int @ D @ I )
=> ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% dvd_imp_le_int
thf(fact_6062_fact__le__power,axiom,
! [N2: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% fact_le_power
thf(fact_6063_fact__le__power,axiom,
! [N2: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% fact_le_power
thf(fact_6064_fact__le__power,axiom,
! [N2: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% fact_le_power
thf(fact_6065_fact__le__power,axiom,
! [N2: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% fact_le_power
thf(fact_6066_abs__mod__less,axiom,
! [L: int,K: int] :
( ( L != zero_zero_int )
=> ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% abs_mod_less
thf(fact_6067_flip__bit__eq__if,axiom,
( bit_se2159334234014336723it_int
= ( ^ [N4: nat,A3: int] : ( if_nat_int_int @ ( bit_se1146084159140164899it_int @ A3 @ N4 ) @ bit_se4203085406695923979it_int @ bit_se7879613467334960850it_int @ N4 @ A3 ) ) ) ).
% flip_bit_eq_if
thf(fact_6068_flip__bit__eq__if,axiom,
( bit_se2161824704523386999it_nat
= ( ^ [N4: nat,A3: nat] : ( if_nat_nat_nat @ ( bit_se1148574629649215175it_nat @ A3 @ N4 ) @ bit_se4205575877204974255it_nat @ bit_se7882103937844011126it_nat @ N4 @ A3 ) ) ) ).
% flip_bit_eq_if
thf(fact_6069_less__mask,axiom,
! [N2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ord_less_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ).
% less_mask
thf(fact_6070_fact__diff__Suc,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ ( suc @ M ) )
=> ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) )
= ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N2 ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_6071_sgn__1__neg,axiom,
! [A: real] :
( ( ( sgn_sgn_real @ A )
= ( uminus_uminus_real @ one_one_real ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% sgn_1_neg
thf(fact_6072_sgn__1__neg,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= ( uminus_uminus_int @ one_one_int ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% sgn_1_neg
thf(fact_6073_sgn__1__neg,axiom,
! [A: code_integer] :
( ( ( sgn_sgn_Code_integer @ A )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% sgn_1_neg
thf(fact_6074_sgn__1__neg,axiom,
! [A: rat] :
( ( ( sgn_sgn_rat @ A )
= ( uminus_uminus_rat @ one_one_rat ) )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% sgn_1_neg
thf(fact_6075_sgn__if,axiom,
( sgn_sgn_real
= ( ^ [X3: real] : ( if_real @ ( X3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% sgn_if
thf(fact_6076_sgn__if,axiom,
( sgn_sgn_int
= ( ^ [X3: int] : ( if_int @ ( X3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% sgn_if
thf(fact_6077_sgn__if,axiom,
( sgn_sgn_Code_integer
= ( ^ [X3: code_integer] : ( if_Code_integer @ ( X3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X3 ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).
% sgn_if
thf(fact_6078_sgn__if,axiom,
( sgn_sgn_rat
= ( ^ [X3: rat] : ( if_rat @ ( X3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% sgn_if
thf(fact_6079_zsgn__def,axiom,
( sgn_sgn_int
= ( ^ [I3: int] : ( if_int @ ( I3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zsgn_def
thf(fact_6080_fact__div__fact__le__pow,axiom,
! [R4: nat,N2: nat] :
( ( ord_less_eq_nat @ R4 @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ R4 ) ) ) @ ( power_power_nat @ N2 @ R4 ) ) ) ).
% fact_div_fact_le_pow
thf(fact_6081_binomial__fact__lemma,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) )
= ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% binomial_fact_lemma
thf(fact_6082_norm__sgn,axiom,
! [X: real] :
( ( ( X = zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
= zero_zero_real ) )
& ( ( X != zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
= one_one_real ) ) ) ).
% norm_sgn
thf(fact_6083_norm__sgn,axiom,
! [X: complex] :
( ( ( X = zero_zero_complex )
=> ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
= zero_zero_real ) )
& ( ( X != zero_zero_complex )
=> ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
= one_one_real ) ) ) ).
% norm_sgn
thf(fact_6084_bit__imp__take__bit__positive,axiom,
! [N2: nat,M: nat,K: int] :
( ( ord_less_nat @ N2 @ M )
=> ( ( bit_se1146084159140164899it_int @ K @ N2 )
=> ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_6085_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L: int,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N2 )
= ( ( ( ord_less_nat @ N2 @ M )
& ( bit_se1146084159140164899it_int @ K @ N2 ) )
| ( ( ord_less_eq_nat @ M @ N2 )
& ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_6086_choose__dvd,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri3624122377584611663nteger @ N2 ) ) ) ).
% choose_dvd
thf(fact_6087_choose__dvd,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% choose_dvd
thf(fact_6088_choose__dvd,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% choose_dvd
thf(fact_6089_choose__dvd,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% choose_dvd
thf(fact_6090_choose__dvd,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% choose_dvd
thf(fact_6091_zdvd__mult__cancel1,axiom,
! [M: int,N2: int] :
( ( M != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ M @ N2 ) @ M )
= ( ( abs_abs_int @ N2 )
= one_one_int ) ) ) ).
% zdvd_mult_cancel1
thf(fact_6092_fact__numeral,axiom,
! [K: num] :
( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
= ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_6093_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_6094_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_6095_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_6096_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_6097_signed__take__bit__eq__concat__bit,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N4: nat,K2: int] : ( bit_concat_bit @ N4 @ K2 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N4 ) ) ) ) ) ) ).
% signed_take_bit_eq_concat_bit
thf(fact_6098_exp__eq__0__imp__not__bit,axiom,
! [N2: nat,A: int] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
= zero_zero_int )
=> ~ ( bit_se1146084159140164899it_int @ A @ N2 ) ) ).
% exp_eq_0_imp_not_bit
thf(fact_6099_exp__eq__0__imp__not__bit,axiom,
! [N2: nat,A: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
= zero_zero_nat )
=> ~ ( bit_se1148574629649215175it_nat @ A @ N2 ) ) ).
% exp_eq_0_imp_not_bit
thf(fact_6100_bit__Suc,axiom,
! [A: int,N2: nat] :
( ( bit_se1146084159140164899it_int @ A @ ( suc @ N2 ) )
= ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N2 ) ) ).
% bit_Suc
thf(fact_6101_bit__Suc,axiom,
! [A: nat,N2: nat] :
( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N2 ) )
= ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 ) ) ).
% bit_Suc
thf(fact_6102_stable__imp__bit__iff__odd,axiom,
! [A: code_integer,N2: nat] :
( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= A )
=> ( ( bit_se9216721137139052372nteger @ A @ N2 )
= ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% stable_imp_bit_iff_odd
thf(fact_6103_stable__imp__bit__iff__odd,axiom,
! [A: int,N2: nat] :
( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A )
=> ( ( bit_se1146084159140164899it_int @ A @ N2 )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% stable_imp_bit_iff_odd
thf(fact_6104_stable__imp__bit__iff__odd,axiom,
! [A: nat,N2: nat] :
( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A )
=> ( ( bit_se1148574629649215175it_nat @ A @ N2 )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% stable_imp_bit_iff_odd
thf(fact_6105_bit__iff__idd__imp__stable,axiom,
! [A: code_integer] :
( ! [N: nat] :
( ( bit_se9216721137139052372nteger @ A @ N )
= ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
=> ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= A ) ) ).
% bit_iff_idd_imp_stable
thf(fact_6106_bit__iff__idd__imp__stable,axiom,
! [A: int] :
( ! [N: nat] :
( ( bit_se1146084159140164899it_int @ A @ N )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
=> ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A ) ) ).
% bit_iff_idd_imp_stable
thf(fact_6107_bit__iff__idd__imp__stable,axiom,
! [A: nat] :
( ! [N: nat] :
( ( bit_se1148574629649215175it_nat @ A @ N )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
=> ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A ) ) ).
% bit_iff_idd_imp_stable
thf(fact_6108_take__bit__eq__mask__iff,axiom,
! [N2: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N2 @ K )
= ( bit_se2000444600071755411sk_int @ N2 ) )
= ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ K @ one_one_int ) )
= zero_zero_int ) ) ).
% take_bit_eq_mask_iff
thf(fact_6109_int__bit__bound,axiom,
! [K: int] :
~ ! [N: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ N @ M3 )
=> ( ( bit_se1146084159140164899it_int @ K @ M3 )
= ( bit_se1146084159140164899it_int @ K @ N ) ) )
=> ~ ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ one_one_nat ) )
= ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ) ) ).
% int_bit_bound
thf(fact_6110_binomial__altdef__nat,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( binomial @ N2 @ K )
= ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_6111_even__abs__add__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% even_abs_add_iff
thf(fact_6112_even__add__abs__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L ) ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% even_add_abs_iff
thf(fact_6113_bit__iff__odd,axiom,
( bit_se9216721137139052372nteger
= ( ^ [A3: code_integer,N4: nat] :
~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% bit_iff_odd
thf(fact_6114_bit__iff__odd,axiom,
( bit_se1146084159140164899it_int
= ( ^ [A3: int,N4: nat] :
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% bit_iff_odd
thf(fact_6115_bit__iff__odd,axiom,
( bit_se1148574629649215175it_nat
= ( ^ [A3: nat,N4: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% bit_iff_odd
thf(fact_6116_square__fact__le__2__fact,axiom,
! [N2: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% square_fact_le_2_fact
thf(fact_6117_Suc__mask__eq__exp,axiom,
! [N2: nat] :
( ( suc @ ( bit_se2002935070580805687sk_nat @ N2 ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% Suc_mask_eq_exp
thf(fact_6118_mask__nat__less__exp,axiom,
! [N2: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% mask_nat_less_exp
thf(fact_6119_bit__int__def,axiom,
( bit_se1146084159140164899it_int
= ( ^ [K2: int,N4: nat] :
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% bit_int_def
thf(fact_6120_nat__intermed__int__val,axiom,
! [M: nat,N2: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_nat @ I2 @ N2 ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_eq_nat @ I2 @ N2 )
& ( ( F @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_6121_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_6122_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_6123_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_6124_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_6125_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_6126_fact__reduce,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( semiri1406184849735516958ct_int @ N2 )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_6127_fact__reduce,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( semiri5044797733671781792omplex @ N2 )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_6128_fact__reduce,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( semiri773545260158071498ct_rat @ N2 )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_6129_fact__reduce,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( semiri1408675320244567234ct_nat @ N2 )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_6130_fact__reduce,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( semiri2265585572941072030t_real @ N2 )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_6131_incr__lemma,axiom,
! [D: int,Z: int,X: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% incr_lemma
thf(fact_6132_decr__lemma,axiom,
! [D: int,X: int,Z: 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 @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% decr_lemma
thf(fact_6133_pochhammer__same,axiom,
! [N2: nat] :
( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ N2 )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( semiri3624122377584611663nteger @ N2 ) ) ) ).
% pochhammer_same
thf(fact_6134_pochhammer__same,axiom,
! [N2: nat] :
( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ N2 )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% pochhammer_same
thf(fact_6135_pochhammer__same,axiom,
! [N2: nat] :
( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ N2 )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).
% pochhammer_same
thf(fact_6136_pochhammer__same,axiom,
! [N2: nat] :
( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ N2 )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% pochhammer_same
thf(fact_6137_pochhammer__same,axiom,
! [N2: nat] :
( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ N2 )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% pochhammer_same
thf(fact_6138_fact__binomial,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K ) ) )
= ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).
% fact_binomial
thf(fact_6139_fact__binomial,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) ) )
= ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).
% fact_binomial
thf(fact_6140_fact__binomial,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) ) )
= ( divide_divide_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).
% fact_binomial
thf(fact_6141_binomial__fact,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K ) )
= ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N2 ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).
% binomial_fact
thf(fact_6142_binomial__fact,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) )
= ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).
% binomial_fact
thf(fact_6143_binomial__fact,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) )
= ( divide_divide_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).
% binomial_fact
thf(fact_6144_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [N2: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% semiring_bit_operations_class.even_mask_iff
thf(fact_6145_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% semiring_bit_operations_class.even_mask_iff
thf(fact_6146_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [N2: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% semiring_bit_operations_class.even_mask_iff
thf(fact_6147_even__bit__succ__iff,axiom,
! [A: code_integer,N2: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N2 )
= ( ( bit_se9216721137139052372nteger @ A @ N2 )
| ( N2 = zero_zero_nat ) ) ) ) ).
% even_bit_succ_iff
thf(fact_6148_even__bit__succ__iff,axiom,
! [A: int,N2: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N2 )
= ( ( bit_se1146084159140164899it_int @ A @ N2 )
| ( N2 = zero_zero_nat ) ) ) ) ).
% even_bit_succ_iff
thf(fact_6149_even__bit__succ__iff,axiom,
! [A: nat,N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N2 )
= ( ( bit_se1148574629649215175it_nat @ A @ N2 )
| ( N2 = zero_zero_nat ) ) ) ) ).
% even_bit_succ_iff
thf(fact_6150_odd__bit__iff__bit__pred,axiom,
! [A: code_integer,N2: nat] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se9216721137139052372nteger @ A @ N2 )
= ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A @ one_one_Code_integer ) @ N2 )
| ( N2 = zero_zero_nat ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_6151_odd__bit__iff__bit__pred,axiom,
! [A: int,N2: nat] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se1146084159140164899it_int @ A @ N2 )
= ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N2 )
| ( N2 = zero_zero_nat ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_6152_odd__bit__iff__bit__pred,axiom,
! [A: nat,N2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se1148574629649215175it_nat @ A @ N2 )
= ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N2 )
| ( N2 = zero_zero_nat ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_6153_mask__nat__def,axiom,
( bit_se2002935070580805687sk_nat
= ( ^ [N4: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) ).
% mask_nat_def
thf(fact_6154_mask__half__int,axiom,
! [N2: nat] :
( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).
% mask_half_int
thf(fact_6155_mask__int__def,axiom,
( bit_se2000444600071755411sk_int
= ( ^ [N4: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) @ one_one_int ) ) ) ).
% mask_int_def
thf(fact_6156_nat__ivt__aux,axiom,
! [N2: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_6157_bit__sum__mult__2__cases,axiom,
! [A: code_integer,B: code_integer,N2: nat] :
( ! [J2: nat] :
~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J2 ) )
=> ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ N2 )
= ( ( ( N2 = zero_zero_nat )
=> ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
& ( ( N2 != zero_zero_nat )
=> ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_6158_bit__sum__mult__2__cases,axiom,
! [A: int,B: int,N2: nat] :
( ! [J2: nat] :
~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J2 ) )
=> ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ N2 )
= ( ( ( N2 = zero_zero_nat )
=> ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
& ( ( N2 != zero_zero_nat )
=> ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_6159_bit__sum__mult__2__cases,axiom,
! [A: nat,B: nat,N2: nat] :
( ! [J2: nat] :
~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J2 ) )
=> ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ N2 )
= ( ( ( N2 = zero_zero_nat )
=> ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
& ( ( N2 != zero_zero_nat )
=> ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_6160_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_6161_bit__rec,axiom,
( bit_se9216721137139052372nteger
= ( ^ [A3: code_integer,N4: nat] :
( ( ( N4 = zero_zero_nat )
=> ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) )
& ( ( N4 != zero_zero_nat )
=> ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% bit_rec
thf(fact_6162_bit__rec,axiom,
( bit_se1146084159140164899it_int
= ( ^ [A3: int,N4: nat] :
( ( ( N4 = zero_zero_nat )
=> ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
& ( ( N4 != zero_zero_nat )
=> ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% bit_rec
thf(fact_6163_bit__rec,axiom,
( bit_se1148574629649215175it_nat
= ( ^ [A3: nat,N4: nat] :
( ( ( N4 = zero_zero_nat )
=> ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) )
& ( ( N4 != zero_zero_nat )
=> ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% bit_rec
thf(fact_6164_mask__eq__exp__minus__1,axiom,
( bit_se2002935070580805687sk_nat
= ( ^ [N4: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_6165_mask__eq__exp__minus__1,axiom,
( bit_se2000444600071755411sk_int
= ( ^ [N4: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) @ one_one_int ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_6166_nat0__intermed__int__val,axiom,
! [N2: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_6167_set__bit__eq,axiom,
( bit_se7879613467334960850it_int
= ( ^ [N4: nat,K2: int] :
( plus_plus_int @ K2
@ ( times_times_int
@ ( zero_n2684676970156552555ol_int
@ ~ ( bit_se1146084159140164899it_int @ K2 @ N4 ) )
@ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% set_bit_eq
thf(fact_6168_unset__bit__eq,axiom,
( bit_se4203085406695923979it_int
= ( ^ [N4: nat,K2: int] : ( minus_minus_int @ K2 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N4 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% unset_bit_eq
thf(fact_6169_take__bit__Suc__from__most,axiom,
! [N2: nat,K: int] :
( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K )
= ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% take_bit_Suc_from_most
thf(fact_6170_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N2: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N2 @ K )
= ( bit_se2000444600071755411sk_int @ N2 ) )
= ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_6171_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N4 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_6172_binomial__code,axiom,
( binomial
= ( ^ [N4: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ N4 @ K2 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( binomial @ N4 @ ( minus_minus_nat @ N4 @ K2 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N4 @ K2 ) @ one_one_nat ) @ N4 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ) ) ).
% binomial_code
thf(fact_6173_cos__coeff__def,axiom,
( cos_coeff
= ( ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N4 ) ) @ zero_zero_real ) ) ) ).
% cos_coeff_def
thf(fact_6174_fact__code,axiom,
( semiri1406184849735516958ct_int
= ( ^ [N4: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_6175_fact__code,axiom,
( semiri5044797733671781792omplex
= ( ^ [N4: nat] : ( semiri8010041392384452111omplex @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_6176_fact__code,axiom,
( semiri773545260158071498ct_rat
= ( ^ [N4: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_6177_fact__code,axiom,
( semiri1408675320244567234ct_nat
= ( ^ [N4: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_6178_fact__code,axiom,
( semiri2265585572941072030t_real
= ( ^ [N4: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_6179_tanh__real__altdef,axiom,
( tanh_real
= ( ^ [X3: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ) ) ).
% tanh_real_altdef
thf(fact_6180_and__int__unfold,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K2: int,L2: int] :
( if_int
@ ( ( K2 = zero_zero_int )
| ( L2 = zero_zero_int ) )
@ zero_zero_int
@ ( if_int
@ ( K2
= ( uminus_uminus_int @ one_one_int ) )
@ L2
@ ( if_int
@ ( L2
= ( uminus_uminus_int @ one_one_int ) )
@ K2
@ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_6181_and_Oidem,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ A @ A )
= A ) ).
% and.idem
thf(fact_6182_and_Oidem,axiom,
! [A: nat] :
( ( bit_se727722235901077358nd_nat @ A @ A )
= A ) ).
% and.idem
thf(fact_6183_and_Oleft__idem,axiom,
! [A: int,B: int] :
( ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ A @ B ) )
= ( bit_se725231765392027082nd_int @ A @ B ) ) ).
% and.left_idem
thf(fact_6184_and_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ A @ B ) )
= ( bit_se727722235901077358nd_nat @ A @ B ) ) ).
% and.left_idem
thf(fact_6185_and_Oright__idem,axiom,
! [A: int,B: int] :
( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ B )
= ( bit_se725231765392027082nd_int @ A @ B ) ) ).
% and.right_idem
thf(fact_6186_and_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ B )
= ( bit_se727722235901077358nd_nat @ A @ B ) ) ).
% and.right_idem
thf(fact_6187_and__zero__eq,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% and_zero_eq
thf(fact_6188_and__zero__eq,axiom,
! [A: nat] :
( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% and_zero_eq
thf(fact_6189_zero__and__eq,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% zero_and_eq
thf(fact_6190_zero__and__eq,axiom,
! [A: nat] :
( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_and_eq
thf(fact_6191_bit_Oconj__zero__left,axiom,
! [X: int] :
( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
= zero_zero_int ) ).
% bit.conj_zero_left
thf(fact_6192_bit_Oconj__zero__right,axiom,
! [X: int] :
( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
= zero_zero_int ) ).
% bit.conj_zero_right
thf(fact_6193_exp__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% exp_less_cancel_iff
thf(fact_6194_exp__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% exp_less_mono
thf(fact_6195_take__bit__and,axiom,
! [N2: nat,A: int,B: int] :
( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se725231765392027082nd_int @ A @ B ) )
= ( bit_se725231765392027082nd_int @ ( bit_se2923211474154528505it_int @ N2 @ A ) @ ( bit_se2923211474154528505it_int @ N2 @ B ) ) ) ).
% take_bit_and
thf(fact_6196_take__bit__and,axiom,
! [N2: nat,A: nat,B: nat] :
( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se727722235901077358nd_nat @ A @ B ) )
= ( bit_se727722235901077358nd_nat @ ( bit_se2925701944663578781it_nat @ N2 @ A ) @ ( bit_se2925701944663578781it_nat @ N2 @ B ) ) ) ).
% take_bit_and
thf(fact_6197_exp__zero,axiom,
( ( exp_complex @ zero_zero_complex )
= one_one_complex ) ).
% exp_zero
thf(fact_6198_exp__zero,axiom,
( ( exp_real @ zero_zero_real )
= one_one_real ) ).
% exp_zero
thf(fact_6199_bit_Oconj__one__right,axiom,
! [X: code_integer] :
( ( bit_se3949692690581998587nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= X ) ).
% bit.conj_one_right
thf(fact_6200_bit_Oconj__one__right,axiom,
! [X: int] :
( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
= X ) ).
% bit.conj_one_right
thf(fact_6201_and_Oright__neutral,axiom,
! [A: code_integer] :
( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= A ) ).
% and.right_neutral
thf(fact_6202_and_Oright__neutral,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= A ) ).
% and.right_neutral
thf(fact_6203_and_Oleft__neutral,axiom,
! [A: code_integer] :
( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
= A ) ).
% and.left_neutral
thf(fact_6204_and_Oleft__neutral,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
= A ) ).
% and.left_neutral
thf(fact_6205_exp__eq__one__iff,axiom,
! [X: real] :
( ( ( exp_real @ X )
= one_one_real )
= ( X = zero_zero_real ) ) ).
% exp_eq_one_iff
thf(fact_6206_and__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
= ( ( ord_less_eq_int @ zero_zero_int @ K )
| ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% and_nonnegative_int_iff
thf(fact_6207_and__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
= ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% and_negative_int_iff
thf(fact_6208_sin__coeff__0,axiom,
( ( sin_coeff @ zero_zero_nat )
= zero_zero_real ) ).
% sin_coeff_0
thf(fact_6209_cos__coeff__0,axiom,
( ( cos_coeff @ zero_zero_nat )
= one_one_real ) ).
% cos_coeff_0
thf(fact_6210_and__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
= one_one_int ) ).
% and_numerals(2)
thf(fact_6211_and__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= one_one_nat ) ).
% and_numerals(2)
thf(fact_6212_and__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
= one_one_int ) ).
% and_numerals(8)
thf(fact_6213_and__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
= one_one_nat ) ).
% and_numerals(8)
thf(fact_6214_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_6215_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_6216_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_6217_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_6218_exp__ln,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( exp_real @ ( ln_ln_real @ X ) )
= X ) ) ).
% exp_ln
thf(fact_6219_exp__ln__iff,axiom,
! [X: real] :
( ( ( exp_real @ ( ln_ln_real @ X ) )
= X )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% exp_ln_iff
thf(fact_6220_and__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
= zero_zero_int ) ).
% and_numerals(5)
thf(fact_6221_and__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
= zero_zero_nat ) ).
% and_numerals(5)
thf(fact_6222_and__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
= zero_zero_int ) ).
% and_numerals(1)
thf(fact_6223_and__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= zero_zero_nat ) ).
% and_numerals(1)
thf(fact_6224_and__numerals_I3_J,axiom,
! [X: num,Y: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% and_numerals(3)
thf(fact_6225_and__numerals_I3_J,axiom,
! [X: num,Y: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% and_numerals(3)
thf(fact_6226_and__minus__numerals_I2_J,axiom,
! [N2: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
= one_one_int ) ).
% and_minus_numerals(2)
thf(fact_6227_and__minus__numerals_I6_J,axiom,
! [N2: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ one_one_int )
= one_one_int ) ).
% and_minus_numerals(6)
thf(fact_6228_and__numerals_I6_J,axiom,
! [X: num,Y: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% and_numerals(6)
thf(fact_6229_and__numerals_I6_J,axiom,
! [X: num,Y: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% and_numerals(6)
thf(fact_6230_and__numerals_I4_J,axiom,
! [X: num,Y: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% and_numerals(4)
thf(fact_6231_and__numerals_I4_J,axiom,
! [X: num,Y: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% and_numerals(4)
thf(fact_6232_and__minus__numerals_I5_J,axiom,
! [N2: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ one_one_int )
= zero_zero_int ) ).
% and_minus_numerals(5)
thf(fact_6233_and__minus__numerals_I1_J,axiom,
! [N2: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
= zero_zero_int ) ).
% and_minus_numerals(1)
thf(fact_6234_and__numerals_I7_J,axiom,
! [X: num,Y: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% and_numerals(7)
thf(fact_6235_and__numerals_I7_J,axiom,
! [X: num,Y: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% and_numerals(7)
thf(fact_6236_of__nat__and__eq,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se727722235901077358nd_nat @ M @ N2 ) )
= ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_and_eq
thf(fact_6237_of__nat__and__eq,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se727722235901077358nd_nat @ M @ N2 ) )
= ( bit_se727722235901077358nd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_and_eq
thf(fact_6238_and_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ C )
= ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).
% and.assoc
thf(fact_6239_and_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ C )
= ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).
% and.assoc
thf(fact_6240_and_Ocommute,axiom,
( bit_se725231765392027082nd_int
= ( ^ [A3: int,B2: int] : ( bit_se725231765392027082nd_int @ B2 @ A3 ) ) ) ).
% and.commute
thf(fact_6241_and_Ocommute,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [A3: nat,B2: nat] : ( bit_se727722235901077358nd_nat @ B2 @ A3 ) ) ) ).
% and.commute
thf(fact_6242_and_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( bit_se725231765392027082nd_int @ B @ ( bit_se725231765392027082nd_int @ A @ C ) )
= ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).
% and.left_commute
thf(fact_6243_and_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( bit_se727722235901077358nd_nat @ B @ ( bit_se727722235901077358nd_nat @ A @ C ) )
= ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).
% and.left_commute
thf(fact_6244_exp__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
% exp_less_cancel
thf(fact_6245_bit__and__iff,axiom,
! [A: int,B: int,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ N2 )
= ( ( bit_se1146084159140164899it_int @ A @ N2 )
& ( bit_se1146084159140164899it_int @ B @ N2 ) ) ) ).
% bit_and_iff
thf(fact_6246_bit__and__iff,axiom,
! [A: nat,B: nat,N2: nat] :
( ( bit_se1148574629649215175it_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ N2 )
= ( ( bit_se1148574629649215175it_nat @ A @ N2 )
& ( bit_se1148574629649215175it_nat @ B @ N2 ) ) ) ).
% bit_and_iff
thf(fact_6247_exp__not__eq__zero,axiom,
! [X: complex] :
( ( exp_complex @ X )
!= zero_zero_complex ) ).
% exp_not_eq_zero
thf(fact_6248_exp__not__eq__zero,axiom,
! [X: real] :
( ( exp_real @ X )
!= zero_zero_real ) ).
% exp_not_eq_zero
thf(fact_6249_bit__and__int__iff,axiom,
! [K: int,L: int,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ N2 )
= ( ( bit_se1146084159140164899it_int @ K @ N2 )
& ( bit_se1146084159140164899it_int @ L @ N2 ) ) ) ).
% bit_and_int_iff
thf(fact_6250_and__eq__minus__1__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ( bit_se3949692690581998587nteger @ A @ B )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( ( A
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
& ( B
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_6251_and__eq__minus__1__iff,axiom,
! [A: int,B: int] :
( ( ( bit_se725231765392027082nd_int @ A @ B )
= ( uminus_uminus_int @ one_one_int ) )
= ( ( A
= ( uminus_uminus_int @ one_one_int ) )
& ( B
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_6252_bit__Suc__0__iff,axiom,
! [N2: nat] :
( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( N2 = zero_zero_nat ) ) ).
% bit_Suc_0_iff
thf(fact_6253_not__bit__Suc__0__Suc,axiom,
! [N2: nat] :
~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N2 ) ) ).
% not_bit_Suc_0_Suc
thf(fact_6254_not__exp__less__zero,axiom,
! [X: real] :
~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% not_exp_less_zero
thf(fact_6255_exp__gt__zero,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% exp_gt_zero
thf(fact_6256_exp__total,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ? [X2: real] :
( ( exp_real @ X2 )
= Y ) ) ).
% exp_total
thf(fact_6257_AND__upper2_H,axiom,
! [Y: int,Z: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).
% AND_upper2'
thf(fact_6258_AND__upper1_H,axiom,
! [Y: int,Z: int,Ya: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).
% AND_upper1'
thf(fact_6259_AND__upper2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Y ) ) ).
% AND_upper2
thf(fact_6260_AND__upper1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ X ) ) ).
% AND_upper1
thf(fact_6261_AND__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y ) ) ) ).
% AND_lower
thf(fact_6262_real__sgn__eq,axiom,
( sgn_sgn_real
= ( ^ [X3: real] : ( divide_divide_real @ X3 @ ( abs_abs_real @ X3 ) ) ) ) ).
% real_sgn_eq
thf(fact_6263_exp__add__commuting,axiom,
! [X: complex,Y: complex] :
( ( ( times_times_complex @ X @ Y )
= ( times_times_complex @ Y @ X ) )
=> ( ( exp_complex @ ( plus_plus_complex @ X @ Y ) )
= ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ) ).
% exp_add_commuting
thf(fact_6264_exp__add__commuting,axiom,
! [X: real,Y: real] :
( ( ( times_times_real @ X @ Y )
= ( times_times_real @ Y @ X ) )
=> ( ( exp_real @ ( plus_plus_real @ X @ Y ) )
= ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ) ).
% exp_add_commuting
thf(fact_6265_mult__exp__exp,axiom,
! [X: complex,Y: complex] :
( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) )
= ( exp_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% mult_exp_exp
thf(fact_6266_mult__exp__exp,axiom,
! [X: real,Y: real] :
( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
= ( exp_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% mult_exp_exp
thf(fact_6267_exp__diff,axiom,
! [X: complex,Y: complex] :
( ( exp_complex @ ( minus_minus_complex @ X @ Y ) )
= ( divide1717551699836669952omplex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ).
% exp_diff
thf(fact_6268_exp__diff,axiom,
! [X: real,Y: real] :
( ( exp_real @ ( minus_minus_real @ X @ Y ) )
= ( divide_divide_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% exp_diff
thf(fact_6269_take__bit__eq__mask,axiom,
( bit_se2923211474154528505it_int
= ( ^ [N4: nat,A3: int] : ( bit_se725231765392027082nd_int @ A3 @ ( bit_se2000444600071755411sk_int @ N4 ) ) ) ) ).
% take_bit_eq_mask
thf(fact_6270_take__bit__eq__mask,axiom,
( bit_se2925701944663578781it_nat
= ( ^ [N4: nat,A3: nat] : ( bit_se727722235901077358nd_nat @ A3 @ ( bit_se2002935070580805687sk_nat @ N4 ) ) ) ) ).
% take_bit_eq_mask
thf(fact_6271_sin__coeff__Suc,axiom,
! [N2: nat] :
( ( sin_coeff @ ( suc @ N2 ) )
= ( divide_divide_real @ ( cos_coeff @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).
% sin_coeff_Suc
thf(fact_6272_not__bit__Suc__0__numeral,axiom,
! [N2: num] :
~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N2 ) ) ).
% not_bit_Suc_0_numeral
thf(fact_6273_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_6274_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_6275_AND__upper2_H_H,axiom,
! [Y: int,Z: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).
% AND_upper2''
thf(fact_6276_AND__upper1_H_H,axiom,
! [Y: int,Z: int,Ya: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).
% AND_upper1''
thf(fact_6277_and__less__eq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% and_less_eq
thf(fact_6278_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_6279_exp__minus__inverse,axiom,
! [X: complex] :
( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
= one_one_complex ) ).
% exp_minus_inverse
thf(fact_6280_exp__of__nat__mult,axiom,
! [N2: nat,X: real] :
( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) )
= ( power_power_real @ ( exp_real @ X ) @ N2 ) ) ).
% exp_of_nat_mult
thf(fact_6281_exp__of__nat__mult,axiom,
! [N2: nat,X: complex] :
( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ X ) )
= ( power_power_complex @ ( exp_complex @ X ) @ N2 ) ) ).
% exp_of_nat_mult
thf(fact_6282_exp__of__nat2__mult,axiom,
! [X: real,N2: nat] :
( ( exp_real @ ( times_times_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) )
= ( power_power_real @ ( exp_real @ X ) @ N2 ) ) ).
% exp_of_nat2_mult
thf(fact_6283_exp__of__nat2__mult,axiom,
! [X: complex,N2: nat] :
( ( exp_complex @ ( times_times_complex @ X @ ( semiri8010041392384452111omplex @ N2 ) ) )
= ( power_power_complex @ ( exp_complex @ X ) @ N2 ) ) ).
% exp_of_nat2_mult
thf(fact_6284_cos__coeff__Suc,axiom,
! [N2: nat] :
( ( cos_coeff @ ( suc @ N2 ) )
= ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N2 ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).
% cos_coeff_Suc
thf(fact_6285_even__and__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ A @ B ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_and_iff
thf(fact_6286_even__and__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_and_iff
thf(fact_6287_even__and__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_and_iff
thf(fact_6288_sgn__real__def,axiom,
( sgn_sgn_real
= ( ^ [A3: real] : ( if_real @ ( A3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% sgn_real_def
thf(fact_6289_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_6290_even__and__iff__int,axiom,
! [K: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).
% even_and_iff_int
thf(fact_6291_lemma__exp__total,axiom,
! [Y: real] :
( ( ord_less_eq_real @ one_one_real @ Y )
=> ? [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_eq_real @ X2 @ ( minus_minus_real @ Y @ one_one_real ) )
& ( ( exp_real @ X2 )
= Y ) ) ) ).
% lemma_exp_total
thf(fact_6292_ln__ge__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X ) )
= ( ord_less_eq_real @ ( exp_real @ Y ) @ X ) ) ) ).
% ln_ge_iff
thf(fact_6293_ln__x__over__x__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).
% ln_x_over_x_mono
thf(fact_6294_and__one__eq,axiom,
! [A: code_integer] :
( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
= ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% and_one_eq
thf(fact_6295_and__one__eq,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ A @ one_one_int )
= ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% and_one_eq
thf(fact_6296_and__one__eq,axiom,
! [A: nat] :
( ( bit_se727722235901077358nd_nat @ A @ one_one_nat )
= ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% and_one_eq
thf(fact_6297_one__and__eq,axiom,
! [A: code_integer] :
( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
= ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% one_and_eq
thf(fact_6298_one__and__eq,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ one_one_int @ A )
= ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% one_and_eq
thf(fact_6299_one__and__eq,axiom,
! [A: nat] :
( ( bit_se727722235901077358nd_nat @ one_one_nat @ A )
= ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% one_and_eq
thf(fact_6300_exp__le,axiom,
ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% exp_le
thf(fact_6301_sgn__power__injE,axiom,
! [A: real,N2: nat,X: real,B: real] :
( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
= X )
=> ( ( X
= ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N2 ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( A = B ) ) ) ) ).
% sgn_power_injE
thf(fact_6302_fold__atLeastAtMost__nat_Osimps,axiom,
( set_fo2584398358068434914at_nat
= ( ^ [F2: nat > nat > nat,A3: nat,B2: nat,Acc: nat] : ( if_nat @ ( ord_less_nat @ B2 @ A3 ) @ Acc @ ( set_fo2584398358068434914at_nat @ F2 @ ( plus_plus_nat @ A3 @ one_one_nat ) @ B2 @ ( F2 @ A3 @ Acc ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_6303_fold__atLeastAtMost__nat_Oelims,axiom,
! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( ord_less_nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less_nat @ Xb @ Xa )
=> ( Y
= ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_6304_exp__divide__power__eq,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
= ( exp_real @ X ) ) ) ).
% exp_divide_power_eq
thf(fact_6305_exp__divide__power__eq,axiom,
! [N2: nat,X: complex] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N2 ) ) ) @ N2 )
= ( exp_complex @ X ) ) ) ).
% exp_divide_power_eq
thf(fact_6306_tanh__altdef,axiom,
( tanh_real
= ( ^ [X3: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) @ ( plus_plus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_6307_tanh__altdef,axiom,
( tanh_complex
= ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_6308_and__exp__eq__0__iff__not__bit,axiom,
! [A: int,N2: nat] :
( ( ( bit_se725231765392027082nd_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
= zero_zero_int )
= ( ~ ( bit_se1146084159140164899it_int @ A @ N2 ) ) ) ).
% and_exp_eq_0_iff_not_bit
thf(fact_6309_and__exp__eq__0__iff__not__bit,axiom,
! [A: nat,N2: nat] :
( ( ( bit_se727722235901077358nd_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= zero_zero_nat )
= ( ~ ( bit_se1148574629649215175it_nat @ A @ N2 ) ) ) ).
% and_exp_eq_0_iff_not_bit
thf(fact_6310_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_6311_bit__nat__def,axiom,
( bit_se1148574629649215175it_nat
= ( ^ [M2: nat,N4: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% bit_nat_def
thf(fact_6312_exp__double,axiom,
! [Z: complex] :
( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
= ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% exp_double
thf(fact_6313_exp__double,axiom,
! [Z: real] :
( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
= ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% exp_double
thf(fact_6314_exp__bound__half,axiom,
! [Z: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% exp_bound_half
thf(fact_6315_exp__bound__half,axiom,
! [Z: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% exp_bound_half
thf(fact_6316_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_6317_and__int__rec,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K2: int,L2: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% and_int_rec
thf(fact_6318_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_6319_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N2: nat,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ X ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_6320_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X: real,N2: nat] :
( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_6321_exp__bound__lemma,axiom,
! [Z: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z ) ) ) ) ) ).
% exp_bound_lemma
thf(fact_6322_exp__bound__lemma,axiom,
! [Z: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ) ).
% exp_bound_lemma
thf(fact_6323_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_6324_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_6325_modulo__int__def,axiom,
( modulo_modulo_int
= ( ^ [K2: int,L2: int] :
( if_int @ ( L2 = zero_zero_int ) @ K2
@ ( if_int
@ ( ( sgn_sgn_int @ K2 )
= ( sgn_sgn_int @ L2 ) )
@ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
@ ( times_times_int @ ( sgn_sgn_int @ L2 )
@ ( minus_minus_int
@ ( times_times_int @ ( abs_abs_int @ L2 )
@ ( zero_n2684676970156552555ol_int
@ ~ ( dvd_dvd_int @ L2 @ K2 ) ) )
@ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).
% modulo_int_def
thf(fact_6326_divide__int__def,axiom,
( divide_divide_int
= ( ^ [K2: int,L2: int] :
( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
@ ( if_int
@ ( ( sgn_sgn_int @ K2 )
= ( sgn_sgn_int @ L2 ) )
@ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
@ ( uminus_uminus_int
@ ( semiri1314217659103216013at_int
@ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_int @ L2 @ K2 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_6327_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_6328_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_6329_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_6330_nat__int,axiom,
! [N2: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N2 ) )
= N2 ) ).
% nat_int
thf(fact_6331_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% nat_numeral
thf(fact_6332_log__one,axiom,
! [A: real] :
( ( log @ A @ one_one_real )
= zero_zero_real ) ).
% log_one
thf(fact_6333_nat__of__bool,axiom,
! [P: $o] :
( ( nat2 @ ( zero_n2684676970156552555ol_int @ P ) )
= ( zero_n2687167440665602831ol_nat @ P ) ) ).
% nat_of_bool
thf(fact_6334_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_6335_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_6336_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_6337_zless__nat__conj,axiom,
! [W: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W @ Z ) ) ) ).
% zless_nat_conj
thf(fact_6338_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= zero_zero_nat ) ).
% nat_neg_numeral
thf(fact_6339_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_6340_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_6341_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_6342_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_6343_log__less__cancel__iff,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_6344_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_6345_nat__zminus__int,axiom,
! [N2: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_6346_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_6347_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% zero_less_nat_eq
thf(fact_6348_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_6349_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_6350_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_6351_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_6352_log__le__cancel__iff,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_6353_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_6354_and__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= zero_zero_nat ) ).
% and_nat_numerals(1)
thf(fact_6355_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_6356_numeral__power__eq__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: int] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= ( nat2 @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_6357_nat__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N2: nat] :
( ( ( nat2 @ Y )
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_6358_dvd__nat__abs__iff,axiom,
! [N2: nat,K: int] :
( ( dvd_dvd_nat @ N2 @ ( nat2 @ ( abs_abs_int @ K ) ) )
= ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ).
% dvd_nat_abs_iff
thf(fact_6359_nat__abs__dvd__iff,axiom,
! [K: int,N2: nat] :
( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N2 )
= ( dvd_dvd_int @ K @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nat_abs_dvd_iff
thf(fact_6360_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% one_less_nat_eq
thf(fact_6361_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_6362_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_6363_and__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= one_one_nat ) ).
% and_nat_numerals(2)
thf(fact_6364_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_6365_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N2: nat,A: int] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) @ ( nat2 @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_6366_nat__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N2: nat] :
( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_6367_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N2: nat,A: int] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) @ ( nat2 @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_6368_nat__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N2: nat] :
( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_6369_and__Suc__0__eq,axiom,
! [N2: nat] :
( ( bit_se727722235901077358nd_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% and_Suc_0_eq
thf(fact_6370_Suc__0__and__eq,axiom,
! [N2: nat] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Suc_0_and_eq
thf(fact_6371_and__nat__def,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M2: nat,N4: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% and_nat_def
thf(fact_6372_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_6373_nat__numeral__as__int,axiom,
( numeral_numeral_nat
= ( ^ [I3: num] : ( nat2 @ ( numeral_numeral_int @ I3 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_6374_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_6375_ex__nat,axiom,
( ( ^ [P5: nat > $o] :
? [X7: nat] : ( P5 @ X7 ) )
= ( ^ [P6: nat > $o] :
? [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
& ( P6 @ ( nat2 @ X3 ) ) ) ) ) ).
% ex_nat
thf(fact_6376_all__nat,axiom,
( ( ^ [P5: nat > $o] :
! [X7: nat] : ( P5 @ X7 ) )
= ( ^ [P6: nat > $o] :
! [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( P6 @ ( nat2 @ X3 ) ) ) ) ) ).
% all_nat
thf(fact_6377_eq__nat__nat__iff,axiom,
! [Z: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z6 ) )
= ( Z = Z6 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_6378_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_6379_log__def,axiom,
( log
= ( ^ [A3: real,X3: real] : ( divide_divide_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ A3 ) ) ) ) ).
% log_def
thf(fact_6380_pi__gt__zero,axiom,
ord_less_real @ zero_zero_real @ pi ).
% pi_gt_zero
thf(fact_6381_pi__not__less__zero,axiom,
~ ( ord_less_real @ pi @ zero_zero_real ) ).
% pi_not_less_zero
thf(fact_6382_unset__bit__nat__def,axiom,
( bit_se4205575877204974255it_nat
= ( ^ [M2: nat,N4: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M2 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% unset_bit_nat_def
thf(fact_6383_nat__mask__eq,axiom,
! [N2: nat] :
( ( nat2 @ ( bit_se2000444600071755411sk_int @ N2 ) )
= ( bit_se2002935070580805687sk_nat @ N2 ) ) ).
% nat_mask_eq
thf(fact_6384_nat__mono__iff,axiom,
! [Z: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W @ Z ) ) ) ).
% nat_mono_iff
thf(fact_6385_zless__nat__eq__int__zless,axiom,
! [M: nat,Z: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_6386_nat__le__iff,axiom,
! [X: int,N2: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N2 )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nat_le_iff
thf(fact_6387_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_6388_int__eq__iff,axiom,
! [M: nat,Z: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z )
= ( ( M
= ( nat2 @ Z ) )
& ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% int_eq_iff
thf(fact_6389_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_6390_nat__abs__mult__distrib,axiom,
! [W: int,Z: int] :
( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
= ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_6391_log__ln,axiom,
( ln_ln_real
= ( log @ ( exp_real @ one_one_real ) ) ) ).
% log_ln
thf(fact_6392_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_6393_less__log__of__power,axiom,
! [B: real,N2: nat,M: real] :
( ( ord_less_real @ ( power_power_real @ B @ N2 ) @ M )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).
% less_log_of_power
thf(fact_6394_log__of__power__eq,axiom,
! [M: nat,B: real,N2: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( power_power_real @ B @ N2 ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( semiri5074537144036343181t_real @ N2 )
= ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% log_of_power_eq
thf(fact_6395_nat__less__eq__zless,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_6396_nat__le__eq__zle,axiom,
! [W: int,Z: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_6397_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat2 @ W )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_6398_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_6399_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N4: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N4 ) )
=> ( P @ N4 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_6400_le__nat__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).
% le_nat_iff
thf(fact_6401_nat__add__distrib,axiom,
! [Z: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
= ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_6402_nat__mult__distrib,axiom,
! [Z: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
= ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).
% nat_mult_distrib
thf(fact_6403_Suc__as__int,axiom,
( suc
= ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_6404_nat__diff__distrib,axiom,
! [Z6: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( ord_less_eq_int @ Z6 @ Z )
=> ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
= ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_6405_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_6406_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_6407_nat__div__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib
thf(fact_6408_nat__div__distrib_H,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib'
thf(fact_6409_nat__power__eq,axiom,
! [Z: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( nat2 @ ( power_power_int @ Z @ N2 ) )
= ( power_power_nat @ ( nat2 @ Z ) @ N2 ) ) ) ).
% nat_power_eq
thf(fact_6410_nat__mod__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( modulo_modulo_int @ X @ Y ) )
= ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_6411_pi__less__4,axiom,
ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% pi_less_4
thf(fact_6412_pi__ge__two,axiom,
ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% pi_ge_two
thf(fact_6413_div__abs__eq__div__nat,axiom,
! [K: int,L: int] :
( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% div_abs_eq_div_nat
thf(fact_6414_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_6415_mod__abs__eq__div__nat,axiom,
! [K: int,L: int] :
( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
= ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% mod_abs_eq_div_nat
thf(fact_6416_nat__take__bit__eq,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
= ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K ) ) ) ) ).
% nat_take_bit_eq
thf(fact_6417_take__bit__nat__eq,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ) ).
% take_bit_nat_eq
thf(fact_6418_bit__nat__iff,axiom,
! [K: int,N2: nat] :
( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N2 )
= ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).
% bit_nat_iff
thf(fact_6419_nat__2,axiom,
( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% nat_2
thf(fact_6420_log__mult,axiom,
! [A: real,X: real,Y: 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 @ Y )
=> ( ( log @ A @ ( times_times_real @ X @ Y ) )
= ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% log_mult
thf(fact_6421_log__divide,axiom,
! [A: real,X: real,Y: 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 @ Y )
=> ( ( log @ A @ ( divide_divide_real @ X @ Y ) )
= ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% log_divide
thf(fact_6422_le__log__of__power,axiom,
! [B: real,N2: nat,M: real] :
( ( ord_less_eq_real @ ( power_power_real @ B @ N2 ) @ M )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).
% le_log_of_power
thf(fact_6423_log__base__pow,axiom,
! [A: real,N2: nat,X: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( log @ ( power_power_real @ A @ N2 ) @ X )
= ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% log_base_pow
thf(fact_6424_log__nat__power,axiom,
! [X: real,B: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ B @ ( power_power_real @ X @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X ) ) ) ) ).
% log_nat_power
thf(fact_6425_Suc__nat__eq__nat__zadd1,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( suc @ ( nat2 @ Z ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_6426_nat__less__iff,axiom,
! [W: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_6427_nat__mult__distrib__neg,axiom,
! [Z: int,Z6: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_6428_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_6429_pi__half__neq__zero,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
!= zero_zero_real ) ).
% pi_half_neq_zero
thf(fact_6430_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_6431_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_6432_log2__of__power__eq,axiom,
! [M: nat,N2: nat] :
( ( M
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
=> ( ( semiri5074537144036343181t_real @ N2 )
= ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% log2_of_power_eq
thf(fact_6433_log__of__power__less,axiom,
! [M: nat,B: real,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% log_of_power_less
thf(fact_6434_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_6435_nat__dvd__iff,axiom,
! [Z: int,M: nat] :
( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_dvd_iff
thf(fact_6436_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_6437_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_6438_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_6439_arctan__ubound,axiom,
! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% arctan_ubound
thf(fact_6440_arctan__one,axiom,
( ( arctan @ one_one_real )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% arctan_one
thf(fact_6441_log__of__power__le,axiom,
! [M: nat,B: real,N2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
=> ( ( 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 @ N2 ) ) ) ) ) ).
% log_of_power_le
thf(fact_6442_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_6443_arctan__lbound,axiom,
! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).
% arctan_lbound
thf(fact_6444_arctan__bounded,axiom,
! [Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% arctan_bounded
thf(fact_6445_and__nat__unfold,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M2: nat,N4: nat] :
( if_nat
@ ( ( M2 = zero_zero_nat )
| ( N4 = zero_zero_nat ) )
@ zero_zero_nat
@ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( 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 @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_6446_less__log2__of__power,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% less_log2_of_power
thf(fact_6447_le__log2__of__power,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% le_log2_of_power
thf(fact_6448_and__nat__rec,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M2: nat,N4: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_6449_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_6450_log2__of__power__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% log2_of_power_less
thf(fact_6451_log2__of__power__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% log2_of_power_le
thf(fact_6452_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_6453_sin__cos__npi,axiom,
! [N2: nat] :
( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).
% sin_cos_npi
thf(fact_6454_ceiling__log__nat__eq__powr__iff,axiom,
! [B: nat,K: nat,N2: 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 @ N2 ) @ one_one_int ) )
= ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_6455_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_6456_ceiling__log__nat__eq__if,axiom,
! [B: nat,N2: nat,K: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
=> ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ 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 @ N2 ) @ one_one_int ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_6457_ceiling__log2__div2,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
= ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% ceiling_log2_div2
thf(fact_6458_floor__log__nat__eq__powr__iff,axiom,
! [B: nat,K: nat,N2: 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 @ N2 ) )
= ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_6459_sin__zero,axiom,
( ( sin_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% sin_zero
thf(fact_6460_sin__zero,axiom,
( ( sin_real @ zero_zero_real )
= zero_zero_real ) ).
% sin_zero
thf(fact_6461_cos__zero,axiom,
( ( cos_complex @ zero_zero_complex )
= one_one_complex ) ).
% cos_zero
thf(fact_6462_cos__zero,axiom,
( ( cos_real @ zero_zero_real )
= one_one_real ) ).
% cos_zero
thf(fact_6463_floor__zero,axiom,
( ( archim6058952711729229775r_real @ zero_zero_real )
= zero_zero_int ) ).
% floor_zero
thf(fact_6464_floor__zero,axiom,
( ( archim3151403230148437115or_rat @ zero_zero_rat )
= zero_zero_int ) ).
% floor_zero
thf(fact_6465_floor__numeral,axiom,
! [V: num] :
( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_int @ V ) ) ).
% floor_numeral
thf(fact_6466_floor__numeral,axiom,
! [V: num] :
( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
= ( numeral_numeral_int @ V ) ) ).
% floor_numeral
thf(fact_6467_ceiling__zero,axiom,
( ( archim2889992004027027881ng_rat @ zero_zero_rat )
= zero_zero_int ) ).
% ceiling_zero
thf(fact_6468_ceiling__zero,axiom,
( ( archim7802044766580827645g_real @ zero_zero_real )
= zero_zero_int ) ).
% ceiling_zero
thf(fact_6469_floor__one,axiom,
( ( archim6058952711729229775r_real @ one_one_real )
= one_one_int ) ).
% floor_one
thf(fact_6470_floor__one,axiom,
( ( archim3151403230148437115or_rat @ one_one_rat )
= one_one_int ) ).
% floor_one
thf(fact_6471_ceiling__numeral,axiom,
! [V: num] :
( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_int @ V ) ) ).
% ceiling_numeral
thf(fact_6472_ceiling__numeral,axiom,
! [V: num] :
( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
= ( numeral_numeral_int @ V ) ) ).
% ceiling_numeral
thf(fact_6473_ceiling__one,axiom,
( ( archim2889992004027027881ng_rat @ one_one_rat )
= one_one_int ) ).
% ceiling_one
thf(fact_6474_ceiling__one,axiom,
( ( archim7802044766580827645g_real @ one_one_real )
= one_one_int ) ).
% ceiling_one
thf(fact_6475_cos__pi,axiom,
( ( cos_real @ pi )
= ( uminus_uminus_real @ one_one_real ) ) ).
% cos_pi
thf(fact_6476_cos__periodic__pi2,axiom,
! [X: real] :
( ( cos_real @ ( plus_plus_real @ pi @ X ) )
= ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% cos_periodic_pi2
thf(fact_6477_cos__periodic__pi,axiom,
! [X: real] :
( ( cos_real @ ( plus_plus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% cos_periodic_pi
thf(fact_6478_sin__periodic__pi2,axiom,
! [X: real] :
( ( sin_real @ ( plus_plus_real @ pi @ X ) )
= ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% sin_periodic_pi2
thf(fact_6479_sin__periodic__pi,axiom,
! [X: real] :
( ( sin_real @ ( plus_plus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% sin_periodic_pi
thf(fact_6480_zero__le__floor,axiom,
! [X: real] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% zero_le_floor
thf(fact_6481_zero__le__floor,axiom,
! [X: rat] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ zero_zero_rat @ X ) ) ).
% zero_le_floor
thf(fact_6482_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_6483_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_6484_floor__less__zero,axiom,
! [X: real] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% floor_less_zero
thf(fact_6485_floor__less__zero,axiom,
! [X: rat] :
( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
= ( ord_less_rat @ X @ zero_zero_rat ) ) ).
% floor_less_zero
thf(fact_6486_numeral__le__floor,axiom,
! [V: num,X: real] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% numeral_le_floor
thf(fact_6487_numeral__le__floor,axiom,
! [V: num,X: rat] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% numeral_le_floor
thf(fact_6488_zero__less__floor,axiom,
! [X: real] :
( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ).
% zero_less_floor
thf(fact_6489_zero__less__floor,axiom,
! [X: rat] :
( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% zero_less_floor
thf(fact_6490_floor__le__zero,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
= ( ord_less_real @ X @ one_one_real ) ) ).
% floor_le_zero
thf(fact_6491_floor__le__zero,axiom,
! [X: rat] :
( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
= ( ord_less_rat @ X @ one_one_rat ) ) ).
% floor_le_zero
thf(fact_6492_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_6493_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_6494_floor__less__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% floor_less_numeral
thf(fact_6495_floor__less__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% floor_less_numeral
thf(fact_6496_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_6497_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_6498_one__le__floor,axiom,
! [X: real] :
( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ).
% one_le_floor
thf(fact_6499_one__le__floor,axiom,
! [X: rat] :
( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% one_le_floor
thf(fact_6500_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_6501_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_6502_floor__less__one,axiom,
! [X: real] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
= ( ord_less_real @ X @ one_one_real ) ) ).
% floor_less_one
thf(fact_6503_floor__less__one,axiom,
! [X: rat] :
( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
= ( ord_less_rat @ X @ one_one_rat ) ) ).
% floor_less_one
thf(fact_6504_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_6505_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_6506_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_6507_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_6508_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_6509_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_6510_floor__neg__numeral,axiom,
! [V: num] :
( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% floor_neg_numeral
thf(fact_6511_floor__neg__numeral,axiom,
! [V: num] :
( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% floor_neg_numeral
thf(fact_6512_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_6513_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_6514_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_6515_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_6516_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_6517_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_6518_floor__diff__numeral,axiom,
! [X: real,V: num] :
( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
= ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% floor_diff_numeral
thf(fact_6519_floor__diff__numeral,axiom,
! [X: rat,V: num] :
( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
= ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% floor_diff_numeral
thf(fact_6520_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_6521_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_6522_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_6523_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_6524_floor__diff__one,axiom,
! [X: real] :
( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ one_one_real ) )
= ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ).
% floor_diff_one
thf(fact_6525_floor__diff__one,axiom,
! [X: rat] :
( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
= ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ).
% floor_diff_one
thf(fact_6526_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_6527_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_6528_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_6529_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_6530_floor__numeral__power,axiom,
! [X: num,N2: nat] :
( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ).
% floor_numeral_power
thf(fact_6531_floor__numeral__power,axiom,
! [X: num,N2: nat] :
( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ).
% floor_numeral_power
thf(fact_6532_ceiling__numeral__power,axiom,
! [X: num,N2: nat] :
( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ).
% ceiling_numeral_power
thf(fact_6533_ceiling__numeral__power,axiom,
! [X: num,N2: nat] :
( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ).
% ceiling_numeral_power
thf(fact_6534_floor__divide__eq__div__numeral,axiom,
! [A: num,B: num] :
( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).
% floor_divide_eq_div_numeral
thf(fact_6535_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_6536_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_6537_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_6538_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_6539_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_6540_ceiling__divide__eq__div__numeral,axiom,
! [A: num,B: num] :
( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
= ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% ceiling_divide_eq_div_numeral
thf(fact_6541_numeral__less__floor,axiom,
! [V: num,X: real] :
( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% numeral_less_floor
thf(fact_6542_numeral__less__floor,axiom,
! [V: num,X: rat] :
( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% numeral_less_floor
thf(fact_6543_floor__le__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_real @ X @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% floor_le_numeral
thf(fact_6544_floor__le__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% floor_le_numeral
thf(fact_6545_one__less__floor,axiom,
! [X: real] :
( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ).
% one_less_floor
thf(fact_6546_one__less__floor,axiom,
! [X: rat] :
( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) ) ).
% one_less_floor
thf(fact_6547_floor__le__one,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
= ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% floor_le_one
thf(fact_6548_floor__le__one,axiom,
! [X: rat] :
( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
= ( ord_less_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% floor_le_one
thf(fact_6549_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_6550_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_6551_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_6552_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_6553_neg__numeral__le__floor,axiom,
! [V: num,X: real] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% neg_numeral_le_floor
thf(fact_6554_neg__numeral__le__floor,axiom,
! [V: num,X: rat] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% neg_numeral_le_floor
thf(fact_6555_floor__less__neg__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_6556_floor__less__neg__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_6557_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_6558_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_6559_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_6560_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_6561_cos__pi__half,axiom,
( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= zero_zero_real ) ).
% cos_pi_half
thf(fact_6562_sin__two__pi,axiom,
( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
= zero_zero_real ) ).
% sin_two_pi
thf(fact_6563_sin__pi__half,axiom,
( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= one_one_real ) ).
% sin_pi_half
thf(fact_6564_cos__two__pi,axiom,
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
= one_one_real ) ).
% cos_two_pi
thf(fact_6565_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_6566_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_6567_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_6568_cos__npi2,axiom,
! [N2: nat] :
( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).
% cos_npi2
thf(fact_6569_cos__npi,axiom,
! [N2: nat] :
( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).
% cos_npi
thf(fact_6570_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_6571_floor__minus__divide__eq__div__numeral,axiom,
! [A: num,B: num] :
( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
= ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).
% floor_minus_divide_eq_div_numeral
thf(fact_6572_ceiling__minus__divide__eq__div__numeral,axiom,
! [A: num,B: num] :
( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
= ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% ceiling_minus_divide_eq_div_numeral
thf(fact_6573_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_6574_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_6575_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_6576_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_6577_sin__2npi,axiom,
! [N2: nat] :
( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) )
= zero_zero_real ) ).
% sin_2npi
thf(fact_6578_cos__2npi,axiom,
! [N2: nat] :
( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) )
= one_one_real ) ).
% cos_2npi
thf(fact_6579_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_6580_neg__numeral__less__floor,axiom,
! [V: num,X: real] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% neg_numeral_less_floor
thf(fact_6581_neg__numeral__less__floor,axiom,
! [V: num,X: rat] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% neg_numeral_less_floor
thf(fact_6582_floor__le__neg__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_real @ X @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% floor_le_neg_numeral
thf(fact_6583_floor__le__neg__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% floor_le_neg_numeral
thf(fact_6584_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_6585_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_6586_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_6587_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_6588_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_6589_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_6590_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_6591_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_6592_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_6593_floor__le__ceiling,axiom,
! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% floor_le_ceiling
thf(fact_6594_floor__le__ceiling,axiom,
! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim2889992004027027881ng_rat @ X ) ) ).
% floor_le_ceiling
thf(fact_6595_sin__add,axiom,
! [X: real,Y: real] :
( ( sin_real @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% sin_add
thf(fact_6596_cos__diff,axiom,
! [X: real,Y: real] :
( ( cos_real @ ( minus_minus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% cos_diff
thf(fact_6597_cos__add,axiom,
! [X: real,Y: real] :
( ( cos_real @ ( plus_plus_real @ X @ Y ) )
= ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% cos_add
thf(fact_6598_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_6599_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_6600_ceiling__diff__floor__le__1,axiom,
! [X: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim6058952711729229775r_real @ X ) ) @ one_one_int ) ).
% ceiling_diff_floor_le_1
thf(fact_6601_ceiling__diff__floor__le__1,axiom,
! [X: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) @ one_one_int ) ).
% ceiling_diff_floor_le_1
thf(fact_6602_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_6603_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_6604_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_6605_sincos__principal__value,axiom,
! [X: real] :
? [Y2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y2 )
& ( ord_less_eq_real @ Y2 @ pi )
& ( ( sin_real @ Y2 )
= ( sin_real @ X ) )
& ( ( cos_real @ Y2 )
= ( cos_real @ X ) ) ) ).
% sincos_principal_value
thf(fact_6606_floor__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ).
% floor_mono
thf(fact_6607_floor__mono,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ).
% floor_mono
thf(fact_6608_floor__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
% floor_less_cancel
thf(fact_6609_floor__less__cancel,axiom,
! [X: rat,Y: rat] :
( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) )
=> ( ord_less_rat @ X @ Y ) ) ).
% floor_less_cancel
thf(fact_6610_ceiling__mono,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% ceiling_mono
thf(fact_6611_ceiling__mono,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X ) ) ) ).
% ceiling_mono
thf(fact_6612_ceiling__less__cancel,axiom,
! [X: rat,Y: rat] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) )
=> ( ord_less_rat @ X @ Y ) ) ).
% ceiling_less_cancel
thf(fact_6613_ceiling__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
% ceiling_less_cancel
thf(fact_6614_sin__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).
% sin_le_one
thf(fact_6615_cos__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).
% cos_le_one
thf(fact_6616_sin__cos__le1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).
% sin_cos_le1
thf(fact_6617_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_6618_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_6619_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_6620_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_6621_le__floor__add,axiom,
! [X: real,Y: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% le_floor_add
thf(fact_6622_le__floor__add,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) ) ) ).
% le_floor_add
thf(fact_6623_of__nat__ceiling,axiom,
! [R4: rat] : ( ord_less_eq_rat @ R4 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R4 ) ) ) ) ).
% of_nat_ceiling
thf(fact_6624_of__nat__ceiling,axiom,
! [R4: real] : ( ord_less_eq_real @ R4 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R4 ) ) ) ) ).
% of_nat_ceiling
thf(fact_6625_sin__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ pi )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% sin_gt_zero
thf(fact_6626_ceiling__add__le,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ Y ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) ) ) ).
% ceiling_add_le
thf(fact_6627_ceiling__add__le,axiom,
! [X: real,Y: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) ) ) ).
% ceiling_add_le
thf(fact_6628_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_6629_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_6630_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_6631_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_6632_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_6633_cos__diff__cos,axiom,
! [W: complex,Z: complex] :
( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z @ W ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_6634_cos__diff__cos,axiom,
! [W: real,Z: real] :
( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z @ W ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_6635_sin__diff__sin,axiom,
! [W: complex,Z: complex] :
( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_6636_sin__diff__sin,axiom,
! [W: real,Z: real] :
( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_6637_sin__plus__sin,axiom,
! [W: complex,Z: complex] :
( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_6638_sin__plus__sin,axiom,
! [W: real,Z: real] :
( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_6639_cos__times__sin,axiom,
! [W: complex,Z: complex] :
( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% cos_times_sin
thf(fact_6640_cos__times__sin,axiom,
! [W: real,Z: real] :
( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% cos_times_sin
thf(fact_6641_sin__times__cos,axiom,
! [W: complex,Z: complex] :
( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% sin_times_cos
thf(fact_6642_sin__times__cos,axiom,
! [W: real,Z: real] :
( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% sin_times_cos
thf(fact_6643_sin__times__sin,axiom,
! [W: complex,Z: complex] :
( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% sin_times_sin
thf(fact_6644_sin__times__sin,axiom,
! [W: real,Z: real] :
( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% sin_times_sin
thf(fact_6645_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_6646_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_6647_cos__double__sin,axiom,
! [W: complex] :
( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
= ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_double_sin
thf(fact_6648_cos__double__sin,axiom,
! [W: real] :
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
= ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_double_sin
thf(fact_6649_of__nat__floor,axiom,
! [R4: real] :
( ( ord_less_eq_real @ zero_zero_real @ R4 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R4 ) ) ) @ R4 ) ) ).
% of_nat_floor
thf(fact_6650_of__nat__floor,axiom,
! [R4: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ R4 )
=> ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R4 ) ) ) @ R4 ) ) ).
% of_nat_floor
thf(fact_6651_one__add__floor,axiom,
! [X: real] :
( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
= ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).
% one_add_floor
thf(fact_6652_one__add__floor,axiom,
! [X: rat] :
( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
= ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ one_one_rat ) ) ) ).
% one_add_floor
thf(fact_6653_le__mult__nat__floor,axiom,
! [A: real,B: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ).
% le_mult_nat_floor
thf(fact_6654_le__mult__nat__floor,axiom,
! [A: rat,B: rat] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim3151403230148437115or_rat @ A ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% le_mult_nat_floor
thf(fact_6655_floor__divide__of__nat__eq,axiom,
! [M: nat,N2: nat] :
( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) ).
% floor_divide_of_nat_eq
thf(fact_6656_floor__divide__of__nat__eq,axiom,
! [M: nat,N2: nat] :
( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) ).
% floor_divide_of_nat_eq
thf(fact_6657_nat__floor__neg,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= zero_zero_nat ) ) ).
% nat_floor_neg
thf(fact_6658_cos__two__neq__zero,axiom,
( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
!= zero_zero_real ) ).
% cos_two_neq_zero
thf(fact_6659_floor__eq3,axiom,
! [N2: nat,X: real] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ X )
=> ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= N2 ) ) ) ).
% floor_eq3
thf(fact_6660_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_6661_cos__mono__less__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ pi )
=> ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
= ( ord_less_real @ Y @ X ) ) ) ) ) ) ).
% cos_mono_less_eq
thf(fact_6662_cos__monotone__0__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi
thf(fact_6663_sin__eq__0__pi,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
=> ( ( ord_less_real @ X @ pi )
=> ( ( ( sin_real @ X )
= zero_zero_real )
=> ( X = zero_zero_real ) ) ) ) ).
% sin_eq_0_pi
thf(fact_6664_sin__zero__pi__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
=> ( ( ( sin_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ) ).
% sin_zero_pi_iff
thf(fact_6665_sincos__total__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ? [T: real] :
( ( ord_less_eq_real @ zero_zero_real @ T )
& ( ord_less_eq_real @ T @ pi )
& ( X
= ( cos_real @ T ) )
& ( Y
= ( sin_real @ T ) ) ) ) ) ).
% sincos_total_pi
thf(fact_6666_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_6667_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_6668_le__mult__floor,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% le_mult_floor
thf(fact_6669_le__mult__floor,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).
% le_mult_floor
thf(fact_6670_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_6671_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_6672_mult__ceiling__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_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_6673_mult__ceiling__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_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_6674_floor__eq4,axiom,
! [N2: nat,X: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ X )
=> ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= N2 ) ) ) ).
% floor_eq4
thf(fact_6675_cos__two__less__zero,axiom,
ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% cos_two_less_zero
thf(fact_6676_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_6677_cos__is__zero,axiom,
? [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 )
& ! [Y3: real] :
( ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
& ( ord_less_eq_real @ Y3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
& ( ( cos_real @ Y3 )
= zero_zero_real ) )
=> ( Y3 = X2 ) ) ) ).
% cos_is_zero
thf(fact_6678_cos__monotone__minus__pi__0,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
=> ( ( ord_less_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0
thf(fact_6679_cos__total,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ? [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_eq_real @ X2 @ pi )
& ( ( cos_real @ X2 )
= Y )
& ! [Y3: real] :
( ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
& ( ord_less_eq_real @ Y3 @ pi )
& ( ( cos_real @ Y3 )
= Y ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% cos_total
thf(fact_6680_sincos__total__pi__half,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ? [T: real] :
( ( ord_less_eq_real @ zero_zero_real @ T )
& ( ord_less_eq_real @ T @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( X
= ( cos_real @ T ) )
& ( Y
= ( sin_real @ T ) ) ) ) ) ) ).
% sincos_total_pi_half
thf(fact_6681_sincos__total__2pi__le,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ? [T: real] :
( ( ord_less_eq_real @ zero_zero_real @ T )
& ( ord_less_eq_real @ T @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
& ( X
= ( cos_real @ T ) )
& ( Y
= ( sin_real @ T ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_6682_sincos__total__2pi,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ~ ! [T: real] :
( ( ord_less_eq_real @ zero_zero_real @ T )
=> ( ( ord_less_real @ T @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( ( X
= ( cos_real @ T ) )
=> ( Y
!= ( sin_real @ T ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_6683_sin__pi__divide__n__ge__0,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% sin_pi_divide_n_ge_0
thf(fact_6684_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_6685_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_6686_cos__plus__cos,axiom,
! [W: complex,Z: complex] :
( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_6687_cos__plus__cos,axiom,
! [W: real,Z: real] :
( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_6688_cos__times__cos,axiom,
! [W: complex,Z: complex] :
( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% cos_times_cos
thf(fact_6689_cos__times__cos,axiom,
! [W: real,Z: real] :
( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% cos_times_cos
thf(fact_6690_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_6691_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_6692_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_6693_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_6694_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_6695_sin__inj__pi,axiom,
! [X: real,Y: 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 ) ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ( sin_real @ X )
= ( sin_real @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% sin_inj_pi
thf(fact_6696_sin__mono__le__eq,axiom,
! [X: real,Y: 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 ) ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% sin_mono_le_eq
thf(fact_6697_sin__monotone__2pi__le,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% sin_monotone_2pi_le
thf(fact_6698_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_6699_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_6700_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_6701_cos__double__cos,axiom,
! [W: complex] :
( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
= ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% cos_double_cos
thf(fact_6702_cos__double__cos,axiom,
! [W: real] :
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).
% cos_double_cos
thf(fact_6703_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_6704_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_6705_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_6706_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_6707_sin__mono__less__eq,axiom,
! [X: real,Y: 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 ) ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_6708_sin__monotone__2pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_6709_sin__total,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_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 )
& ! [Y3: real] :
( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
& ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ Y3 )
= Y ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% sin_total
thf(fact_6710_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_6711_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_6712_cos__one__2pi,axiom,
! [X: real] :
( ( ( cos_real @ X )
= one_one_real )
= ( ? [X3: nat] :
( X
= ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
| ? [X3: nat] :
( X
= ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% cos_one_2pi
thf(fact_6713_sin__pi__divide__n__gt__0,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% sin_pi_divide_n_gt_0
thf(fact_6714_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_6715_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_6716_floor__log2__div2,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
= ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% floor_log2_div2
thf(fact_6717_floor__log__nat__eq__if,axiom,
! [B: nat,N2: nat,K: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
=> ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_6718_sin__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( sin_real @ X )
= zero_zero_real )
=> ? [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( X
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_6719_sin__zero__iff,axiom,
! [X: real] :
( ( ( sin_real @ X )
= zero_zero_real )
= ( ? [N4: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
& ( X
= ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
| ? [N4: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
& ( X
= ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% sin_zero_iff
thf(fact_6720_cos__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( cos_real @ X )
= zero_zero_real )
=> ? [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( X
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_6721_cos__zero__iff,axiom,
! [X: real] :
( ( ( cos_real @ X )
= zero_zero_real )
= ( ? [N4: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
& ( X
= ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
| ? [N4: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
& ( X
= ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% cos_zero_iff
thf(fact_6722_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_6723_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_6724_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_6725_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_6726_complex__unimodular__polar,axiom,
! [Z: complex] :
( ( ( real_V1022390504157884413omplex @ Z )
= one_one_real )
=> ~ ! [T: real] :
( ( ord_less_eq_real @ zero_zero_real @ T )
=> ( ( ord_less_real @ T @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( Z
!= ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_6727_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_6728_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_6729_powr__eq__0__iff,axiom,
! [W: real,Z: real] :
( ( ( powr_real @ W @ Z )
= zero_zero_real )
= ( W = zero_zero_real ) ) ).
% powr_eq_0_iff
thf(fact_6730_powr__0,axiom,
! [Z: real] :
( ( powr_real @ zero_zero_real @ Z )
= zero_zero_real ) ).
% powr_0
thf(fact_6731_powr__one__eq__one,axiom,
! [A: real] :
( ( powr_real @ one_one_real @ A )
= one_one_real ) ).
% powr_one_eq_one
thf(fact_6732_tan__zero,axiom,
( ( tan_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% tan_zero
thf(fact_6733_tan__zero,axiom,
( ( tan_real @ zero_zero_real )
= zero_zero_real ) ).
% tan_zero
thf(fact_6734_powr__zero__eq__one,axiom,
! [X: real] :
( ( ( X = zero_zero_real )
=> ( ( powr_real @ X @ zero_zero_real )
= zero_zero_real ) )
& ( ( X != zero_zero_real )
=> ( ( powr_real @ X @ zero_zero_real )
= one_one_real ) ) ) ).
% powr_zero_eq_one
thf(fact_6735_powr__gt__zero,axiom,
! [X: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
= ( X != zero_zero_real ) ) ).
% powr_gt_zero
thf(fact_6736_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_6737_tan__periodic__pi,axiom,
! [X: real] :
( ( tan_real @ ( plus_plus_real @ X @ pi ) )
= ( tan_real @ X ) ) ).
% tan_periodic_pi
thf(fact_6738_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_6739_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_6740_powr__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ one_one_real )
= X ) ) ).
% powr_one
thf(fact_6741_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_6742_numeral__powr__numeral__real,axiom,
! [M: num,N2: num] :
( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
= ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% numeral_powr_numeral_real
thf(fact_6743_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_6744_log__powr__cancel,axiom,
! [A: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ ( powr_real @ A @ Y ) )
= Y ) ) ) ).
% log_powr_cancel
thf(fact_6745_tan__periodic__n,axiom,
! [X: real,N2: num] :
( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N2 ) @ pi ) ) )
= ( tan_real @ X ) ) ).
% tan_periodic_n
thf(fact_6746_sin__arcsin,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( sin_real @ ( arcsin @ Y ) )
= Y ) ) ) ).
% sin_arcsin
thf(fact_6747_tan__periodic__nat,axiom,
! [X: real,N2: nat] :
( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) ) )
= ( tan_real @ X ) ) ).
% tan_periodic_nat
thf(fact_6748_norm__cos__sin,axiom,
! [T2: real] :
( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T2 ) @ ( sin_real @ T2 ) ) )
= one_one_real ) ).
% norm_cos_sin
thf(fact_6749_powr__numeral,axiom,
! [X: real,N2: num] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( numeral_numeral_real @ N2 ) )
= ( power_power_real @ X @ ( numeral_numeral_nat @ N2 ) ) ) ) ).
% powr_numeral
thf(fact_6750_arcsin__1,axiom,
( ( arcsin @ one_one_real )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% arcsin_1
thf(fact_6751_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_6752_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_6753_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_6754_powr__non__neg,axiom,
! [A: real,X: real] :
~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).
% powr_non_neg
thf(fact_6755_powr__less__mono2__neg,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_6756_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_6757_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_6758_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_6759_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_6760_one__complex_Ocode,axiom,
( one_one_complex
= ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% one_complex.code
thf(fact_6761_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_6762_powr__less__mono2,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_less_mono2
thf(fact_6763_powr__mono2_H,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% powr_mono2'
thf(fact_6764_gr__one__powr,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).
% gr_one_powr
thf(fact_6765_powr__inj,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ( powr_real @ A @ X )
= ( powr_real @ A @ Y ) )
= ( X = Y ) ) ) ) ).
% powr_inj
thf(fact_6766_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_6767_powr__mono__both,axiom,
! [A: real,B: real,X: real,Y: 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 @ Y )
=> ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_6768_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_6769_powr__divide,axiom,
! [X: real,Y: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( powr_real @ ( divide_divide_real @ X @ Y ) @ A )
= ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_divide
thf(fact_6770_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_6771_log__base__powr,axiom,
! [A: real,B: real,X: real] :
( ( A != zero_zero_real )
=> ( ( log @ ( powr_real @ A @ B ) @ X )
= ( divide_divide_real @ ( log @ A @ X ) @ B ) ) ) ).
% log_base_powr
thf(fact_6772_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_6773_powr__add,axiom,
! [X: real,A: real,B: real] :
( ( powr_real @ X @ ( plus_plus_real @ A @ B ) )
= ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ).
% powr_add
thf(fact_6774_powr__diff,axiom,
! [W: real,Z1: real,Z22: real] :
( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
= ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).
% powr_diff
thf(fact_6775_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_6776_arcsin__le__arcsin,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_le_arcsin
thf(fact_6777_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_6778_arcsin__eq__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ( arcsin @ X )
= ( arcsin @ Y ) )
= ( X = Y ) ) ) ) ).
% arcsin_eq_iff
thf(fact_6779_arcsin__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% arcsin_le_mono
thf(fact_6780_tan__def,axiom,
( tan_complex
= ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X3 ) @ ( cos_complex @ X3 ) ) ) ) ).
% tan_def
thf(fact_6781_tan__def,axiom,
( tan_real
= ( ^ [X3: real] : ( divide_divide_real @ ( sin_real @ X3 ) @ ( cos_real @ X3 ) ) ) ) ).
% tan_def
thf(fact_6782_powr__realpow,axiom,
! [X: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N2 ) )
= ( power_power_real @ X @ N2 ) ) ) ).
% powr_realpow
thf(fact_6783_powr__less__iff,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X )
= ( ord_less_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
% powr_less_iff
thf(fact_6784_less__powr__iff,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( powr_real @ B @ Y ) )
= ( ord_less_real @ ( log @ B @ X ) @ Y ) ) ) ) ).
% less_powr_iff
thf(fact_6785_log__less__iff,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ B @ X ) @ Y )
= ( ord_less_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% log_less_iff
thf(fact_6786_less__log__iff,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ ( log @ B @ X ) )
= ( ord_less_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% less_log_iff
thf(fact_6787_powr__minus__divide,axiom,
! [X: real,A: real] :
( ( powr_real @ X @ ( uminus_uminus_real @ A ) )
= ( divide_divide_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ).
% powr_minus_divide
thf(fact_6788_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_6789_arcsin__less__arcsin,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_less_arcsin
thf(fact_6790_powr__mult__base,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
= ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).
% powr_mult_base
thf(fact_6791_arcsin__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% arcsin_less_mono
thf(fact_6792_powr__le__iff,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X )
= ( ord_less_eq_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
% powr_le_iff
thf(fact_6793_le__powr__iff,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) )
= ( ord_less_eq_real @ ( log @ B @ X ) @ Y ) ) ) ) ).
% le_powr_iff
thf(fact_6794_log__le__iff,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y )
= ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% log_le_iff
thf(fact_6795_le__log__iff,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ ( log @ B @ X ) )
= ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% le_log_iff
thf(fact_6796_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_6797_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_6798_tan__45,axiom,
( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
= one_one_real ) ).
% tan_45
thf(fact_6799_log__add__eq__powr,axiom,
! [B: real,X: real,Y: 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 ) @ Y )
= ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_6800_add__log__eq__powr,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( plus_plus_real @ Y @ ( log @ B @ X ) )
= ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_6801_minus__log__eq__powr,axiom,
! [B: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( minus_minus_real @ Y @ ( log @ B @ X ) )
= ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_6802_tan__60,axiom,
( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
= ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% tan_60
thf(fact_6803_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_6804_powr__def,axiom,
( powr_real
= ( ^ [X3: real,A3: real] : ( if_real @ ( X3 = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A3 @ ( ln_ln_real @ X3 ) ) ) ) ) ) ).
% powr_def
thf(fact_6805_lemma__tan__total,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ? [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
& ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ord_less_real @ Y @ ( tan_real @ X2 ) ) ) ) ).
% lemma_tan_total
thf(fact_6806_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_6807_tan__total,axiom,
! [Y: 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 )
& ! [Y3: real] :
( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
& ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ Y3 )
= Y ) )
=> ( Y3 = X2 ) ) ) ).
% tan_total
thf(fact_6808_tan__monotone,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ).
% tan_monotone
thf(fact_6809_tan__monotone_H,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ ( 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 @ Y @ X )
= ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ) ) ).
% tan_monotone'
thf(fact_6810_tan__mono__lt__eq,axiom,
! [X: real,Y: 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 ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_lt_eq
thf(fact_6811_lemma__tan__total1,axiom,
! [Y: 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 ) ) ).
% lemma_tan_total1
thf(fact_6812_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_6813_tan__inverse,axiom,
! [Y: real] :
( ( divide_divide_real @ one_one_real @ ( tan_real @ Y ) )
= ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y ) ) ) ).
% tan_inverse
thf(fact_6814_log__minus__eq__powr,axiom,
! [B: real,X: real,Y: 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 ) @ Y )
= ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_6815_complex__norm,axiom,
! [X: real,Y: real] :
( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y ) )
= ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% complex_norm
thf(fact_6816_add__tan__eq,axiom,
! [X: complex,Y: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y )
!= zero_zero_complex )
=> ( ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) )
= ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_6817_add__tan__eq,axiom,
! [X: real,Y: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ Y )
!= zero_zero_real )
=> ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
= ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_6818_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_6819_powr__neg__numeral,axiom,
! [X: real,N2: num] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_6820_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_6821_tan__total__pos,axiom,
! [Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ? [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X2 )
= Y ) ) ) ).
% tan_total_pos
thf(fact_6822_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_6823_tan__mono__le__eq,axiom,
! [X: real,Y: 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 ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_6824_tan__mono__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).
% tan_mono_le
thf(fact_6825_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_6826_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_6827_arctan,axiom,
! [Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ ( arctan @ Y ) )
= Y ) ) ).
% arctan
thf(fact_6828_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_6829_arctan__unique,axiom,
! [X: real,Y: 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 )
= Y )
=> ( ( arctan @ Y )
= X ) ) ) ) ).
% arctan_unique
thf(fact_6830_tan__add,axiom,
! [X: complex,Y: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y )
!= zero_zero_complex )
=> ( ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
!= zero_zero_complex )
=> ( ( tan_complex @ ( plus_plus_complex @ X @ Y ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_6831_tan__add,axiom,
! [X: real,Y: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ Y )
!= zero_zero_real )
=> ( ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
!= zero_zero_real )
=> ( ( tan_real @ ( plus_plus_real @ X @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_6832_tan__diff,axiom,
! [X: complex,Y: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y )
!= zero_zero_complex )
=> ( ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
!= zero_zero_complex )
=> ( ( tan_complex @ ( minus_minus_complex @ X @ Y ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_6833_tan__diff,axiom,
! [X: real,Y: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ Y )
!= zero_zero_real )
=> ( ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
!= zero_zero_real )
=> ( ( tan_real @ ( minus_minus_real @ X @ Y ) )
= ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_6834_lemma__tan__add1,axiom,
! [X: complex,Y: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y )
!= zero_zero_complex )
=> ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) )
= ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_6835_lemma__tan__add1,axiom,
! [X: real,Y: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ Y )
!= zero_zero_real )
=> ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) )
= ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_6836_tan__total__pi4,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ? [Z2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z2 )
& ( ord_less_real @ Z2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
& ( ( tan_real @ Z2 )
= X ) ) ) ).
% tan_total_pi4
thf(fact_6837_arcsin__lt__bounded,axiom,
! [Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_real @ Y @ one_one_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arcsin_lt_bounded
thf(fact_6838_arcsin__bounded,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arcsin_bounded
thf(fact_6839_arcsin__ubound,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% arcsin_ubound
thf(fact_6840_arcsin__lbound,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).
% arcsin_lbound
thf(fact_6841_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_6842_tan__half,axiom,
( tan_complex
= ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_complex ) ) ) ) ).
% tan_half
thf(fact_6843_tan__half,axiom,
( tan_real
= ( ^ [X3: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_real ) ) ) ) ).
% tan_half
thf(fact_6844_le__arcsin__iff,axiom,
! [X: real,Y: 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 ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ Y @ ( arcsin @ X ) )
= ( ord_less_eq_real @ ( sin_real @ Y ) @ X ) ) ) ) ) ) ).
% le_arcsin_iff
thf(fact_6845_arcsin__le__iff,axiom,
! [X: real,Y: 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 ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y )
= ( ord_less_eq_real @ X @ ( sin_real @ Y ) ) ) ) ) ) ) ).
% arcsin_le_iff
thf(fact_6846_arcsin__pi,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
& ( ( sin_real @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
% arcsin_pi
thf(fact_6847_arcsin,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
% arcsin
thf(fact_6848_arcosh__def,axiom,
( arcosh_real
= ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% arcosh_def
thf(fact_6849_sin__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( sin_real @ ( arccos @ Y ) )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% sin_arccos_abs
thf(fact_6850_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_6851_arsinh__def,axiom,
( arsinh_real
= ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% arsinh_def
thf(fact_6852_cos__npi__int,axiom,
! [N2: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
=> ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N2 ) ) )
= one_one_real ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
=> ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N2 ) ) )
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% cos_npi_int
thf(fact_6853_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( ( ring_1_of_int_real @ W )
= ( ring_1_of_int_real @ Z ) )
= ( W = Z ) ) ).
% of_int_eq_iff
thf(fact_6854_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( ( ring_1_of_int_rat @ W )
= ( ring_1_of_int_rat @ Z ) )
= ( W = Z ) ) ).
% of_int_eq_iff
thf(fact_6855_of__int__floor__cancel,axiom,
! [X: real] :
( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) )
= X )
= ( ? [N4: int] :
( X
= ( ring_1_of_int_real @ N4 ) ) ) ) ).
% of_int_floor_cancel
thf(fact_6856_of__int__floor__cancel,axiom,
! [X: rat] :
( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) )
= X )
= ( ? [N4: int] :
( X
= ( ring_1_of_int_rat @ N4 ) ) ) ) ).
% of_int_floor_cancel
thf(fact_6857_of__int__ceiling__cancel,axiom,
! [X: rat] :
( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) )
= X )
= ( ? [N4: int] :
( X
= ( ring_1_of_int_rat @ N4 ) ) ) ) ).
% of_int_ceiling_cancel
thf(fact_6858_of__int__ceiling__cancel,axiom,
! [X: real] :
( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) )
= X )
= ( ? [N4: int] :
( X
= ( ring_1_of_int_real @ N4 ) ) ) ) ).
% of_int_ceiling_cancel
thf(fact_6859_of__int__0,axiom,
( ( ring_17405671764205052669omplex @ zero_zero_int )
= zero_zero_complex ) ).
% of_int_0
thf(fact_6860_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_6861_of__int__0,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_0
thf(fact_6862_of__int__0,axiom,
( ( ring_1_of_int_rat @ zero_zero_int )
= zero_zero_rat ) ).
% of_int_0
thf(fact_6863_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_complex
= ( ring_17405671764205052669omplex @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_6864_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_6865_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_6866_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_rat
= ( ring_1_of_int_rat @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_6867_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_17405671764205052669omplex @ Z )
= zero_zero_complex )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_6868_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= zero_zero_int )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_6869_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= zero_zero_real )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_6870_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_rat @ Z )
= zero_zero_rat )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_6871_of__real__0,axiom,
( ( real_V1803761363581548252l_real @ zero_zero_real )
= zero_zero_real ) ).
% of_real_0
thf(fact_6872_of__real__0,axiom,
( ( real_V4546457046886955230omplex @ zero_zero_real )
= zero_zero_complex ) ).
% of_real_0
thf(fact_6873_of__real__eq__0__iff,axiom,
! [X: real] :
( ( ( real_V1803761363581548252l_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% of_real_eq_0_iff
thf(fact_6874_of__real__eq__0__iff,axiom,
! [X: real] :
( ( ( real_V4546457046886955230omplex @ X )
= zero_zero_complex )
= ( X = zero_zero_real ) ) ).
% of_real_eq_0_iff
thf(fact_6875_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_6876_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_6877_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_6878_of__int__numeral,axiom,
! [K: num] :
( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
= ( numera6690914467698888265omplex @ K ) ) ).
% of_int_numeral
thf(fact_6879_of__int__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_real @ K ) ) ).
% of_int_numeral
thf(fact_6880_of__int__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_rat @ K ) ) ).
% of_int_numeral
thf(fact_6881_of__int__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ K ) ) ).
% of_int_numeral
thf(fact_6882_of__int__eq__numeral__iff,axiom,
! [Z: int,N2: num] :
( ( ( ring_17405671764205052669omplex @ Z )
= ( numera6690914467698888265omplex @ N2 ) )
= ( Z
= ( numeral_numeral_int @ N2 ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_6883_of__int__eq__numeral__iff,axiom,
! [Z: int,N2: num] :
( ( ( ring_1_of_int_real @ Z )
= ( numeral_numeral_real @ N2 ) )
= ( Z
= ( numeral_numeral_int @ N2 ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_6884_of__int__eq__numeral__iff,axiom,
! [Z: int,N2: num] :
( ( ( ring_1_of_int_rat @ Z )
= ( numeral_numeral_rat @ N2 ) )
= ( Z
= ( numeral_numeral_int @ N2 ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_6885_of__int__eq__numeral__iff,axiom,
! [Z: int,N2: num] :
( ( ( ring_1_of_int_int @ Z )
= ( numeral_numeral_int @ N2 ) )
= ( Z
= ( numeral_numeral_int @ N2 ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_6886_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_6887_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_6888_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_6889_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_17405671764205052669omplex @ Z )
= one_one_complex )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6890_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= one_one_int )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6891_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= one_one_real )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6892_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_rat @ Z )
= one_one_rat )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6893_of__int__1,axiom,
( ( ring_17405671764205052669omplex @ one_one_int )
= one_one_complex ) ).
% of_int_1
thf(fact_6894_of__int__1,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_1
thf(fact_6895_of__int__1,axiom,
( ( ring_1_of_int_real @ one_one_int )
= one_one_real ) ).
% of_int_1
thf(fact_6896_of__int__1,axiom,
( ( ring_1_of_int_rat @ one_one_int )
= one_one_rat ) ).
% of_int_1
thf(fact_6897_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
= ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_mult
thf(fact_6898_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
= ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_mult
thf(fact_6899_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
= ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_mult
thf(fact_6900_of__int__add,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
= ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_add
thf(fact_6901_of__int__add,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_add
thf(fact_6902_of__int__add,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
= ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_add
thf(fact_6903_of__real__eq__1__iff,axiom,
! [X: real] :
( ( ( real_V1803761363581548252l_real @ X )
= one_one_real )
= ( X = one_one_real ) ) ).
% of_real_eq_1_iff
thf(fact_6904_of__real__eq__1__iff,axiom,
! [X: real] :
( ( ( real_V4546457046886955230omplex @ X )
= one_one_complex )
= ( X = one_one_real ) ) ).
% of_real_eq_1_iff
thf(fact_6905_of__real__1,axiom,
( ( real_V1803761363581548252l_real @ one_one_real )
= one_one_real ) ).
% of_real_1
thf(fact_6906_of__real__1,axiom,
( ( real_V4546457046886955230omplex @ one_one_real )
= one_one_complex ) ).
% of_real_1
thf(fact_6907_of__int__minus,axiom,
! [Z: int] :
( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z ) )
= ( uminus_uminus_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_minus
thf(fact_6908_of__int__minus,axiom,
! [Z: int] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z ) )
= ( uminus_uminus_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_minus
thf(fact_6909_of__int__minus,axiom,
! [Z: int] :
( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ Z ) )
= ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ Z ) ) ) ).
% of_int_minus
thf(fact_6910_of__int__minus,axiom,
! [Z: int] :
( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ Z ) )
= ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% of_int_minus
thf(fact_6911_of__int__minus,axiom,
! [Z: int] :
( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z ) )
= ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_minus
thf(fact_6912_of__real__numeral,axiom,
! [W: num] :
( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
= ( numeral_numeral_real @ W ) ) ).
% of_real_numeral
thf(fact_6913_of__real__numeral,axiom,
! [W: num] :
( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
= ( numera6690914467698888265omplex @ W ) ) ).
% of_real_numeral
thf(fact_6914_of__real__divide,axiom,
! [X: real,Y: real] :
( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y ) )
= ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% of_real_divide
thf(fact_6915_of__real__divide,axiom,
! [X: real,Y: real] :
( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y ) )
= ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% of_real_divide
thf(fact_6916_of__int__diff,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
= ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_diff
thf(fact_6917_of__int__diff,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
= ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_diff
thf(fact_6918_of__int__diff,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
= ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_diff
thf(fact_6919_of__int__of__nat__eq,axiom,
! [N2: nat] :
( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri5074537144036343181t_real @ N2 ) ) ).
% of_int_of_nat_eq
thf(fact_6920_of__int__of__nat__eq,axiom,
! [N2: nat] :
( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri1314217659103216013at_int @ N2 ) ) ).
% of_int_of_nat_eq
thf(fact_6921_of__int__of__nat__eq,axiom,
! [N2: nat] :
( ( ring_17405671764205052669omplex @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri8010041392384452111omplex @ N2 ) ) ).
% of_int_of_nat_eq
thf(fact_6922_of__int__of__nat__eq,axiom,
! [N2: nat] :
( ( ring_1_of_int_rat @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri681578069525770553at_rat @ N2 ) ) ).
% of_int_of_nat_eq
thf(fact_6923_of__real__add,axiom,
! [X: real,Y: real] :
( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% of_real_add
thf(fact_6924_of__real__add,axiom,
! [X: real,Y: real] :
( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% of_real_add
thf(fact_6925_of__real__power,axiom,
! [X: real,N2: nat] :
( ( real_V1803761363581548252l_real @ ( power_power_real @ X @ N2 ) )
= ( power_power_real @ ( real_V1803761363581548252l_real @ X ) @ N2 ) ) ).
% of_real_power
thf(fact_6926_of__real__power,axiom,
! [X: real,N2: nat] :
( ( real_V4546457046886955230omplex @ ( power_power_real @ X @ N2 ) )
= ( power_power_complex @ ( real_V4546457046886955230omplex @ X ) @ N2 ) ) ).
% of_real_power
thf(fact_6927_of__int__abs,axiom,
! [X: int] :
( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
= ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).
% of_int_abs
thf(fact_6928_of__int__abs,axiom,
! [X: int] :
( ( ring_18347121197199848620nteger @ ( abs_abs_int @ X ) )
= ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ X ) ) ) ).
% of_int_abs
thf(fact_6929_of__int__abs,axiom,
! [X: int] :
( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
= ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).
% of_int_abs
thf(fact_6930_of__int__abs,axiom,
! [X: int] :
( ( ring_1_of_int_rat @ ( abs_abs_int @ X ) )
= ( abs_abs_rat @ ( ring_1_of_int_rat @ X ) ) ) ).
% of_int_abs
thf(fact_6931_of__int__power,axiom,
! [Z: int,N2: nat] :
( ( ring_1_of_int_int @ ( power_power_int @ Z @ N2 ) )
= ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N2 ) ) ).
% of_int_power
thf(fact_6932_of__int__power,axiom,
! [Z: int,N2: nat] :
( ( ring_1_of_int_real @ ( power_power_int @ Z @ N2 ) )
= ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N2 ) ) ).
% of_int_power
thf(fact_6933_of__int__power,axiom,
! [Z: int,N2: nat] :
( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N2 ) )
= ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N2 ) ) ).
% of_int_power
thf(fact_6934_of__int__power,axiom,
! [Z: int,N2: nat] :
( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N2 ) )
= ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N2 ) ) ).
% of_int_power
thf(fact_6935_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
= ( ring_1_of_int_int @ X ) )
= ( ( power_power_int @ B @ W )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_6936_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
= ( ring_1_of_int_real @ X ) )
= ( ( power_power_int @ B @ W )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_6937_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
= ( ring_17405671764205052669omplex @ X ) )
= ( ( power_power_int @ B @ W )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_6938_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
= ( ring_1_of_int_rat @ X ) )
= ( ( power_power_int @ B @ W )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_6939_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ( ring_1_of_int_int @ X )
= ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
= ( X
= ( power_power_int @ B @ W ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_6940_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ( ring_1_of_int_real @ X )
= ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
= ( X
= ( power_power_int @ B @ W ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_6941_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ( ring_17405671764205052669omplex @ X )
= ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
= ( X
= ( power_power_int @ B @ W ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_6942_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ( ring_1_of_int_rat @ X )
= ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
= ( X
= ( power_power_int @ B @ W ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_6943_arccos__1,axiom,
( ( arccos @ one_one_real )
= zero_zero_real ) ).
% arccos_1
thf(fact_6944_of__int__of__bool,axiom,
! [P: $o] :
( ( ring_1_of_int_real @ ( zero_n2684676970156552555ol_int @ P ) )
= ( zero_n3304061248610475627l_real @ P ) ) ).
% of_int_of_bool
thf(fact_6945_of__int__of__bool,axiom,
! [P: $o] :
( ( ring_1_of_int_rat @ ( zero_n2684676970156552555ol_int @ P ) )
= ( zero_n2052037380579107095ol_rat @ P ) ) ).
% of_int_of_bool
thf(fact_6946_of__int__of__bool,axiom,
! [P: $o] :
( ( ring_1_of_int_int @ ( zero_n2684676970156552555ol_int @ P ) )
= ( zero_n2684676970156552555ol_int @ P ) ) ).
% of_int_of_bool
thf(fact_6947_of__int__of__bool,axiom,
! [P: $o] :
( ( ring_18347121197199848620nteger @ ( zero_n2684676970156552555ol_int @ P ) )
= ( zero_n356916108424825756nteger @ P ) ) ).
% of_int_of_bool
thf(fact_6948_sin__of__real__pi,axiom,
( ( sin_real @ ( real_V1803761363581548252l_real @ pi ) )
= zero_zero_real ) ).
% sin_of_real_pi
thf(fact_6949_sin__of__real__pi,axiom,
( ( sin_complex @ ( real_V4546457046886955230omplex @ pi ) )
= zero_zero_complex ) ).
% sin_of_real_pi
thf(fact_6950_ceiling__add__of__int,axiom,
! [X: rat,Z: int] :
( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
= ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z ) ) ).
% ceiling_add_of_int
thf(fact_6951_ceiling__add__of__int,axiom,
! [X: real,Z: int] :
( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
= ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ Z ) ) ).
% ceiling_add_of_int
thf(fact_6952_of__nat__nat__take__bit__eq,axiom,
! [N2: nat,K: int] :
( ( semiri5074537144036343181t_real @ ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) )
= ( ring_1_of_int_real @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% of_nat_nat_take_bit_eq
thf(fact_6953_of__nat__nat__take__bit__eq,axiom,
! [N2: nat,K: int] :
( ( semiri1314217659103216013at_int @ ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) )
= ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% of_nat_nat_take_bit_eq
thf(fact_6954_of__nat__nat__take__bit__eq,axiom,
! [N2: nat,K: int] :
( ( semiri8010041392384452111omplex @ ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) )
= ( ring_17405671764205052669omplex @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% of_nat_nat_take_bit_eq
thf(fact_6955_of__nat__nat__take__bit__eq,axiom,
! [N2: nat,K: int] :
( ( semiri681578069525770553at_rat @ ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) )
= ( ring_1_of_int_rat @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% of_nat_nat_take_bit_eq
thf(fact_6956_arccos__minus__1,axiom,
( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
= pi ) ).
% arccos_minus_1
thf(fact_6957_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_6958_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_6959_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_6960_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_6961_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_6962_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_6963_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_6964_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_6965_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_6966_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_6967_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_6968_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_6969_of__int__le__numeral__iff,axiom,
! [Z: int,N2: num] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N2 ) )
= ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% of_int_le_numeral_iff
thf(fact_6970_of__int__le__numeral__iff,axiom,
! [Z: int,N2: num] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N2 ) )
= ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% of_int_le_numeral_iff
thf(fact_6971_of__int__le__numeral__iff,axiom,
! [Z: int,N2: num] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N2 ) )
= ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% of_int_le_numeral_iff
thf(fact_6972_of__int__numeral__le__iff,axiom,
! [N2: num,Z: int] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% of_int_numeral_le_iff
thf(fact_6973_of__int__numeral__le__iff,axiom,
! [N2: num,Z: int] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% of_int_numeral_le_iff
thf(fact_6974_of__int__numeral__le__iff,axiom,
! [N2: num,Z: int] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% of_int_numeral_le_iff
thf(fact_6975_of__int__less__numeral__iff,axiom,
! [Z: int,N2: num] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N2 ) )
= ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% of_int_less_numeral_iff
thf(fact_6976_of__int__less__numeral__iff,axiom,
! [Z: int,N2: num] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N2 ) )
= ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% of_int_less_numeral_iff
thf(fact_6977_of__int__less__numeral__iff,axiom,
! [Z: int,N2: num] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N2 ) )
= ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% of_int_less_numeral_iff
thf(fact_6978_of__int__numeral__less__iff,axiom,
! [N2: num,Z: int] :
( ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% of_int_numeral_less_iff
thf(fact_6979_of__int__numeral__less__iff,axiom,
! [N2: num,Z: int] :
( ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% of_int_numeral_less_iff
thf(fact_6980_of__int__numeral__less__iff,axiom,
! [N2: num,Z: int] :
( ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% of_int_numeral_less_iff
thf(fact_6981_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_6982_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_6983_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_6984_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_6985_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_6986_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_6987_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_6988_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_6989_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_6990_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_6991_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_6992_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_6993_of__real__neg__numeral,axiom,
! [W: num] :
( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).
% of_real_neg_numeral
thf(fact_6994_of__real__neg__numeral,axiom,
! [W: num] :
( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% of_real_neg_numeral
thf(fact_6995_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_6996_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_6997_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_6998_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_6999_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_7000_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_7001_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N2: nat,Y: int] :
( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 )
= ( ring_17405671764205052669omplex @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_7002_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N2: nat,Y: int] :
( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 )
= ( ring_1_of_int_real @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_7003_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N2: nat,Y: int] :
( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 )
= ( ring_1_of_int_rat @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_7004_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N2: nat,Y: int] :
( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
= ( ring_1_of_int_int @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_7005_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N2: nat] :
( ( ( ring_17405671764205052669omplex @ Y )
= ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_7006_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N2: nat] :
( ( ( ring_1_of_int_real @ Y )
= ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_7007_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N2: nat] :
( ( ( ring_1_of_int_rat @ Y )
= ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_7008_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N2: nat] :
( ( ( ring_1_of_int_int @ Y )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_7009_cos__of__real__pi,axiom,
( ( cos_real @ ( real_V1803761363581548252l_real @ pi ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% cos_of_real_pi
thf(fact_7010_cos__of__real__pi,axiom,
( ( cos_complex @ ( real_V4546457046886955230omplex @ pi ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% cos_of_real_pi
thf(fact_7011_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_7012_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_7013_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_7014_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_7015_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_7016_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_7017_of__nat__nat,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri5074537144036343181t_real @ ( nat2 @ Z ) )
= ( ring_1_of_int_real @ Z ) ) ) ).
% of_nat_nat
thf(fact_7018_of__nat__nat,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= ( ring_1_of_int_int @ Z ) ) ) ).
% of_nat_nat
thf(fact_7019_of__nat__nat,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri8010041392384452111omplex @ ( nat2 @ Z ) )
= ( ring_17405671764205052669omplex @ Z ) ) ) ).
% of_nat_nat
thf(fact_7020_of__nat__nat,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri681578069525770553at_rat @ ( nat2 @ Z ) )
= ( ring_1_of_int_rat @ Z ) ) ) ).
% of_nat_nat
thf(fact_7021_cos__arccos,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( cos_real @ ( arccos @ Y ) )
= Y ) ) ) ).
% cos_arccos
thf(fact_7022_tan__periodic__int,axiom,
! [X: real,I: int] :
( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
= ( tan_real @ X ) ) ).
% tan_periodic_int
thf(fact_7023_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N2: nat,A: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_7024_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N2: nat,A: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_7025_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N2: nat,A: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_7026_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N2: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_7027_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N2: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_7028_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N2: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_7029_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N2: nat,A: int] :
( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_7030_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N2: nat,A: int] :
( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_7031_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N2: nat,A: int] :
( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_7032_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N2: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_7033_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N2: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_7034_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N2: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_7035_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N2: nat,Y: int] :
( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 )
= ( ring_1_of_int_real @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7036_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N2: nat,Y: int] :
( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
= ( ring_1_of_int_int @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7037_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N2: nat,Y: int] :
( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N2 )
= ( ring_17405671764205052669omplex @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7038_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N2: nat,Y: int] :
( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 )
= ( ring_18347121197199848620nteger @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7039_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N2: nat,Y: int] :
( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 )
= ( ring_1_of_int_rat @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7040_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N2: nat] :
( ( ( ring_1_of_int_real @ Y )
= ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7041_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N2: nat] :
( ( ( ring_1_of_int_int @ Y )
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7042_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N2: nat] :
( ( ( ring_17405671764205052669omplex @ Y )
= ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N2 ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7043_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N2: nat] :
( ( ( ring_18347121197199848620nteger @ Y )
= ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7044_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N2: nat] :
( ( ( ring_1_of_int_rat @ Y )
= ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7045_norm__of__real__add1,axiom,
! [X: real] :
( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ one_one_real ) )
= ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).
% norm_of_real_add1
thf(fact_7046_norm__of__real__add1,axiom,
! [X: real] :
( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ one_one_complex ) )
= ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).
% norm_of_real_add1
thf(fact_7047_norm__of__real__addn,axiom,
! [X: real,B: num] :
( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( numeral_numeral_real @ B ) ) )
= ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).
% norm_of_real_addn
thf(fact_7048_norm__of__real__addn,axiom,
! [X: real,B: num] :
( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( numera6690914467698888265omplex @ B ) ) )
= ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).
% norm_of_real_addn
thf(fact_7049_arccos__0,axiom,
( ( arccos @ zero_zero_real )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% arccos_0
thf(fact_7050_cos__of__real__pi__half,axiom,
( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= zero_zero_real ) ).
% cos_of_real_pi_half
thf(fact_7051_cos__of__real__pi__half,axiom,
( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
= zero_zero_complex ) ).
% cos_of_real_pi_half
thf(fact_7052_sin__of__real__pi__half,axiom,
( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= one_one_real ) ).
% sin_of_real_pi_half
thf(fact_7053_sin__of__real__pi__half,axiom,
( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
= one_one_complex ) ).
% sin_of_real_pi_half
thf(fact_7054_sin__int__2pin,axiom,
! [N2: int] :
( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N2 ) ) )
= zero_zero_real ) ).
% sin_int_2pin
thf(fact_7055_cos__int__2pin,axiom,
! [N2: int] :
( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N2 ) ) )
= one_one_real ) ).
% cos_int_2pin
thf(fact_7056_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N2: nat,A: int] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7057_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N2: nat,A: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7058_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N2: nat,A: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7059_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N2: nat,A: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7060_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N2: nat] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7061_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N2: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7062_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N2: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7063_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N2: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7064_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N2: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7065_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N2: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7066_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N2: nat] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7067_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N2: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7068_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N2: nat,A: int] :
( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7069_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N2: nat,A: int] :
( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7070_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N2: nat,A: int] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7071_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N2: nat,A: int] :
( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7072_ex__le__of__int,axiom,
! [X: rat] :
? [Z2: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ).
% ex_le_of_int
thf(fact_7073_ex__le__of__int,axiom,
! [X: real] :
? [Z2: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).
% ex_le_of_int
thf(fact_7074_ex__of__int__less,axiom,
! [X: real] :
? [Z2: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ).
% ex_of_int_less
thf(fact_7075_ex__of__int__less,axiom,
! [X: rat] :
? [Z2: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ).
% ex_of_int_less
thf(fact_7076_ex__less__of__int,axiom,
! [X: real] :
? [Z2: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).
% ex_less_of_int
thf(fact_7077_ex__less__of__int,axiom,
! [X: rat] :
? [Z2: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ).
% ex_less_of_int
thf(fact_7078_mult__of__int__commute,axiom,
! [X: int,Y: real] :
( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
= ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).
% mult_of_int_commute
thf(fact_7079_mult__of__int__commute,axiom,
! [X: int,Y: rat] :
( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y )
= ( times_times_rat @ Y @ ( ring_1_of_int_rat @ X ) ) ) ).
% mult_of_int_commute
thf(fact_7080_mult__of__int__commute,axiom,
! [X: int,Y: int] :
( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
= ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).
% mult_of_int_commute
thf(fact_7081_sgn__eq,axiom,
( sgn_sgn_complex
= ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ Z5 @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ Z5 ) ) ) ) ) ).
% sgn_eq
thf(fact_7082_of__int__floor__le,axiom,
! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X ) ).
% of_int_floor_le
thf(fact_7083_of__int__floor__le,axiom,
! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X ) ).
% of_int_floor_le
thf(fact_7084_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_7085_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_7086_take__bit__of__int,axiom,
! [N2: nat,K: int] :
( ( bit_se2923211474154528505it_int @ N2 @ ( ring_1_of_int_int @ K ) )
= ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% take_bit_of_int
thf(fact_7087_of__int__and__eq,axiom,
! [K: int,L: int] :
( ( ring_1_of_int_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
= ( bit_se725231765392027082nd_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% of_int_and_eq
thf(fact_7088_of__int__mask__eq,axiom,
! [N2: nat] :
( ( ring_1_of_int_int @ ( bit_se2000444600071755411sk_int @ N2 ) )
= ( bit_se2000444600071755411sk_int @ N2 ) ) ).
% of_int_mask_eq
thf(fact_7089_nonzero__of__real__divide,axiom,
! [Y: real,X: real] :
( ( Y != zero_zero_real )
=> ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y ) )
= ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ) ).
% nonzero_of_real_divide
thf(fact_7090_nonzero__of__real__divide,axiom,
! [Y: real,X: real] :
( ( Y != zero_zero_real )
=> ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y ) )
= ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ) ).
% nonzero_of_real_divide
thf(fact_7091_le__floor__iff,axiom,
! [Z: int,X: real] :
( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% le_floor_iff
thf(fact_7092_le__floor__iff,axiom,
! [Z: int,X: rat] :
( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).
% le_floor_iff
thf(fact_7093_floor__less__iff,axiom,
! [X: real,Z: int] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ Z )
= ( ord_less_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% floor_less_iff
thf(fact_7094_floor__less__iff,axiom,
! [X: rat,Z: int] :
( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
= ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).
% floor_less_iff
thf(fact_7095_ceiling__le,axiom,
! [X: rat,A: int] :
( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) )
=> ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ A ) ) ).
% ceiling_le
thf(fact_7096_ceiling__le,axiom,
! [X: real,A: int] :
( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A ) ) ).
% ceiling_le
thf(fact_7097_ceiling__le__iff,axiom,
! [X: rat,Z: int] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
= ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).
% ceiling_le_iff
thf(fact_7098_ceiling__le__iff,axiom,
! [X: real,Z: int] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z )
= ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% ceiling_le_iff
thf(fact_7099_less__ceiling__iff,axiom,
! [Z: int,X: rat] :
( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).
% less_ceiling_iff
thf(fact_7100_less__ceiling__iff,axiom,
! [Z: int,X: real] :
( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% less_ceiling_iff
thf(fact_7101_floor__add__int,axiom,
! [X: real,Z: int] :
( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ Z )
= ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ) ).
% floor_add_int
thf(fact_7102_floor__add__int,axiom,
! [X: rat,Z: int] :
( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
= ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ) ).
% floor_add_int
thf(fact_7103_int__add__floor,axiom,
! [Z: int,X: real] :
( ( plus_plus_int @ Z @ ( archim6058952711729229775r_real @ X ) )
= ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ) ).
% int_add_floor
thf(fact_7104_int__add__floor,axiom,
! [Z: int,X: rat] :
( ( plus_plus_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
= ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ) ).
% int_add_floor
thf(fact_7105_real__of__int__div4,axiom,
! [N2: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X ) ) ) ).
% real_of_int_div4
thf(fact_7106_floor__divide__of__int__eq,axiom,
! [K: int,L: int] :
( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K ) @ ( ring_1_of_int_real @ L ) ) )
= ( divide_divide_int @ K @ L ) ) ).
% floor_divide_of_int_eq
thf(fact_7107_floor__divide__of__int__eq,axiom,
! [K: int,L: int] :
( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L ) ) )
= ( divide_divide_int @ K @ L ) ) ).
% floor_divide_of_int_eq
thf(fact_7108_floor__power,axiom,
! [X: real,N2: nat] :
( ( X
= ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
=> ( ( archim6058952711729229775r_real @ ( power_power_real @ X @ N2 ) )
= ( power_power_int @ ( archim6058952711729229775r_real @ X ) @ N2 ) ) ) ).
% floor_power
thf(fact_7109_floor__power,axiom,
! [X: rat,N2: nat] :
( ( X
= ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
=> ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X @ N2 ) )
= ( power_power_int @ ( archim3151403230148437115or_rat @ X ) @ N2 ) ) ) ).
% floor_power
thf(fact_7110_real__of__int__div,axiom,
! [D: int,N2: int] :
( ( dvd_dvd_int @ D @ N2 )
=> ( ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ D ) )
= ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% real_of_int_div
thf(fact_7111_complex__of__real__add__Complex,axiom,
! [R4: real,X: real,Y: real] :
( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R4 ) @ ( complex2 @ X @ Y ) )
= ( complex2 @ ( plus_plus_real @ R4 @ X ) @ Y ) ) ).
% complex_of_real_add_Complex
thf(fact_7112_Complex__add__complex__of__real,axiom,
! [X: real,Y: real,R4: real] :
( ( plus_plus_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R4 ) )
= ( complex2 @ ( plus_plus_real @ X @ R4 ) @ Y ) ) ).
% Complex_add_complex_of_real
thf(fact_7113_arccos__le__arccos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_le_arccos
thf(fact_7114_arccos__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% arccos_le_mono
thf(fact_7115_arccos__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
& ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
=> ( ( ( arccos @ X )
= ( arccos @ Y ) )
= ( X = Y ) ) ) ).
% arccos_eq_iff
thf(fact_7116_norm__less__p1,axiom,
! [X: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X ) ) @ one_one_real ) ) ) ).
% norm_less_p1
thf(fact_7117_norm__less__p1,axiom,
! [X: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X ) ) @ one_one_complex ) ) ) ).
% norm_less_p1
thf(fact_7118_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_nonneg
thf(fact_7119_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_nonneg
thf(fact_7120_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_nonneg
thf(fact_7121_of__int__leD,axiom,
! [N2: int,X: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X )
=> ( ( N2 = zero_zero_int )
| ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).
% of_int_leD
thf(fact_7122_of__int__leD,axiom,
! [N2: int,X: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X )
=> ( ( N2 = zero_zero_int )
| ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% of_int_leD
thf(fact_7123_of__int__leD,axiom,
! [N2: int,X: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
=> ( ( N2 = zero_zero_int )
| ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% of_int_leD
thf(fact_7124_of__int__leD,axiom,
! [N2: int,X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X )
=> ( ( N2 = zero_zero_int )
| ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% of_int_leD
thf(fact_7125_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_pos
thf(fact_7126_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_pos
thf(fact_7127_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_pos
thf(fact_7128_of__int__lessD,axiom,
! [N2: int,X: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X )
=> ( ( N2 = zero_zero_int )
| ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).
% of_int_lessD
thf(fact_7129_of__int__lessD,axiom,
! [N2: int,X: real] :
( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X )
=> ( ( N2 = zero_zero_int )
| ( ord_less_real @ one_one_real @ X ) ) ) ).
% of_int_lessD
thf(fact_7130_of__int__lessD,axiom,
! [N2: int,X: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X )
=> ( ( N2 = zero_zero_int )
| ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% of_int_lessD
thf(fact_7131_of__int__lessD,axiom,
! [N2: int,X: int] :
( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
=> ( ( N2 = zero_zero_int )
| ( ord_less_int @ one_one_int @ X ) ) ) ).
% of_int_lessD
thf(fact_7132_floor__exists1,axiom,
! [X: rat] :
? [X2: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X2 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X2 @ one_one_int ) ) )
& ! [Y3: int] :
( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y3 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y3 @ one_one_int ) ) ) )
=> ( Y3 = X2 ) ) ) ).
% floor_exists1
thf(fact_7133_floor__exists1,axiom,
! [X: real] :
? [X2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X2 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X2 @ one_one_int ) ) )
& ! [Y3: int] :
( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y3 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y3 @ one_one_int ) ) ) )
=> ( Y3 = X2 ) ) ) ).
% floor_exists1
thf(fact_7134_floor__exists,axiom,
! [X: rat] :
? [Z2: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).
% floor_exists
thf(fact_7135_floor__exists,axiom,
! [X: real] :
? [Z2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).
% floor_exists
thf(fact_7136_of__int__ceiling__le__add__one,axiom,
! [R4: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R4 ) ) @ ( plus_plus_rat @ R4 @ one_one_rat ) ) ).
% of_int_ceiling_le_add_one
thf(fact_7137_of__int__ceiling__le__add__one,axiom,
! [R4: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R4 ) ) @ ( plus_plus_real @ R4 @ one_one_real ) ) ).
% of_int_ceiling_le_add_one
thf(fact_7138_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_7139_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_7140_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_7141_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_7142_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_7143_of__int__ceiling__diff__one__le,axiom,
! [R4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R4 ) ) @ one_one_rat ) @ R4 ) ).
% of_int_ceiling_diff_one_le
thf(fact_7144_of__int__ceiling__diff__one__le,axiom,
! [R4: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R4 ) ) @ one_one_real ) @ R4 ) ).
% of_int_ceiling_diff_one_le
thf(fact_7145_of__nat__less__of__int__iff,axiom,
! [N2: nat,X: int] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_7146_of__nat__less__of__int__iff,axiom,
! [N2: nat,X: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_7147_of__nat__less__of__int__iff,axiom,
! [N2: nat,X: int] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( ring_1_of_int_rat @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_7148_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N4: int,M2: int] : ( ord_less_real @ ( ring_1_of_int_real @ N4 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M2 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_7149_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N4: int,M2: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N4 ) @ one_one_real ) @ ( ring_1_of_int_real @ M2 ) ) ) ) ).
% int_less_real_le
thf(fact_7150_ceiling__divide__eq__div,axiom,
! [A: int,B: int] :
( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) )
= ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% ceiling_divide_eq_div
thf(fact_7151_ceiling__divide__eq__div,axiom,
! [A: int,B: int] :
( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) )
= ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% ceiling_divide_eq_div
thf(fact_7152_ceiling__altdef,axiom,
( archim7802044766580827645g_real
= ( ^ [X3: real] :
( if_int
@ ( X3
= ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X3 ) ) )
@ ( archim6058952711729229775r_real @ X3 )
@ ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int ) ) ) ) ).
% ceiling_altdef
thf(fact_7153_ceiling__altdef,axiom,
( archim2889992004027027881ng_rat
= ( ^ [X3: rat] :
( if_int
@ ( X3
= ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) )
@ ( archim3151403230148437115or_rat @ X3 )
@ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int ) ) ) ) ).
% ceiling_altdef
thf(fact_7154_arccos__cos__eq__abs__2pi,axiom,
! [Theta: real] :
~ ! [K3: int] :
( ( arccos @ ( cos_real @ Theta ) )
!= ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).
% arccos_cos_eq_abs_2pi
thf(fact_7155_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_7156_floor__eq,axiom,
! [N2: int,X: real] :
( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= N2 ) ) ) ).
% floor_eq
thf(fact_7157_real__of__int__floor__add__one__gt,axiom,
! [R4: real] : ( ord_less_real @ R4 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R4 ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_gt
thf(fact_7158_real__of__int__floor__add__one__ge,axiom,
! [R4: real] : ( ord_less_eq_real @ R4 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R4 ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_ge
thf(fact_7159_real__of__int__floor__gt__diff__one,axiom,
! [R4: real] : ( ord_less_real @ ( minus_minus_real @ R4 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R4 ) ) ) ).
% real_of_int_floor_gt_diff_one
thf(fact_7160_real__of__int__floor__ge__diff__one,axiom,
! [R4: real] : ( ord_less_eq_real @ ( minus_minus_real @ R4 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R4 ) ) ) ).
% real_of_int_floor_ge_diff_one
thf(fact_7161_arccos__lbound,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).
% arccos_lbound
thf(fact_7162_arccos__less__arccos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_less_arccos
thf(fact_7163_arccos__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
= ( ord_less_real @ Y @ X ) ) ) ) ).
% arccos_less_mono
thf(fact_7164_arccos__ubound,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).
% arccos_ubound
thf(fact_7165_floor__unique,axiom,
! [Z: int,X: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= Z ) ) ) ).
% floor_unique
thf(fact_7166_floor__unique,axiom,
! [Z: int,X: rat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X )
=> ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
=> ( ( archim3151403230148437115or_rat @ X )
= Z ) ) ) ).
% floor_unique
thf(fact_7167_floor__eq__iff,axiom,
! [X: real,A: int] :
( ( ( archim6058952711729229775r_real @ X )
= A )
= ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X )
& ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).
% floor_eq_iff
thf(fact_7168_floor__eq__iff,axiom,
! [X: rat,A: int] :
( ( ( archim3151403230148437115or_rat @ X )
= A )
= ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X )
& ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).
% floor_eq_iff
thf(fact_7169_floor__split,axiom,
! [P: int > $o,T2: real] :
( ( P @ ( archim6058952711729229775r_real @ T2 ) )
= ( ! [I3: int] :
( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I3 ) @ T2 )
& ( ord_less_real @ T2 @ ( plus_plus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) ) )
=> ( P @ I3 ) ) ) ) ).
% floor_split
thf(fact_7170_floor__split,axiom,
! [P: int > $o,T2: rat] :
( ( P @ ( archim3151403230148437115or_rat @ T2 ) )
= ( ! [I3: int] :
( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I3 ) @ T2 )
& ( ord_less_rat @ T2 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) ) )
=> ( P @ I3 ) ) ) ) ).
% floor_split
thf(fact_7171_cos__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( cos_real @ ( arccos @ Y ) )
= Y ) ) ).
% cos_arccos_abs
thf(fact_7172_ceiling__correct,axiom,
! [X: rat] :
( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
& ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).
% ceiling_correct
thf(fact_7173_ceiling__correct,axiom,
! [X: real] :
( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) @ one_one_real ) @ X )
& ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% ceiling_correct
thf(fact_7174_ceiling__unique,axiom,
! [Z: int,X: rat] :
( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X )
=> ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) )
=> ( ( archim2889992004027027881ng_rat @ X )
= Z ) ) ) ).
% ceiling_unique
thf(fact_7175_ceiling__unique,axiom,
! [Z: int,X: real] :
( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) )
=> ( ( archim7802044766580827645g_real @ X )
= Z ) ) ) ).
% ceiling_unique
thf(fact_7176_ceiling__eq__iff,axiom,
! [X: rat,A: int] :
( ( ( archim2889992004027027881ng_rat @ X )
= A )
= ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X )
& ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) ) ) ) ).
% ceiling_eq_iff
thf(fact_7177_ceiling__eq__iff,axiom,
! [X: real,A: int] :
( ( ( archim7802044766580827645g_real @ X )
= A )
= ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X )
& ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) ) ) ) ).
% ceiling_eq_iff
thf(fact_7178_ceiling__split,axiom,
! [P: int > $o,T2: rat] :
( ( P @ ( archim2889992004027027881ng_rat @ T2 ) )
= ( ! [I3: int] :
( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) @ T2 )
& ( ord_less_eq_rat @ T2 @ ( ring_1_of_int_rat @ I3 ) ) )
=> ( P @ I3 ) ) ) ) ).
% ceiling_split
thf(fact_7179_ceiling__split,axiom,
! [P: int > $o,T2: real] :
( ( P @ ( archim7802044766580827645g_real @ T2 ) )
= ( ! [I3: int] :
( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) @ T2 )
& ( ord_less_eq_real @ T2 @ ( ring_1_of_int_real @ I3 ) ) )
=> ( P @ I3 ) ) ) ) ).
% ceiling_split
thf(fact_7180_less__floor__iff,axiom,
! [Z: int,X: real] :
( ( ord_less_int @ Z @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).
% less_floor_iff
thf(fact_7181_less__floor__iff,axiom,
! [Z: int,X: rat] :
( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).
% less_floor_iff
thf(fact_7182_floor__le__iff,axiom,
! [X: real,Z: int] :
( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z )
= ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% floor_le_iff
thf(fact_7183_floor__le__iff,axiom,
! [X: rat,Z: int] :
( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
= ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% floor_le_iff
thf(fact_7184_ceiling__less__iff,axiom,
! [X: rat,Z: int] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
= ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% ceiling_less_iff
thf(fact_7185_ceiling__less__iff,axiom,
! [X: real,Z: int] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z )
= ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% ceiling_less_iff
thf(fact_7186_le__ceiling__iff,axiom,
! [Z: int,X: rat] :
( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).
% le_ceiling_iff
thf(fact_7187_le__ceiling__iff,axiom,
! [Z: int,X: real] :
( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).
% le_ceiling_iff
thf(fact_7188_floor__correct,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).
% floor_correct
thf(fact_7189_floor__correct,axiom,
! [X: rat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).
% floor_correct
thf(fact_7190_real__of__int__div2,axiom,
! [N2: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X ) ) ) ) ).
% real_of_int_div2
thf(fact_7191_real__of__int__div3,axiom,
! [N2: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X ) ) ) @ one_one_real ) ).
% real_of_int_div3
thf(fact_7192_floor__eq2,axiom,
! [N2: int,X: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ N2 ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= N2 ) ) ) ).
% floor_eq2
thf(fact_7193_floor__divide__real__eq__div,axiom,
! [B: int,A: real] :
( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
= ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% floor_divide_real_eq_div
thf(fact_7194_floor__divide__lower,axiom,
! [Q3: real,P3: real] :
( ( ord_less_real @ zero_zero_real @ Q3 )
=> ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P3 @ Q3 ) ) ) @ Q3 ) @ P3 ) ) ).
% floor_divide_lower
thf(fact_7195_floor__divide__lower,axiom,
! [Q3: rat,P3: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q3 )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P3 @ Q3 ) ) ) @ Q3 ) @ P3 ) ) ).
% floor_divide_lower
thf(fact_7196_ceiling__divide__upper,axiom,
! [Q3: rat,P3: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q3 )
=> ( ord_less_eq_rat @ P3 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P3 @ Q3 ) ) ) @ Q3 ) ) ) ).
% ceiling_divide_upper
thf(fact_7197_ceiling__divide__upper,axiom,
! [Q3: real,P3: real] :
( ( ord_less_real @ zero_zero_real @ Q3 )
=> ( ord_less_eq_real @ P3 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P3 @ Q3 ) ) ) @ Q3 ) ) ) ).
% ceiling_divide_upper
thf(fact_7198_arccos__lt__bounded,axiom,
! [Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_real @ Y @ one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
& ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% arccos_lt_bounded
thf(fact_7199_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_7200_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_7201_arccos__bounded,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
& ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% arccos_bounded
thf(fact_7202_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_7203_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_7204_of__int__of__nat,axiom,
( ring_18347121197199848620nteger
= ( ^ [K2: int] : ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri4939895301339042750nteger @ ( nat2 @ K2 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_7205_of__int__of__nat,axiom,
( ring_1_of_int_real
= ( ^ [K2: int] : ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K2 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_7206_of__int__of__nat,axiom,
( ring_1_of_int_int
= ( ^ [K2: int] : ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K2 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_7207_of__int__of__nat,axiom,
( ring_17405671764205052669omplex
= ( ^ [K2: int] : ( if_complex @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri8010041392384452111omplex @ ( nat2 @ K2 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_7208_of__int__of__nat,axiom,
( ring_1_of_int_rat
= ( ^ [K2: int] : ( if_rat @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri681578069525770553at_rat @ ( nat2 @ K2 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_7209_floor__divide__upper,axiom,
! [Q3: real,P3: real] :
( ( ord_less_real @ zero_zero_real @ Q3 )
=> ( ord_less_real @ P3 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P3 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) ) ) ).
% floor_divide_upper
thf(fact_7210_floor__divide__upper,axiom,
! [Q3: rat,P3: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q3 )
=> ( ord_less_rat @ P3 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P3 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) ) ) ).
% floor_divide_upper
thf(fact_7211_ceiling__divide__lower,axiom,
! [Q3: real,P3: real] :
( ( ord_less_real @ zero_zero_real @ Q3 )
=> ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P3 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) @ P3 ) ) ).
% ceiling_divide_lower
thf(fact_7212_ceiling__divide__lower,axiom,
! [Q3: rat,P3: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q3 )
=> ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P3 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) @ P3 ) ) ).
% ceiling_divide_lower
thf(fact_7213_ceiling__eq,axiom,
! [N2: int,X: rat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ N2 ) @ X )
=> ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N2 ) @ one_one_rat ) )
=> ( ( archim2889992004027027881ng_rat @ X )
= ( plus_plus_int @ N2 @ one_one_int ) ) ) ) ).
% ceiling_eq
thf(fact_7214_ceiling__eq,axiom,
! [N2: int,X: real] :
( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X )
=> ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
=> ( ( archim7802044766580827645g_real @ X )
= ( plus_plus_int @ N2 @ one_one_int ) ) ) ) ).
% ceiling_eq
thf(fact_7215_cos__one__2pi__int,axiom,
! [X: real] :
( ( ( cos_real @ X )
= one_one_real )
= ( ? [X3: int] :
( X
= ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% cos_one_2pi_int
thf(fact_7216_arccos,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
& ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
& ( ( cos_real @ ( arccos @ Y ) )
= Y ) ) ) ) ).
% arccos
thf(fact_7217_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_7218_sin__cos__eq,axiom,
( sin_real
= ( ^ [X3: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% sin_cos_eq
thf(fact_7219_sin__cos__eq,axiom,
( sin_complex
= ( ^ [X3: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% sin_cos_eq
thf(fact_7220_cos__sin__eq,axiom,
( cos_real
= ( ^ [X3: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% cos_sin_eq
thf(fact_7221_cos__sin__eq,axiom,
( cos_complex
= ( ^ [X3: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ).
% cos_sin_eq
thf(fact_7222_minus__sin__cos__eq,axiom,
! [X: real] :
( ( uminus_uminus_real @ ( sin_real @ X ) )
= ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% minus_sin_cos_eq
thf(fact_7223_minus__sin__cos__eq,axiom,
! [X: complex] :
( ( uminus1482373934393186551omplex @ ( sin_complex @ X ) )
= ( cos_complex @ ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% minus_sin_cos_eq
thf(fact_7224_arccos__le__pi2,axiom,
! [Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% arccos_le_pi2
thf(fact_7225_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_7226_cos__zero__iff__int,axiom,
! [X: real] :
( ( ( cos_real @ X )
= zero_zero_real )
= ( ? [I3: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
& ( X
= ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% cos_zero_iff_int
thf(fact_7227_sin__zero__iff__int,axiom,
! [X: real] :
( ( ( sin_real @ X )
= zero_zero_real )
= ( ? [I3: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
& ( X
= ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% sin_zero_iff_int
thf(fact_7228_powr__int,axiom,
! [X: real,I: int] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
= ( power_power_real @ X @ ( nat2 @ I ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_7229_round__unique,axiom,
! [X: rat,Y: int] :
( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
=> ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
=> ( ( archim7778729529865785530nd_rat @ X )
= Y ) ) ) ).
% round_unique
thf(fact_7230_round__unique,axiom,
! [X: real,Y: int] :
( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
=> ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( archim8280529875227126926d_real @ X )
= Y ) ) ) ).
% round_unique
thf(fact_7231_round__unique_H,axiom,
! [X: real,N2: int] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N2 ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( archim8280529875227126926d_real @ X )
= N2 ) ) ).
% round_unique'
thf(fact_7232_round__unique_H,axiom,
! [X: rat,N2: int] :
( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N2 ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
=> ( ( archim7778729529865785530nd_rat @ X )
= N2 ) ) ).
% round_unique'
thf(fact_7233_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_7234_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_7235_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_7236_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_7237_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_7238_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_7239_round__0,axiom,
( ( archim8280529875227126926d_real @ zero_zero_real )
= zero_zero_int ) ).
% round_0
thf(fact_7240_round__0,axiom,
( ( archim7778729529865785530nd_rat @ zero_zero_rat )
= zero_zero_int ) ).
% round_0
thf(fact_7241_round__numeral,axiom,
! [N2: num] :
( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N2 ) )
= ( numeral_numeral_int @ N2 ) ) ).
% round_numeral
thf(fact_7242_round__numeral,axiom,
! [N2: num] :
( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N2 ) )
= ( numeral_numeral_int @ N2 ) ) ).
% round_numeral
thf(fact_7243_round__1,axiom,
( ( archim8280529875227126926d_real @ one_one_real )
= one_one_int ) ).
% round_1
thf(fact_7244_round__1,axiom,
( ( archim7778729529865785530nd_rat @ one_one_rat )
= one_one_int ) ).
% round_1
thf(fact_7245_round__neg__numeral,axiom,
! [N2: num] :
( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% round_neg_numeral
thf(fact_7246_round__neg__numeral,axiom,
! [N2: num] :
( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% round_neg_numeral
thf(fact_7247_round__mono,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).
% round_mono
thf(fact_7248_round__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim8280529875227126926d_real @ Y ) ) ) ).
% round_mono
thf(fact_7249_floor__le__round,axiom,
! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim8280529875227126926d_real @ X ) ) ).
% floor_le_round
thf(fact_7250_floor__le__round,axiom,
! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim7778729529865785530nd_rat @ X ) ) ).
% floor_le_round
thf(fact_7251_ceiling__ge__round,axiom,
! [X: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% ceiling_ge_round
thf(fact_7252_round__diff__minimal,axiom,
! [Z: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ M ) ) ) ) ).
% round_diff_minimal
thf(fact_7253_round__diff__minimal,axiom,
! [Z: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ M ) ) ) ) ).
% round_diff_minimal
thf(fact_7254_round__def,axiom,
( archim8280529875227126926d_real
= ( ^ [X3: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% round_def
thf(fact_7255_round__def,axiom,
( archim7778729529865785530nd_rat
= ( ^ [X3: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% round_def
thf(fact_7256_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_7257_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_7258_round__altdef,axiom,
( archim8280529875227126926d_real
= ( ^ [X3: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X3 ) ) @ ( archim7802044766580827645g_real @ X3 ) @ ( archim6058952711729229775r_real @ X3 ) ) ) ) ).
% round_altdef
thf(fact_7259_round__altdef,axiom,
( archim7778729529865785530nd_rat
= ( ^ [X3: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X3 ) ) @ ( archim2889992004027027881ng_rat @ X3 ) @ ( archim3151403230148437115or_rat @ X3 ) ) ) ) ).
% round_altdef
thf(fact_7260_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_7261_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_7262_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_7263_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_7264_cot__zero,axiom,
( ( cot_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% cot_zero
thf(fact_7265_cot__zero,axiom,
( ( cot_real @ zero_zero_real )
= zero_zero_real ) ).
% cot_zero
thf(fact_7266_frac__of__int,axiom,
! [Z: int] :
( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z ) )
= zero_zero_real ) ).
% frac_of_int
thf(fact_7267_frac__of__int,axiom,
! [Z: int] :
( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z ) )
= zero_zero_rat ) ).
% frac_of_int
thf(fact_7268_frac__ge__0,axiom,
! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).
% frac_ge_0
thf(fact_7269_frac__ge__0,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).
% frac_ge_0
thf(fact_7270_frac__lt__1,axiom,
! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).
% frac_lt_1
thf(fact_7271_frac__lt__1,axiom,
! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).
% frac_lt_1
thf(fact_7272_frac__1__eq,axiom,
! [X: real] :
( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ one_one_real ) )
= ( archim2898591450579166408c_real @ X ) ) ).
% frac_1_eq
thf(fact_7273_frac__1__eq,axiom,
! [X: rat] :
( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
= ( archimedean_frac_rat @ X ) ) ).
% frac_1_eq
thf(fact_7274_cot__def,axiom,
( cot_complex
= ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( cos_complex @ X3 ) @ ( sin_complex @ X3 ) ) ) ) ).
% cot_def
thf(fact_7275_cot__def,axiom,
( cot_real
= ( ^ [X3: real] : ( divide_divide_real @ ( cos_real @ X3 ) @ ( sin_real @ X3 ) ) ) ) ).
% cot_def
thf(fact_7276_frac__eq,axiom,
! [X: rat] :
( ( ( archimedean_frac_rat @ X )
= X )
= ( ( ord_less_eq_rat @ zero_zero_rat @ X )
& ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% frac_eq
thf(fact_7277_frac__eq,axiom,
! [X: real] :
( ( ( archim2898591450579166408c_real @ X )
= X )
= ( ( ord_less_eq_real @ zero_zero_real @ X )
& ( ord_less_real @ X @ one_one_real ) ) ) ).
% frac_eq
thf(fact_7278_frac__add,axiom,
! [X: real,Y: real] :
( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
=> ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) ) )
& ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
=> ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
= ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real ) ) ) ) ).
% frac_add
thf(fact_7279_frac__add,axiom,
! [X: rat,Y: rat] :
( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
=> ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
= ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) ) )
& ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
=> ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
= ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat ) ) ) ) ).
% frac_add
thf(fact_7280_floor__add,axiom,
! [X: real,Y: real] :
( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
=> ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) )
& ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
=> ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ one_one_int ) ) ) ) ).
% floor_add
thf(fact_7281_floor__add,axiom,
! [X: rat,Y: rat] :
( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
=> ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
= ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) )
& ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
=> ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
= ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ one_one_int ) ) ) ) ).
% floor_add
thf(fact_7282_powr__real__of__int,axiom,
! [X: real,N2: int] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ N2 )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ N2 ) )
= ( power_power_real @ X @ ( nat2 @ N2 ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ N2 )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ N2 ) )
= ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N2 ) ) ) ) ) ) ) ) ).
% powr_real_of_int
thf(fact_7283_cis__2pi,axiom,
( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
= one_one_complex ) ).
% cis_2pi
thf(fact_7284_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_7285_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_7286_cmod__complex__polar,axiom,
! [R4: real,A: real] :
( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R4 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
= ( abs_abs_real @ R4 ) ) ).
% cmod_complex_polar
thf(fact_7287_inverse__inverse__eq,axiom,
! [A: real] :
( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_7288_inverse__inverse__eq,axiom,
! [A: complex] :
( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_7289_inverse__inverse__eq,axiom,
! [A: rat] :
( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_7290_inverse__eq__iff__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
= ( A = B ) ) ).
% inverse_eq_iff_eq
thf(fact_7291_inverse__eq__iff__eq,axiom,
! [A: complex,B: complex] :
( ( ( invers8013647133539491842omplex @ A )
= ( invers8013647133539491842omplex @ B ) )
= ( A = B ) ) ).
% inverse_eq_iff_eq
thf(fact_7292_inverse__eq__iff__eq,axiom,
! [A: rat,B: rat] :
( ( ( inverse_inverse_rat @ A )
= ( inverse_inverse_rat @ B ) )
= ( A = B ) ) ).
% inverse_eq_iff_eq
thf(fact_7293_inverse__nonzero__iff__nonzero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_7294_inverse__nonzero__iff__nonzero,axiom,
! [A: complex] :
( ( ( invers8013647133539491842omplex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_7295_inverse__nonzero__iff__nonzero,axiom,
! [A: rat] :
( ( ( inverse_inverse_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_7296_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_7297_inverse__zero,axiom,
( ( invers8013647133539491842omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% inverse_zero
thf(fact_7298_inverse__zero,axiom,
( ( inverse_inverse_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% inverse_zero
thf(fact_7299_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_7300_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_7301_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_7302_inverse__eq__1__iff,axiom,
! [X: real] :
( ( ( inverse_inverse_real @ X )
= one_one_real )
= ( X = one_one_real ) ) ).
% inverse_eq_1_iff
thf(fact_7303_inverse__eq__1__iff,axiom,
! [X: complex] :
( ( ( invers8013647133539491842omplex @ X )
= one_one_complex )
= ( X = one_one_complex ) ) ).
% inverse_eq_1_iff
thf(fact_7304_inverse__eq__1__iff,axiom,
! [X: rat] :
( ( ( inverse_inverse_rat @ X )
= one_one_rat )
= ( X = one_one_rat ) ) ).
% inverse_eq_1_iff
thf(fact_7305_inverse__1,axiom,
( ( inverse_inverse_real @ one_one_real )
= one_one_real ) ).
% inverse_1
thf(fact_7306_inverse__1,axiom,
( ( invers8013647133539491842omplex @ one_one_complex )
= one_one_complex ) ).
% inverse_1
thf(fact_7307_inverse__1,axiom,
( ( inverse_inverse_rat @ one_one_rat )
= one_one_rat ) ).
% inverse_1
thf(fact_7308_inverse__divide,axiom,
! [A: real,B: real] :
( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ B @ A ) ) ).
% inverse_divide
thf(fact_7309_inverse__divide,axiom,
! [A: complex,B: complex] :
( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ B @ A ) ) ).
% inverse_divide
thf(fact_7310_inverse__divide,axiom,
! [A: rat,B: rat] :
( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ B @ A ) ) ).
% inverse_divide
thf(fact_7311_inverse__minus__eq,axiom,
! [A: real] :
( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
= ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ).
% inverse_minus_eq
thf(fact_7312_inverse__minus__eq,axiom,
! [A: complex] :
( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
= ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ).
% inverse_minus_eq
thf(fact_7313_inverse__minus__eq,axiom,
! [A: rat] :
( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
= ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% inverse_minus_eq
thf(fact_7314_abs__inverse,axiom,
! [A: real] :
( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
= ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ).
% abs_inverse
thf(fact_7315_abs__inverse,axiom,
! [A: complex] :
( ( abs_abs_complex @ ( invers8013647133539491842omplex @ A ) )
= ( invers8013647133539491842omplex @ ( abs_abs_complex @ A ) ) ) ).
% abs_inverse
thf(fact_7316_abs__inverse,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
= ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ).
% abs_inverse
thf(fact_7317_inverse__sgn,axiom,
! [A: real] :
( ( inverse_inverse_real @ ( sgn_sgn_real @ A ) )
= ( sgn_sgn_real @ A ) ) ).
% inverse_sgn
thf(fact_7318_inverse__sgn,axiom,
! [A: rat] :
( ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) )
= ( sgn_sgn_rat @ A ) ) ).
% inverse_sgn
thf(fact_7319_sgn__inverse,axiom,
! [A: real] :
( ( sgn_sgn_real @ ( inverse_inverse_real @ A ) )
= ( inverse_inverse_real @ ( sgn_sgn_real @ A ) ) ) ).
% sgn_inverse
thf(fact_7320_sgn__inverse,axiom,
! [A: complex] :
( ( sgn_sgn_complex @ ( invers8013647133539491842omplex @ A ) )
= ( invers8013647133539491842omplex @ ( sgn_sgn_complex @ A ) ) ) ).
% sgn_inverse
thf(fact_7321_sgn__inverse,axiom,
! [A: rat] :
( ( sgn_sgn_rat @ ( inverse_inverse_rat @ A ) )
= ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) ) ) ).
% sgn_inverse
thf(fact_7322_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_7323_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_7324_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_7325_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_7326_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_7327_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_7328_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_7329_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_7330_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_7331_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_7332_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_7333_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_7334_norm__ii,axiom,
( ( real_V1022390504157884413omplex @ imaginary_unit )
= one_one_real ) ).
% norm_ii
thf(fact_7335_norm__cis,axiom,
! [A: real] :
( ( real_V1022390504157884413omplex @ ( cis @ A ) )
= one_one_real ) ).
% norm_cis
thf(fact_7336_cis__zero,axiom,
( ( cis @ zero_zero_real )
= one_one_complex ) ).
% cis_zero
thf(fact_7337_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_7338_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_7339_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_7340_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_7341_left__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
= one_one_real ) ) ).
% left_inverse
thf(fact_7342_left__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
= one_one_complex ) ) ).
% left_inverse
thf(fact_7343_left__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
= one_one_rat ) ) ).
% left_inverse
thf(fact_7344_right__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
= one_one_real ) ) ).
% right_inverse
thf(fact_7345_right__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
= one_one_complex ) ) ).
% right_inverse
thf(fact_7346_right__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
= one_one_rat ) ) ).
% right_inverse
thf(fact_7347_inverse__eq__divide__numeral,axiom,
! [W: num] :
( ( inverse_inverse_real @ ( numeral_numeral_real @ W ) )
= ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ W ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_7348_inverse__eq__divide__numeral,axiom,
! [W: num] :
( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ W ) )
= ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_7349_inverse__eq__divide__numeral,axiom,
! [W: num] :
( ( inverse_inverse_rat @ ( numeral_numeral_rat @ W ) )
= ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_7350_cis__pi,axiom,
( ( cis @ pi )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% cis_pi
thf(fact_7351_divide__i,axiom,
! [X: complex] :
( ( divide1717551699836669952omplex @ X @ imaginary_unit )
= ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).
% divide_i
thf(fact_7352_i__squared,axiom,
( ( times_times_complex @ imaginary_unit @ imaginary_unit )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% i_squared
thf(fact_7353_inverse__eq__divide__neg__numeral,axiom,
! [W: num] :
( ( inverse_inverse_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
= ( divide_divide_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_7354_inverse__eq__divide__neg__numeral,axiom,
! [W: num] :
( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
= ( divide1717551699836669952omplex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_7355_inverse__eq__divide__neg__numeral,axiom,
! [W: num] :
( ( inverse_inverse_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
= ( divide_divide_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_7356_divide__numeral__i,axiom,
! [Z: complex,N2: num] :
( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N2 ) @ imaginary_unit ) )
= ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% divide_numeral_i
thf(fact_7357_power2__i,axiom,
( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% power2_i
thf(fact_7358_cis__pi__half,axiom,
( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= imaginary_unit ) ).
% cis_pi_half
thf(fact_7359_i__even__power,axiom,
! [N2: nat] :
( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) ) ).
% i_even_power
thf(fact_7360_exp__pi__i,axiom,
( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% exp_pi_i
thf(fact_7361_exp__pi__i_H,axiom,
( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% exp_pi_i'
thf(fact_7362_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_7363_mult__commute__imp__mult__inverse__commute,axiom,
! [Y: real,X: real] :
( ( ( times_times_real @ Y @ X )
= ( times_times_real @ X @ Y ) )
=> ( ( times_times_real @ ( inverse_inverse_real @ Y ) @ X )
= ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_7364_mult__commute__imp__mult__inverse__commute,axiom,
! [Y: complex,X: complex] :
( ( ( times_times_complex @ Y @ X )
= ( times_times_complex @ X @ Y ) )
=> ( ( times_times_complex @ ( invers8013647133539491842omplex @ Y ) @ X )
= ( times_times_complex @ X @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_7365_mult__commute__imp__mult__inverse__commute,axiom,
! [Y: rat,X: rat] :
( ( ( times_times_rat @ Y @ X )
= ( times_times_rat @ X @ Y ) )
=> ( ( times_times_rat @ ( inverse_inverse_rat @ Y ) @ X )
= ( times_times_rat @ X @ ( inverse_inverse_rat @ Y ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_7366_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_7367_field__class_Ofield__inverse__zero,axiom,
( ( invers8013647133539491842omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% field_class.field_inverse_zero
thf(fact_7368_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% field_class.field_inverse_zero
thf(fact_7369_inverse__zero__imp__zero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_7370_inverse__zero__imp__zero,axiom,
! [A: complex] :
( ( ( invers8013647133539491842omplex @ A )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ).
% inverse_zero_imp_zero
thf(fact_7371_inverse__zero__imp__zero,axiom,
! [A: rat] :
( ( ( inverse_inverse_rat @ A )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ).
% inverse_zero_imp_zero
thf(fact_7372_nonzero__inverse__eq__imp__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
=> ( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_7373_nonzero__inverse__eq__imp__eq,axiom,
! [A: complex,B: complex] :
( ( ( invers8013647133539491842omplex @ A )
= ( invers8013647133539491842omplex @ B ) )
=> ( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_7374_nonzero__inverse__eq__imp__eq,axiom,
! [A: rat,B: rat] :
( ( ( inverse_inverse_rat @ A )
= ( inverse_inverse_rat @ B ) )
=> ( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_7375_nonzero__inverse__inverse__eq,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_7376_nonzero__inverse__inverse__eq,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_7377_nonzero__inverse__inverse__eq,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_7378_nonzero__imp__inverse__nonzero,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_7379_nonzero__imp__inverse__nonzero,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ A )
!= zero_zero_complex ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_7380_nonzero__imp__inverse__nonzero,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( inverse_inverse_rat @ A )
!= zero_zero_rat ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_7381_nonzero__norm__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
= ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_7382_nonzero__norm__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
= ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_7383_inverse__eq__imp__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
=> ( A = B ) ) ).
% inverse_eq_imp_eq
thf(fact_7384_inverse__eq__imp__eq,axiom,
! [A: complex,B: complex] :
( ( ( invers8013647133539491842omplex @ A )
= ( invers8013647133539491842omplex @ B ) )
=> ( A = B ) ) ).
% inverse_eq_imp_eq
thf(fact_7385_inverse__eq__imp__eq,axiom,
! [A: rat,B: rat] :
( ( ( inverse_inverse_rat @ A )
= ( inverse_inverse_rat @ B ) )
=> ( A = B ) ) ).
% inverse_eq_imp_eq
thf(fact_7386_power__inverse,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ ( inverse_inverse_real @ A ) @ N2 )
= ( inverse_inverse_real @ ( power_power_real @ A @ N2 ) ) ) ).
% power_inverse
thf(fact_7387_power__inverse,axiom,
! [A: complex,N2: nat] :
( ( power_power_complex @ ( invers8013647133539491842omplex @ A ) @ N2 )
= ( invers8013647133539491842omplex @ ( power_power_complex @ A @ N2 ) ) ) ).
% power_inverse
thf(fact_7388_power__inverse,axiom,
! [A: rat,N2: nat] :
( ( power_power_rat @ ( inverse_inverse_rat @ A ) @ N2 )
= ( inverse_inverse_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% power_inverse
thf(fact_7389_real__sqrt__inverse,axiom,
! [X: real] :
( ( sqrt @ ( inverse_inverse_real @ X ) )
= ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).
% real_sqrt_inverse
thf(fact_7390_complex__i__not__one,axiom,
imaginary_unit != one_one_complex ).
% complex_i_not_one
thf(fact_7391_norm__inverse__le__norm,axiom,
! [R4: real,X: real] :
( ( ord_less_eq_real @ R4 @ ( real_V7735802525324610683m_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ R4 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ X ) ) @ ( inverse_inverse_real @ R4 ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_7392_norm__inverse__le__norm,axiom,
! [R4: real,X: complex] :
( ( ord_less_eq_real @ R4 @ ( real_V1022390504157884413omplex @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ R4 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ X ) ) @ ( inverse_inverse_real @ R4 ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_7393_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_7394_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_7395_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_7396_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_7397_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_7398_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_7399_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_7400_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_7401_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_7402_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_7403_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_7404_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_7405_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_7406_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_7407_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_7408_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_7409_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_7410_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_7411_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_7412_nonzero__inverse__minus__eq,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
= ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_7413_nonzero__inverse__minus__eq,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
= ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_7414_nonzero__inverse__minus__eq,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
= ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_7415_inverse__numeral__1,axiom,
( ( inverse_inverse_real @ ( numeral_numeral_real @ one ) )
= ( numeral_numeral_real @ one ) ) ).
% inverse_numeral_1
thf(fact_7416_inverse__numeral__1,axiom,
( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ one ) )
= ( numera6690914467698888265omplex @ one ) ) ).
% inverse_numeral_1
thf(fact_7417_inverse__numeral__1,axiom,
( ( inverse_inverse_rat @ ( numeral_numeral_rat @ one ) )
= ( numeral_numeral_rat @ one ) ) ).
% inverse_numeral_1
thf(fact_7418_inverse__unique,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= one_one_real )
=> ( ( inverse_inverse_real @ A )
= B ) ) ).
% inverse_unique
thf(fact_7419_inverse__unique,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= one_one_complex )
=> ( ( invers8013647133539491842omplex @ A )
= B ) ) ).
% inverse_unique
thf(fact_7420_inverse__unique,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= one_one_rat )
=> ( ( inverse_inverse_rat @ A )
= B ) ) ).
% inverse_unique
thf(fact_7421_field__class_Ofield__divide__inverse,axiom,
( divide_divide_real
= ( ^ [A3: real,B2: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B2 ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_7422_field__class_Ofield__divide__inverse,axiom,
( divide1717551699836669952omplex
= ( ^ [A3: complex,B2: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_7423_field__class_Ofield__divide__inverse,axiom,
( divide_divide_rat
= ( ^ [A3: rat,B2: rat] : ( times_times_rat @ A3 @ ( inverse_inverse_rat @ B2 ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_7424_divide__inverse,axiom,
( divide_divide_real
= ( ^ [A3: real,B2: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B2 ) ) ) ) ).
% divide_inverse
thf(fact_7425_divide__inverse,axiom,
( divide1717551699836669952omplex
= ( ^ [A3: complex,B2: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ).
% divide_inverse
thf(fact_7426_divide__inverse,axiom,
( divide_divide_rat
= ( ^ [A3: rat,B2: rat] : ( times_times_rat @ A3 @ ( inverse_inverse_rat @ B2 ) ) ) ) ).
% divide_inverse
thf(fact_7427_divide__inverse__commute,axiom,
( divide_divide_real
= ( ^ [A3: real,B2: real] : ( times_times_real @ ( inverse_inverse_real @ B2 ) @ A3 ) ) ) ).
% divide_inverse_commute
thf(fact_7428_divide__inverse__commute,axiom,
( divide1717551699836669952omplex
= ( ^ [A3: complex,B2: complex] : ( times_times_complex @ ( invers8013647133539491842omplex @ B2 ) @ A3 ) ) ) ).
% divide_inverse_commute
thf(fact_7429_divide__inverse__commute,axiom,
( divide_divide_rat
= ( ^ [A3: rat,B2: rat] : ( times_times_rat @ ( inverse_inverse_rat @ B2 ) @ A3 ) ) ) ).
% divide_inverse_commute
thf(fact_7430_inverse__eq__divide,axiom,
( inverse_inverse_real
= ( divide_divide_real @ one_one_real ) ) ).
% inverse_eq_divide
thf(fact_7431_inverse__eq__divide,axiom,
( invers8013647133539491842omplex
= ( divide1717551699836669952omplex @ one_one_complex ) ) ).
% inverse_eq_divide
thf(fact_7432_inverse__eq__divide,axiom,
( inverse_inverse_rat
= ( divide_divide_rat @ one_one_rat ) ) ).
% inverse_eq_divide
thf(fact_7433_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_7434_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_7435_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_7436_power__mult__power__inverse__commute,axiom,
! [X: real,M: nat,N2: nat] :
( ( times_times_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N2 ) )
= ( times_times_real @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N2 ) @ ( power_power_real @ X @ M ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_7437_power__mult__power__inverse__commute,axiom,
! [X: complex,M: nat,N2: nat] :
( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N2 ) )
= ( times_times_complex @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N2 ) @ ( power_power_complex @ X @ M ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_7438_power__mult__power__inverse__commute,axiom,
! [X: rat,M: nat,N2: nat] :
( ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N2 ) )
= ( times_times_rat @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N2 ) @ ( power_power_rat @ X @ M ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_7439_mult__inverse__of__nat__commute,axiom,
! [Xa: nat,X: real] :
( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) @ X )
= ( times_times_real @ X @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_7440_mult__inverse__of__nat__commute,axiom,
! [Xa: nat,X: complex] :
( ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) @ X )
= ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_7441_mult__inverse__of__nat__commute,axiom,
! [Xa: nat,X: rat] :
( ( times_times_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) @ X )
= ( times_times_rat @ X @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_7442_nonzero__abs__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
= ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ) ).
% nonzero_abs_inverse
thf(fact_7443_nonzero__abs__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
= ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ) ).
% nonzero_abs_inverse
thf(fact_7444_mult__inverse__of__int__commute,axiom,
! [Xa: int,X: real] :
( ( times_times_real @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) @ X )
= ( times_times_real @ X @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_7445_mult__inverse__of__int__commute,axiom,
! [Xa: int,X: complex] :
( ( times_times_complex @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) @ X )
= ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_7446_mult__inverse__of__int__commute,axiom,
! [Xa: int,X: rat] :
( ( times_times_rat @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) @ X )
= ( times_times_rat @ X @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_7447_divide__real__def,axiom,
( divide_divide_real
= ( ^ [X3: real,Y4: real] : ( times_times_real @ X3 @ ( inverse_inverse_real @ Y4 ) ) ) ) ).
% divide_real_def
thf(fact_7448_cis__mult,axiom,
! [A: real,B: real] :
( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
= ( cis @ ( plus_plus_real @ A @ B ) ) ) ).
% cis_mult
thf(fact_7449_cis__divide,axiom,
! [A: real,B: real] :
( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
= ( cis @ ( minus_minus_real @ A @ B ) ) ) ).
% cis_divide
thf(fact_7450_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_7451_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_7452_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_7453_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_7454_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_7455_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_7456_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_7457_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_7458_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_7459_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_7460_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_7461_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_7462_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_7463_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_7464_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_7465_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_7466_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_7467_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_7468_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_7469_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_7470_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_7471_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_7472_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_7473_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_7474_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_7475_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_7476_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_7477_nonzero__inverse__eq__divide,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ A )
= ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_7478_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_7479_Complex__eq__i,axiom,
! [X: real,Y: real] :
( ( ( complex2 @ X @ Y )
= imaginary_unit )
= ( ( X = zero_zero_real )
& ( Y = one_one_real ) ) ) ).
% Complex_eq_i
thf(fact_7480_imaginary__unit_Ocode,axiom,
( imaginary_unit
= ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% imaginary_unit.code
thf(fact_7481_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_7482_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_7483_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_7484_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_7485_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_7486_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_7487_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_7488_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_7489_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_7490_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_7491_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_7492_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_7493_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_7494_reals__Archimedean,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ X ) ) ).
% reals_Archimedean
thf(fact_7495_reals__Archimedean,axiom,
! [X: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ? [N: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) ) @ X ) ) ).
% reals_Archimedean
thf(fact_7496_forall__pos__mono__1,axiom,
! [P: real > $o,E2: real] :
( ! [D4: real,E: real] :
( ( ord_less_real @ D4 @ E )
=> ( ( P @ D4 )
=> ( P @ E ) ) )
=> ( ! [N: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_7497_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D4: real,E: real] :
( ( ord_less_real @ D4 @ E )
=> ( ( P @ D4 )
=> ( P @ E ) ) )
=> ( ! [N: nat] :
( ( N != zero_zero_nat )
=> ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_7498_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
= ( ? [N4: nat] :
( ( N4 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_7499_sqrt__divide__self__eq,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( divide_divide_real @ ( sqrt @ X ) @ X )
= ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).
% sqrt_divide_self_eq
thf(fact_7500_ln__inverse,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
= ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).
% ln_inverse
thf(fact_7501_ex__inverse__of__nat__less,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) @ X ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_7502_ex__inverse__of__nat__less,axiom,
! [X: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ? [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
& ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N ) ) @ X ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_7503_power__diff__conv__inverse,axiom,
! [X: real,M: nat,N2: nat] :
( ( X != zero_zero_real )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( ( power_power_real @ X @ ( minus_minus_nat @ N2 @ M ) )
= ( times_times_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ M ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_7504_power__diff__conv__inverse,axiom,
! [X: complex,M: nat,N2: nat] :
( ( X != zero_zero_complex )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( ( power_power_complex @ X @ ( minus_minus_nat @ N2 @ M ) )
= ( times_times_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ M ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_7505_power__diff__conv__inverse,axiom,
! [X: rat,M: nat,N2: nat] :
( ( X != zero_zero_rat )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( ( power_power_rat @ X @ ( minus_minus_nat @ N2 @ M ) )
= ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ M ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_7506_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_7507_Complex__eq,axiom,
( complex2
= ( ^ [A3: real,B2: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A3 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B2 ) ) ) ) ) ).
% Complex_eq
thf(fact_7508_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_7509_complex__split__polar,axiom,
! [Z: complex] :
? [R2: real,A5: real] :
( Z
= ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A5 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A5 ) ) ) ) ) ) ).
% complex_split_polar
thf(fact_7510_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_7511_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_7512_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_7513_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_7514_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_7515_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_7516_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_7517_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_7518_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_7519_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_7520_Arg__ii,axiom,
( ( arg @ imaginary_unit )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% Arg_ii
thf(fact_7521_Arg__correct,axiom,
! [Z: complex] :
( ( Z != zero_zero_complex )
=> ( ( ( sgn_sgn_complex @ Z )
= ( cis @ ( arg @ Z ) ) )
& ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
& ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% Arg_correct
thf(fact_7522_cis__Arg__unique,axiom,
! [Z: complex,X: real] :
( ( ( sgn_sgn_complex @ Z )
= ( cis @ X ) )
=> ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ( arg @ Z )
= X ) ) ) ) ).
% cis_Arg_unique
thf(fact_7523_csqrt__eq__1,axiom,
! [Z: complex] :
( ( ( csqrt @ Z )
= one_one_complex )
= ( Z = one_one_complex ) ) ).
% csqrt_eq_1
thf(fact_7524_csqrt__1,axiom,
( ( csqrt @ one_one_complex )
= one_one_complex ) ).
% csqrt_1
thf(fact_7525_power2__csqrt,axiom,
! [Z: complex] :
( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= Z ) ).
% power2_csqrt
thf(fact_7526_divide__complex__def,axiom,
( divide1717551699836669952omplex
= ( ^ [X3: complex,Y4: complex] : ( times_times_complex @ X3 @ ( invers8013647133539491842omplex @ Y4 ) ) ) ) ).
% divide_complex_def
thf(fact_7527_Arg__bounded,axiom,
! [Z: complex] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
& ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% Arg_bounded
thf(fact_7528_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_7529_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_7530_cis__multiple__2pi,axiom,
! [N2: real] :
( ( member_real @ N2 @ ring_1_Ints_real )
=> ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
= one_one_complex ) ) ).
% cis_multiple_2pi
thf(fact_7531_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_7532_Suc__0__xor__eq,axiom,
! [N2: nat] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% Suc_0_xor_eq
thf(fact_7533_xor__Suc__0__eq,axiom,
! [N2: nat] :
( ( bit_se6528837805403552850or_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% xor_Suc_0_eq
thf(fact_7534_bit_Oxor__left__self,axiom,
! [X: int,Y: int] :
( ( bit_se6526347334894502574or_int @ X @ ( bit_se6526347334894502574or_int @ X @ Y ) )
= Y ) ).
% bit.xor_left_self
thf(fact_7535_xor_Oright__neutral,axiom,
! [A: nat] :
( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
= A ) ).
% xor.right_neutral
thf(fact_7536_xor_Oright__neutral,axiom,
! [A: int] :
( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
= A ) ).
% xor.right_neutral
thf(fact_7537_xor_Oleft__neutral,axiom,
! [A: nat] :
( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
= A ) ).
% xor.left_neutral
thf(fact_7538_xor_Oleft__neutral,axiom,
! [A: int] :
( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
= A ) ).
% xor.left_neutral
thf(fact_7539_xor__self__eq,axiom,
! [A: nat] :
( ( bit_se6528837805403552850or_nat @ A @ A )
= zero_zero_nat ) ).
% xor_self_eq
thf(fact_7540_xor__self__eq,axiom,
! [A: int] :
( ( bit_se6526347334894502574or_int @ A @ A )
= zero_zero_int ) ).
% xor_self_eq
thf(fact_7541_bit_Oxor__self,axiom,
! [X: int] :
( ( bit_se6526347334894502574or_int @ X @ X )
= zero_zero_int ) ).
% bit.xor_self
thf(fact_7542_sinh__0,axiom,
( ( sinh_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% sinh_0
thf(fact_7543_sinh__0,axiom,
( ( sinh_real @ zero_zero_real )
= zero_zero_real ) ).
% sinh_0
thf(fact_7544_take__bit__xor,axiom,
! [N2: nat,A: int,B: int] :
( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se6526347334894502574or_int @ A @ B ) )
= ( bit_se6526347334894502574or_int @ ( bit_se2923211474154528505it_int @ N2 @ A ) @ ( bit_se2923211474154528505it_int @ N2 @ B ) ) ) ).
% take_bit_xor
thf(fact_7545_take__bit__xor,axiom,
! [N2: nat,A: nat,B: nat] :
( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se6528837805403552850or_nat @ A @ B ) )
= ( bit_se6528837805403552850or_nat @ ( bit_se2925701944663578781it_nat @ N2 @ A ) @ ( bit_se2925701944663578781it_nat @ N2 @ B ) ) ) ).
% take_bit_xor
thf(fact_7546_sinh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% sinh_real_less_iff
thf(fact_7547_cosh__0,axiom,
( ( cosh_complex @ zero_zero_complex )
= one_one_complex ) ).
% cosh_0
thf(fact_7548_cosh__0,axiom,
( ( cosh_real @ zero_zero_real )
= one_one_real ) ).
% cosh_0
thf(fact_7549_sinh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% sinh_real_neg_iff
thf(fact_7550_sinh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% sinh_real_pos_iff
thf(fact_7551_frac__eq__0__iff,axiom,
! [X: real] :
( ( ( archim2898591450579166408c_real @ X )
= zero_zero_real )
= ( member_real @ X @ ring_1_Ints_real ) ) ).
% frac_eq_0_iff
thf(fact_7552_frac__eq__0__iff,axiom,
! [X: rat] :
( ( ( archimedean_frac_rat @ X )
= zero_zero_rat )
= ( member_rat @ X @ ring_1_Ints_rat ) ) ).
% frac_eq_0_iff
thf(fact_7553_floor__add2,axiom,
! [X: real,Y: real] :
( ( ( member_real @ X @ ring_1_Ints_real )
| ( member_real @ Y @ ring_1_Ints_real ) )
=> ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ) ).
% floor_add2
thf(fact_7554_floor__add2,axiom,
! [X: rat,Y: rat] :
( ( ( member_rat @ X @ ring_1_Ints_rat )
| ( member_rat @ Y @ ring_1_Ints_rat ) )
=> ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
= ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ) ).
% floor_add2
thf(fact_7555_frac__gt__0__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) )
= ( ~ ( member_real @ X @ ring_1_Ints_real ) ) ) ).
% frac_gt_0_iff
thf(fact_7556_frac__gt__0__iff,axiom,
! [X: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) )
= ( ~ ( member_rat @ X @ ring_1_Ints_rat ) ) ) ).
% frac_gt_0_iff
thf(fact_7557_xor__numerals_I3_J,axiom,
! [X: num,Y: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% xor_numerals(3)
thf(fact_7558_xor__numerals_I3_J,axiom,
! [X: num,Y: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% xor_numerals(3)
thf(fact_7559_xor__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% xor_numerals(1)
thf(fact_7560_xor__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
= ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).
% xor_numerals(1)
thf(fact_7561_xor__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% xor_numerals(2)
thf(fact_7562_xor__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
= ( numeral_numeral_int @ ( bit0 @ Y ) ) ) ).
% xor_numerals(2)
thf(fact_7563_xor__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% xor_numerals(5)
thf(fact_7564_xor__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% xor_numerals(5)
thf(fact_7565_xor__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% xor_numerals(8)
thf(fact_7566_xor__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ X ) ) ) ).
% xor_numerals(8)
thf(fact_7567_xor__numerals_I7_J,axiom,
! [X: num,Y: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% xor_numerals(7)
thf(fact_7568_xor__numerals_I7_J,axiom,
! [X: num,Y: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% xor_numerals(7)
thf(fact_7569_xor__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% xor_nat_numerals(1)
thf(fact_7570_xor__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% xor_nat_numerals(2)
thf(fact_7571_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_7572_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_7573_xor__numerals_I4_J,axiom,
! [X: num,Y: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_7574_xor__numerals_I4_J,axiom,
! [X: num,Y: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_7575_xor__numerals_I6_J,axiom,
! [X: num,Y: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_7576_xor__numerals_I6_J,axiom,
! [X: num,Y: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_7577_Ints__0,axiom,
member_complex @ zero_zero_complex @ ring_1_Ints_complex ).
% Ints_0
thf(fact_7578_Ints__0,axiom,
member_real @ zero_zero_real @ ring_1_Ints_real ).
% Ints_0
thf(fact_7579_Ints__0,axiom,
member_rat @ zero_zero_rat @ ring_1_Ints_rat ).
% Ints_0
thf(fact_7580_Ints__0,axiom,
member_int @ zero_zero_int @ ring_1_Ints_int ).
% Ints_0
thf(fact_7581_Ints__power,axiom,
! [A: int,N2: nat] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( member_int @ ( power_power_int @ A @ N2 ) @ ring_1_Ints_int ) ) ).
% Ints_power
thf(fact_7582_Ints__power,axiom,
! [A: real,N2: nat] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( member_real @ ( power_power_real @ A @ N2 ) @ ring_1_Ints_real ) ) ).
% Ints_power
thf(fact_7583_Ints__power,axiom,
! [A: complex,N2: nat] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( member_complex @ ( power_power_complex @ A @ N2 ) @ ring_1_Ints_complex ) ) ).
% Ints_power
thf(fact_7584_Ints__power,axiom,
! [A: rat,N2: nat] :
( ( member_rat @ A @ ring_1_Ints_rat )
=> ( member_rat @ ( power_power_rat @ A @ N2 ) @ ring_1_Ints_rat ) ) ).
% Ints_power
thf(fact_7585_Ints__of__nat,axiom,
! [N2: nat] : ( member_real @ ( semiri5074537144036343181t_real @ N2 ) @ ring_1_Ints_real ) ).
% Ints_of_nat
thf(fact_7586_Ints__of__nat,axiom,
! [N2: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N2 ) @ ring_1_Ints_int ) ).
% Ints_of_nat
thf(fact_7587_Ints__of__nat,axiom,
! [N2: nat] : ( member_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ring_1_Ints_complex ) ).
% Ints_of_nat
thf(fact_7588_Ints__of__nat,axiom,
! [N2: nat] : ( member_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ring_1_Ints_rat ) ).
% Ints_of_nat
thf(fact_7589_bit_Oconj__xor__distrib,axiom,
! [X: int,Y: int,Z: int] :
( ( bit_se725231765392027082nd_int @ X @ ( bit_se6526347334894502574or_int @ Y @ Z ) )
= ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se725231765392027082nd_int @ X @ Z ) ) ) ).
% bit.conj_xor_distrib
thf(fact_7590_bit_Oconj__xor__distrib2,axiom,
! [Y: int,Z: int,X: int] :
( ( bit_se725231765392027082nd_int @ ( bit_se6526347334894502574or_int @ Y @ Z ) @ X )
= ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ Y @ X ) @ ( bit_se725231765392027082nd_int @ Z @ X ) ) ) ).
% bit.conj_xor_distrib2
thf(fact_7591_bit__xor__iff,axiom,
! [A: nat,B: nat,N2: nat] :
( ( bit_se1148574629649215175it_nat @ ( bit_se6528837805403552850or_nat @ A @ B ) @ N2 )
= ( ( bit_se1148574629649215175it_nat @ A @ N2 )
!= ( bit_se1148574629649215175it_nat @ B @ N2 ) ) ) ).
% bit_xor_iff
thf(fact_7592_bit__xor__iff,axiom,
! [A: int,B: int,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ A @ B ) @ N2 )
= ( ( bit_se1146084159140164899it_int @ A @ N2 )
!= ( bit_se1146084159140164899it_int @ B @ N2 ) ) ) ).
% bit_xor_iff
thf(fact_7593_Ints__abs,axiom,
! [A: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( member_int @ ( abs_abs_int @ A ) @ ring_1_Ints_int ) ) ).
% Ints_abs
thf(fact_7594_Ints__abs,axiom,
! [A: code_integer] :
( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
=> ( member_Code_integer @ ( abs_abs_Code_integer @ A ) @ ring_11222124179247155820nteger ) ) ).
% Ints_abs
thf(fact_7595_Ints__abs,axiom,
! [A: rat] :
( ( member_rat @ A @ ring_1_Ints_rat )
=> ( member_rat @ ( abs_abs_rat @ A ) @ ring_1_Ints_rat ) ) ).
% Ints_abs
thf(fact_7596_Ints__abs,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( member_real @ ( abs_abs_real @ A ) @ ring_1_Ints_real ) ) ).
% Ints_abs
thf(fact_7597_cosh__add,axiom,
! [X: real,Y: real] :
( ( cosh_real @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% cosh_add
thf(fact_7598_sinh__add,axiom,
! [X: real,Y: real] :
( ( sinh_real @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% sinh_add
thf(fact_7599_sinh__less__cosh__real,axiom,
! [X: real] : ( ord_less_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% sinh_less_cosh_real
thf(fact_7600_xor_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( bit_se6528837805403552850or_nat @ B @ ( bit_se6528837805403552850or_nat @ A @ C ) )
= ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B @ C ) ) ) ).
% xor.left_commute
thf(fact_7601_xor_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( bit_se6526347334894502574or_int @ B @ ( bit_se6526347334894502574or_int @ A @ C ) )
= ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B @ C ) ) ) ).
% xor.left_commute
thf(fact_7602_xor_Ocommute,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [A3: nat,B2: nat] : ( bit_se6528837805403552850or_nat @ B2 @ A3 ) ) ) ).
% xor.commute
thf(fact_7603_xor_Ocommute,axiom,
( bit_se6526347334894502574or_int
= ( ^ [A3: int,B2: int] : ( bit_se6526347334894502574or_int @ B2 @ A3 ) ) ) ).
% xor.commute
thf(fact_7604_xor_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( bit_se6528837805403552850or_nat @ ( bit_se6528837805403552850or_nat @ A @ B ) @ C )
= ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B @ C ) ) ) ).
% xor.assoc
thf(fact_7605_xor_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( bit_se6526347334894502574or_int @ ( bit_se6526347334894502574or_int @ A @ B ) @ C )
= ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B @ C ) ) ) ).
% xor.assoc
thf(fact_7606_of__nat__xor__eq,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se6528837805403552850or_nat @ M @ N2 ) )
= ( bit_se6528837805403552850or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_xor_eq
thf(fact_7607_of__nat__xor__eq,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se6528837805403552850or_nat @ M @ N2 ) )
= ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_xor_eq
thf(fact_7608_Ints__diff,axiom,
! [A: complex,B: complex] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( ( member_complex @ B @ ring_1_Ints_complex )
=> ( member_complex @ ( minus_minus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% Ints_diff
thf(fact_7609_Ints__diff,axiom,
! [A: real,B: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( member_real @ B @ ring_1_Ints_real )
=> ( member_real @ ( minus_minus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% Ints_diff
thf(fact_7610_Ints__diff,axiom,
! [A: rat,B: rat] :
( ( member_rat @ A @ ring_1_Ints_rat )
=> ( ( member_rat @ B @ ring_1_Ints_rat )
=> ( member_rat @ ( minus_minus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% Ints_diff
thf(fact_7611_Ints__diff,axiom,
! [A: int,B: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( member_int @ B @ ring_1_Ints_int )
=> ( member_int @ ( minus_minus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% Ints_diff
thf(fact_7612_Ints__add,axiom,
! [A: complex,B: complex] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( ( member_complex @ B @ ring_1_Ints_complex )
=> ( member_complex @ ( plus_plus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% Ints_add
thf(fact_7613_Ints__add,axiom,
! [A: real,B: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( member_real @ B @ ring_1_Ints_real )
=> ( member_real @ ( plus_plus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% Ints_add
thf(fact_7614_Ints__add,axiom,
! [A: rat,B: rat] :
( ( member_rat @ A @ ring_1_Ints_rat )
=> ( ( member_rat @ B @ ring_1_Ints_rat )
=> ( member_rat @ ( plus_plus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% Ints_add
thf(fact_7615_Ints__add,axiom,
! [A: int,B: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( member_int @ B @ ring_1_Ints_int )
=> ( member_int @ ( plus_plus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% Ints_add
thf(fact_7616_Ints__1,axiom,
member_complex @ one_one_complex @ ring_1_Ints_complex ).
% Ints_1
thf(fact_7617_Ints__1,axiom,
member_rat @ one_one_rat @ ring_1_Ints_rat ).
% Ints_1
thf(fact_7618_Ints__1,axiom,
member_int @ one_one_int @ ring_1_Ints_int ).
% Ints_1
thf(fact_7619_Ints__1,axiom,
member_real @ one_one_real @ ring_1_Ints_real ).
% Ints_1
thf(fact_7620_Ints__mult,axiom,
! [A: complex,B: complex] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( ( member_complex @ B @ ring_1_Ints_complex )
=> ( member_complex @ ( times_times_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% Ints_mult
thf(fact_7621_Ints__mult,axiom,
! [A: real,B: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( member_real @ B @ ring_1_Ints_real )
=> ( member_real @ ( times_times_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% Ints_mult
thf(fact_7622_Ints__mult,axiom,
! [A: rat,B: rat] :
( ( member_rat @ A @ ring_1_Ints_rat )
=> ( ( member_rat @ B @ ring_1_Ints_rat )
=> ( member_rat @ ( times_times_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).
% Ints_mult
thf(fact_7623_Ints__mult,axiom,
! [A: int,B: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( member_int @ B @ ring_1_Ints_int )
=> ( member_int @ ( times_times_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% Ints_mult
thf(fact_7624_of__int__xor__eq,axiom,
! [K: int,L: int] :
( ( ring_1_of_int_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
= ( bit_se6526347334894502574or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).
% of_int_xor_eq
thf(fact_7625_tanh__def,axiom,
( tanh_complex
= ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( sinh_complex @ X3 ) @ ( cosh_complex @ X3 ) ) ) ) ).
% tanh_def
thf(fact_7626_tanh__def,axiom,
( tanh_real
= ( ^ [X3: real] : ( divide_divide_real @ ( sinh_real @ X3 ) @ ( cosh_real @ X3 ) ) ) ) ).
% tanh_def
thf(fact_7627_Ints__minus,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( member_real @ ( uminus_uminus_real @ A ) @ ring_1_Ints_real ) ) ).
% Ints_minus
thf(fact_7628_Ints__minus,axiom,
! [A: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( member_int @ ( uminus_uminus_int @ A ) @ ring_1_Ints_int ) ) ).
% Ints_minus
thf(fact_7629_Ints__minus,axiom,
! [A: complex] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( member_complex @ ( uminus1482373934393186551omplex @ A ) @ ring_1_Ints_complex ) ) ).
% Ints_minus
thf(fact_7630_Ints__minus,axiom,
! [A: code_integer] :
( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
=> ( member_Code_integer @ ( uminus1351360451143612070nteger @ A ) @ ring_11222124179247155820nteger ) ) ).
% Ints_minus
thf(fact_7631_Ints__minus,axiom,
! [A: rat] :
( ( member_rat @ A @ ring_1_Ints_rat )
=> ( member_rat @ ( uminus_uminus_rat @ A ) @ ring_1_Ints_rat ) ) ).
% Ints_minus
thf(fact_7632_minus__in__Ints__iff,axiom,
! [X: real] :
( ( member_real @ ( uminus_uminus_real @ X ) @ ring_1_Ints_real )
= ( member_real @ X @ ring_1_Ints_real ) ) ).
% minus_in_Ints_iff
thf(fact_7633_minus__in__Ints__iff,axiom,
! [X: int] :
( ( member_int @ ( uminus_uminus_int @ X ) @ ring_1_Ints_int )
= ( member_int @ X @ ring_1_Ints_int ) ) ).
% minus_in_Ints_iff
thf(fact_7634_minus__in__Ints__iff,axiom,
! [X: complex] :
( ( member_complex @ ( uminus1482373934393186551omplex @ X ) @ ring_1_Ints_complex )
= ( member_complex @ X @ ring_1_Ints_complex ) ) ).
% minus_in_Ints_iff
thf(fact_7635_minus__in__Ints__iff,axiom,
! [X: code_integer] :
( ( member_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ ring_11222124179247155820nteger )
= ( member_Code_integer @ X @ ring_11222124179247155820nteger ) ) ).
% minus_in_Ints_iff
thf(fact_7636_minus__in__Ints__iff,axiom,
! [X: rat] :
( ( member_rat @ ( uminus_uminus_rat @ X ) @ ring_1_Ints_rat )
= ( member_rat @ X @ ring_1_Ints_rat ) ) ).
% minus_in_Ints_iff
thf(fact_7637_Ints__numeral,axiom,
! [N2: num] : ( member_complex @ ( numera6690914467698888265omplex @ N2 ) @ ring_1_Ints_complex ) ).
% Ints_numeral
thf(fact_7638_Ints__numeral,axiom,
! [N2: num] : ( member_real @ ( numeral_numeral_real @ N2 ) @ ring_1_Ints_real ) ).
% Ints_numeral
thf(fact_7639_Ints__numeral,axiom,
! [N2: num] : ( member_rat @ ( numeral_numeral_rat @ N2 ) @ ring_1_Ints_rat ) ).
% Ints_numeral
thf(fact_7640_Ints__numeral,axiom,
! [N2: num] : ( member_int @ ( numeral_numeral_int @ N2 ) @ ring_1_Ints_int ) ).
% Ints_numeral
thf(fact_7641_cosh__plus__sinh,axiom,
! [X: complex] :
( ( plus_plus_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ X ) )
= ( exp_complex @ X ) ) ).
% cosh_plus_sinh
thf(fact_7642_cosh__plus__sinh,axiom,
! [X: real] :
( ( plus_plus_real @ ( cosh_real @ X ) @ ( sinh_real @ X ) )
= ( exp_real @ X ) ) ).
% cosh_plus_sinh
thf(fact_7643_sinh__plus__cosh,axiom,
! [X: complex] :
( ( plus_plus_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) )
= ( exp_complex @ X ) ) ).
% sinh_plus_cosh
thf(fact_7644_sinh__plus__cosh,axiom,
! [X: real] :
( ( plus_plus_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) )
= ( exp_real @ X ) ) ).
% sinh_plus_cosh
thf(fact_7645_Ints__of__int,axiom,
! [Z: int] : ( member_complex @ ( ring_17405671764205052669omplex @ Z ) @ ring_1_Ints_complex ) ).
% Ints_of_int
thf(fact_7646_Ints__of__int,axiom,
! [Z: int] : ( member_int @ ( ring_1_of_int_int @ Z ) @ ring_1_Ints_int ) ).
% Ints_of_int
thf(fact_7647_Ints__of__int,axiom,
! [Z: int] : ( member_real @ ( ring_1_of_int_real @ Z ) @ ring_1_Ints_real ) ).
% Ints_of_int
thf(fact_7648_Ints__of__int,axiom,
! [Z: int] : ( member_rat @ ( ring_1_of_int_rat @ Z ) @ ring_1_Ints_rat ) ).
% Ints_of_int
thf(fact_7649_Ints__induct,axiom,
! [Q3: complex,P: complex > $o] :
( ( member_complex @ Q3 @ ring_1_Ints_complex )
=> ( ! [Z2: int] : ( P @ ( ring_17405671764205052669omplex @ Z2 ) )
=> ( P @ Q3 ) ) ) ).
% Ints_induct
thf(fact_7650_Ints__induct,axiom,
! [Q3: int,P: int > $o] :
( ( member_int @ Q3 @ ring_1_Ints_int )
=> ( ! [Z2: int] : ( P @ ( ring_1_of_int_int @ Z2 ) )
=> ( P @ Q3 ) ) ) ).
% Ints_induct
thf(fact_7651_Ints__induct,axiom,
! [Q3: real,P: real > $o] :
( ( member_real @ Q3 @ ring_1_Ints_real )
=> ( ! [Z2: int] : ( P @ ( ring_1_of_int_real @ Z2 ) )
=> ( P @ Q3 ) ) ) ).
% Ints_induct
thf(fact_7652_Ints__induct,axiom,
! [Q3: rat,P: rat > $o] :
( ( member_rat @ Q3 @ ring_1_Ints_rat )
=> ( ! [Z2: int] : ( P @ ( ring_1_of_int_rat @ Z2 ) )
=> ( P @ Q3 ) ) ) ).
% Ints_induct
thf(fact_7653_Ints__cases,axiom,
! [Q3: complex] :
( ( member_complex @ Q3 @ ring_1_Ints_complex )
=> ~ ! [Z2: int] :
( Q3
!= ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% Ints_cases
thf(fact_7654_Ints__cases,axiom,
! [Q3: int] :
( ( member_int @ Q3 @ ring_1_Ints_int )
=> ~ ! [Z2: int] :
( Q3
!= ( ring_1_of_int_int @ Z2 ) ) ) ).
% Ints_cases
thf(fact_7655_Ints__cases,axiom,
! [Q3: real] :
( ( member_real @ Q3 @ ring_1_Ints_real )
=> ~ ! [Z2: int] :
( Q3
!= ( ring_1_of_int_real @ Z2 ) ) ) ).
% Ints_cases
thf(fact_7656_Ints__cases,axiom,
! [Q3: rat] :
( ( member_rat @ Q3 @ ring_1_Ints_rat )
=> ~ ! [Z2: int] :
( Q3
!= ( ring_1_of_int_rat @ Z2 ) ) ) ).
% Ints_cases
thf(fact_7657_cosh__real__pos,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% cosh_real_pos
thf(fact_7658_cosh__real__ge__1,axiom,
! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).
% cosh_real_ge_1
thf(fact_7659_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_7660_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_7661_Ints__double__eq__0__iff,axiom,
! [A: complex] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( ( ( plus_plus_complex @ A @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_7662_Ints__double__eq__0__iff,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_7663_Ints__double__eq__0__iff,axiom,
! [A: rat] :
( ( member_rat @ A @ ring_1_Ints_rat )
=> ( ( ( plus_plus_rat @ A @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_7664_Ints__double__eq__0__iff,axiom,
! [A: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_7665_cosh__real__nonpos__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
= ( ord_less_real @ Y @ X ) ) ) ) ).
% cosh_real_nonpos_less_iff
thf(fact_7666_cosh__real__nonneg__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% cosh_real_nonneg_less_iff
thf(fact_7667_cosh__real__strict__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) ) ) ).
% cosh_real_strict_mono
thf(fact_7668_Ints__odd__nonzero,axiom,
! [A: complex] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A ) @ A )
!= zero_zero_complex ) ) ).
% Ints_odd_nonzero
thf(fact_7669_Ints__odd__nonzero,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
!= zero_zero_real ) ) ).
% Ints_odd_nonzero
thf(fact_7670_Ints__odd__nonzero,axiom,
! [A: rat] :
( ( member_rat @ A @ ring_1_Ints_rat )
=> ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A )
!= zero_zero_rat ) ) ).
% Ints_odd_nonzero
thf(fact_7671_Ints__odd__nonzero,axiom,
! [A: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
!= zero_zero_int ) ) ).
% Ints_odd_nonzero
thf(fact_7672_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_7673_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_7674_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_7675_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_7676_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_7677_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_7678_of__int__divide__in__Ints,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( member_complex @ ( divide1717551699836669952omplex @ ( ring_17405671764205052669omplex @ A ) @ ( ring_17405671764205052669omplex @ B ) ) @ ring_1_Ints_complex ) ) ).
% of_int_divide_in_Ints
thf(fact_7679_of__int__divide__in__Ints,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( member_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) @ ring_1_Ints_real ) ) ).
% of_int_divide_in_Ints
thf(fact_7680_of__int__divide__in__Ints,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( member_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) @ ring_1_Ints_rat ) ) ).
% of_int_divide_in_Ints
thf(fact_7681_of__int__divide__in__Ints,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( member_int @ ( divide_divide_int @ ( ring_1_of_int_int @ A ) @ ( ring_1_of_int_int @ B ) ) @ ring_1_Ints_int ) ) ).
% of_int_divide_in_Ints
thf(fact_7682_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_7683_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_7684_even__xor__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ A @ B ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_xor_iff
thf(fact_7685_even__xor__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_xor_iff
thf(fact_7686_even__xor__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_xor_iff
thf(fact_7687_Ints__odd__less__0,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% Ints_odd_less_0
thf(fact_7688_Ints__odd__less__0,axiom,
! [A: rat] :
( ( member_rat @ A @ ring_1_Ints_rat )
=> ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% Ints_odd_less_0
thf(fact_7689_Ints__odd__less__0,axiom,
! [A: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% Ints_odd_less_0
thf(fact_7690_Ints__nonzero__abs__ge1,axiom,
! [X: code_integer] :
( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
=> ( ( X != zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_7691_Ints__nonzero__abs__ge1,axiom,
! [X: rat] :
( ( member_rat @ X @ ring_1_Ints_rat )
=> ( ( X != zero_zero_rat )
=> ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_7692_Ints__nonzero__abs__ge1,axiom,
! [X: int] :
( ( member_int @ X @ ring_1_Ints_int )
=> ( ( X != zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_7693_Ints__nonzero__abs__ge1,axiom,
! [X: real] :
( ( member_real @ X @ ring_1_Ints_real )
=> ( ( X != zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_7694_Ints__nonzero__abs__less1,axiom,
! [X: code_integer] :
( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
=> ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer )
=> ( X = zero_z3403309356797280102nteger ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_7695_Ints__nonzero__abs__less1,axiom,
! [X: real] :
( ( member_real @ X @ ring_1_Ints_real )
=> ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( X = zero_zero_real ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_7696_Ints__nonzero__abs__less1,axiom,
! [X: rat] :
( ( member_rat @ X @ ring_1_Ints_rat )
=> ( ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat )
=> ( X = zero_zero_rat ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_7697_Ints__nonzero__abs__less1,axiom,
! [X: int] :
( ( member_int @ X @ ring_1_Ints_int )
=> ( ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int )
=> ( X = zero_zero_int ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_7698_Ints__eq__abs__less1,axiom,
! [X: code_integer,Y: code_integer] :
( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
=> ( ( member_Code_integer @ Y @ ring_11222124179247155820nteger )
=> ( ( X = Y )
= ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ Y ) ) @ one_one_Code_integer ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_7699_Ints__eq__abs__less1,axiom,
! [X: real,Y: real] :
( ( member_real @ X @ ring_1_Ints_real )
=> ( ( member_real @ Y @ ring_1_Ints_real )
=> ( ( X = Y )
= ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) @ one_one_real ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_7700_Ints__eq__abs__less1,axiom,
! [X: rat,Y: rat] :
( ( member_rat @ X @ ring_1_Ints_rat )
=> ( ( member_rat @ Y @ ring_1_Ints_rat )
=> ( ( X = Y )
= ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ Y ) ) @ one_one_rat ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_7701_Ints__eq__abs__less1,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ ring_1_Ints_int )
=> ( ( member_int @ Y @ ring_1_Ints_int )
=> ( ( X = Y )
= ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Y ) ) @ one_one_int ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_7702_frac__neg,axiom,
! [X: real] :
( ( ( member_real @ X @ ring_1_Ints_real )
=> ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
= zero_zero_real ) )
& ( ~ ( member_real @ X @ ring_1_Ints_real )
=> ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
= ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X ) ) ) ) ) ).
% frac_neg
thf(fact_7703_frac__neg,axiom,
! [X: rat] :
( ( ( member_rat @ X @ ring_1_Ints_rat )
=> ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
= zero_zero_rat ) )
& ( ~ ( member_rat @ X @ ring_1_Ints_rat )
=> ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
= ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X ) ) ) ) ) ).
% frac_neg
thf(fact_7704_le__mult__floor__Ints,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( member_real @ A @ ring_1_Ints_real )
=> ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_7705_le__mult__floor__Ints,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( member_real @ A @ ring_1_Ints_real )
=> ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_int @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_7706_le__mult__floor__Ints,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( member_real @ A @ ring_1_Ints_real )
=> ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_7707_le__mult__floor__Ints,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( member_rat @ A @ ring_1_Ints_rat )
=> ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_7708_le__mult__floor__Ints,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( member_rat @ A @ ring_1_Ints_rat )
=> ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_int @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_7709_le__mult__floor__Ints,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( member_rat @ A @ ring_1_Ints_rat )
=> ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_real @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_7710_frac__unique__iff,axiom,
! [X: rat,A: rat] :
( ( ( archimedean_frac_rat @ X )
= A )
= ( ( member_rat @ ( minus_minus_rat @ X @ A ) @ ring_1_Ints_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ A @ one_one_rat ) ) ) ).
% frac_unique_iff
thf(fact_7711_frac__unique__iff,axiom,
! [X: real,A: real] :
( ( ( archim2898591450579166408c_real @ X )
= A )
= ( ( member_real @ ( minus_minus_real @ X @ A ) @ ring_1_Ints_real )
& ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_real @ A @ one_one_real ) ) ) ).
% frac_unique_iff
thf(fact_7712_mult__ceiling__le__Ints,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( member_rat @ A @ ring_1_Ints_rat )
=> ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_7713_mult__ceiling__le__Ints,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( member_rat @ A @ ring_1_Ints_rat )
=> ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_7714_mult__ceiling__le__Ints,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( member_rat @ A @ ring_1_Ints_rat )
=> ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_7715_mult__ceiling__le__Ints,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( member_real @ A @ ring_1_Ints_real )
=> ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_7716_mult__ceiling__le__Ints,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( member_real @ A @ ring_1_Ints_real )
=> ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_7717_mult__ceiling__le__Ints,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( member_real @ A @ ring_1_Ints_real )
=> ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_7718_tanh__add,axiom,
! [X: complex,Y: complex] :
( ( ( cosh_complex @ X )
!= zero_zero_complex )
=> ( ( ( cosh_complex @ Y )
!= zero_zero_complex )
=> ( ( tanh_complex @ ( plus_plus_complex @ X @ Y ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y ) ) ) ) ) ) ) ).
% tanh_add
thf(fact_7719_tanh__add,axiom,
! [X: real,Y: real] :
( ( ( cosh_real @ X )
!= zero_zero_real )
=> ( ( ( cosh_real @ Y )
!= zero_zero_real )
=> ( ( tanh_real @ ( plus_plus_real @ X @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) ) ) ) ) ) ) ).
% tanh_add
thf(fact_7720_sin__integer__2pi,axiom,
! [N2: real] :
( ( member_real @ N2 @ ring_1_Ints_real )
=> ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
= zero_zero_real ) ) ).
% sin_integer_2pi
thf(fact_7721_cos__integer__2pi,axiom,
! [N2: real] :
( ( member_real @ N2 @ ring_1_Ints_real )
=> ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
= one_one_real ) ) ).
% cos_integer_2pi
thf(fact_7722_cosh__field__def,axiom,
( cosh_real
= ( ^ [Z5: real] : ( divide_divide_real @ ( plus_plus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% cosh_field_def
thf(fact_7723_cosh__field__def,axiom,
( cosh_complex
= ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% cosh_field_def
thf(fact_7724_xor__nat__unfold,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M2: nat,N4: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = 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 @ N4 @ ( 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 @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_7725_xor__nat__rec,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M2: nat,N4: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_7726_xor__one__eq,axiom,
! [A: code_integer] :
( ( bit_se3222712562003087583nteger @ A @ one_one_Code_integer )
= ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
@ ( zero_n356916108424825756nteger
@ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% xor_one_eq
thf(fact_7727_xor__one__eq,axiom,
! [A: nat] :
( ( bit_se6528837805403552850or_nat @ A @ one_one_nat )
= ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% xor_one_eq
thf(fact_7728_xor__one__eq,axiom,
! [A: int] :
( ( bit_se6526347334894502574or_int @ A @ one_one_int )
= ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
@ ( zero_n2684676970156552555ol_int
@ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% xor_one_eq
thf(fact_7729_one__xor__eq,axiom,
! [A: code_integer] :
( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ A )
= ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
@ ( zero_n356916108424825756nteger
@ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% one_xor_eq
thf(fact_7730_one__xor__eq,axiom,
! [A: nat] :
( ( bit_se6528837805403552850or_nat @ one_one_nat @ A )
= ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% one_xor_eq
thf(fact_7731_one__xor__eq,axiom,
! [A: int] :
( ( bit_se6526347334894502574or_int @ one_one_int @ A )
= ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
@ ( zero_n2684676970156552555ol_int
@ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% one_xor_eq
thf(fact_7732_sinh__field__def,axiom,
( sinh_real
= ( ^ [Z5: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% sinh_field_def
thf(fact_7733_sinh__field__def,axiom,
( sinh_complex
= ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% sinh_field_def
thf(fact_7734_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_7735_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_7736_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_7737_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_7738_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_7739_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_7740_push__bit__numeral__minus__1,axiom,
! [N2: num] :
( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ N2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_7741_push__bit__numeral__minus__1,axiom,
! [N2: num] :
( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N2 ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_7742_psubsetI,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_nat @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_7743_vebt__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa )
=> ( ! [A5: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( ( Xa = zero_zero_nat )
=> A5 )
& ( ( Xa != zero_zero_nat )
=> ( ( ( Xa = one_one_nat )
=> B5 )
& ( Xa = one_one_nat ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ 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 ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less_nat @ Xa @ Mi )
& ( ~ ( ord_less_nat @ Xa @ Mi )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa )
& ( ~ ( ord_less_nat @ Ma2 @ Xa )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_7744_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% set_vebt'_def
thf(fact_7745_push__bit__nonnegative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N2 @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% push_bit_nonnegative_int_iff
thf(fact_7746_push__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_int @ ( bit_se545348938243370406it_int @ N2 @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% push_bit_negative_int_iff
thf(fact_7747_push__bit__of__0,axiom,
! [N2: nat] :
( ( bit_se545348938243370406it_int @ N2 @ zero_zero_int )
= zero_zero_int ) ).
% push_bit_of_0
thf(fact_7748_push__bit__of__0,axiom,
! [N2: nat] :
( ( bit_se547839408752420682it_nat @ N2 @ zero_zero_nat )
= zero_zero_nat ) ).
% push_bit_of_0
thf(fact_7749_push__bit__eq__0__iff,axiom,
! [N2: nat,A: int] :
( ( ( bit_se545348938243370406it_int @ N2 @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% push_bit_eq_0_iff
thf(fact_7750_push__bit__eq__0__iff,axiom,
! [N2: nat,A: nat] :
( ( ( bit_se547839408752420682it_nat @ N2 @ A )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% push_bit_eq_0_iff
thf(fact_7751_push__bit__push__bit,axiom,
! [M: nat,N2: nat,A: int] :
( ( bit_se545348938243370406it_int @ M @ ( bit_se545348938243370406it_int @ N2 @ A ) )
= ( bit_se545348938243370406it_int @ ( plus_plus_nat @ M @ N2 ) @ A ) ) ).
% push_bit_push_bit
thf(fact_7752_push__bit__push__bit,axiom,
! [M: nat,N2: nat,A: nat] :
( ( bit_se547839408752420682it_nat @ M @ ( bit_se547839408752420682it_nat @ N2 @ A ) )
= ( bit_se547839408752420682it_nat @ ( plus_plus_nat @ M @ N2 ) @ A ) ) ).
% push_bit_push_bit
thf(fact_7753_push__bit__and,axiom,
! [N2: nat,A: int,B: int] :
( ( bit_se545348938243370406it_int @ N2 @ ( bit_se725231765392027082nd_int @ A @ B ) )
= ( bit_se725231765392027082nd_int @ ( bit_se545348938243370406it_int @ N2 @ A ) @ ( bit_se545348938243370406it_int @ N2 @ B ) ) ) ).
% push_bit_and
thf(fact_7754_push__bit__and,axiom,
! [N2: nat,A: nat,B: nat] :
( ( bit_se547839408752420682it_nat @ N2 @ ( bit_se727722235901077358nd_nat @ A @ B ) )
= ( bit_se727722235901077358nd_nat @ ( bit_se547839408752420682it_nat @ N2 @ A ) @ ( bit_se547839408752420682it_nat @ N2 @ B ) ) ) ).
% push_bit_and
thf(fact_7755_push__bit__xor,axiom,
! [N2: nat,A: int,B: int] :
( ( bit_se545348938243370406it_int @ N2 @ ( bit_se6526347334894502574or_int @ A @ B ) )
= ( bit_se6526347334894502574or_int @ ( bit_se545348938243370406it_int @ N2 @ A ) @ ( bit_se545348938243370406it_int @ N2 @ B ) ) ) ).
% push_bit_xor
thf(fact_7756_push__bit__xor,axiom,
! [N2: nat,A: nat,B: nat] :
( ( bit_se547839408752420682it_nat @ N2 @ ( bit_se6528837805403552850or_nat @ A @ B ) )
= ( bit_se6528837805403552850or_nat @ ( bit_se547839408752420682it_nat @ N2 @ A ) @ ( bit_se547839408752420682it_nat @ N2 @ B ) ) ) ).
% push_bit_xor
thf(fact_7757_concat__bit__of__zero__1,axiom,
! [N2: nat,L: int] :
( ( bit_concat_bit @ N2 @ zero_zero_int @ L )
= ( bit_se545348938243370406it_int @ N2 @ L ) ) ).
% concat_bit_of_zero_1
thf(fact_7758_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K ) )
= zero_zero_complex ) ).
% gbinomial_0(2)
thf(fact_7759_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
= zero_zero_real ) ).
% gbinomial_0(2)
thf(fact_7760_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
= zero_zero_rat ) ).
% gbinomial_0(2)
thf(fact_7761_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% gbinomial_0(2)
thf(fact_7762_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
= zero_zero_int ) ).
% gbinomial_0(2)
thf(fact_7763_gbinomial__0_I1_J,axiom,
! [A: complex] :
( ( gbinomial_complex @ A @ zero_zero_nat )
= one_one_complex ) ).
% gbinomial_0(1)
thf(fact_7764_gbinomial__0_I1_J,axiom,
! [A: real] :
( ( gbinomial_real @ A @ zero_zero_nat )
= one_one_real ) ).
% gbinomial_0(1)
thf(fact_7765_gbinomial__0_I1_J,axiom,
! [A: rat] :
( ( gbinomial_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% gbinomial_0(1)
thf(fact_7766_gbinomial__0_I1_J,axiom,
! [A: nat] :
( ( gbinomial_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% gbinomial_0(1)
thf(fact_7767_gbinomial__0_I1_J,axiom,
! [A: int] :
( ( gbinomial_int @ A @ zero_zero_nat )
= one_one_int ) ).
% gbinomial_0(1)
thf(fact_7768_xor__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
= ( ( ord_less_eq_int @ zero_zero_int @ K )
= ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% xor_nonnegative_int_iff
thf(fact_7769_xor__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
= ( ( ord_less_int @ K @ zero_zero_int )
!= ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% xor_negative_int_iff
thf(fact_7770_push__bit__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( bit_se545348938243370406it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ K ) )
= ( bit_se545348938243370406it_int @ N2 @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% push_bit_Suc_numeral
thf(fact_7771_push__bit__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( bit_se547839408752420682it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
= ( bit_se547839408752420682it_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% push_bit_Suc_numeral
thf(fact_7772_push__bit__Suc__minus__numeral,axiom,
! [N2: nat,K: num] :
( ( bit_se7788150548672797655nteger @ ( suc @ N2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( bit_se7788150548672797655nteger @ N2 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_Suc_minus_numeral
thf(fact_7773_push__bit__Suc__minus__numeral,axiom,
! [N2: nat,K: num] :
( ( bit_se545348938243370406it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( bit_se545348938243370406it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_Suc_minus_numeral
thf(fact_7774_push__bit__numeral,axiom,
! [L: num,K: num] :
( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ K ) )
= ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% push_bit_numeral
thf(fact_7775_push__bit__numeral,axiom,
! [L: num,K: num] :
( ( bit_se547839408752420682it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ K ) )
= ( bit_se547839408752420682it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% push_bit_numeral
thf(fact_7776_push__bit__Suc,axiom,
! [N2: nat,A: int] :
( ( bit_se545348938243370406it_int @ ( suc @ N2 ) @ A )
= ( bit_se545348938243370406it_int @ N2 @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% push_bit_Suc
thf(fact_7777_push__bit__Suc,axiom,
! [N2: nat,A: nat] :
( ( bit_se547839408752420682it_nat @ ( suc @ N2 ) @ A )
= ( bit_se547839408752420682it_nat @ N2 @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% push_bit_Suc
thf(fact_7778_push__bit__of__1,axiom,
! [N2: nat] :
( ( bit_se545348938243370406it_int @ N2 @ one_one_int )
= ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).
% push_bit_of_1
thf(fact_7779_push__bit__of__1,axiom,
! [N2: nat] :
( ( bit_se547839408752420682it_nat @ N2 @ one_one_nat )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% push_bit_of_1
thf(fact_7780_push__bit__of__Suc__0,axiom,
! [N2: nat] :
( ( bit_se547839408752420682it_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% push_bit_of_Suc_0
thf(fact_7781_even__push__bit__iff,axiom,
! [N2: nat,A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se7788150548672797655nteger @ N2 @ A ) )
= ( ( N2 != zero_zero_nat )
| ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_push_bit_iff
thf(fact_7782_even__push__bit__iff,axiom,
! [N2: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se545348938243370406it_int @ N2 @ A ) )
= ( ( N2 != zero_zero_nat )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_push_bit_iff
thf(fact_7783_even__push__bit__iff,axiom,
! [N2: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se547839408752420682it_nat @ N2 @ A ) )
= ( ( N2 != zero_zero_nat )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_push_bit_iff
thf(fact_7784_push__bit__minus__numeral,axiom,
! [L: num,K: num] :
( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( bit_se7788150548672797655nteger @ ( pred_numeral @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_minus_numeral
thf(fact_7785_push__bit__minus__numeral,axiom,
! [L: num,K: num] :
( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_minus_numeral
thf(fact_7786_bit__xor__int__iff,axiom,
! [K: int,L: int,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ N2 )
= ( ( bit_se1146084159140164899it_int @ K @ N2 )
!= ( bit_se1146084159140164899it_int @ L @ N2 ) ) ) ).
% bit_xor_int_iff
thf(fact_7787_flip__bit__int__def,axiom,
( bit_se2159334234014336723it_int
= ( ^ [N4: nat,K2: int] : ( bit_se6526347334894502574or_int @ K2 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).
% flip_bit_int_def
thf(fact_7788_push__bit__of__int,axiom,
! [N2: nat,K: int] :
( ( bit_se545348938243370406it_int @ N2 @ ( ring_1_of_int_int @ K ) )
= ( ring_1_of_int_int @ ( bit_se545348938243370406it_int @ N2 @ K ) ) ) ).
% push_bit_of_int
thf(fact_7789_push__bit__add,axiom,
! [N2: nat,A: int,B: int] :
( ( bit_se545348938243370406it_int @ N2 @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( bit_se545348938243370406it_int @ N2 @ A ) @ ( bit_se545348938243370406it_int @ N2 @ B ) ) ) ).
% push_bit_add
thf(fact_7790_push__bit__add,axiom,
! [N2: nat,A: nat,B: nat] :
( ( bit_se547839408752420682it_nat @ N2 @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N2 @ A ) @ ( bit_se547839408752420682it_nat @ N2 @ B ) ) ) ).
% push_bit_add
thf(fact_7791_push__bit__minus,axiom,
! [N2: nat,A: code_integer] :
( ( bit_se7788150548672797655nteger @ N2 @ ( uminus1351360451143612070nteger @ A ) )
= ( uminus1351360451143612070nteger @ ( bit_se7788150548672797655nteger @ N2 @ A ) ) ) ).
% push_bit_minus
thf(fact_7792_push__bit__minus,axiom,
! [N2: nat,A: int] :
( ( bit_se545348938243370406it_int @ N2 @ ( uminus_uminus_int @ A ) )
= ( uminus_uminus_int @ ( bit_se545348938243370406it_int @ N2 @ A ) ) ) ).
% push_bit_minus
thf(fact_7793_push__bit__nat__eq,axiom,
! [N2: nat,K: int] :
( ( bit_se547839408752420682it_nat @ N2 @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se545348938243370406it_int @ N2 @ K ) ) ) ).
% push_bit_nat_eq
thf(fact_7794_subset__divisors__dvd,axiom,
! [A: real,B: real] :
( ( ord_less_eq_set_real
@ ( collect_real
@ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
@ ( collect_real
@ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B ) ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_7795_subset__divisors__dvd,axiom,
! [A: int,B: int] :
( ( ord_less_eq_set_int
@ ( collect_int
@ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
@ ( collect_int
@ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_7796_subset__divisors__dvd,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le7084787975880047091nteger
@ ( collect_Code_integer
@ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ A ) )
@ ( collect_Code_integer
@ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ B ) ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_7797_subset__divisors__dvd,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
@ ( collect_nat
@ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_7798_push__bit__of__nat,axiom,
! [N2: nat,M: nat] :
( ( bit_se545348938243370406it_int @ N2 @ ( semiri1314217659103216013at_int @ M ) )
= ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ N2 @ M ) ) ) ).
% push_bit_of_nat
thf(fact_7799_push__bit__of__nat,axiom,
! [N2: nat,M: nat] :
( ( bit_se547839408752420682it_nat @ N2 @ ( semiri1316708129612266289at_nat @ M ) )
= ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ N2 @ M ) ) ) ).
% push_bit_of_nat
thf(fact_7800_of__nat__push__bit,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ M @ N2 ) )
= ( bit_se545348938243370406it_int @ M @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_push_bit
thf(fact_7801_of__nat__push__bit,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ M @ N2 ) )
= ( bit_se547839408752420682it_nat @ M @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_push_bit
thf(fact_7802_strict__subset__divisors__dvd,axiom,
! [A: real,B: real] :
( ( ord_less_set_real
@ ( collect_real
@ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
@ ( collect_real
@ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B ) ) )
= ( ( dvd_dvd_real @ A @ B )
& ~ ( dvd_dvd_real @ B @ A ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_7803_strict__subset__divisors__dvd,axiom,
! [A: nat,B: nat] :
( ( ord_less_set_nat
@ ( collect_nat
@ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
@ ( collect_nat
@ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
= ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_7804_strict__subset__divisors__dvd,axiom,
! [A: int,B: int] :
( ( ord_less_set_int
@ ( collect_int
@ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
@ ( collect_int
@ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
= ( ( dvd_dvd_int @ A @ B )
& ~ ( dvd_dvd_int @ B @ A ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_7805_strict__subset__divisors__dvd,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le1307284697595431911nteger
@ ( collect_Code_integer
@ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ A ) )
@ ( collect_Code_integer
@ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ B ) ) )
= ( ( dvd_dvd_Code_integer @ A @ B )
& ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_7806_less__set__def,axiom,
( ord_less_set_complex
= ( ^ [A6: set_complex,B6: set_complex] :
( ord_less_complex_o
@ ^ [X3: complex] : ( member_complex @ X3 @ A6 )
@ ^ [X3: complex] : ( member_complex @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_7807_less__set__def,axiom,
( ord_less_set_real
= ( ^ [A6: set_real,B6: set_real] :
( ord_less_real_o
@ ^ [X3: real] : ( member_real @ X3 @ A6 )
@ ^ [X3: real] : ( member_real @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_7808_less__set__def,axiom,
( ord_less_set_set_nat
= ( ^ [A6: set_set_nat,B6: set_set_nat] :
( ord_less_set_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_7809_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_7810_less__set__def,axiom,
( ord_less_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ord_less_int_o
@ ^ [X3: int] : ( member_int @ X3 @ A6 )
@ ^ [X3: int] : ( member_int @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_7811_psubsetD,axiom,
! [A2: set_complex,B3: set_complex,C: complex] :
( ( ord_less_set_complex @ A2 @ B3 )
=> ( ( member_complex @ C @ A2 )
=> ( member_complex @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_7812_psubsetD,axiom,
! [A2: set_real,B3: set_real,C: real] :
( ( ord_less_set_real @ A2 @ B3 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_7813_psubsetD,axiom,
! [A2: set_set_nat,B3: set_set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A2 @ B3 )
=> ( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_7814_psubsetD,axiom,
! [A2: set_nat,B3: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_7815_psubsetD,axiom,
! [A2: set_int,B3: set_int,C: int] :
( ( ord_less_set_int @ A2 @ B3 )
=> ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_7816_psubset__imp__ex__mem,axiom,
! [A2: set_complex,B3: set_complex] :
( ( ord_less_set_complex @ A2 @ B3 )
=> ? [B5: complex] : ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_7817_psubset__imp__ex__mem,axiom,
! [A2: set_real,B3: set_real] :
( ( ord_less_set_real @ A2 @ B3 )
=> ? [B5: real] : ( member_real @ B5 @ ( minus_minus_set_real @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_7818_psubset__imp__ex__mem,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B3 )
=> ? [B5: set_nat] : ( member_set_nat @ B5 @ ( minus_2163939370556025621et_nat @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_7819_psubset__imp__ex__mem,axiom,
! [A2: set_int,B3: set_int] :
( ( ord_less_set_int @ A2 @ B3 )
=> ? [B5: int] : ( member_int @ B5 @ ( minus_minus_set_int @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_7820_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_7821_lambda__one,axiom,
( ( ^ [X3: complex] : X3 )
= ( times_times_complex @ one_one_complex ) ) ).
% lambda_one
thf(fact_7822_lambda__one,axiom,
( ( ^ [X3: real] : X3 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_7823_lambda__one,axiom,
( ( ^ [X3: rat] : X3 )
= ( times_times_rat @ one_one_rat ) ) ).
% lambda_one
thf(fact_7824_lambda__one,axiom,
( ( ^ [X3: nat] : X3 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_7825_lambda__one,axiom,
( ( ^ [X3: int] : X3 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_7826_lambda__zero,axiom,
( ( ^ [H: complex] : zero_zero_complex )
= ( times_times_complex @ zero_zero_complex ) ) ).
% lambda_zero
thf(fact_7827_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_7828_lambda__zero,axiom,
( ( ^ [H: rat] : zero_zero_rat )
= ( times_times_rat @ zero_zero_rat ) ) ).
% lambda_zero
thf(fact_7829_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_7830_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_7831_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numera6690914467698888265omplex @ ( bit0 @ N2 ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_7832_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numeral_numeral_real @ ( bit0 @ N2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_7833_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_7834_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_7835_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numeral_numeral_int @ ( bit0 @ N2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_7836_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_less_as_int
thf(fact_7837_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_leq_as_int
thf(fact_7838_numeral__code_I3_J,axiom,
! [N2: num] :
( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).
% numeral_code(3)
thf(fact_7839_numeral__code_I3_J,axiom,
! [N2: num] :
( ( numeral_numeral_real @ ( bit1 @ N2 ) )
= ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).
% numeral_code(3)
thf(fact_7840_numeral__code_I3_J,axiom,
! [N2: num] :
( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).
% numeral_code(3)
thf(fact_7841_numeral__code_I3_J,axiom,
! [N2: num] :
( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).
% numeral_code(3)
thf(fact_7842_numeral__code_I3_J,axiom,
! [N2: num] :
( ( numeral_numeral_int @ ( bit1 @ N2 ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).
% numeral_code(3)
thf(fact_7843_power__numeral__even,axiom,
! [Z: complex,W: num] :
( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_7844_power__numeral__even,axiom,
! [Z: real,W: num] :
( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_7845_power__numeral__even,axiom,
! [Z: rat,W: num] :
( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_7846_power__numeral__even,axiom,
! [Z: nat,W: num] :
( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_7847_power__numeral__even,axiom,
! [Z: int,W: num] :
( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_7848_power__numeral__odd,axiom,
! [Z: complex,W: num] :
( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_7849_power__numeral__odd,axiom,
! [Z: real,W: num] :
( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_7850_power__numeral__odd,axiom,
! [Z: rat,W: num] :
( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_7851_power__numeral__odd,axiom,
! [Z: nat,W: num] :
( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_7852_power__numeral__odd,axiom,
! [Z: int,W: num] :
( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_7853_nat__plus__as__int,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_7854_nat__div__as__int,axiom,
( divide_divide_nat
= ( ^ [A3: nat,B2: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_7855_XOR__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ) ).
% XOR_lower
thf(fact_7856_signed__take__bit__code,axiom,
( bit_ri6519982836138164636nteger
= ( ^ [N4: nat,A3: code_integer] : ( if_Code_integer @ ( bit_se9216721137139052372nteger @ ( bit_se1745604003318907178nteger @ ( suc @ N4 ) @ A3 ) @ N4 ) @ ( plus_p5714425477246183910nteger @ ( bit_se1745604003318907178nteger @ ( suc @ N4 ) @ A3 ) @ ( bit_se7788150548672797655nteger @ ( suc @ N4 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) @ ( bit_se1745604003318907178nteger @ ( suc @ N4 ) @ A3 ) ) ) ) ).
% signed_take_bit_code
thf(fact_7857_signed__take__bit__code,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N4: nat,A3: int] : ( if_int @ ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ A3 ) @ N4 ) @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ A3 ) @ ( bit_se545348938243370406it_int @ ( suc @ N4 ) @ ( uminus_uminus_int @ one_one_int ) ) ) @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ A3 ) ) ) ) ).
% signed_take_bit_code
thf(fact_7858_set__conv__nth,axiom,
( set_real2
= ( ^ [Xs3: list_real] :
( collect_real
@ ^ [Uu3: real] :
? [I3: nat] :
( ( Uu3
= ( nth_real @ Xs3 @ I3 ) )
& ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs3 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_7859_set__conv__nth,axiom,
( set_list_nat2
= ( ^ [Xs3: list_list_nat] :
( collect_list_nat
@ ^ [Uu3: list_nat] :
? [I3: nat] :
( ( Uu3
= ( nth_list_nat @ Xs3 @ I3 ) )
& ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs3 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_7860_set__conv__nth,axiom,
( set_set_nat2
= ( ^ [Xs3: list_set_nat] :
( collect_set_nat
@ ^ [Uu3: set_nat] :
? [I3: nat] :
( ( Uu3
= ( nth_set_nat @ Xs3 @ I3 ) )
& ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs3 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_7861_set__conv__nth,axiom,
( set_VEBT_VEBT2
= ( ^ [Xs3: list_VEBT_VEBT] :
( collect_VEBT_VEBT
@ ^ [Uu3: vEBT_VEBT] :
? [I3: nat] :
( ( Uu3
= ( nth_VEBT_VEBT @ Xs3 @ I3 ) )
& ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs3 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_7862_set__conv__nth,axiom,
( set_o2
= ( ^ [Xs3: list_o] :
( collect_o
@ ^ [Uu3: $o] :
? [I3: nat] :
( ( Uu3
= ( nth_o @ Xs3 @ I3 ) )
& ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs3 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_7863_set__conv__nth,axiom,
( set_nat2
= ( ^ [Xs3: list_nat] :
( collect_nat
@ ^ [Uu3: nat] :
? [I3: nat] :
( ( Uu3
= ( nth_nat @ Xs3 @ I3 ) )
& ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_7864_set__conv__nth,axiom,
( set_int2
= ( ^ [Xs3: list_int] :
( collect_int
@ ^ [Uu3: int] :
? [I3: nat] :
( ( Uu3
= ( nth_int @ Xs3 @ I3 ) )
& ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs3 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_7865_push__bit__take__bit,axiom,
! [M: nat,N2: nat,A: int] :
( ( bit_se545348938243370406it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
= ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N2 ) @ ( bit_se545348938243370406it_int @ M @ A ) ) ) ).
% push_bit_take_bit
thf(fact_7866_push__bit__take__bit,axiom,
! [M: nat,N2: nat,A: nat] :
( ( bit_se547839408752420682it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) )
= ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N2 ) @ ( bit_se547839408752420682it_nat @ M @ A ) ) ) ).
% push_bit_take_bit
thf(fact_7867_take__bit__push__bit,axiom,
! [M: nat,N2: nat,A: int] :
( ( bit_se2923211474154528505it_int @ M @ ( bit_se545348938243370406it_int @ N2 @ A ) )
= ( bit_se545348938243370406it_int @ N2 @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N2 ) @ A ) ) ) ).
% take_bit_push_bit
thf(fact_7868_take__bit__push__bit,axiom,
! [M: nat,N2: nat,A: nat] :
( ( bit_se2925701944663578781it_nat @ M @ ( bit_se547839408752420682it_nat @ N2 @ A ) )
= ( bit_se547839408752420682it_nat @ N2 @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ M @ N2 ) @ A ) ) ) ).
% take_bit_push_bit
thf(fact_7869_flip__bit__nat__def,axiom,
( bit_se2161824704523386999it_nat
= ( ^ [M2: nat,N4: nat] : ( bit_se6528837805403552850or_nat @ N4 @ ( bit_se547839408752420682it_nat @ M2 @ one_one_nat ) ) ) ) ).
% flip_bit_nat_def
thf(fact_7870_diff__nat__eq__if,axiom,
! [Z6: int,Z: int] :
( ( ( ord_less_int @ Z6 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
= ( nat2 @ Z ) ) )
& ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
= ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_7871_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_7872_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_7873_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_7874_gbinomial__code,axiom,
( gbinomial_complex
= ( ^ [A3: complex,K2: nat] :
( if_complex @ ( K2 = zero_zero_nat ) @ one_one_complex
@ ( divide1717551699836669952omplex
@ ( set_fo1517530859248394432omplex
@ ^ [L2: nat] : ( times_times_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ L2 ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ K2 @ one_one_nat )
@ one_one_complex )
@ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ) ).
% gbinomial_code
thf(fact_7875_gbinomial__code,axiom,
( gbinomial_rat
= ( ^ [A3: rat,K2: nat] :
( if_rat @ ( K2 = zero_zero_nat ) @ one_one_rat
@ ( divide_divide_rat
@ ( set_fo1949268297981939178at_rat
@ ^ [L2: nat] : ( times_times_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ L2 ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ K2 @ one_one_nat )
@ one_one_rat )
@ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ) ).
% gbinomial_code
thf(fact_7876_gbinomial__code,axiom,
( gbinomial_real
= ( ^ [A3: real,K2: nat] :
( if_real @ ( K2 = zero_zero_nat ) @ one_one_real
@ ( divide_divide_real
@ ( set_fo3111899725591712190t_real
@ ^ [L2: nat] : ( times_times_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ L2 ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ K2 @ one_one_nat )
@ one_one_real )
@ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ) ).
% gbinomial_code
thf(fact_7877_gbinomial__of__nat__symmetric,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N2 ) @ K )
= ( gbinomial_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_7878_gbinomial__of__nat__symmetric,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N2 ) @ K )
= ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_7879_gbinomial__of__nat__symmetric,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N2 ) @ K )
= ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_7880_bit__push__bit__iff__int,axiom,
! [M: nat,K: int,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N2 )
= ( ( ord_less_eq_nat @ M @ N2 )
& ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_7881_xor__nat__def,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M2: nat,N4: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% xor_nat_def
thf(fact_7882_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X3: nat] :
( collect_nat
@ ^ [N4: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_7883_psubsetE,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_7884_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_7885_psubset__imp__subset,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_7886_psubset__subset__trans,axiom,
! [A2: set_nat,B3: set_nat,C4: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C4 )
=> ( ord_less_set_nat @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_7887_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B6 )
& ~ ( ord_less_eq_set_nat @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_7888_subset__psubset__trans,axiom,
! [A2: set_nat,B3: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_set_nat @ B3 @ C4 )
=> ( ord_less_set_nat @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_7889_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_set_nat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_7890_bit__push__bit__iff__nat,axiom,
! [M: nat,Q3: nat,N2: nat] :
( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q3 ) @ N2 )
= ( ( ord_less_eq_nat @ M @ N2 )
& ( bit_se1148574629649215175it_nat @ Q3 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_7891_concat__bit__eq,axiom,
( bit_concat_bit
= ( ^ [N4: nat,K2: int,L2: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N4 @ K2 ) @ ( bit_se545348938243370406it_int @ N4 @ L2 ) ) ) ) ).
% concat_bit_eq
thf(fact_7892_flip__bit__eq__xor,axiom,
( bit_se2159334234014336723it_int
= ( ^ [N4: nat,A3: int] : ( bit_se6526347334894502574or_int @ A3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_7893_flip__bit__eq__xor,axiom,
( bit_se2161824704523386999it_nat
= ( ^ [N4: nat,A3: nat] : ( bit_se6528837805403552850or_nat @ A3 @ ( bit_se547839408752420682it_nat @ N4 @ one_one_nat ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_7894_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_7895_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_7896_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_7897_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_7898_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_7899_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_7900_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_7901_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_7902_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_7903_gbinomial__mult__1,axiom,
! [A: complex,K: nat] :
( ( times_times_complex @ A @ ( gbinomial_complex @ A @ K ) )
= ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_7904_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_7905_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_7906_gbinomial__mult__1_H,axiom,
! [A: complex,K: nat] :
( ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ A )
= ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_7907_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_7908_push__bit__double,axiom,
! [N2: nat,A: int] :
( ( bit_se545348938243370406it_int @ N2 @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( times_times_int @ ( bit_se545348938243370406it_int @ N2 @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% push_bit_double
thf(fact_7909_push__bit__double,axiom,
! [N2: nat,A: nat] :
( ( bit_se547839408752420682it_nat @ N2 @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( times_times_nat @ ( bit_se547839408752420682it_nat @ N2 @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% push_bit_double
thf(fact_7910_pochhammer__code,axiom,
( comm_s7457072308508201937r_real
= ( ^ [A3: real,N4: nat] :
( if_real @ ( N4 = zero_zero_nat ) @ one_one_real
@ ( set_fo3111899725591712190t_real
@ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N4 @ one_one_nat )
@ one_one_real ) ) ) ) ).
% pochhammer_code
thf(fact_7911_pochhammer__code,axiom,
( comm_s4660882817536571857er_int
= ( ^ [A3: int,N4: nat] :
( if_int @ ( N4 = zero_zero_nat ) @ one_one_int
@ ( set_fo2581907887559384638at_int
@ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N4 @ one_one_nat )
@ one_one_int ) ) ) ) ).
% pochhammer_code
thf(fact_7912_pochhammer__code,axiom,
( comm_s2602460028002588243omplex
= ( ^ [A3: complex,N4: nat] :
( if_complex @ ( N4 = zero_zero_nat ) @ one_one_complex
@ ( set_fo1517530859248394432omplex
@ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A3 @ ( semiri8010041392384452111omplex @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N4 @ one_one_nat )
@ one_one_complex ) ) ) ) ).
% pochhammer_code
thf(fact_7913_pochhammer__code,axiom,
( comm_s4028243227959126397er_rat
= ( ^ [A3: rat,N4: nat] :
( if_rat @ ( N4 = zero_zero_nat ) @ one_one_rat
@ ( set_fo1949268297981939178at_rat
@ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A3 @ ( semiri681578069525770553at_rat @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N4 @ one_one_nat )
@ one_one_rat ) ) ) ) ).
% pochhammer_code
thf(fact_7914_pochhammer__code,axiom,
( comm_s4663373288045622133er_nat
= ( ^ [A3: nat,N4: nat] :
( if_nat @ ( N4 = zero_zero_nat ) @ one_one_nat
@ ( set_fo2584398358068434914at_nat
@ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N4 @ one_one_nat )
@ one_one_nat ) ) ) ) ).
% pochhammer_code
thf(fact_7915_bit__iff__and__push__bit__not__eq__0,axiom,
( bit_se1146084159140164899it_int
= ( ^ [A3: int,N4: nat] :
( ( bit_se725231765392027082nd_int @ A3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) )
!= zero_zero_int ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_7916_bit__iff__and__push__bit__not__eq__0,axiom,
( bit_se1148574629649215175it_nat
= ( ^ [A3: nat,N4: nat] :
( ( bit_se727722235901077358nd_nat @ A3 @ ( bit_se547839408752420682it_nat @ N4 @ one_one_nat ) )
!= zero_zero_nat ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_7917_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_7918_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_7919_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_7920_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_7921_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_7922_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_7923_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_7924_gbinomial__trinomial__revision,axiom,
! [K: nat,M: nat,A: complex] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
= ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_7925_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_7926_push__bit__int__def,axiom,
( bit_se545348938243370406it_int
= ( ^ [N4: nat,K2: int] : ( times_times_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% push_bit_int_def
thf(fact_7927_push__bit__nat__def,axiom,
( bit_se547839408752420682it_nat
= ( ^ [N4: nat,M2: nat] : ( times_times_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% push_bit_nat_def
thf(fact_7928_push__bit__eq__mult,axiom,
( bit_se545348938243370406it_int
= ( ^ [N4: nat,A3: int] : ( times_times_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% push_bit_eq_mult
thf(fact_7929_push__bit__eq__mult,axiom,
( bit_se547839408752420682it_nat
= ( ^ [N4: nat,A3: nat] : ( times_times_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% push_bit_eq_mult
thf(fact_7930_exp__dvdE,axiom,
! [N2: nat,A: code_integer] :
( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ A )
=> ~ ! [B5: code_integer] :
( A
!= ( bit_se7788150548672797655nteger @ N2 @ B5 ) ) ) ).
% exp_dvdE
thf(fact_7931_exp__dvdE,axiom,
! [N2: nat,A: int] :
( ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ A )
=> ~ ! [B5: int] :
( A
!= ( bit_se545348938243370406it_int @ N2 @ B5 ) ) ) ).
% exp_dvdE
thf(fact_7932_exp__dvdE,axiom,
! [N2: nat,A: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ A )
=> ~ ! [B5: nat] :
( A
!= ( bit_se547839408752420682it_nat @ N2 @ B5 ) ) ) ).
% exp_dvdE
thf(fact_7933_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ X )
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_7934_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_7935_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_7936_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_7937_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_7938_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_7939_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_7940_gbinomial__index__swap,axiom,
! [K: nat,N2: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ one_one_real ) @ K ) )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ N2 ) ) ) ).
% gbinomial_index_swap
thf(fact_7941_gbinomial__index__swap,axiom,
! [K: nat,N2: nat] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ one_one_complex ) @ K ) )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N2 ) ) ) ).
% gbinomial_index_swap
thf(fact_7942_gbinomial__index__swap,axiom,
! [K: nat,N2: nat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ one_one_rat ) @ K ) )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ N2 ) ) ) ).
% gbinomial_index_swap
thf(fact_7943_gbinomial__negated__upper,axiom,
( gbinomial_real
= ( ^ [A3: real,K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K2 ) @ A3 ) @ one_one_real ) @ K2 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_7944_gbinomial__negated__upper,axiom,
( gbinomial_complex
= ( ^ [A3: complex,K2: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K2 ) @ A3 ) @ one_one_complex ) @ K2 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_7945_gbinomial__negated__upper,axiom,
( gbinomial_rat
= ( ^ [A3: rat,K2: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K2 ) @ A3 ) @ one_one_rat ) @ K2 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_7946_push__bit__minus__one,axiom,
! [N2: nat] :
( ( bit_se545348938243370406it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% push_bit_minus_one
thf(fact_7947_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd2 ) @ X )
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_7948_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi3: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc ) @ X )
= ( ( X = Mi3 )
| ( X = Ma )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_7949_vebt__member_Osimps_I5_J,axiom,
! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( ( X != Mi3 )
=> ( ( X != Ma )
=> ( ~ ( ord_less_nat @ X @ Mi3 )
& ( ~ ( ord_less_nat @ X @ Mi3 )
=> ( ~ ( ord_less_nat @ Ma @ X )
& ( ~ ( ord_less_nat @ Ma @ X )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_7950_XOR__upper,axiom,
! [X: int,N2: nat,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
=> ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
=> ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% XOR_upper
thf(fact_7951_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_7952_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_7953_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_7954_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_7955_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_7956_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_7957_gbinomial__pochhammer,axiom,
( gbinomial_complex
= ( ^ [A3: complex,K2: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A3 ) @ K2 ) ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_7958_gbinomial__pochhammer,axiom,
( gbinomial_rat
= ( ^ [A3: rat,K2: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A3 ) @ K2 ) ) @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_7959_gbinomial__pochhammer,axiom,
( gbinomial_real
= ( ^ [A3: real,K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A3 ) @ K2 ) ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_7960_gbinomial__pochhammer_H,axiom,
( gbinomial_complex
= ( ^ [A3: complex,K2: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_7961_gbinomial__pochhammer_H,axiom,
( gbinomial_rat
= ( ^ [A3: rat,K2: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 ) @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_7962_gbinomial__pochhammer_H,axiom,
( gbinomial_real
= ( ^ [A3: real,K2: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_7963_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa )
= Y )
=> ( ! [A5: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( Y
= ( ~ ( ( ( Xa = zero_zero_nat )
=> A5 )
& ( ( Xa != zero_zero_nat )
=> ( ( ( Xa = one_one_nat )
=> B5 )
& ( Xa = one_one_nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> Y )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
=> ( Y
= ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_7964_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ! [A5: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ~ ( ( ( Xa = zero_zero_nat )
=> A5 )
& ( ( Xa != zero_zero_nat )
=> ( ( ( Xa = one_one_nat )
=> B5 )
& ( Xa = one_one_nat ) ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_7965_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ! [A5: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( ( Xa = zero_zero_nat )
=> A5 )
& ( ( Xa != zero_zero_nat )
=> ( ( ( Xa = one_one_nat )
=> B5 )
& ( Xa = one_one_nat ) ) ) ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_7966_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ! [Mi: nat,Ma2: nat] :
( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ~ ( ( Xa = Mi )
| ( Xa = Ma2 ) ) )
=> ( ! [Mi: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ~ ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_7967_xor__int__rec,axiom,
( bit_se6526347334894502574or_int
= ( ^ [K2: int,L2: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% xor_int_rec
thf(fact_7968_vebt__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_vebt_member @ X @ Xa )
=> ( ! [A5: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ~ ( ( ( Xa = zero_zero_nat )
=> A5 )
& ( ( Xa != zero_zero_nat )
=> ( ( ( Xa = one_one_nat )
=> B5 )
& ( Xa = one_one_nat ) ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ~ ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less_nat @ Xa @ Mi )
& ( ~ ( ord_less_nat @ Xa @ Mi )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa )
& ( ~ ( ord_less_nat @ Ma2 @ Xa )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_7969_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> Y )
=> ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> Y )
=> ( ! [Mi: nat,Ma2: nat] :
( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( Y
= ( ~ ( ( Xa = Mi )
| ( Xa = Ma2 ) ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ( Y
= ( ~ ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
=> ( Y
= ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_7970_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ! [Uu2: $o,Uv2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> ( ! [Mi: nat,Ma2: nat] :
( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( ( Xa = Mi )
| ( Xa = Ma2 ) ) )
=> ( ! [Mi: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_7971_bit__horner__sum__bit__iff,axiom,
! [Bs: list_o,N2: nat] :
( ( bit_se9216721137139052372nteger @ ( groups3417619833198082522nteger @ zero_n356916108424825756nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Bs ) @ N2 )
= ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs ) )
& ( nth_o @ Bs @ N2 ) ) ) ).
% bit_horner_sum_bit_iff
thf(fact_7972_bit__horner__sum__bit__iff,axiom,
! [Bs: list_o,N2: nat] :
( ( bit_se1148574629649215175it_nat @ ( groups9119017779487936845_o_nat @ zero_n2687167440665602831ol_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Bs ) @ N2 )
= ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs ) )
& ( nth_o @ Bs @ N2 ) ) ) ).
% bit_horner_sum_bit_iff
thf(fact_7973_bit__horner__sum__bit__iff,axiom,
! [Bs: list_o,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ N2 )
= ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs ) )
& ( nth_o @ Bs @ N2 ) ) ) ).
% bit_horner_sum_bit_iff
thf(fact_7974_vebt__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa )
= Y )
=> ( ! [A5: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( Y
= ( ~ ( ( ( Xa = zero_zero_nat )
=> A5 )
& ( ( Xa != zero_zero_nat )
=> ( ( ( Xa = one_one_nat )
=> B5 )
& ( Xa = one_one_nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> Y )
=> ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> Y )
=> ( ( ? [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 ) )
=> Y )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less_nat @ Xa @ Mi )
& ( ~ ( ord_less_nat @ Xa @ Mi )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa )
& ( ~ ( ord_less_nat @ Ma2 @ Xa )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_7975_of__int__code__if,axiom,
( ring_1_of_int_real
= ( ^ [K2: int] :
( if_real @ ( K2 = zero_zero_int ) @ zero_zero_real
@ ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K2 ) ) )
@ ( if_real
@ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_7976_of__int__code__if,axiom,
( ring_1_of_int_int
= ( ^ [K2: int] :
( if_int @ ( K2 = zero_zero_int ) @ zero_zero_int
@ ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K2 ) ) )
@ ( if_int
@ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_7977_of__int__code__if,axiom,
( ring_17405671764205052669omplex
= ( ^ [K2: int] :
( if_complex @ ( K2 = zero_zero_int ) @ zero_zero_complex
@ ( if_complex @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K2 ) ) )
@ ( if_complex
@ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_7978_of__int__code__if,axiom,
( ring_18347121197199848620nteger
= ( ^ [K2: int] :
( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
@ ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K2 ) ) )
@ ( if_Code_integer
@ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_7979_of__int__code__if,axiom,
( ring_1_of_int_rat
= ( ^ [K2: int] :
( if_rat @ ( K2 = zero_zero_int ) @ zero_zero_rat
@ ( if_rat @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K2 ) ) )
@ ( if_rat
@ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_7980_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( topolo6980174941875973593q_real
@ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_7981_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
@ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N4 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_7982_arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( arctan @ X )
= ( suminf_real
@ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_7983_summable__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( summable_real
@ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_7984_powser__zero,axiom,
! [F: nat > complex] :
( ( suminf_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) ) )
= ( F @ zero_zero_nat ) ) ).
% powser_zero
thf(fact_7985_powser__zero,axiom,
! [F: nat > real] :
( ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) ) )
= ( F @ zero_zero_nat ) ) ).
% powser_zero
thf(fact_7986_powser__insidea,axiom,
! [F: nat > real,X: real,Z: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ X @ N4 ) ) )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
=> ( summable_real
@ ^ [N4: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) ) ) ) ) ).
% powser_insidea
thf(fact_7987_powser__insidea,axiom,
! [F: nat > complex,X: complex,Z: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ X @ N4 ) ) )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
=> ( summable_real
@ ^ [N4: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) ) ) ) ) ).
% powser_insidea
thf(fact_7988_powser__inside,axiom,
! [F: nat > real,X: real,Z: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ X @ N4 ) ) )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
=> ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) ) ) ) ).
% powser_inside
thf(fact_7989_powser__inside,axiom,
! [F: nat > complex,X: complex,Z: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ X @ N4 ) ) )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
=> ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) ) ) ) ).
% powser_inside
thf(fact_7990_summable__exp,axiom,
! [X: complex] :
( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri5044797733671781792omplex @ N4 ) ) @ ( power_power_complex @ X @ N4 ) ) ) ).
% summable_exp
thf(fact_7991_summable__exp,axiom,
! [X: real] :
( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N4 ) ) @ ( power_power_real @ X @ N4 ) ) ) ).
% summable_exp
thf(fact_7992_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_7993_pi__series,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( suminf_real
@ ^ [K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% pi_series
thf(fact_7994_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_7995_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_7996_summable__divide__iff,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( divide1717551699836669952omplex @ ( F @ N4 ) @ C ) )
= ( ( C = zero_zero_complex )
| ( summable_complex @ F ) ) ) ).
% summable_divide_iff
thf(fact_7997_summable__divide__iff,axiom,
! [F: nat > real,C: real] :
( ( summable_real
@ ^ [N4: nat] : ( divide_divide_real @ ( F @ N4 ) @ C ) )
= ( ( C = zero_zero_real )
| ( summable_real @ F ) ) ) ).
% summable_divide_iff
thf(fact_7998_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_7999_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_8000_summable__cmult__iff,axiom,
! [C: complex,F: nat > complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ C @ ( F @ N4 ) ) )
= ( ( C = zero_zero_complex )
| ( summable_complex @ F ) ) ) ).
% summable_cmult_iff
thf(fact_8001_summable__cmult__iff,axiom,
! [C: real,F: nat > real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ C @ ( F @ N4 ) ) )
= ( ( C = zero_zero_real )
| ( summable_real @ F ) ) ) ).
% summable_cmult_iff
thf(fact_8002_summable__single,axiom,
! [I: nat,F: nat > complex] :
( summable_complex
@ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex ) ) ).
% summable_single
thf(fact_8003_summable__single,axiom,
! [I: nat,F: nat > real] :
( summable_real
@ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real ) ) ).
% summable_single
thf(fact_8004_summable__single,axiom,
! [I: nat,F: nat > nat] :
( summable_nat
@ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).
% summable_single
thf(fact_8005_summable__single,axiom,
! [I: nat,F: nat > int] :
( summable_int
@ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int ) ) ).
% summable_single
thf(fact_8006_summable__zero,axiom,
( summable_complex
@ ^ [N4: nat] : zero_zero_complex ) ).
% summable_zero
thf(fact_8007_summable__zero,axiom,
( summable_real
@ ^ [N4: nat] : zero_zero_real ) ).
% summable_zero
thf(fact_8008_summable__zero,axiom,
( summable_nat
@ ^ [N4: nat] : zero_zero_nat ) ).
% summable_zero
thf(fact_8009_summable__zero,axiom,
( summable_int
@ ^ [N4: nat] : zero_zero_int ) ).
% summable_zero
thf(fact_8010_suminf__zero,axiom,
( ( suminf_complex
@ ^ [N4: nat] : zero_zero_complex )
= zero_zero_complex ) ).
% suminf_zero
thf(fact_8011_suminf__zero,axiom,
( ( suminf_real
@ ^ [N4: nat] : zero_zero_real )
= zero_zero_real ) ).
% suminf_zero
thf(fact_8012_suminf__zero,axiom,
( ( suminf_nat
@ ^ [N4: nat] : zero_zero_nat )
= zero_zero_nat ) ).
% suminf_zero
thf(fact_8013_suminf__zero,axiom,
( ( suminf_int
@ ^ [N4: nat] : zero_zero_int )
= zero_zero_int ) ).
% suminf_zero
thf(fact_8014_summable__iff__shift,axiom,
! [F: nat > real,K: nat] :
( ( summable_real
@ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ K ) ) )
= ( summable_real @ F ) ) ).
% summable_iff_shift
thf(fact_8015_summable__const__iff,axiom,
! [C: complex] :
( ( summable_complex
@ ^ [Uu3: nat] : C )
= ( C = zero_zero_complex ) ) ).
% summable_const_iff
thf(fact_8016_summable__const__iff,axiom,
! [C: real] :
( ( summable_real
@ ^ [Uu3: nat] : C )
= ( C = zero_zero_real ) ) ).
% summable_const_iff
thf(fact_8017_summable__add,axiom,
! [F: nat > real,G: nat > real] :
( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N4: nat] : ( plus_plus_real @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ).
% summable_add
thf(fact_8018_summable__add,axiom,
! [F: nat > nat,G: nat > nat] :
( ( summable_nat @ F )
=> ( ( summable_nat @ G )
=> ( summable_nat
@ ^ [N4: nat] : ( plus_plus_nat @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ).
% summable_add
thf(fact_8019_summable__add,axiom,
! [F: nat > int,G: nat > int] :
( ( summable_int @ F )
=> ( ( summable_int @ G )
=> ( summable_int
@ ^ [N4: nat] : ( plus_plus_int @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ).
% summable_add
thf(fact_8020_summable__divide,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex @ F )
=> ( summable_complex
@ ^ [N4: nat] : ( divide1717551699836669952omplex @ ( F @ N4 ) @ C ) ) ) ).
% summable_divide
thf(fact_8021_summable__divide,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( summable_real
@ ^ [N4: nat] : ( divide_divide_real @ ( F @ N4 ) @ C ) ) ) ).
% summable_divide
thf(fact_8022_summable__Suc__iff,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N4: nat] : ( F @ ( suc @ N4 ) ) )
= ( summable_real @ F ) ) ).
% summable_Suc_iff
thf(fact_8023_summable__ignore__initial__segment,axiom,
! [F: nat > real,K: nat] :
( ( summable_real @ F )
=> ( summable_real
@ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ K ) ) ) ) ).
% summable_ignore_initial_segment
thf(fact_8024_suminf__le,axiom,
! [F: nat > nat,G: nat > nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( G @ N ) )
=> ( ( summable_nat @ F )
=> ( ( summable_nat @ G )
=> ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).
% suminf_le
thf(fact_8025_suminf__le,axiom,
! [F: nat > int,G: nat > int] :
( ! [N: nat] : ( ord_less_eq_int @ ( F @ N ) @ ( G @ N ) )
=> ( ( summable_int @ F )
=> ( ( summable_int @ G )
=> ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).
% suminf_le
thf(fact_8026_suminf__le,axiom,
! [F: nat > real,G: nat > real] :
( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( G @ N ) )
=> ( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
% suminf_le
thf(fact_8027_summable__mult__D,axiom,
! [C: complex,F: nat > complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ C @ ( F @ N4 ) ) )
=> ( ( C != zero_zero_complex )
=> ( summable_complex @ F ) ) ) ).
% summable_mult_D
thf(fact_8028_summable__mult__D,axiom,
! [C: real,F: nat > real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ C @ ( F @ N4 ) ) )
=> ( ( C != zero_zero_real )
=> ( summable_real @ F ) ) ) ).
% summable_mult_D
thf(fact_8029_summable__zero__power,axiom,
summable_int @ ( power_power_int @ zero_zero_int ) ).
% summable_zero_power
thf(fact_8030_summable__zero__power,axiom,
summable_real @ ( power_power_real @ zero_zero_real ) ).
% summable_zero_power
thf(fact_8031_summable__zero__power,axiom,
summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% summable_zero_power
thf(fact_8032_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
@ ^ [N4: nat] : ( plus_plus_real @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).
% suminf_add
thf(fact_8033_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
@ ^ [N4: nat] : ( plus_plus_nat @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).
% suminf_add
thf(fact_8034_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
@ ^ [N4: nat] : ( plus_plus_int @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).
% suminf_add
thf(fact_8035_suminf__divide,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex @ F )
=> ( ( suminf_complex
@ ^ [N4: nat] : ( divide1717551699836669952omplex @ ( F @ N4 ) @ C ) )
= ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).
% suminf_divide
thf(fact_8036_suminf__divide,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N4: nat] : ( divide_divide_real @ ( F @ N4 ) @ C ) )
= ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).
% suminf_divide
thf(fact_8037_suminf__eq__zero__iff,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N ) )
=> ( ( ( suminf_nat @ F )
= zero_zero_nat )
= ( ! [N4: nat] :
( ( F @ N4 )
= zero_zero_nat ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_8038_suminf__eq__zero__iff,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N ) )
=> ( ( ( suminf_int @ F )
= zero_zero_int )
= ( ! [N4: nat] :
( ( F @ N4 )
= zero_zero_int ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_8039_suminf__eq__zero__iff,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N ) )
=> ( ( ( suminf_real @ F )
= zero_zero_real )
= ( ! [N4: nat] :
( ( F @ N4 )
= zero_zero_real ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_8040_suminf__nonneg,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_8041_suminf__nonneg,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_8042_suminf__nonneg,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_8043_suminf__pos,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N ) )
=> ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% suminf_pos
thf(fact_8044_suminf__pos,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N ) )
=> ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% suminf_pos
thf(fact_8045_suminf__pos,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N ) )
=> ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% suminf_pos
thf(fact_8046_summable__zero__power_H,axiom,
! [F: nat > complex] :
( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) ) ) ).
% summable_zero_power'
thf(fact_8047_summable__zero__power_H,axiom,
! [F: nat > real] :
( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) ) ) ).
% summable_zero_power'
thf(fact_8048_summable__zero__power_H,axiom,
! [F: nat > int] :
( summable_int
@ ^ [N4: nat] : ( times_times_int @ ( F @ N4 ) @ ( power_power_int @ zero_zero_int @ N4 ) ) ) ).
% summable_zero_power'
thf(fact_8049_summable__0__powser,axiom,
! [F: nat > complex] :
( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) ) ) ).
% summable_0_powser
thf(fact_8050_summable__0__powser,axiom,
! [F: nat > real] :
( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) ) ) ).
% summable_0_powser
thf(fact_8051_powser__split__head_I3_J,axiom,
! [F: nat > complex,Z: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
=> ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z @ N4 ) ) ) ) ).
% powser_split_head(3)
thf(fact_8052_powser__split__head_I3_J,axiom,
! [F: nat > real,Z: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
=> ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z @ N4 ) ) ) ) ).
% powser_split_head(3)
thf(fact_8053_summable__powser__split__head,axiom,
! [F: nat > complex,Z: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z @ N4 ) ) )
= ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) ) ) ).
% summable_powser_split_head
thf(fact_8054_summable__powser__split__head,axiom,
! [F: nat > real,Z: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z @ N4 ) ) )
= ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) ) ) ).
% summable_powser_split_head
thf(fact_8055_summable__powser__ignore__initial__segment,axiom,
! [F: nat > complex,M: nat,Z: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N4 @ M ) ) @ ( power_power_complex @ Z @ N4 ) ) )
= ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_8056_summable__powser__ignore__initial__segment,axiom,
! [F: nat > real,M: nat,Z: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N4 @ M ) ) @ ( power_power_real @ Z @ N4 ) ) )
= ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_8057_summable__norm__comparison__test,axiom,
! [F: nat > complex,G: nat > real] :
( ? [N6: nat] :
! [N: nat] :
( ( ord_less_eq_nat @ N6 @ N )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N ) ) @ ( G @ N ) ) )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N4: nat] : ( real_V1022390504157884413omplex @ ( F @ N4 ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_8058_summable__rabs__comparison__test,axiom,
! [F: nat > real,G: nat > real] :
( ? [N6: nat] :
! [N: nat] :
( ( ord_less_eq_nat @ N6 @ N )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N ) ) @ ( G @ N ) ) )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_8059_suminf__pos2,axiom,
! [F: nat > nat,I: nat] :
( ( summable_nat @ F )
=> ( ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_8060_suminf__pos2,axiom,
! [F: nat > int,I: nat] :
( ( summable_int @ F )
=> ( ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
=> ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_8061_suminf__pos2,axiom,
! [F: nat > real,I: nat] :
( ( summable_real @ F )
=> ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
=> ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_8062_suminf__pos__iff,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
= ( ? [I3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_8063_suminf__pos__iff,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N ) )
=> ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
= ( ? [I3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_8064_suminf__pos__iff,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N ) )
=> ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
= ( ? [I3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_8065_summable__comparison__test_H,axiom,
! [G: nat > real,N3: nat,F: nat > real] :
( ( summable_real @ G )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ N3 @ N )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N ) ) @ ( G @ N ) ) )
=> ( summable_real @ F ) ) ) ).
% summable_comparison_test'
thf(fact_8066_summable__comparison__test_H,axiom,
! [G: nat > real,N3: nat,F: nat > complex] :
( ( summable_real @ G )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ N3 @ N )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N ) ) @ ( G @ N ) ) )
=> ( summable_complex @ F ) ) ) ).
% summable_comparison_test'
thf(fact_8067_summable__comparison__test,axiom,
! [F: nat > real,G: nat > real] :
( ? [N6: nat] :
! [N: nat] :
( ( ord_less_eq_nat @ N6 @ N )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N ) ) @ ( G @ N ) ) )
=> ( ( summable_real @ G )
=> ( summable_real @ F ) ) ) ).
% summable_comparison_test
thf(fact_8068_summable__comparison__test,axiom,
! [F: nat > complex,G: nat > real] :
( ? [N6: nat] :
! [N: nat] :
( ( ord_less_eq_nat @ N6 @ N )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N ) ) @ ( G @ N ) ) )
=> ( ( summable_real @ G )
=> ( summable_complex @ F ) ) ) ).
% summable_comparison_test
thf(fact_8069_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_8070_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_8071_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_8072_summable__geometric,axiom,
! [C: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
=> ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% summable_geometric
thf(fact_8073_suminf__split__head,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N4: nat] : ( F @ ( suc @ N4 ) ) )
= ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% suminf_split_head
thf(fact_8074_powser__split__head_I1_J,axiom,
! [F: nat > complex,Z: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
=> ( ( suminf_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
= ( plus_plus_complex @ ( F @ zero_zero_nat )
@ ( times_times_complex
@ ( suminf_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z @ N4 ) ) )
@ Z ) ) ) ) ).
% powser_split_head(1)
thf(fact_8075_powser__split__head_I1_J,axiom,
! [F: nat > real,Z: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
=> ( ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
= ( plus_plus_real @ ( F @ zero_zero_nat )
@ ( times_times_real
@ ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z @ N4 ) ) )
@ Z ) ) ) ) ).
% powser_split_head(1)
thf(fact_8076_powser__split__head_I2_J,axiom,
! [F: nat > complex,Z: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
=> ( ( times_times_complex
@ ( suminf_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z @ N4 ) ) )
@ Z )
= ( minus_minus_complex
@ ( suminf_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% powser_split_head(2)
thf(fact_8077_powser__split__head_I2_J,axiom,
! [F: nat > real,Z: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
=> ( ( times_times_real
@ ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z @ N4 ) ) )
@ Z )
= ( minus_minus_real
@ ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% powser_split_head(2)
thf(fact_8078_suminf__exist__split,axiom,
! [R4: real,F: nat > real] :
( ( ord_less_real @ zero_zero_real @ R4 )
=> ( ( summable_real @ F )
=> ? [N7: nat] :
! [N8: nat] :
( ( ord_less_eq_nat @ N7 @ N8 )
=> ( ord_less_real
@ ( real_V7735802525324610683m_real
@ ( suminf_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N8 ) ) ) )
@ R4 ) ) ) ) ).
% suminf_exist_split
thf(fact_8079_suminf__exist__split,axiom,
! [R4: real,F: nat > complex] :
( ( ord_less_real @ zero_zero_real @ R4 )
=> ( ( summable_complex @ F )
=> ? [N7: nat] :
! [N8: nat] :
( ( ord_less_eq_nat @ N7 @ N8 )
=> ( ord_less_real
@ ( real_V1022390504157884413omplex
@ ( suminf_complex
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N8 ) ) ) )
@ R4 ) ) ) ) ).
% suminf_exist_split
thf(fact_8080_summable__power__series,axiom,
! [F: nat > real,Z: real] :
( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
=> ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ Z @ one_one_real )
=> ( summable_real
@ ^ [I3: nat] : ( times_times_real @ ( F @ I3 ) @ ( power_power_real @ Z @ I3 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_8081_Abel__lemma,axiom,
! [R4: real,R0: real,A: nat > complex,M7: real] :
( ( ord_less_eq_real @ zero_zero_real @ R4 )
=> ( ( ord_less_real @ R4 @ R0 )
=> ( ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N ) ) @ ( power_power_real @ R0 @ N ) ) @ M7 )
=> ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N4 ) ) @ ( power_power_real @ R4 @ N4 ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_8082_summable__ratio__test,axiom,
! [C: real,N3: nat,F: nat > real] :
( ( ord_less_real @ C @ one_one_real )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ N3 @ N )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N ) ) ) ) )
=> ( summable_real @ F ) ) ) ).
% summable_ratio_test
thf(fact_8083_summable__ratio__test,axiom,
! [C: real,N3: nat,F: nat > complex] :
( ( ord_less_real @ C @ one_one_real )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ N3 @ N )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N ) ) ) ) )
=> ( summable_complex @ F ) ) ) ).
% summable_ratio_test
thf(fact_8084_xor__int__unfold,axiom,
( bit_se6526347334894502574or_int
= ( ^ [K2: int,L2: int] :
( if_int
@ ( K2
= ( uminus_uminus_int @ one_one_int ) )
@ ( bit_ri7919022796975470100ot_int @ L2 )
@ ( if_int
@ ( L2
= ( uminus_uminus_int @ one_one_int ) )
@ ( bit_ri7919022796975470100ot_int @ K2 )
@ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_8085_vebt__buildup_Oelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( ( X = zero_zero_nat )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X
= ( suc @ zero_zero_nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( 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 ) ) )
=> ( Y
= ( 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.elims
thf(fact_8086_sin__paired,axiom,
! [X: real] :
( sums_real
@ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
@ ( sin_real @ X ) ) ).
% sin_paired
thf(fact_8087_vebt__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
=> ( ! [A5: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
=> ( ( ( Xa = zero_zero_nat )
=> A5 )
& ( ( Xa != zero_zero_nat )
=> ( ( ( Xa = one_one_nat )
=> B5 )
& ( Xa = one_one_nat ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) )
=> ( ! [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 ) @ Xa ) ) )
=> ( ! [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 ) @ Xa ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) )
=> ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less_nat @ Xa @ Mi )
& ( ~ ( ord_less_nat @ Xa @ Mi )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa )
& ( ~ ( ord_less_nat @ Ma2 @ Xa )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_8088_intind,axiom,
! [I: nat,N2: nat,P: nat > $o,X: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( ( P @ X )
=> ( P @ ( nth_nat @ ( replicate_nat @ N2 @ X ) @ I ) ) ) ) ).
% intind
thf(fact_8089_intind,axiom,
! [I: nat,N2: nat,P: int > $o,X: int] :
( ( ord_less_nat @ I @ N2 )
=> ( ( P @ X )
=> ( P @ ( nth_int @ ( replicate_int @ N2 @ X ) @ I ) ) ) ) ).
% intind
thf(fact_8090_intind,axiom,
! [I: nat,N2: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
( ( ord_less_nat @ I @ N2 )
=> ( ( P @ X )
=> ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) @ I ) ) ) ) ).
% intind
thf(fact_8091_bit_Ocompl__eq__compl__iff,axiom,
! [X: int,Y: int] :
( ( ( bit_ri7919022796975470100ot_int @ X )
= ( bit_ri7919022796975470100ot_int @ Y ) )
= ( X = Y ) ) ).
% bit.compl_eq_compl_iff
thf(fact_8092_bit_Odouble__compl,axiom,
! [X: int] :
( ( bit_ri7919022796975470100ot_int @ ( bit_ri7919022796975470100ot_int @ X ) )
= X ) ).
% bit.double_compl
thf(fact_8093_replicate__eq__replicate,axiom,
! [M: nat,X: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
( ( ( replicate_VEBT_VEBT @ M @ X )
= ( replicate_VEBT_VEBT @ N2 @ Y ) )
= ( ( M = N2 )
& ( ( M != zero_zero_nat )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_8094_length__replicate,axiom,
! [N2: nat,X: vEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) )
= N2 ) ).
% length_replicate
thf(fact_8095_length__replicate,axiom,
! [N2: nat,X: $o] :
( ( size_size_list_o @ ( replicate_o @ N2 @ X ) )
= N2 ) ).
% length_replicate
thf(fact_8096_length__replicate,axiom,
! [N2: nat,X: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N2 @ X ) )
= N2 ) ).
% length_replicate
thf(fact_8097_length__replicate,axiom,
! [N2: nat,X: int] :
( ( size_size_list_int @ ( replicate_int @ N2 @ X ) )
= N2 ) ).
% length_replicate
thf(fact_8098_bit_Oxor__compl__right,axiom,
! [X: int,Y: int] :
( ( bit_se6526347334894502574or_int @ X @ ( bit_ri7919022796975470100ot_int @ Y ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ).
% bit.xor_compl_right
thf(fact_8099_bit_Oxor__compl__left,axiom,
! [X: int,Y: int] :
( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ Y )
= ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ).
% bit.xor_compl_left
thf(fact_8100_sums__zero,axiom,
( sums_complex
@ ^ [N4: nat] : zero_zero_complex
@ zero_zero_complex ) ).
% sums_zero
thf(fact_8101_sums__zero,axiom,
( sums_real
@ ^ [N4: nat] : zero_zero_real
@ zero_zero_real ) ).
% sums_zero
thf(fact_8102_sums__zero,axiom,
( sums_nat
@ ^ [N4: nat] : zero_zero_nat
@ zero_zero_nat ) ).
% sums_zero
thf(fact_8103_sums__zero,axiom,
( sums_int
@ ^ [N4: nat] : zero_zero_int
@ zero_zero_int ) ).
% sums_zero
thf(fact_8104_bit_Oconj__cancel__left,axiom,
! [X: int] :
( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
= zero_zero_int ) ).
% bit.conj_cancel_left
thf(fact_8105_bit_Oconj__cancel__right,axiom,
! [X: int] :
( ( bit_se725231765392027082nd_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
= zero_zero_int ) ).
% bit.conj_cancel_right
thf(fact_8106_Ball__set__replicate,axiom,
! [N2: nat,A: int,P: int > $o] :
( ( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
=> ( P @ X3 ) ) )
= ( ( P @ A )
| ( N2 = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_8107_Ball__set__replicate,axiom,
! [N2: nat,A: nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
=> ( P @ X3 ) ) )
= ( ( P @ A )
| ( N2 = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_8108_Ball__set__replicate,axiom,
! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
=> ( P @ X3 ) ) )
= ( ( P @ A )
| ( N2 = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_8109_Bex__set__replicate,axiom,
! [N2: nat,A: int,P: int > $o] :
( ( ? [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
& ( P @ X3 ) ) )
= ( ( P @ A )
& ( N2 != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_8110_Bex__set__replicate,axiom,
! [N2: nat,A: nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
& ( P @ X3 ) ) )
= ( ( P @ A )
& ( N2 != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_8111_Bex__set__replicate,axiom,
! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ? [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
& ( P @ X3 ) ) )
= ( ( P @ A )
& ( N2 != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_8112_in__set__replicate,axiom,
! [X: complex,N2: nat,Y: complex] :
( ( member_complex @ X @ ( set_complex2 @ ( replicate_complex @ N2 @ Y ) ) )
= ( ( X = Y )
& ( N2 != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_8113_in__set__replicate,axiom,
! [X: real,N2: nat,Y: real] :
( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N2 @ Y ) ) )
= ( ( X = Y )
& ( N2 != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_8114_in__set__replicate,axiom,
! [X: set_nat,N2: nat,Y: set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N2 @ Y ) ) )
= ( ( X = Y )
& ( N2 != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_8115_in__set__replicate,axiom,
! [X: int,N2: nat,Y: int] :
( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ Y ) ) )
= ( ( X = Y )
& ( N2 != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_8116_in__set__replicate,axiom,
! [X: nat,N2: nat,Y: nat] :
( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ Y ) ) )
= ( ( X = Y )
& ( N2 != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_8117_in__set__replicate,axiom,
! [X: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ Y ) ) )
= ( ( X = Y )
& ( N2 != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_8118_nth__replicate,axiom,
! [I: nat,N2: nat,X: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( ( nth_nat @ ( replicate_nat @ N2 @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_8119_nth__replicate,axiom,
! [I: nat,N2: nat,X: int] :
( ( ord_less_nat @ I @ N2 )
=> ( ( nth_int @ ( replicate_int @ N2 @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_8120_nth__replicate,axiom,
! [I: nat,N2: nat,X: vEBT_VEBT] :
( ( ord_less_nat @ I @ N2 )
=> ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_8121_bit_Ocompl__one,axiom,
( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% bit.compl_one
thf(fact_8122_bit_Ocompl__one,axiom,
( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% bit.compl_one
thf(fact_8123_bit_Ocompl__zero,axiom,
( ( bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% bit.compl_zero
thf(fact_8124_bit_Ocompl__zero,axiom,
( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% bit.compl_zero
thf(fact_8125_bit_Oxor__cancel__right,axiom,
! [X: code_integer] :
( ( bit_se3222712562003087583nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% bit.xor_cancel_right
thf(fact_8126_bit_Oxor__cancel__right,axiom,
! [X: int] :
( ( bit_se6526347334894502574or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% bit.xor_cancel_right
thf(fact_8127_bit_Oxor__cancel__left,axiom,
! [X: code_integer] :
( ( bit_se3222712562003087583nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% bit.xor_cancel_left
thf(fact_8128_bit_Oxor__cancel__left,axiom,
! [X: int] :
( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
= ( uminus_uminus_int @ one_one_int ) ) ).
% bit.xor_cancel_left
thf(fact_8129_bit_Oxor__one__right,axiom,
! [X: code_integer] :
( ( bit_se3222712562003087583nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( bit_ri7632146776885996613nteger @ X ) ) ).
% bit.xor_one_right
thf(fact_8130_bit_Oxor__one__right,axiom,
! [X: int] :
( ( bit_se6526347334894502574or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ X ) ) ).
% bit.xor_one_right
thf(fact_8131_bit_Oxor__one__left,axiom,
! [X: code_integer] :
( ( bit_se3222712562003087583nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
= ( bit_ri7632146776885996613nteger @ X ) ) ).
% bit.xor_one_left
thf(fact_8132_bit_Oxor__one__left,axiom,
! [X: int] :
( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
= ( bit_ri7919022796975470100ot_int @ X ) ) ).
% bit.xor_one_left
thf(fact_8133_not__negative__int__iff,axiom,
! [K: int] :
( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% not_negative_int_iff
thf(fact_8134_not__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% not_nonnegative_int_iff
thf(fact_8135_minus__not__numeral__eq,axiom,
! [N2: num] :
( ( uminus1351360451143612070nteger @ ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
= ( numera6620942414471956472nteger @ ( inc @ N2 ) ) ) ).
% minus_not_numeral_eq
thf(fact_8136_minus__not__numeral__eq,axiom,
! [N2: num] :
( ( uminus_uminus_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
= ( numeral_numeral_int @ ( inc @ N2 ) ) ) ).
% minus_not_numeral_eq
thf(fact_8137_even__not__iff,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri7632146776885996613nteger @ A ) )
= ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_not_iff
thf(fact_8138_even__not__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ A ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_not_iff
thf(fact_8139_push__bit__minus__one__eq__not__mask,axiom,
! [N2: nat] :
( ( bit_se7788150548672797655nteger @ N2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N2 ) ) ) ).
% push_bit_minus_one_eq_not_mask
thf(fact_8140_push__bit__minus__one__eq__not__mask,axiom,
! [N2: nat] :
( ( bit_se545348938243370406it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ) ).
% push_bit_minus_one_eq_not_mask
thf(fact_8141_powser__sums__zero__iff,axiom,
! [A: nat > complex,X: complex] :
( ( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ ( A @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) )
@ X )
= ( ( A @ zero_zero_nat )
= X ) ) ).
% powser_sums_zero_iff
thf(fact_8142_powser__sums__zero__iff,axiom,
! [A: nat > real,X: real] :
( ( sums_real
@ ^ [N4: nat] : ( times_times_real @ ( A @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) )
@ X )
= ( ( A @ zero_zero_nat )
= X ) ) ).
% powser_sums_zero_iff
thf(fact_8143_not__one__eq,axiom,
( ( bit_ri7632146776885996613nteger @ one_one_Code_integer )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% not_one_eq
thf(fact_8144_not__one__eq,axiom,
( ( bit_ri7919022796975470100ot_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% not_one_eq
thf(fact_8145_sums__le,axiom,
! [F: nat > nat,G: nat > nat,S: nat,T2: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( G @ N ) )
=> ( ( sums_nat @ F @ S )
=> ( ( sums_nat @ G @ T2 )
=> ( ord_less_eq_nat @ S @ T2 ) ) ) ) ).
% sums_le
thf(fact_8146_sums__le,axiom,
! [F: nat > int,G: nat > int,S: int,T2: int] :
( ! [N: nat] : ( ord_less_eq_int @ ( F @ N ) @ ( G @ N ) )
=> ( ( sums_int @ F @ S )
=> ( ( sums_int @ G @ T2 )
=> ( ord_less_eq_int @ S @ T2 ) ) ) ) ).
% sums_le
thf(fact_8147_sums__le,axiom,
! [F: nat > real,G: nat > real,S: real,T2: real] :
( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( G @ N ) )
=> ( ( sums_real @ F @ S )
=> ( ( sums_real @ G @ T2 )
=> ( ord_less_eq_real @ S @ T2 ) ) ) ) ).
% sums_le
thf(fact_8148_sums__0,axiom,
! [F: nat > complex] :
( ! [N: nat] :
( ( F @ N )
= zero_zero_complex )
=> ( sums_complex @ F @ zero_zero_complex ) ) ).
% sums_0
thf(fact_8149_sums__0,axiom,
! [F: nat > real] :
( ! [N: nat] :
( ( F @ N )
= zero_zero_real )
=> ( sums_real @ F @ zero_zero_real ) ) ).
% sums_0
thf(fact_8150_sums__0,axiom,
! [F: nat > nat] :
( ! [N: nat] :
( ( F @ N )
= zero_zero_nat )
=> ( sums_nat @ F @ zero_zero_nat ) ) ).
% sums_0
thf(fact_8151_sums__0,axiom,
! [F: nat > int] :
( ! [N: nat] :
( ( F @ N )
= zero_zero_int )
=> ( sums_int @ F @ zero_zero_int ) ) ).
% sums_0
thf(fact_8152_sums__single,axiom,
! [I: nat,F: nat > complex] :
( sums_complex
@ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex )
@ ( F @ I ) ) ).
% sums_single
thf(fact_8153_sums__single,axiom,
! [I: nat,F: nat > real] :
( sums_real
@ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real )
@ ( F @ I ) ) ).
% sums_single
thf(fact_8154_sums__single,axiom,
! [I: nat,F: nat > nat] :
( sums_nat
@ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat )
@ ( F @ I ) ) ).
% sums_single
thf(fact_8155_sums__single,axiom,
! [I: nat,F: nat > int] :
( sums_int
@ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int )
@ ( F @ I ) ) ).
% sums_single
thf(fact_8156_sums__add,axiom,
! [F: nat > real,A: real,G: nat > real,B: real] :
( ( sums_real @ F @ A )
=> ( ( sums_real @ G @ B )
=> ( sums_real
@ ^ [N4: nat] : ( plus_plus_real @ ( F @ N4 ) @ ( G @ N4 ) )
@ ( plus_plus_real @ A @ B ) ) ) ) ).
% sums_add
thf(fact_8157_sums__add,axiom,
! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
( ( sums_nat @ F @ A )
=> ( ( sums_nat @ G @ B )
=> ( sums_nat
@ ^ [N4: nat] : ( plus_plus_nat @ ( F @ N4 ) @ ( G @ N4 ) )
@ ( plus_plus_nat @ A @ B ) ) ) ) ).
% sums_add
thf(fact_8158_sums__add,axiom,
! [F: nat > int,A: int,G: nat > int,B: int] :
( ( sums_int @ F @ A )
=> ( ( sums_int @ G @ B )
=> ( sums_int
@ ^ [N4: nat] : ( plus_plus_int @ ( F @ N4 ) @ ( G @ N4 ) )
@ ( plus_plus_int @ A @ B ) ) ) ) ).
% sums_add
thf(fact_8159_sums__divide,axiom,
! [F: nat > complex,A: complex,C: complex] :
( ( sums_complex @ F @ A )
=> ( sums_complex
@ ^ [N4: nat] : ( divide1717551699836669952omplex @ ( F @ N4 ) @ C )
@ ( divide1717551699836669952omplex @ A @ C ) ) ) ).
% sums_divide
thf(fact_8160_sums__divide,axiom,
! [F: nat > real,A: real,C: real] :
( ( sums_real @ F @ A )
=> ( sums_real
@ ^ [N4: nat] : ( divide_divide_real @ ( F @ N4 ) @ C )
@ ( divide_divide_real @ A @ C ) ) ) ).
% sums_divide
thf(fact_8161_bit__not__int__iff,axiom,
! [K: int,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N2 )
= ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).
% bit_not_int_iff
thf(fact_8162_take__bit__not__iff,axiom,
! [N2: nat,A: int,B: int] :
( ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_ri7919022796975470100ot_int @ A ) )
= ( bit_se2923211474154528505it_int @ N2 @ ( bit_ri7919022796975470100ot_int @ B ) ) )
= ( ( bit_se2923211474154528505it_int @ N2 @ A )
= ( bit_se2923211474154528505it_int @ N2 @ B ) ) ) ).
% take_bit_not_iff
thf(fact_8163_take__bit__not__take__bit,axiom,
! [N2: nat,A: int] :
( ( bit_se2923211474154528505it_int @ N2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
= ( bit_se2923211474154528505it_int @ N2 @ ( bit_ri7919022796975470100ot_int @ A ) ) ) ).
% take_bit_not_take_bit
thf(fact_8164_of__int__not__eq,axiom,
! [K: int] :
( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ K ) )
= ( bit_ri7919022796975470100ot_int @ ( ring_1_of_int_int @ K ) ) ) ).
% of_int_not_eq
thf(fact_8165_of__int__not__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) )
= ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) ) ).
% of_int_not_numeral
thf(fact_8166_not__add__distrib,axiom,
! [A: int,B: int] :
( ( bit_ri7919022796975470100ot_int @ ( plus_plus_int @ A @ B ) )
= ( minus_minus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).
% not_add_distrib
thf(fact_8167_not__diff__distrib,axiom,
! [A: int,B: int] :
( ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A @ B ) )
= ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).
% not_diff_distrib
thf(fact_8168_sums__mult__iff,axiom,
! [C: complex,F: nat > complex,D: complex] :
( ( C != zero_zero_complex )
=> ( ( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ C @ ( F @ N4 ) )
@ ( times_times_complex @ C @ D ) )
= ( sums_complex @ F @ D ) ) ) ).
% sums_mult_iff
thf(fact_8169_sums__mult__iff,axiom,
! [C: real,F: nat > real,D: real] :
( ( C != zero_zero_real )
=> ( ( sums_real
@ ^ [N4: nat] : ( times_times_real @ C @ ( F @ N4 ) )
@ ( times_times_real @ C @ D ) )
= ( sums_real @ F @ D ) ) ) ).
% sums_mult_iff
thf(fact_8170_sums__mult2__iff,axiom,
! [C: complex,F: nat > complex,D: complex] :
( ( C != zero_zero_complex )
=> ( ( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ C )
@ ( times_times_complex @ D @ C ) )
= ( sums_complex @ F @ D ) ) ) ).
% sums_mult2_iff
thf(fact_8171_sums__mult2__iff,axiom,
! [C: real,F: nat > real,D: real] :
( ( C != zero_zero_real )
=> ( ( sums_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ C )
@ ( times_times_real @ D @ C ) )
= ( sums_real @ F @ D ) ) ) ).
% sums_mult2_iff
thf(fact_8172_replicate__length__same,axiom,
! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
( ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( X2 = X ) )
=> ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_8173_replicate__length__same,axiom,
! [Xs: list_o,X: $o] :
( ! [X2: $o] :
( ( member_o @ X2 @ ( set_o2 @ Xs ) )
=> ( X2 = X ) )
=> ( ( replicate_o @ ( size_size_list_o @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_8174_replicate__length__same,axiom,
! [Xs: list_nat,X: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( X2 = X ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_8175_replicate__length__same,axiom,
! [Xs: list_int,X: int] :
( ! [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
=> ( X2 = X ) )
=> ( ( replicate_int @ ( size_size_list_int @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_8176_replicate__eqI,axiom,
! [Xs: list_complex,N2: nat,X: complex] :
( ( ( size_s3451745648224563538omplex @ Xs )
= N2 )
=> ( ! [Y2: complex] :
( ( member_complex @ Y2 @ ( set_complex2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_complex @ N2 @ X ) ) ) ) ).
% replicate_eqI
thf(fact_8177_replicate__eqI,axiom,
! [Xs: list_real,N2: nat,X: real] :
( ( ( size_size_list_real @ Xs )
= N2 )
=> ( ! [Y2: real] :
( ( member_real @ Y2 @ ( set_real2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_real @ N2 @ X ) ) ) ) ).
% replicate_eqI
thf(fact_8178_replicate__eqI,axiom,
! [Xs: list_set_nat,N2: nat,X: set_nat] :
( ( ( size_s3254054031482475050et_nat @ Xs )
= N2 )
=> ( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ ( set_set_nat2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_set_nat @ N2 @ X ) ) ) ) ).
% replicate_eqI
thf(fact_8179_replicate__eqI,axiom,
! [Xs: list_VEBT_VEBT,N2: nat,X: vEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= N2 )
=> ( ! [Y2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ Y2 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_VEBT_VEBT @ N2 @ X ) ) ) ) ).
% replicate_eqI
thf(fact_8180_replicate__eqI,axiom,
! [Xs: list_o,N2: nat,X: $o] :
( ( ( size_size_list_o @ Xs )
= N2 )
=> ( ! [Y2: $o] :
( ( member_o @ Y2 @ ( set_o2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_o @ N2 @ X ) ) ) ) ).
% replicate_eqI
thf(fact_8181_replicate__eqI,axiom,
! [Xs: list_nat,N2: nat,X: nat] :
( ( ( size_size_list_nat @ Xs )
= N2 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_nat @ N2 @ X ) ) ) ) ).
% replicate_eqI
thf(fact_8182_replicate__eqI,axiom,
! [Xs: list_int,N2: nat,X: int] :
( ( ( size_size_list_int @ Xs )
= N2 )
=> ( ! [Y2: int] :
( ( member_int @ Y2 @ ( set_int2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_int @ N2 @ X ) ) ) ) ).
% replicate_eqI
thf(fact_8183_sums__mult__D,axiom,
! [C: complex,F: nat > complex,A: complex] :
( ( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ C @ ( F @ N4 ) )
@ A )
=> ( ( C != zero_zero_complex )
=> ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).
% sums_mult_D
thf(fact_8184_sums__mult__D,axiom,
! [C: real,F: nat > real,A: real] :
( ( sums_real
@ ^ [N4: nat] : ( times_times_real @ C @ ( F @ N4 ) )
@ A )
=> ( ( C != zero_zero_real )
=> ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).
% sums_mult_D
thf(fact_8185_sums__Suc__imp,axiom,
! [F: nat > complex,S: complex] :
( ( ( F @ zero_zero_nat )
= zero_zero_complex )
=> ( ( sums_complex
@ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
@ S )
=> ( sums_complex @ F @ S ) ) ) ).
% sums_Suc_imp
thf(fact_8186_sums__Suc__imp,axiom,
! [F: nat > real,S: real] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( sums_real
@ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
@ S )
=> ( sums_real @ F @ S ) ) ) ).
% sums_Suc_imp
thf(fact_8187_sums__Suc__iff,axiom,
! [F: nat > real,S: real] :
( ( sums_real
@ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
@ S )
= ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc_iff
thf(fact_8188_sums__Suc,axiom,
! [F: nat > real,L: real] :
( ( sums_real
@ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
@ L )
=> ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_8189_sums__Suc,axiom,
! [F: nat > nat,L: nat] :
( ( sums_nat
@ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
@ L )
=> ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_8190_sums__Suc,axiom,
! [F: nat > int,L: int] :
( ( sums_int
@ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
@ L )
=> ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_8191_sums__zero__iff__shift,axiom,
! [N2: nat,F: nat > complex,S: complex] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ( F @ I2 )
= zero_zero_complex ) )
=> ( ( sums_complex
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
@ S )
= ( sums_complex @ F @ S ) ) ) ).
% sums_zero_iff_shift
thf(fact_8192_sums__zero__iff__shift,axiom,
! [N2: nat,F: nat > real,S: real] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ( F @ I2 )
= zero_zero_real ) )
=> ( ( sums_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
@ S )
= ( sums_real @ F @ S ) ) ) ).
% sums_zero_iff_shift
thf(fact_8193_minus__eq__not__plus__1,axiom,
( uminus1351360451143612070nteger
= ( ^ [A3: code_integer] : ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A3 ) @ one_one_Code_integer ) ) ) ).
% minus_eq_not_plus_1
thf(fact_8194_minus__eq__not__plus__1,axiom,
( uminus_uminus_int
= ( ^ [A3: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A3 ) @ one_one_int ) ) ) ).
% minus_eq_not_plus_1
thf(fact_8195_minus__eq__not__minus__1,axiom,
( uminus1351360451143612070nteger
= ( ^ [A3: code_integer] : ( bit_ri7632146776885996613nteger @ ( minus_8373710615458151222nteger @ A3 @ one_one_Code_integer ) ) ) ) ).
% minus_eq_not_minus_1
thf(fact_8196_minus__eq__not__minus__1,axiom,
( uminus_uminus_int
= ( ^ [A3: int] : ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A3 @ one_one_int ) ) ) ) ).
% minus_eq_not_minus_1
thf(fact_8197_not__eq__complement,axiom,
( bit_ri7632146776885996613nteger
= ( ^ [A3: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A3 ) @ one_one_Code_integer ) ) ) ).
% not_eq_complement
thf(fact_8198_not__eq__complement,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [A3: int] : ( minus_minus_int @ ( uminus_uminus_int @ A3 ) @ one_one_int ) ) ) ).
% not_eq_complement
thf(fact_8199_not__int__def,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [K2: int] : ( minus_minus_int @ ( uminus_uminus_int @ K2 ) @ one_one_int ) ) ) ).
% not_int_def
thf(fact_8200_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_8201_disjunctive__diff,axiom,
! [B: int,A: int] :
( ! [N: nat] :
( ( bit_se1146084159140164899it_int @ B @ N )
=> ( bit_se1146084159140164899it_int @ A @ N ) )
=> ( ( minus_minus_int @ A @ B )
= ( bit_se725231765392027082nd_int @ A @ ( bit_ri7919022796975470100ot_int @ B ) ) ) ) ).
% disjunctive_diff
thf(fact_8202_take__bit__not__eq__mask__diff,axiom,
! [N2: nat,A: int] :
( ( bit_se2923211474154528505it_int @ N2 @ ( bit_ri7919022796975470100ot_int @ A ) )
= ( minus_minus_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ).
% take_bit_not_eq_mask_diff
thf(fact_8203_minus__numeral__inc__eq,axiom,
! [N2: num] :
( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N2 ) ) )
= ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% minus_numeral_inc_eq
thf(fact_8204_minus__numeral__inc__eq,axiom,
! [N2: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N2 ) ) )
= ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% minus_numeral_inc_eq
thf(fact_8205_powser__sums__if,axiom,
! [M: nat,Z: complex] :
( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ ( if_complex @ ( N4 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N4 ) )
@ ( power_power_complex @ Z @ M ) ) ).
% powser_sums_if
thf(fact_8206_powser__sums__if,axiom,
! [M: nat,Z: real] :
( sums_real
@ ^ [N4: nat] : ( times_times_real @ ( if_real @ ( N4 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N4 ) )
@ ( power_power_real @ Z @ M ) ) ).
% powser_sums_if
thf(fact_8207_powser__sums__if,axiom,
! [M: nat,Z: int] :
( sums_int
@ ^ [N4: nat] : ( times_times_int @ ( if_int @ ( N4 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N4 ) )
@ ( power_power_int @ Z @ M ) ) ).
% powser_sums_if
thf(fact_8208_powser__sums__zero,axiom,
! [A: nat > complex] :
( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ ( A @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) )
@ ( A @ zero_zero_nat ) ) ).
% powser_sums_zero
thf(fact_8209_powser__sums__zero,axiom,
! [A: nat > real] :
( sums_real
@ ^ [N4: nat] : ( times_times_real @ ( A @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) )
@ ( A @ zero_zero_nat ) ) ).
% powser_sums_zero
thf(fact_8210_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_8211_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_8212_and__not__numerals_I2_J,axiom,
! [N2: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
= one_one_int ) ).
% and_not_numerals(2)
thf(fact_8213_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_8214_not__numeral__Bit0__eq,axiom,
! [N2: num] :
( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) ) ) ).
% not_numeral_Bit0_eq
thf(fact_8215_not__numeral__Bit0__eq,axiom,
! [N2: num] :
( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ) ).
% not_numeral_Bit0_eq
thf(fact_8216_bit__minus__int__iff,axiom,
! [K: int,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N2 )
= ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N2 ) ) ).
% bit_minus_int_iff
thf(fact_8217_not__numeral__BitM__eq,axiom,
! [N2: num] :
( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bitM @ N2 ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) ) ) ).
% not_numeral_BitM_eq
thf(fact_8218_not__numeral__BitM__eq,axiom,
! [N2: num] :
( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bitM @ N2 ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ) ).
% not_numeral_BitM_eq
thf(fact_8219_take__bit__not__mask__eq__0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) )
= zero_zero_int ) ) ).
% take_bit_not_mask_eq_0
thf(fact_8220_push__bit__mask__eq,axiom,
! [M: nat,N2: nat] :
( ( bit_se545348938243370406it_int @ M @ ( bit_se2000444600071755411sk_int @ N2 ) )
= ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ N2 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ M ) ) ) ) ).
% push_bit_mask_eq
thf(fact_8221_unset__bit__eq__and__not,axiom,
( bit_se4203085406695923979it_int
= ( ^ [N4: nat,A3: int] : ( bit_se725231765392027082nd_int @ A3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ) ).
% unset_bit_eq_and_not
thf(fact_8222_unset__bit__int__def,axiom,
( bit_se4203085406695923979it_int
= ( ^ [N4: nat,K2: int] : ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ) ).
% unset_bit_int_def
thf(fact_8223_and__not__numerals_I5_J,axiom,
! [M: num,N2: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% and_not_numerals(5)
thf(fact_8224_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_8225_and__not__numerals_I3_J,axiom,
! [N2: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
= zero_zero_int ) ).
% and_not_numerals(3)
thf(fact_8226_and__not__numerals_I6_J,axiom,
! [M: num,N2: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% and_not_numerals(6)
thf(fact_8227_and__not__numerals_I9_J,axiom,
! [M: num,N2: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% and_not_numerals(9)
thf(fact_8228_bit__not__iff__eq,axiom,
! [A: int,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ A ) @ N2 )
= ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
!= zero_zero_int )
& ~ ( bit_se1146084159140164899it_int @ A @ N2 ) ) ) ).
% bit_not_iff_eq
thf(fact_8229_minus__exp__eq__not__mask,axiom,
! [N2: nat] :
( ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
= ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N2 ) ) ) ).
% minus_exp_eq_not_mask
thf(fact_8230_minus__exp__eq__not__mask,axiom,
! [N2: nat] :
( ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ) ).
% minus_exp_eq_not_mask
thf(fact_8231_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_8232_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_8233_power__half__series,axiom,
( sums_real
@ ^ [N4: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N4 ) )
@ one_one_real ) ).
% power_half_series
thf(fact_8234_sums__if_H,axiom,
! [G: nat > real,X: real] :
( ( sums_real @ G @ X )
=> ( sums_real
@ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ X ) ) ).
% sums_if'
thf(fact_8235_sums__if,axiom,
! [G: nat > real,X: real,F: nat > real,Y: real] :
( ( sums_real @ G @ X )
=> ( ( sums_real @ F @ Y )
=> ( sums_real
@ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( F @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_real @ X @ Y ) ) ) ) ).
% sums_if
thf(fact_8236_and__not__numerals_I8_J,axiom,
! [M: num,N2: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
= ( 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 @ N2 ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_8237_not__int__rec,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [K2: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% not_int_rec
thf(fact_8238_cos__paired,axiom,
! [X: real] :
( sums_real
@ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
@ ( cos_real @ X ) ) ).
% cos_paired
thf(fact_8239_vebt__buildup_Osimps_I3_J,axiom,
! [Va2: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.simps(3)
thf(fact_8240_vebt__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_vebt_member @ X @ Xa )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
=> ( ! [A5: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
=> ~ ( ( ( Xa = zero_zero_nat )
=> A5 )
& ( ( Xa != zero_zero_nat )
=> ( ( ( Xa = one_one_nat )
=> B5 )
& ( Xa = one_one_nat ) ) ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) )
=> ~ ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less_nat @ Xa @ Mi )
& ( ~ ( ord_less_nat @ Xa @ Mi )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa )
& ( ~ ( ord_less_nat @ Ma2 @ Xa )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_8241_geometric__deriv__sums,axiom,
! [Z: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
=> ( sums_real
@ ^ [N4: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) @ ( power_power_real @ Z @ N4 ) )
@ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_8242_geometric__deriv__sums,axiom,
! [Z: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
=> ( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N4 ) ) @ ( power_power_complex @ Z @ N4 ) )
@ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_8243_vebt__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
=> ( ! [A5: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( Y
= ( ( ( Xa = zero_zero_nat )
=> A5 )
& ( ( Xa != zero_zero_nat )
=> ( ( ( Xa = one_one_nat )
=> B5 )
& ( Xa = one_one_nat ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) ) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [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 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y
= ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less_nat @ Xa @ Mi )
& ( ~ ( ord_less_nat @ Xa @ Mi )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa )
& ( ~ ( ord_less_nat @ Ma2 @ Xa )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_8244_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
=> ( ! [A5: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( Y
= ( ( ( Xa = zero_zero_nat )
=> A5 )
& ( ( Xa != zero_zero_nat )
=> ( ( ( Xa = one_one_nat )
=> B5 )
& ( Xa = one_one_nat ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) ) ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
=> ( ( Y
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( 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 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_8245_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
=> ( ! [A5: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
=> ~ ( ( ( Xa = zero_zero_nat )
=> A5 )
& ( ( Xa != zero_zero_nat )
=> ( ( ( Xa = one_one_nat )
=> B5 )
& ( Xa = one_one_nat ) ) ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_8246_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
=> ( ! [A5: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B5 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
=> ( ( ( Xa = zero_zero_nat )
=> A5 )
& ( ( Xa != zero_zero_nat )
=> ( ( ( Xa = one_one_nat )
=> B5 )
& ( Xa = one_one_nat ) ) ) ) ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_8247_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) )
=> ( ! [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 ) @ Xa ) ) )
=> ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) )
=> ( ( Xa = Mi )
| ( Xa = Ma2 ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa ) )
=> ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ Xa ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_8248_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
=> ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) )
=> ~ ( ( Xa = Mi )
| ( Xa = Ma2 ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa ) )
=> ~ ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ Xa ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_8249_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( ( Y
= ( ( Xa = Mi )
| ( Xa = Ma2 ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ( ( Y
= ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Mi @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
=> ( ( Y
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Vd ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_8250_diffs__equiv,axiom,
! [C: nat > real,X: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( diffs_real @ C @ N4 ) @ ( power_power_real @ X @ N4 ) ) )
=> ( sums_real
@ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( C @ N4 ) ) @ ( power_power_real @ X @ ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) ) )
@ ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( diffs_real @ C @ N4 ) @ ( power_power_real @ X @ N4 ) ) ) ) ) ).
% diffs_equiv
thf(fact_8251_diffs__equiv,axiom,
! [C: nat > complex,X: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( diffs_complex @ C @ N4 ) @ ( power_power_complex @ X @ N4 ) ) )
=> ( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N4 ) @ ( C @ N4 ) ) @ ( power_power_complex @ X @ ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) ) )
@ ( suminf_complex
@ ^ [N4: nat] : ( times_times_complex @ ( diffs_complex @ C @ N4 ) @ ( power_power_complex @ X @ N4 ) ) ) ) ) ).
% diffs_equiv
thf(fact_8252_exp__first__two__terms,axiom,
( exp_real
= ( ^ [X3: real] :
( plus_plus_real @ ( plus_plus_real @ one_one_real @ X3 )
@ ( suminf_real
@ ^ [N4: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% exp_first_two_terms
thf(fact_8253_exp__first__two__terms,axiom,
( exp_complex
= ( ^ [X3: complex] :
( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ X3 )
@ ( suminf_complex
@ ^ [N4: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_complex @ X3 @ ( plus_plus_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% exp_first_two_terms
thf(fact_8254_monoI1,axiom,
! [X8: nat > set_nat] :
( ! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_set_nat @ ( X8 @ M5 ) @ ( X8 @ N ) ) )
=> ( topolo7278393974255667507et_nat @ X8 ) ) ).
% monoI1
thf(fact_8255_monoI1,axiom,
! [X8: nat > rat] :
( ! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_rat @ ( X8 @ M5 ) @ ( X8 @ N ) ) )
=> ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% monoI1
thf(fact_8256_monoI1,axiom,
! [X8: nat > num] :
( ! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_num @ ( X8 @ M5 ) @ ( X8 @ N ) ) )
=> ( topolo1459490580787246023eq_num @ X8 ) ) ).
% monoI1
thf(fact_8257_monoI1,axiom,
! [X8: nat > nat] :
( ! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_nat @ ( X8 @ M5 ) @ ( X8 @ N ) ) )
=> ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% monoI1
thf(fact_8258_monoI1,axiom,
! [X8: nat > int] :
( ! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_int @ ( X8 @ M5 ) @ ( X8 @ N ) ) )
=> ( topolo4899668324122417113eq_int @ X8 ) ) ).
% monoI1
thf(fact_8259_monoI1,axiom,
! [X8: nat > real] :
( ! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_real @ ( X8 @ M5 ) @ ( X8 @ N ) ) )
=> ( topolo6980174941875973593q_real @ X8 ) ) ).
% monoI1
thf(fact_8260_monoI2,axiom,
! [X8: nat > set_nat] :
( ! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_set_nat @ ( X8 @ N ) @ ( X8 @ M5 ) ) )
=> ( topolo7278393974255667507et_nat @ X8 ) ) ).
% monoI2
thf(fact_8261_monoI2,axiom,
! [X8: nat > rat] :
( ! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_rat @ ( X8 @ N ) @ ( X8 @ M5 ) ) )
=> ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% monoI2
thf(fact_8262_monoI2,axiom,
! [X8: nat > num] :
( ! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_num @ ( X8 @ N ) @ ( X8 @ M5 ) ) )
=> ( topolo1459490580787246023eq_num @ X8 ) ) ).
% monoI2
thf(fact_8263_monoI2,axiom,
! [X8: nat > nat] :
( ! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_nat @ ( X8 @ N ) @ ( X8 @ M5 ) ) )
=> ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% monoI2
thf(fact_8264_monoI2,axiom,
! [X8: nat > int] :
( ! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_int @ ( X8 @ N ) @ ( X8 @ M5 ) ) )
=> ( topolo4899668324122417113eq_int @ X8 ) ) ).
% monoI2
thf(fact_8265_monoI2,axiom,
! [X8: nat > real] :
( ! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_real @ ( X8 @ N ) @ ( X8 @ M5 ) ) )
=> ( topolo6980174941875973593q_real @ X8 ) ) ).
% monoI2
thf(fact_8266_scaleR__cancel__right,axiom,
! [A: real,X: complex,B: real] :
( ( ( real_V2046097035970521341omplex @ A @ X )
= ( real_V2046097035970521341omplex @ B @ X ) )
= ( ( A = B )
| ( X = zero_zero_complex ) ) ) ).
% scaleR_cancel_right
thf(fact_8267_scaleR__cancel__right,axiom,
! [A: real,X: real,B: real] :
( ( ( real_V1485227260804924795R_real @ A @ X )
= ( real_V1485227260804924795R_real @ B @ X ) )
= ( ( A = B )
| ( X = zero_zero_real ) ) ) ).
% scaleR_cancel_right
thf(fact_8268_scaleR__zero__right,axiom,
! [A: real] :
( ( real_V2046097035970521341omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% scaleR_zero_right
thf(fact_8269_scaleR__zero__right,axiom,
! [A: real] :
( ( real_V1485227260804924795R_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% scaleR_zero_right
thf(fact_8270_scaleR__zero__left,axiom,
! [X: complex] :
( ( real_V2046097035970521341omplex @ zero_zero_real @ X )
= zero_zero_complex ) ).
% scaleR_zero_left
thf(fact_8271_scaleR__zero__left,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% scaleR_zero_left
thf(fact_8272_scaleR__eq__0__iff,axiom,
! [A: real,X: complex] :
( ( ( real_V2046097035970521341omplex @ A @ X )
= zero_zero_complex )
= ( ( A = zero_zero_real )
| ( X = zero_zero_complex ) ) ) ).
% scaleR_eq_0_iff
thf(fact_8273_scaleR__eq__0__iff,axiom,
! [A: real,X: real] :
( ( ( real_V1485227260804924795R_real @ A @ X )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% scaleR_eq_0_iff
thf(fact_8274_scaleR__eq__iff,axiom,
! [B: real,U: real,A: real] :
( ( ( plus_plus_real @ B @ ( real_V1485227260804924795R_real @ U @ A ) )
= ( plus_plus_real @ A @ ( real_V1485227260804924795R_real @ U @ B ) ) )
= ( ( A = B )
| ( U = one_one_real ) ) ) ).
% scaleR_eq_iff
thf(fact_8275_scaleR__power,axiom,
! [X: real,Y: real,N2: nat] :
( ( power_power_real @ ( real_V1485227260804924795R_real @ X @ Y ) @ N2 )
= ( real_V1485227260804924795R_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) ) ) ).
% scaleR_power
thf(fact_8276_scaleR__power,axiom,
! [X: real,Y: complex,N2: nat] :
( ( power_power_complex @ ( real_V2046097035970521341omplex @ X @ Y ) @ N2 )
= ( real_V2046097035970521341omplex @ ( power_power_real @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) ) ) ).
% scaleR_power
thf(fact_8277_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_8278_scaleR__minus1__left,axiom,
! [X: complex] :
( ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ one_one_real ) @ X )
= ( uminus1482373934393186551omplex @ X ) ) ).
% scaleR_minus1_left
thf(fact_8279_scaleR__collapse,axiom,
! [U: real,A: real] :
( ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V1485227260804924795R_real @ U @ A ) )
= A ) ).
% scaleR_collapse
thf(fact_8280_scaleR__times,axiom,
! [U: num,W: num,A: complex] :
( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ U ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
= ( real_V2046097035970521341omplex @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).
% scaleR_times
thf(fact_8281_scaleR__times,axiom,
! [U: num,W: num,A: real] :
( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ U ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
= ( real_V1485227260804924795R_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).
% scaleR_times
thf(fact_8282_inverse__scaleR__times,axiom,
! [V: num,W: num,A: complex] :
( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
= ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% inverse_scaleR_times
thf(fact_8283_inverse__scaleR__times,axiom,
! [V: num,W: num,A: real] :
( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
= ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% inverse_scaleR_times
thf(fact_8284_fraction__scaleR__times,axiom,
! [U: num,V: num,W: num,A: complex] :
( ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
= ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% fraction_scaleR_times
thf(fact_8285_fraction__scaleR__times,axiom,
! [U: num,V: num,W: num,A: real] :
( ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
= ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% fraction_scaleR_times
thf(fact_8286_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_8287_scaleR__right__imp__eq,axiom,
! [X: complex,A: real,B: real] :
( ( X != zero_zero_complex )
=> ( ( ( real_V2046097035970521341omplex @ A @ X )
= ( real_V2046097035970521341omplex @ B @ X ) )
=> ( A = B ) ) ) ).
% scaleR_right_imp_eq
thf(fact_8288_scaleR__right__imp__eq,axiom,
! [X: real,A: real,B: real] :
( ( X != zero_zero_real )
=> ( ( ( real_V1485227260804924795R_real @ A @ X )
= ( real_V1485227260804924795R_real @ B @ X ) )
=> ( A = B ) ) ) ).
% scaleR_right_imp_eq
thf(fact_8289_scaleR__right__distrib,axiom,
! [A: real,X: real,Y: real] :
( ( real_V1485227260804924795R_real @ A @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ).
% scaleR_right_distrib
thf(fact_8290_scaleR__left_Oadd,axiom,
! [X: real,Y: real,Xa: real] :
( ( real_V1485227260804924795R_real @ ( plus_plus_real @ X @ Y ) @ Xa )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ X @ Xa ) @ ( real_V1485227260804924795R_real @ Y @ Xa ) ) ) ).
% scaleR_left.add
thf(fact_8291_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_8292_of__real__def,axiom,
( real_V1803761363581548252l_real
= ( ^ [R5: real] : ( real_V1485227260804924795R_real @ R5 @ one_one_real ) ) ) ).
% of_real_def
thf(fact_8293_of__real__def,axiom,
( real_V4546457046886955230omplex
= ( ^ [R5: real] : ( real_V2046097035970521341omplex @ R5 @ one_one_complex ) ) ) ).
% of_real_def
thf(fact_8294_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_8295_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_8296_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_8297_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_8298_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_8299_scaleR__left__mono,axiom,
! [X: real,Y: real,A: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ) ).
% scaleR_left_mono
thf(fact_8300_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_8301_eq__vector__fraction__iff,axiom,
! [X: complex,U: real,V: real,A: complex] :
( ( X
= ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A ) )
= ( ( ( V = zero_zero_real )
=> ( X = zero_zero_complex ) )
& ( ( V != zero_zero_real )
=> ( ( real_V2046097035970521341omplex @ V @ X )
= ( real_V2046097035970521341omplex @ U @ A ) ) ) ) ) ).
% eq_vector_fraction_iff
thf(fact_8302_eq__vector__fraction__iff,axiom,
! [X: real,U: real,V: real,A: real] :
( ( X
= ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A ) )
= ( ( ( V = zero_zero_real )
=> ( X = zero_zero_real ) )
& ( ( V != zero_zero_real )
=> ( ( real_V1485227260804924795R_real @ V @ X )
= ( real_V1485227260804924795R_real @ U @ A ) ) ) ) ) ).
% eq_vector_fraction_iff
thf(fact_8303_vector__fraction__eq__iff,axiom,
! [U: real,V: real,A: complex,X: complex] :
( ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A )
= X )
= ( ( ( V = zero_zero_real )
=> ( X = zero_zero_complex ) )
& ( ( V != zero_zero_real )
=> ( ( real_V2046097035970521341omplex @ U @ A )
= ( real_V2046097035970521341omplex @ V @ X ) ) ) ) ) ).
% vector_fraction_eq_iff
thf(fact_8304_vector__fraction__eq__iff,axiom,
! [U: real,V: real,A: real,X: real] :
( ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A )
= X )
= ( ( ( V = zero_zero_real )
=> ( X = zero_zero_real ) )
& ( ( V != zero_zero_real )
=> ( ( real_V1485227260804924795R_real @ U @ A )
= ( real_V1485227260804924795R_real @ V @ X ) ) ) ) ) ).
% vector_fraction_eq_iff
thf(fact_8305_Real__Vector__Spaces_Ole__add__iff2,axiom,
! [A: real,E2: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E2 ) @ D ) )
= ( ord_less_eq_real @ C @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% Real_Vector_Spaces.le_add_iff2
thf(fact_8306_Real__Vector__Spaces_Ole__add__iff1,axiom,
! [A: real,E2: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E2 ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% Real_Vector_Spaces.le_add_iff1
thf(fact_8307_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_8308_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_8309_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_8310_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_8311_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_8312_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_8313_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_8314_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_8315_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_8316_scaleR__mono,axiom,
! [A: real,B: real,X: real,Y: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( 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 @ Y ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_8317_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_8318_scaleR__2,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
= ( plus_plus_real @ X @ X ) ) ).
% scaleR_2
thf(fact_8319_real__vector__affinity__eq,axiom,
! [M: real,X: real,C: real,Y: real] :
( ( M != zero_zero_real )
=> ( ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X ) @ C )
= Y )
= ( X
= ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) ) ) ) ) ).
% real_vector_affinity_eq
thf(fact_8320_real__vector__eq__affinity,axiom,
! [M: real,Y: real,X: real,C: real] :
( ( M != zero_zero_real )
=> ( ( Y
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X ) @ C ) )
= ( ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) )
= X ) ) ) ).
% real_vector_eq_affinity
thf(fact_8321_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_8322_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_8323_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_8324_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_8325_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_8326_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_8327_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_8328_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_8329_nonzero__inverse__scaleR__distrib,axiom,
! [A: real,X: real] :
( ( A != zero_zero_real )
=> ( ( X != zero_zero_real )
=> ( ( inverse_inverse_real @ ( real_V1485227260804924795R_real @ A @ X ) )
= ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ X ) ) ) ) ) ).
% nonzero_inverse_scaleR_distrib
thf(fact_8330_nonzero__inverse__scaleR__distrib,axiom,
! [A: real,X: complex] :
( ( A != zero_zero_real )
=> ( ( X != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( real_V2046097035970521341omplex @ A @ X ) )
= ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ A ) @ ( invers8013647133539491842omplex @ X ) ) ) ) ) ).
% nonzero_inverse_scaleR_distrib
thf(fact_8331_diffs__def,axiom,
( diffs_real
= ( ^ [C2: nat > real,N4: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) @ ( C2 @ ( suc @ N4 ) ) ) ) ) ).
% diffs_def
thf(fact_8332_diffs__def,axiom,
( diffs_int
= ( ^ [C2: nat > int,N4: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) @ ( C2 @ ( suc @ N4 ) ) ) ) ) ).
% diffs_def
thf(fact_8333_diffs__def,axiom,
( diffs_complex
= ( ^ [C2: nat > complex,N4: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N4 ) ) @ ( C2 @ ( suc @ N4 ) ) ) ) ) ).
% diffs_def
thf(fact_8334_diffs__def,axiom,
( diffs_rat
= ( ^ [C2: nat > rat,N4: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N4 ) ) @ ( C2 @ ( suc @ N4 ) ) ) ) ) ).
% diffs_def
thf(fact_8335_summable__exp__generic,axiom,
! [X: real] :
( summable_real
@ ^ [N4: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N4 ) ) @ ( power_power_real @ X @ N4 ) ) ) ).
% summable_exp_generic
thf(fact_8336_summable__exp__generic,axiom,
! [X: complex] :
( summable_complex
@ ^ [N4: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N4 ) ) @ ( power_power_complex @ X @ N4 ) ) ) ).
% summable_exp_generic
thf(fact_8337_termdiff__converges__all,axiom,
! [C: nat > complex,X: complex] :
( ! [X2: complex] :
( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( C @ N4 ) @ ( power_power_complex @ X2 @ N4 ) ) )
=> ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( diffs_complex @ C @ N4 ) @ ( power_power_complex @ X @ N4 ) ) ) ) ).
% termdiff_converges_all
thf(fact_8338_termdiff__converges__all,axiom,
! [C: nat > real,X: real] :
( ! [X2: real] :
( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( C @ N4 ) @ ( power_power_real @ X2 @ N4 ) ) )
=> ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( diffs_real @ C @ N4 ) @ ( power_power_real @ X @ N4 ) ) ) ) ).
% termdiff_converges_all
thf(fact_8339_sin__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N4: nat] : ( real_V1485227260804924795R_real @ ( sin_coeff @ N4 ) @ ( power_power_real @ X @ N4 ) )
@ ( sin_real @ X ) ) ).
% sin_converges
thf(fact_8340_sin__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N4: nat] : ( real_V2046097035970521341omplex @ ( sin_coeff @ N4 ) @ ( power_power_complex @ X @ N4 ) )
@ ( sin_complex @ X ) ) ).
% sin_converges
thf(fact_8341_sin__def,axiom,
( sin_real
= ( ^ [X3: real] :
( suminf_real
@ ^ [N4: nat] : ( real_V1485227260804924795R_real @ ( sin_coeff @ N4 ) @ ( power_power_real @ X3 @ N4 ) ) ) ) ) ).
% sin_def
thf(fact_8342_sin__def,axiom,
( sin_complex
= ( ^ [X3: complex] :
( suminf_complex
@ ^ [N4: nat] : ( real_V2046097035970521341omplex @ ( sin_coeff @ N4 ) @ ( power_power_complex @ X3 @ N4 ) ) ) ) ) ).
% sin_def
thf(fact_8343_cos__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N4: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N4 ) @ ( power_power_real @ X @ N4 ) )
@ ( cos_real @ X ) ) ).
% cos_converges
thf(fact_8344_cos__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N4: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N4 ) @ ( power_power_complex @ X @ N4 ) )
@ ( cos_complex @ X ) ) ).
% cos_converges
thf(fact_8345_cos__def,axiom,
( cos_real
= ( ^ [X3: real] :
( suminf_real
@ ^ [N4: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N4 ) @ ( power_power_real @ X3 @ N4 ) ) ) ) ) ).
% cos_def
thf(fact_8346_cos__def,axiom,
( cos_complex
= ( ^ [X3: complex] :
( suminf_complex
@ ^ [N4: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N4 ) @ ( power_power_complex @ X3 @ N4 ) ) ) ) ) ).
% cos_def
thf(fact_8347_summable__norm__sin,axiom,
! [X: real] :
( summable_real
@ ^ [N4: nat] : ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ ( sin_coeff @ N4 ) @ ( power_power_real @ X @ N4 ) ) ) ) ).
% summable_norm_sin
thf(fact_8348_summable__norm__sin,axiom,
! [X: complex] :
( summable_real
@ ^ [N4: nat] : ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ ( sin_coeff @ N4 ) @ ( power_power_complex @ X @ N4 ) ) ) ) ).
% summable_norm_sin
thf(fact_8349_summable__norm__cos,axiom,
! [X: real] :
( summable_real
@ ^ [N4: nat] : ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ ( cos_coeff @ N4 ) @ ( power_power_real @ X @ N4 ) ) ) ) ).
% summable_norm_cos
thf(fact_8350_summable__norm__cos,axiom,
! [X: complex] :
( summable_real
@ ^ [N4: nat] : ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ ( cos_coeff @ N4 ) @ ( power_power_complex @ X @ N4 ) ) ) ) ).
% summable_norm_cos
thf(fact_8351_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_8352_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_8353_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_8354_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_8355_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_8356_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_8357_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_8358_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_8359_exp__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N4: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N4 ) ) @ ( power_power_real @ X @ N4 ) )
@ ( exp_real @ X ) ) ).
% exp_converges
thf(fact_8360_exp__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N4: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N4 ) ) @ ( power_power_complex @ X @ N4 ) )
@ ( exp_complex @ X ) ) ).
% exp_converges
thf(fact_8361_exp__def,axiom,
( exp_real
= ( ^ [X3: real] :
( suminf_real
@ ^ [N4: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N4 ) ) @ ( power_power_real @ X3 @ N4 ) ) ) ) ) ).
% exp_def
thf(fact_8362_exp__def,axiom,
( exp_complex
= ( ^ [X3: complex] :
( suminf_complex
@ ^ [N4: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N4 ) ) @ ( power_power_complex @ X3 @ N4 ) ) ) ) ) ).
% exp_def
thf(fact_8363_summable__norm__exp,axiom,
! [X: real] :
( summable_real
@ ^ [N4: nat] : ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N4 ) ) @ ( power_power_real @ X @ N4 ) ) ) ) ).
% summable_norm_exp
thf(fact_8364_summable__norm__exp,axiom,
! [X: complex] :
( summable_real
@ ^ [N4: nat] : ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N4 ) ) @ ( power_power_complex @ X @ N4 ) ) ) ) ).
% summable_norm_exp
thf(fact_8365_sin__minus__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N4: nat] : ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( sin_coeff @ N4 ) @ ( power_power_real @ ( uminus_uminus_real @ X ) @ N4 ) ) )
@ ( sin_real @ X ) ) ).
% sin_minus_converges
thf(fact_8366_sin__minus__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N4: nat] : ( uminus1482373934393186551omplex @ ( real_V2046097035970521341omplex @ ( sin_coeff @ N4 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ N4 ) ) )
@ ( sin_complex @ X ) ) ).
% sin_minus_converges
thf(fact_8367_cos__minus__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N4: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N4 ) @ ( power_power_real @ ( uminus_uminus_real @ X ) @ N4 ) )
@ ( cos_real @ X ) ) ).
% cos_minus_converges
thf(fact_8368_cos__minus__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N4: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N4 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ N4 ) )
@ ( cos_complex @ X ) ) ).
% cos_minus_converges
thf(fact_8369_termdiff__converges,axiom,
! [X: real,K5: real,C: nat > real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ K5 )
=> ( ! [X2: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ K5 )
=> ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( C @ N4 ) @ ( power_power_real @ X2 @ N4 ) ) ) )
=> ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( diffs_real @ C @ N4 ) @ ( power_power_real @ X @ N4 ) ) ) ) ) ).
% termdiff_converges
thf(fact_8370_termdiff__converges,axiom,
! [X: complex,K5: real,C: nat > complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ K5 )
=> ( ! [X2: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ K5 )
=> ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( C @ N4 ) @ ( power_power_complex @ X2 @ N4 ) ) ) )
=> ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( diffs_complex @ C @ N4 ) @ ( power_power_complex @ X @ N4 ) ) ) ) ) ).
% termdiff_converges
thf(fact_8371_cosh__def,axiom,
( cosh_real
= ( ^ [X3: real] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ) ).
% cosh_def
thf(fact_8372_cosh__def,axiom,
( cosh_complex
= ( ^ [X3: complex] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) ) ) ) ).
% cosh_def
thf(fact_8373_sinh__def,axiom,
( sinh_real
= ( ^ [X3: real] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ) ).
% sinh_def
thf(fact_8374_sinh__def,axiom,
( sinh_complex
= ( ^ [X3: complex] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( minus_minus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) ) ) ) ).
% sinh_def
thf(fact_8375_exp__first__term,axiom,
( exp_real
= ( ^ [X3: real] :
( plus_plus_real @ one_one_real
@ ( suminf_real
@ ^ [N4: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( suc @ N4 ) ) ) @ ( power_power_real @ X3 @ ( suc @ N4 ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_8376_exp__first__term,axiom,
( exp_complex
= ( ^ [X3: complex] :
( plus_plus_complex @ one_one_complex
@ ( suminf_complex
@ ^ [N4: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( suc @ N4 ) ) ) @ ( power_power_complex @ X3 @ ( suc @ N4 ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_8377_cosh__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N4 ) ) @ ( power_power_real @ X @ N4 ) ) @ zero_zero_real )
@ ( cosh_real @ X ) ) ).
% cosh_converges
thf(fact_8378_cosh__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N4: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N4 ) ) @ ( power_power_complex @ X @ N4 ) ) @ zero_zero_complex )
@ ( cosh_complex @ X ) ) ).
% cosh_converges
thf(fact_8379_sinh__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N4 ) ) @ ( power_power_real @ X @ N4 ) ) )
@ ( sinh_real @ X ) ) ).
% sinh_converges
thf(fact_8380_sinh__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N4: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N4 ) ) @ ( power_power_complex @ X @ N4 ) ) )
@ ( sinh_complex @ X ) ) ).
% sinh_converges
thf(fact_8381_mono__SucI1,axiom,
! [X8: nat > set_nat] :
( ! [N: nat] : ( ord_less_eq_set_nat @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
=> ( topolo7278393974255667507et_nat @ X8 ) ) ).
% mono_SucI1
thf(fact_8382_mono__SucI1,axiom,
! [X8: nat > rat] :
( ! [N: nat] : ( ord_less_eq_rat @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
=> ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% mono_SucI1
thf(fact_8383_mono__SucI1,axiom,
! [X8: nat > num] :
( ! [N: nat] : ( ord_less_eq_num @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
=> ( topolo1459490580787246023eq_num @ X8 ) ) ).
% mono_SucI1
thf(fact_8384_mono__SucI1,axiom,
! [X8: nat > nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
=> ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% mono_SucI1
thf(fact_8385_mono__SucI1,axiom,
! [X8: nat > int] :
( ! [N: nat] : ( ord_less_eq_int @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
=> ( topolo4899668324122417113eq_int @ X8 ) ) ).
% mono_SucI1
thf(fact_8386_mono__SucI1,axiom,
! [X8: nat > real] :
( ! [N: nat] : ( ord_less_eq_real @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
=> ( topolo6980174941875973593q_real @ X8 ) ) ).
% mono_SucI1
thf(fact_8387_mono__SucI2,axiom,
! [X8: nat > set_nat] :
( ! [N: nat] : ( ord_less_eq_set_nat @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) )
=> ( topolo7278393974255667507et_nat @ X8 ) ) ).
% mono_SucI2
thf(fact_8388_mono__SucI2,axiom,
! [X8: nat > rat] :
( ! [N: nat] : ( ord_less_eq_rat @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) )
=> ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% mono_SucI2
thf(fact_8389_mono__SucI2,axiom,
! [X8: nat > num] :
( ! [N: nat] : ( ord_less_eq_num @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) )
=> ( topolo1459490580787246023eq_num @ X8 ) ) ).
% mono_SucI2
thf(fact_8390_mono__SucI2,axiom,
! [X8: nat > nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) )
=> ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% mono_SucI2
thf(fact_8391_mono__SucI2,axiom,
! [X8: nat > int] :
( ! [N: nat] : ( ord_less_eq_int @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) )
=> ( topolo4899668324122417113eq_int @ X8 ) ) ).
% mono_SucI2
thf(fact_8392_mono__SucI2,axiom,
! [X8: nat > real] :
( ! [N: nat] : ( ord_less_eq_real @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) )
=> ( topolo6980174941875973593q_real @ X8 ) ) ).
% mono_SucI2
thf(fact_8393_monoseq__Suc,axiom,
( topolo7278393974255667507et_nat
= ( ^ [X6: nat > set_nat] :
( ! [N4: nat] : ( ord_less_eq_set_nat @ ( X6 @ N4 ) @ ( X6 @ ( suc @ N4 ) ) )
| ! [N4: nat] : ( ord_less_eq_set_nat @ ( X6 @ ( suc @ N4 ) ) @ ( X6 @ N4 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_8394_monoseq__Suc,axiom,
( topolo4267028734544971653eq_rat
= ( ^ [X6: nat > rat] :
( ! [N4: nat] : ( ord_less_eq_rat @ ( X6 @ N4 ) @ ( X6 @ ( suc @ N4 ) ) )
| ! [N4: nat] : ( ord_less_eq_rat @ ( X6 @ ( suc @ N4 ) ) @ ( X6 @ N4 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_8395_monoseq__Suc,axiom,
( topolo1459490580787246023eq_num
= ( ^ [X6: nat > num] :
( ! [N4: nat] : ( ord_less_eq_num @ ( X6 @ N4 ) @ ( X6 @ ( suc @ N4 ) ) )
| ! [N4: nat] : ( ord_less_eq_num @ ( X6 @ ( suc @ N4 ) ) @ ( X6 @ N4 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_8396_monoseq__Suc,axiom,
( topolo4902158794631467389eq_nat
= ( ^ [X6: nat > nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( X6 @ N4 ) @ ( X6 @ ( suc @ N4 ) ) )
| ! [N4: nat] : ( ord_less_eq_nat @ ( X6 @ ( suc @ N4 ) ) @ ( X6 @ N4 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_8397_monoseq__Suc,axiom,
( topolo4899668324122417113eq_int
= ( ^ [X6: nat > int] :
( ! [N4: nat] : ( ord_less_eq_int @ ( X6 @ N4 ) @ ( X6 @ ( suc @ N4 ) ) )
| ! [N4: nat] : ( ord_less_eq_int @ ( X6 @ ( suc @ N4 ) ) @ ( X6 @ N4 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_8398_monoseq__Suc,axiom,
( topolo6980174941875973593q_real
= ( ^ [X6: nat > real] :
( ! [N4: nat] : ( ord_less_eq_real @ ( X6 @ N4 ) @ ( X6 @ ( suc @ N4 ) ) )
| ! [N4: nat] : ( ord_less_eq_real @ ( X6 @ ( suc @ N4 ) ) @ ( X6 @ N4 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_8399_monoseq__def,axiom,
( topolo7278393974255667507et_nat
= ( ^ [X6: nat > set_nat] :
( ! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_eq_set_nat @ ( X6 @ M2 ) @ ( X6 @ N4 ) ) )
| ! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_eq_set_nat @ ( X6 @ N4 ) @ ( X6 @ M2 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_8400_monoseq__def,axiom,
( topolo4267028734544971653eq_rat
= ( ^ [X6: nat > rat] :
( ! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_eq_rat @ ( X6 @ M2 ) @ ( X6 @ N4 ) ) )
| ! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_eq_rat @ ( X6 @ N4 ) @ ( X6 @ M2 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_8401_monoseq__def,axiom,
( topolo1459490580787246023eq_num
= ( ^ [X6: nat > num] :
( ! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_eq_num @ ( X6 @ M2 ) @ ( X6 @ N4 ) ) )
| ! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_eq_num @ ( X6 @ N4 ) @ ( X6 @ M2 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_8402_monoseq__def,axiom,
( topolo4902158794631467389eq_nat
= ( ^ [X6: nat > nat] :
( ! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_eq_nat @ ( X6 @ M2 ) @ ( X6 @ N4 ) ) )
| ! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_eq_nat @ ( X6 @ N4 ) @ ( X6 @ M2 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_8403_monoseq__def,axiom,
( topolo4899668324122417113eq_int
= ( ^ [X6: nat > int] :
( ! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_eq_int @ ( X6 @ M2 ) @ ( X6 @ N4 ) ) )
| ! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_eq_int @ ( X6 @ N4 ) @ ( X6 @ M2 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_8404_monoseq__def,axiom,
( topolo6980174941875973593q_real
= ( ^ [X6: nat > real] :
( ! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_eq_real @ ( X6 @ M2 ) @ ( X6 @ N4 ) ) )
| ! [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_eq_real @ ( X6 @ N4 ) @ ( X6 @ M2 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_8405_of__nat__code,axiom,
( semiri5074537144036343181t_real
= ( ^ [N4: nat] :
( semiri7260567687927622513x_real
@ ^ [I3: real] : ( plus_plus_real @ I3 @ one_one_real )
@ N4
@ zero_zero_real ) ) ) ).
% of_nat_code
thf(fact_8406_of__nat__code,axiom,
( semiri1314217659103216013at_int
= ( ^ [N4: nat] :
( semiri8420488043553186161ux_int
@ ^ [I3: int] : ( plus_plus_int @ I3 @ one_one_int )
@ N4
@ zero_zero_int ) ) ) ).
% of_nat_code
thf(fact_8407_of__nat__code,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N4: nat] :
( semiri8422978514062236437ux_nat
@ ^ [I3: nat] : ( plus_plus_nat @ I3 @ one_one_nat )
@ N4
@ zero_zero_nat ) ) ) ).
% of_nat_code
thf(fact_8408_of__nat__code,axiom,
( semiri8010041392384452111omplex
= ( ^ [N4: nat] :
( semiri2816024913162550771omplex
@ ^ [I3: complex] : ( plus_plus_complex @ I3 @ one_one_complex )
@ N4
@ zero_zero_complex ) ) ) ).
% of_nat_code
thf(fact_8409_of__nat__code,axiom,
( semiri681578069525770553at_rat
= ( ^ [N4: nat] :
( semiri7787848453975740701ux_rat
@ ^ [I3: rat] : ( plus_plus_rat @ I3 @ one_one_rat )
@ N4
@ zero_zero_rat ) ) ) ).
% of_nat_code
thf(fact_8410_Arg__def,axiom,
( arg
= ( ^ [Z5: complex] :
( if_real @ ( Z5 = zero_zero_complex ) @ zero_zero_real
@ ( fChoice_real
@ ^ [A3: real] :
( ( ( sgn_sgn_complex @ Z5 )
= ( cis @ A3 ) )
& ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A3 )
& ( ord_less_eq_real @ A3 @ pi ) ) ) ) ) ) ).
% Arg_def
thf(fact_8411_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).
% set_vebt_def
thf(fact_8412_sin__x__sin__y,axiom,
! [X: complex,Y: complex] :
( sums_complex
@ ^ [P4: nat] :
( groups2073611262835488442omplex
@ ^ [N4: nat] :
( if_complex
@ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P4 )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
@ ( 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 @ N4 ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) ) @ ( power_power_complex @ X @ N4 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P4 @ N4 ) ) )
@ zero_zero_complex )
@ ( set_ord_atMost_nat @ P4 ) )
@ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y ) ) ) ).
% sin_x_sin_y
thf(fact_8413_sin__x__sin__y,axiom,
! [X: real,Y: real] :
( sums_real
@ ^ [P4: nat] :
( groups6591440286371151544t_real
@ ^ [N4: nat] :
( if_real
@ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P4 )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
@ ( 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 @ N4 ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) ) @ ( power_power_real @ X @ N4 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P4 @ N4 ) ) )
@ zero_zero_real )
@ ( set_ord_atMost_nat @ P4 ) )
@ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ).
% sin_x_sin_y
thf(fact_8414_atMost__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_8415_atMost__iff,axiom,
! [I: rat,K: rat] :
( ( member_rat @ I @ ( set_ord_atMost_rat @ K ) )
= ( ord_less_eq_rat @ I @ K ) ) ).
% atMost_iff
thf(fact_8416_atMost__iff,axiom,
! [I: num,K: num] :
( ( member_num @ I @ ( set_ord_atMost_num @ K ) )
= ( ord_less_eq_num @ I @ K ) ) ).
% atMost_iff
thf(fact_8417_atMost__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
= ( ord_less_eq_int @ I @ K ) ) ).
% atMost_iff
thf(fact_8418_atMost__iff,axiom,
! [I: real,K: real] :
( ( member_real @ I @ ( set_ord_atMost_real @ K ) )
= ( ord_less_eq_real @ I @ K ) ) ).
% atMost_iff
thf(fact_8419_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_8420_Ints__sum,axiom,
! [A2: set_complex,F: complex > int] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( member_int @ ( F @ X2 ) @ ring_1_Ints_int ) )
=> ( member_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ring_1_Ints_int ) ) ).
% Ints_sum
thf(fact_8421_Ints__sum,axiom,
! [A2: set_real,F: real > complex] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( member_complex @ ( F @ X2 ) @ ring_1_Ints_complex ) )
=> ( member_complex @ ( groups5754745047067104278omplex @ F @ A2 ) @ ring_1_Ints_complex ) ) ).
% Ints_sum
thf(fact_8422_Ints__sum,axiom,
! [A2: set_real,F: real > int] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( member_int @ ( F @ X2 ) @ ring_1_Ints_int ) )
=> ( member_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ring_1_Ints_int ) ) ).
% Ints_sum
thf(fact_8423_Ints__sum,axiom,
! [A2: set_nat,F: nat > complex] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_complex @ ( F @ X2 ) @ ring_1_Ints_complex ) )
=> ( member_complex @ ( groups2073611262835488442omplex @ F @ A2 ) @ ring_1_Ints_complex ) ) ).
% Ints_sum
thf(fact_8424_Ints__sum,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_int @ ( F @ X2 ) @ ring_1_Ints_int ) )
=> ( member_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ring_1_Ints_int ) ) ).
% Ints_sum
thf(fact_8425_Ints__sum,axiom,
! [A2: set_int,F: int > complex] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( member_complex @ ( F @ X2 ) @ ring_1_Ints_complex ) )
=> ( member_complex @ ( groups3049146728041665814omplex @ F @ A2 ) @ ring_1_Ints_complex ) ) ).
% Ints_sum
thf(fact_8426_Ints__sum,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( member_real @ ( F @ X2 ) @ ring_1_Ints_real ) )
=> ( member_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ring_1_Ints_real ) ) ).
% Ints_sum
thf(fact_8427_Ints__sum,axiom,
! [A2: set_real,F: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( member_real @ ( F @ X2 ) @ ring_1_Ints_real ) )
=> ( member_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ring_1_Ints_real ) ) ).
% Ints_sum
thf(fact_8428_Ints__sum,axiom,
! [A2: set_int,F: int > real] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( member_real @ ( F @ X2 ) @ ring_1_Ints_real ) )
=> ( member_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ring_1_Ints_real ) ) ).
% Ints_sum
thf(fact_8429_Ints__sum,axiom,
! [A2: set_complex,F: complex > complex] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( member_complex @ ( F @ X2 ) @ ring_1_Ints_complex ) )
=> ( member_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ring_1_Ints_complex ) ) ).
% Ints_sum
thf(fact_8430_of__nat__sum,axiom,
! [F: complex > nat,A2: set_complex] :
( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A2 ) )
= ( groups7754918857620584856omplex
@ ^ [X3: complex] : ( semiri8010041392384452111omplex @ ( F @ X3 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_8431_of__nat__sum,axiom,
! [F: int > nat,A2: set_int] :
( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
= ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_8432_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3539618377306564664at_int
@ ^ [X3: nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_8433_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri8010041392384452111omplex @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups2073611262835488442omplex
@ ^ [X3: nat] : ( semiri8010041392384452111omplex @ ( F @ X3 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_8434_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri681578069525770553at_rat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups2906978787729119204at_rat
@ ^ [X3: nat] : ( semiri681578069525770553at_rat @ ( F @ X3 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_8435_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( semiri1316708129612266289at_nat @ ( F @ X3 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_8436_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [X3: nat] : ( semiri5074537144036343181t_real @ ( F @ X3 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_8437_of__int__sum,axiom,
! [F: complex > int,A2: set_complex] :
( ( ring_17405671764205052669omplex @ ( groups5690904116761175830ex_int @ F @ A2 ) )
= ( groups7754918857620584856omplex
@ ^ [X3: complex] : ( ring_17405671764205052669omplex @ ( F @ X3 ) )
@ A2 ) ) ).
% of_int_sum
thf(fact_8438_of__int__sum,axiom,
! [F: nat > int,A2: set_nat] :
( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [X3: nat] : ( ring_1_of_int_real @ ( F @ X3 ) )
@ A2 ) ) ).
% of_int_sum
thf(fact_8439_of__int__sum,axiom,
! [F: int > int,A2: set_int] :
( ( ring_1_of_int_real @ ( groups4538972089207619220nt_int @ F @ A2 ) )
= ( groups8778361861064173332t_real
@ ^ [X3: int] : ( ring_1_of_int_real @ ( F @ X3 ) )
@ A2 ) ) ).
% of_int_sum
thf(fact_8440_of__int__sum,axiom,
! [F: int > int,A2: set_int] :
( ( ring_1_of_int_rat @ ( groups4538972089207619220nt_int @ F @ A2 ) )
= ( groups3906332499630173760nt_rat
@ ^ [X3: int] : ( ring_1_of_int_rat @ ( F @ X3 ) )
@ A2 ) ) ).
% of_int_sum
thf(fact_8441_of__int__sum,axiom,
! [F: int > int,A2: set_int] :
( ( ring_1_of_int_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
= ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( ring_1_of_int_int @ ( F @ X3 ) )
@ A2 ) ) ).
% of_int_sum
thf(fact_8442_atMost__subset__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X ) @ ( set_or4236626031148496127et_nat @ Y ) )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_8443_atMost__subset__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X ) @ ( set_ord_atMost_rat @ Y ) )
= ( ord_less_eq_rat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_8444_atMost__subset__iff,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X ) @ ( set_ord_atMost_num @ Y ) )
= ( ord_less_eq_num @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_8445_atMost__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_8446_atMost__subset__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ X ) @ ( set_ord_atMost_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_8447_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_8448_sum_OatMost__Suc,axiom,
! [G: nat > rat,N2: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% sum.atMost_Suc
thf(fact_8449_sum_OatMost__Suc,axiom,
! [G: nat > int,N2: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% sum.atMost_Suc
thf(fact_8450_sum_OatMost__Suc,axiom,
! [G: nat > nat,N2: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% sum.atMost_Suc
thf(fact_8451_sum_OatMost__Suc,axiom,
! [G: nat > real,N2: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% sum.atMost_Suc
thf(fact_8452_sum__choose__upper,axiom,
! [M: nat,N2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( binomial @ K2 @ M )
@ ( set_ord_atMost_nat @ N2 ) )
= ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ).
% sum_choose_upper
thf(fact_8453_mod__sum__eq,axiom,
! [F: nat > nat,A: nat,A2: set_nat] :
( ( modulo_modulo_nat
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ A ) ) ).
% mod_sum_eq
thf(fact_8454_mod__sum__eq,axiom,
! [F: int > int,A: int,A2: set_int] :
( ( modulo_modulo_int
@ ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ A ) ) ).
% mod_sum_eq
thf(fact_8455_sum_OatMost__Suc__shift,axiom,
! [G: nat > rat,N2: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( plus_plus_rat @ ( G @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_8456_sum_OatMost__Suc__shift,axiom,
! [G: nat > int,N2: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_8457_sum_OatMost__Suc__shift,axiom,
! [G: nat > nat,N2: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_8458_sum_OatMost__Suc__shift,axiom,
! [G: nat > real,N2: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_8459_sum__telescope,axiom,
! [F: nat > rat,I: nat] :
( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
@ ( set_ord_atMost_nat @ I ) )
= ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% sum_telescope
thf(fact_8460_sum__telescope,axiom,
! [F: nat > int,I: nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
@ ( set_ord_atMost_nat @ I ) )
= ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% sum_telescope
thf(fact_8461_sum__telescope,axiom,
! [F: nat > real,I: nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( minus_minus_real @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
@ ( set_ord_atMost_nat @ I ) )
= ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% sum_telescope
thf(fact_8462_polyfun__eq__coeffs,axiom,
! [C: nat > complex,N2: nat,D: nat > complex] :
( ( ! [X3: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( D @ I3 ) @ ( power_power_complex @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( C @ I3 )
= ( D @ I3 ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_8463_polyfun__eq__coeffs,axiom,
! [C: nat > real,N2: nat,D: nat > real] :
( ( ! [X3: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( D @ I3 ) @ ( power_power_real @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( C @ I3 )
= ( D @ I3 ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_8464_bounded__imp__summable,axiom,
! [A: nat > int,B3: int] :
( ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( A @ N ) )
=> ( ! [N: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ A @ ( set_ord_atMost_nat @ N ) ) @ B3 )
=> ( summable_int @ A ) ) ) ).
% bounded_imp_summable
thf(fact_8465_bounded__imp__summable,axiom,
! [A: nat > nat,B3: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( A @ N ) )
=> ( ! [N: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ A @ ( set_ord_atMost_nat @ N ) ) @ B3 )
=> ( summable_nat @ A ) ) ) ).
% bounded_imp_summable
thf(fact_8466_bounded__imp__summable,axiom,
! [A: nat > real,B3: real] :
( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
=> ( ! [N: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ A @ ( set_ord_atMost_nat @ N ) ) @ B3 )
=> ( summable_real @ A ) ) ) ).
% bounded_imp_summable
thf(fact_8467_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U2: set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8468_atMost__def,axiom,
( set_ord_atMost_rat
= ( ^ [U2: rat] :
( collect_rat
@ ^ [X3: rat] : ( ord_less_eq_rat @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8469_atMost__def,axiom,
( set_ord_atMost_num
= ( ^ [U2: num] :
( collect_num
@ ^ [X3: num] : ( ord_less_eq_num @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8470_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X3: int] : ( ord_less_eq_int @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8471_atMost__def,axiom,
( set_ord_atMost_real
= ( ^ [U2: real] :
( collect_real
@ ^ [X3: real] : ( ord_less_eq_real @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8472_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_eq_nat @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8473_sum__choose__lower,axiom,
! [R4: nat,N2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( binomial @ ( plus_plus_nat @ R4 @ K2 ) @ K2 )
@ ( set_ord_atMost_nat @ N2 ) )
= ( binomial @ ( suc @ ( plus_plus_nat @ R4 @ N2 ) ) @ N2 ) ) ).
% sum_choose_lower
thf(fact_8474_choose__rising__sum_I1_J,axiom,
! [N2: nat,M: nat] :
( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
@ ( set_ord_atMost_nat @ M ) )
= ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% choose_rising_sum(1)
thf(fact_8475_choose__rising__sum_I2_J,axiom,
! [N2: nat,M: nat] :
( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
@ ( set_ord_atMost_nat @ M ) )
= ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ M ) ) ).
% choose_rising_sum(2)
thf(fact_8476_zero__polynom__imp__zero__coeffs,axiom,
! [C: nat > complex,N2: nat,K: nat] :
( ! [W3: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ W3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_complex )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( C @ K )
= zero_zero_complex ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_8477_zero__polynom__imp__zero__coeffs,axiom,
! [C: nat > real,N2: nat,K: nat] :
( ! [W3: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ W3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( C @ K )
= zero_zero_real ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_8478_polyfun__eq__0,axiom,
! [C: nat > complex,N2: nat] :
( ( ! [X3: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_complex ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( C @ I3 )
= zero_zero_complex ) ) ) ) ).
% polyfun_eq_0
thf(fact_8479_polyfun__eq__0,axiom,
! [C: nat > real,N2: nat] :
( ( ! [X3: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_real ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( C @ I3 )
= zero_zero_real ) ) ) ) ).
% polyfun_eq_0
thf(fact_8480_gbinomial__parallel__sum,axiom,
! [A: complex,N2: nat] :
( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( gbinomial_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K2 ) ) @ K2 )
@ ( set_ord_atMost_nat @ N2 ) )
= ( gbinomial_complex @ ( plus_plus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N2 ) ) @ one_one_complex ) @ N2 ) ) ).
% gbinomial_parallel_sum
thf(fact_8481_gbinomial__parallel__sum,axiom,
! [A: rat,N2: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K2: nat] : ( gbinomial_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ K2 )
@ ( set_ord_atMost_nat @ N2 ) )
= ( gbinomial_rat @ ( plus_plus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N2 ) ) @ one_one_rat ) @ N2 ) ) ).
% gbinomial_parallel_sum
thf(fact_8482_gbinomial__parallel__sum,axiom,
! [A: real,N2: nat] :
( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( gbinomial_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ K2 )
@ ( set_ord_atMost_nat @ N2 ) )
= ( gbinomial_real @ ( plus_plus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N2 ) ) @ one_one_real ) @ N2 ) ) ).
% gbinomial_parallel_sum
thf(fact_8483_sum__choose__diagonal,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( binomial @ ( minus_minus_nat @ N2 @ K2 ) @ ( minus_minus_nat @ M @ K2 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( binomial @ ( suc @ N2 ) @ M ) ) ) ).
% sum_choose_diagonal
thf(fact_8484_vandermonde,axiom,
! [M: nat,N2: nat,R4: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( times_times_nat @ ( binomial @ M @ K2 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R4 @ K2 ) ) )
@ ( set_ord_atMost_nat @ R4 ) )
= ( binomial @ ( plus_plus_nat @ M @ N2 ) @ R4 ) ) ).
% vandermonde
thf(fact_8485_sum__gp__basic,axiom,
! [X: complex,N2: nat] :
( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
= ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) ).
% sum_gp_basic
thf(fact_8486_sum__gp__basic,axiom,
! [X: rat,N2: nat] :
( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
= ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) ).
% sum_gp_basic
thf(fact_8487_sum__gp__basic,axiom,
! [X: int,N2: nat] :
( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
= ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ ( suc @ N2 ) ) ) ) ).
% sum_gp_basic
thf(fact_8488_sum__gp__basic,axiom,
! [X: real,N2: nat] :
( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
= ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) ).
% sum_gp_basic
thf(fact_8489_choose__row__sum,axiom,
! [N2: nat] :
( ( groups3542108847815614940at_nat @ ( binomial @ N2 ) @ ( set_ord_atMost_nat @ N2 ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% choose_row_sum
thf(fact_8490_binomial,axiom,
! [A: nat,B: nat,N2: nat] :
( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
= ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% binomial
thf(fact_8491_sum_Oin__pairs__0,axiom,
! [G: nat > rat,N2: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
= ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% sum.in_pairs_0
thf(fact_8492_sum_Oin__pairs__0,axiom,
! [G: nat > int,N2: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% sum.in_pairs_0
thf(fact_8493_sum_Oin__pairs__0,axiom,
! [G: nat > nat,N2: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% sum.in_pairs_0
thf(fact_8494_sum_Oin__pairs__0,axiom,
! [G: nat > real,N2: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% sum.in_pairs_0
thf(fact_8495_polynomial__product,axiom,
! [M: nat,A: nat > complex,N2: nat,B: nat > complex,X: complex] :
( ! [I2: nat] :
( ( ord_less_nat @ M @ I2 )
=> ( ( A @ I2 )
= zero_zero_complex ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N2 @ J2 )
=> ( ( B @ J2 )
= zero_zero_complex ) )
=> ( ( times_times_complex
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( times_times_complex @ ( B @ J3 ) @ ( power_power_complex @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( groups2073611262835488442omplex
@ ^ [R5: nat] :
( times_times_complex
@ ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( times_times_complex @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_complex @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).
% polynomial_product
thf(fact_8496_polynomial__product,axiom,
! [M: nat,A: nat > rat,N2: nat,B: nat > rat,X: rat] :
( ! [I2: nat] :
( ( ord_less_nat @ M @ I2 )
=> ( ( A @ I2 )
= zero_zero_rat ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N2 @ J2 )
=> ( ( B @ J2 )
= zero_zero_rat ) )
=> ( ( times_times_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X @ I3 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( times_times_rat @ ( B @ J3 ) @ ( power_power_rat @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( groups2906978787729119204at_rat
@ ^ [R5: nat] :
( times_times_rat
@ ( groups2906978787729119204at_rat
@ ^ [K2: nat] : ( times_times_rat @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_rat @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).
% polynomial_product
thf(fact_8497_polynomial__product,axiom,
! [M: nat,A: nat > int,N2: nat,B: nat > int,X: int] :
( ! [I2: nat] :
( ( ord_less_nat @ M @ I2 )
=> ( ( A @ I2 )
= zero_zero_int ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N2 @ J2 )
=> ( ( B @ J2 )
= zero_zero_int ) )
=> ( ( times_times_int
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( times_times_int @ ( B @ J3 ) @ ( power_power_int @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( groups3539618377306564664at_int
@ ^ [R5: nat] :
( times_times_int
@ ( groups3539618377306564664at_int
@ ^ [K2: nat] : ( times_times_int @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_int @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).
% polynomial_product
thf(fact_8498_polynomial__product,axiom,
! [M: nat,A: nat > real,N2: nat,B: nat > real,X: real] :
( ! [I2: nat] :
( ( ord_less_nat @ M @ I2 )
=> ( ( A @ I2 )
= zero_zero_real ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N2 @ J2 )
=> ( ( B @ J2 )
= zero_zero_real ) )
=> ( ( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( times_times_real @ ( B @ J3 ) @ ( power_power_real @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( groups6591440286371151544t_real
@ ^ [R5: nat] :
( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( times_times_real @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_real @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).
% polynomial_product
thf(fact_8499_gbinomial__sum__lower__neg,axiom,
! [A: complex,M: nat] :
( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) )
@ ( 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_8500_gbinomial__sum__lower__neg,axiom,
! [A: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K2: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) )
@ ( 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_8501_gbinomial__sum__lower__neg,axiom,
! [A: real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) )
@ ( 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_8502_binomial__ring,axiom,
! [A: int,B: int,N2: nat] :
( ( power_power_int @ ( plus_plus_int @ A @ B ) @ N2 )
= ( groups3539618377306564664at_int
@ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ K2 ) ) @ ( power_power_int @ A @ K2 ) ) @ ( power_power_int @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% binomial_ring
thf(fact_8503_binomial__ring,axiom,
! [A: complex,B: complex,N2: nat] :
( ( power_power_complex @ ( plus_plus_complex @ A @ B ) @ N2 )
= ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K2 ) ) @ ( power_power_complex @ A @ K2 ) ) @ ( power_power_complex @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% binomial_ring
thf(fact_8504_binomial__ring,axiom,
! [A: rat,B: rat,N2: nat] :
( ( power_power_rat @ ( plus_plus_rat @ A @ B ) @ N2 )
= ( groups2906978787729119204at_rat
@ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K2 ) ) @ ( power_power_rat @ A @ K2 ) ) @ ( power_power_rat @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% binomial_ring
thf(fact_8505_binomial__ring,axiom,
! [A: nat,B: nat,N2: nat] :
( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
= ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% binomial_ring
thf(fact_8506_binomial__ring,axiom,
! [A: real,B: real,N2: nat] :
( ( power_power_real @ ( plus_plus_real @ A @ B ) @ N2 )
= ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K2 ) ) @ ( power_power_real @ A @ K2 ) ) @ ( power_power_real @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% binomial_ring
thf(fact_8507_polynomial__product__nat,axiom,
! [M: nat,A: nat > nat,N2: nat,B: nat > nat,X: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ M @ I2 )
=> ( ( A @ I2 )
= zero_zero_nat ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N2 @ J2 )
=> ( ( B @ J2 )
= zero_zero_nat ) )
=> ( ( times_times_nat
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( power_power_nat @ X @ I3 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( groups3542108847815614940at_nat
@ ^ [R5: nat] :
( times_times_nat
@ ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( times_times_nat @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_nat @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_8508_choose__square__sum,axiom,
! [N2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( power_power_nat @ ( binomial @ N2 @ K2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).
% choose_square_sum
thf(fact_8509_pochhammer__binomial__sum,axiom,
! [A: int,B: int,N2: nat] :
( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B ) @ N2 )
= ( groups3539618377306564664at_int
@ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ K2 ) ) @ ( comm_s4660882817536571857er_int @ A @ K2 ) ) @ ( comm_s4660882817536571857er_int @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% pochhammer_binomial_sum
thf(fact_8510_pochhammer__binomial__sum,axiom,
! [A: complex,B: complex,N2: nat] :
( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ B ) @ N2 )
= ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K2 ) ) @ ( comm_s2602460028002588243omplex @ A @ K2 ) ) @ ( comm_s2602460028002588243omplex @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% pochhammer_binomial_sum
thf(fact_8511_pochhammer__binomial__sum,axiom,
! [A: rat,B: rat,N2: nat] :
( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B ) @ N2 )
= ( groups2906978787729119204at_rat
@ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% pochhammer_binomial_sum
thf(fact_8512_pochhammer__binomial__sum,axiom,
! [A: real,B: real,N2: nat] :
( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B ) @ N2 )
= ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K2 ) ) @ ( comm_s7457072308508201937r_real @ A @ K2 ) ) @ ( comm_s7457072308508201937r_real @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% pochhammer_binomial_sum
thf(fact_8513_sum__power__add,axiom,
! [X: complex,M: nat,I5: set_nat] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M @ I3 ) )
@ I5 )
= ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I5 ) ) ) ).
% sum_power_add
thf(fact_8514_sum__power__add,axiom,
! [X: rat,M: nat,I5: set_nat] :
( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M @ I3 ) )
@ I5 )
= ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I5 ) ) ) ).
% sum_power_add
thf(fact_8515_sum__power__add,axiom,
! [X: int,M: nat,I5: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I3 ) )
@ I5 )
= ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I5 ) ) ) ).
% sum_power_add
thf(fact_8516_sum__power__add,axiom,
! [X: real,M: nat,I5: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M @ I3 ) )
@ I5 )
= ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I5 ) ) ) ).
% sum_power_add
thf(fact_8517_sum_Ozero__middle,axiom,
! [P3: nat,K: nat,G: nat > complex,H2: nat > complex] :
( ( ord_less_eq_nat @ one_one_nat @ P3 )
=> ( ( ord_less_eq_nat @ K @ P3 )
=> ( ( 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 @ P3 ) )
= ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8518_sum_Ozero__middle,axiom,
! [P3: nat,K: nat,G: nat > rat,H2: nat > rat] :
( ( ord_less_eq_nat @ one_one_nat @ P3 )
=> ( ( ord_less_eq_nat @ K @ P3 )
=> ( ( 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 @ P3 ) )
= ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8519_sum_Ozero__middle,axiom,
! [P3: nat,K: nat,G: nat > int,H2: nat > int] :
( ( ord_less_eq_nat @ one_one_nat @ P3 )
=> ( ( ord_less_eq_nat @ K @ P3 )
=> ( ( 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 @ P3 ) )
= ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8520_sum_Ozero__middle,axiom,
! [P3: nat,K: nat,G: nat > nat,H2: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P3 )
=> ( ( ord_less_eq_nat @ K @ P3 )
=> ( ( 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 @ P3 ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8521_sum_Ozero__middle,axiom,
! [P3: nat,K: nat,G: nat > real,H2: nat > real] :
( ( ord_less_eq_nat @ one_one_nat @ P3 )
=> ( ( ord_less_eq_nat @ K @ P3 )
=> ( ( 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 @ P3 ) )
= ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8522_gbinomial__partial__sum__poly,axiom,
! [M: nat,A: complex,X: complex,Y: complex] :
( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K2 ) @ ( power_power_complex @ X @ K2 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ M @ K2 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ K2 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_nat @ M @ K2 ) ) )
@ ( set_ord_atMost_nat @ M ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_8523_gbinomial__partial__sum__poly,axiom,
! [M: nat,A: rat,X: rat,Y: rat] :
( ( groups2906978787729119204at_rat
@ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K2 ) @ ( power_power_rat @ X @ K2 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M @ K2 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( groups2906978787729119204at_rat
@ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ X ) @ K2 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_nat @ M @ K2 ) ) )
@ ( set_ord_atMost_nat @ M ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_8524_gbinomial__partial__sum__poly,axiom,
! [M: nat,A: real,X: real,Y: real] :
( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K2 ) @ ( power_power_real @ X @ K2 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ M @ K2 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K2 ) @ ( power_power_real @ ( uminus_uminus_real @ X ) @ K2 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_nat @ M @ K2 ) ) )
@ ( set_ord_atMost_nat @ M ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_8525_exp__series__add__commuting,axiom,
! [X: complex,Y: complex,N2: nat] :
( ( ( times_times_complex @ X @ Y )
= ( times_times_complex @ Y @ X ) )
=> ( ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ N2 ) )
= ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ I3 ) ) @ ( power_power_complex @ X @ I3 ) ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ I3 ) ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% exp_series_add_commuting
thf(fact_8526_exp__series__add__commuting,axiom,
! [X: real,Y: real,N2: nat] :
( ( ( times_times_real @ X @ Y )
= ( times_times_real @ Y @ X ) )
=> ( ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y ) @ N2 ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ I3 ) ) @ ( power_power_real @ X @ I3 ) ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ I3 ) ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% exp_series_add_commuting
thf(fact_8527_root__polyfun,axiom,
! [N2: nat,Z: int,A: int] :
( ( ord_less_eq_nat @ one_one_nat @ N2 )
=> ( ( ( power_power_int @ Z @ N2 )
= A )
= ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( if_int @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_int @ A ) @ ( if_int @ ( I3 = N2 ) @ one_one_int @ zero_zero_int ) ) @ ( power_power_int @ Z @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_int ) ) ) ).
% root_polyfun
thf(fact_8528_root__polyfun,axiom,
! [N2: nat,Z: complex,A: complex] :
( ( ord_less_eq_nat @ one_one_nat @ N2 )
=> ( ( ( power_power_complex @ Z @ N2 )
= A )
= ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( if_complex @ ( I3 = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A ) @ ( if_complex @ ( I3 = N2 ) @ one_one_complex @ zero_zero_complex ) ) @ ( power_power_complex @ Z @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_complex ) ) ) ).
% root_polyfun
thf(fact_8529_root__polyfun,axiom,
! [N2: nat,Z: code_integer,A: code_integer] :
( ( ord_less_eq_nat @ one_one_nat @ N2 )
=> ( ( ( power_8256067586552552935nteger @ Z @ N2 )
= A )
= ( ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( if_Code_integer @ ( I3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ A ) @ ( if_Code_integer @ ( I3 = N2 ) @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) @ ( power_8256067586552552935nteger @ Z @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_z3403309356797280102nteger ) ) ) ).
% root_polyfun
thf(fact_8530_root__polyfun,axiom,
! [N2: nat,Z: rat,A: rat] :
( ( ord_less_eq_nat @ one_one_nat @ N2 )
=> ( ( ( power_power_rat @ Z @ N2 )
= A )
= ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( if_rat @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_rat @ A ) @ ( if_rat @ ( I3 = N2 ) @ one_one_rat @ zero_zero_rat ) ) @ ( power_power_rat @ Z @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_rat ) ) ) ).
% root_polyfun
thf(fact_8531_root__polyfun,axiom,
! [N2: nat,Z: real,A: real] :
( ( ord_less_eq_nat @ one_one_nat @ N2 )
=> ( ( ( power_power_real @ Z @ N2 )
= A )
= ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( if_real @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_real @ A ) @ ( if_real @ ( I3 = N2 ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_real ) ) ) ).
% root_polyfun
thf(fact_8532_sum__gp0,axiom,
! [X: complex,N2: nat] :
( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N2 ) )
= ( semiri8010041392384452111omplex @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% sum_gp0
thf(fact_8533_sum__gp0,axiom,
! [X: rat,N2: nat] :
( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N2 ) )
= ( semiri681578069525770553at_rat @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N2 ) )
= ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% sum_gp0
thf(fact_8534_sum__gp0,axiom,
! [X: real,N2: nat] :
( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N2 ) )
= ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N2 ) )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% sum_gp0
thf(fact_8535_choose__alternating__linear__sum,axiom,
! [N2: nat] :
( ( N2 != one_one_nat )
=> ( ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ I3 ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I3 ) ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_z3403309356797280102nteger ) ) ).
% choose_alternating_linear_sum
thf(fact_8536_choose__alternating__linear__sum,axiom,
! [N2: nat] :
( ( N2 != one_one_nat )
=> ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ I3 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_int ) ) ).
% choose_alternating_linear_sum
thf(fact_8537_choose__alternating__linear__sum,axiom,
! [N2: nat] :
( ( N2 != one_one_nat )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I3 ) @ ( semiri8010041392384452111omplex @ I3 ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_complex ) ) ).
% choose_alternating_linear_sum
thf(fact_8538_choose__alternating__linear__sum,axiom,
! [N2: nat] :
( ( N2 != one_one_nat )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ I3 ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_rat ) ) ).
% choose_alternating_linear_sum
thf(fact_8539_choose__alternating__linear__sum,axiom,
! [N2: nat] :
( ( N2 != one_one_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( semiri5074537144036343181t_real @ I3 ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_real ) ) ).
% choose_alternating_linear_sum
thf(fact_8540_gbinomial__sum__nat__pow2,axiom,
! [M: nat] :
( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( divide1717551699836669952omplex @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ K2 ) ) @ K2 ) @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ K2 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ M ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_8541_gbinomial__sum__nat__pow2,axiom,
! [M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K2: nat] : ( divide_divide_rat @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ K2 ) ) @ K2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ K2 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ M ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_8542_gbinomial__sum__nat__pow2,axiom,
! [M: nat] :
( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( divide_divide_real @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ K2 ) ) @ K2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ K2 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ M ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_8543_gbinomial__partial__sum__poly__xpos,axiom,
! [M: nat,A: complex,X: complex,Y: complex] :
( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K2 ) @ ( power_power_complex @ X @ K2 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ M @ K2 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ K2 ) @ A ) @ one_one_complex ) @ K2 ) @ ( power_power_complex @ X @ K2 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_nat @ M @ K2 ) ) )
@ ( set_ord_atMost_nat @ M ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_8544_gbinomial__partial__sum__poly__xpos,axiom,
! [M: nat,A: rat,X: rat,Y: rat] :
( ( groups2906978787729119204at_rat
@ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K2 ) @ ( power_power_rat @ X @ K2 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M @ K2 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( groups2906978787729119204at_rat
@ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ K2 ) @ A ) @ one_one_rat ) @ K2 ) @ ( power_power_rat @ X @ K2 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_nat @ M @ K2 ) ) )
@ ( set_ord_atMost_nat @ M ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_8545_gbinomial__partial__sum__poly__xpos,axiom,
! [M: nat,A: real,X: real,Y: real] :
( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K2 ) @ ( power_power_real @ X @ K2 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ M @ K2 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ K2 ) @ A ) @ one_one_real ) @ K2 ) @ ( power_power_real @ X @ K2 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_nat @ M @ K2 ) ) )
@ ( set_ord_atMost_nat @ M ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_8546_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_8547_choose__linear__sum,axiom,
! [N2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( times_times_nat @ I3 @ ( binomial @ N2 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( times_times_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% choose_linear_sum
thf(fact_8548_choose__alternating__sum,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I3 ) ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_z3403309356797280102nteger ) ) ).
% choose_alternating_sum
thf(fact_8549_choose__alternating__sum,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_int ) ) ).
% choose_alternating_sum
thf(fact_8550_choose__alternating__sum,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_complex ) ) ).
% choose_alternating_sum
thf(fact_8551_choose__alternating__sum,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_rat ) ) ).
% choose_alternating_sum
thf(fact_8552_choose__alternating__sum,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_real ) ) ).
% choose_alternating_sum
thf(fact_8553_polyfun__extremal__lemma,axiom,
! [E2: real,C: nat > complex,N2: nat] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [M8: real] :
! [Z3: complex] :
( ( ord_less_eq_real @ M8 @ ( real_V1022390504157884413omplex @ Z3 ) )
=> ( ord_less_eq_real
@ ( real_V1022390504157884413omplex
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) )
@ ( times_times_real @ E2 @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( suc @ N2 ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_8554_polyfun__extremal__lemma,axiom,
! [E2: real,C: nat > real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [M8: real] :
! [Z3: real] :
( ( ord_less_eq_real @ M8 @ ( real_V7735802525324610683m_real @ Z3 ) )
=> ( ord_less_eq_real
@ ( real_V7735802525324610683m_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) )
@ ( times_times_real @ E2 @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z3 ) @ ( suc @ N2 ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_8555_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_8556_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_8557_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_8558_choose__odd__sum,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] :
( if_int
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
@ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) )
@ zero_zero_int )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% choose_odd_sum
thf(fact_8559_choose__odd__sum,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] :
( if_complex
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
@ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) )
@ zero_zero_complex )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% choose_odd_sum
thf(fact_8560_choose__odd__sum,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] :
( if_rat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
@ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) )
@ zero_zero_rat )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% choose_odd_sum
thf(fact_8561_choose__odd__sum,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] :
( if_real
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
@ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) )
@ zero_zero_real )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% choose_odd_sum
thf(fact_8562_choose__even__sum,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) ) @ zero_zero_int )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% choose_even_sum
thf(fact_8563_choose__even__sum,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) ) @ zero_zero_complex )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% choose_even_sum
thf(fact_8564_choose__even__sum,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) ) @ zero_zero_rat )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% choose_even_sum
thf(fact_8565_choose__even__sum,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) ) @ zero_zero_real )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% choose_even_sum
thf(fact_8566_gbinomial__partial__row__sum,axiom,
! [A: complex,M: nat] :
( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K2 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K2 ) ) )
@ ( 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_8567_gbinomial__partial__row__sum,axiom,
! [A: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K2: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K2 ) ) )
@ ( 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_8568_gbinomial__partial__row__sum,axiom,
! [A: real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ ( minus_minus_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
@ ( 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_8569_mask__eq__sum__exp,axiom,
! [N2: nat] :
( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int )
= ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N2 ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_8570_mask__eq__sum__exp,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N2 ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_8571_mask__eq__sum__exp__nat,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( suc @ zero_zero_nat ) )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N2 ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_8572_cos__x__cos__y,axiom,
! [X: complex,Y: complex] :
( sums_complex
@ ^ [P4: nat] :
( groups2073611262835488442omplex
@ ^ [N4: nat] :
( if_complex
@ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P4 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
@ ( 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 @ N4 ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_complex @ X @ N4 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P4 @ N4 ) ) )
@ zero_zero_complex )
@ ( set_ord_atMost_nat @ P4 ) )
@ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ).
% cos_x_cos_y
thf(fact_8573_cos__x__cos__y,axiom,
! [X: real,Y: real] :
( sums_real
@ ^ [P4: nat] :
( groups6591440286371151544t_real
@ ^ [N4: nat] :
( if_real
@ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P4 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
@ ( 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 @ N4 ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ X @ N4 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P4 @ N4 ) ) )
@ zero_zero_real )
@ ( set_ord_atMost_nat @ P4 ) )
@ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ).
% cos_x_cos_y
thf(fact_8574_sums__cos__x__plus__y,axiom,
! [X: complex,Y: complex] :
( sums_complex
@ ^ [P4: nat] :
( groups2073611262835488442omplex
@ ^ [N4: 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 @ N4 ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_complex @ X @ N4 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P4 @ N4 ) ) ) @ zero_zero_complex )
@ ( set_ord_atMost_nat @ P4 ) )
@ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% sums_cos_x_plus_y
thf(fact_8575_sums__cos__x__plus__y,axiom,
! [X: real,Y: real] :
( sums_real
@ ^ [P4: nat] :
( groups6591440286371151544t_real
@ ^ [N4: 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 @ N4 ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ X @ N4 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P4 @ N4 ) ) ) @ zero_zero_real )
@ ( set_ord_atMost_nat @ P4 ) )
@ ( cos_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% sums_cos_x_plus_y
thf(fact_8576_sum__abs__ge__zero,axiom,
! [F: nat > real,A2: set_nat] :
( ord_less_eq_real @ zero_zero_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
@ A2 ) ) ).
% sum_abs_ge_zero
thf(fact_8577_sum__abs__ge__zero,axiom,
! [F: int > int,A2: set_int] :
( ord_less_eq_int @ zero_zero_int
@ ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
@ A2 ) ) ).
% sum_abs_ge_zero
thf(fact_8578_sum__abs,axiom,
! [F: nat > real,A2: set_nat] :
( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
@ A2 ) ) ).
% sum_abs
thf(fact_8579_sum__abs,axiom,
! [F: int > int,A2: set_int] :
( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
@ ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
@ A2 ) ) ).
% sum_abs
thf(fact_8580_convex__sum__bound__le,axiom,
! [I5: set_complex,X: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
=> ( ( ( groups6621422865394947399nteger @ X @ I5 )
= one_one_Code_integer )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups6621422865394947399nteger
@ ^ [I3: complex] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_8581_convex__sum__bound__le,axiom,
! [I5: set_real,X: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
=> ( ( ( groups7713935264441627589nteger @ X @ I5 )
= one_one_Code_integer )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups7713935264441627589nteger
@ ^ [I3: real] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_8582_convex__sum__bound__le,axiom,
! [I5: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
=> ( ( ( groups7501900531339628137nteger @ X @ I5 )
= one_one_Code_integer )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_8583_convex__sum__bound__le,axiom,
! [I5: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
=> ( ( ( groups7873554091576472773nteger @ X @ I5 )
= one_one_Code_integer )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups7873554091576472773nteger
@ ^ [I3: int] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_8584_convex__sum__bound__le,axiom,
! [I5: set_complex,X: complex > rat,A: complex > rat,B: rat,Delta: rat] :
( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
=> ( ( ( groups5058264527183730370ex_rat @ X @ I5 )
= one_one_rat )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups5058264527183730370ex_rat
@ ^ [I3: complex] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_8585_convex__sum__bound__le,axiom,
! [I5: set_real,X: real > rat,A: real > rat,B: rat,Delta: rat] :
( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ X @ I5 )
= one_one_rat )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups1300246762558778688al_rat
@ ^ [I3: real] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_8586_convex__sum__bound__le,axiom,
! [I5: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ X @ I5 )
= one_one_rat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_8587_convex__sum__bound__le,axiom,
! [I5: set_int,X: int > rat,A: int > rat,B: rat,Delta: rat] :
( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ X @ I5 )
= one_one_rat )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups3906332499630173760nt_rat
@ ^ [I3: int] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_8588_convex__sum__bound__le,axiom,
! [I5: set_complex,X: complex > int,A: complex > int,B: int,Delta: int] :
( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( X @ I2 ) ) )
=> ( ( ( groups5690904116761175830ex_int @ X @ I5 )
= one_one_int )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_int
@ ( abs_abs_int
@ ( minus_minus_int
@ ( groups5690904116761175830ex_int
@ ^ [I3: complex] : ( times_times_int @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_8589_convex__sum__bound__le,axiom,
! [I5: set_real,X: real > int,A: real > int,B: int,Delta: int] :
( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( X @ I2 ) ) )
=> ( ( ( groups1932886352136224148al_int @ X @ I5 )
= one_one_int )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_int
@ ( abs_abs_int
@ ( minus_minus_int
@ ( groups1932886352136224148al_int
@ ^ [I3: real] : ( times_times_int @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_8590_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu3: nat] : zero_zero_nat
@ A2 )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_8591_sum_Oneutral__const,axiom,
! [A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [Uu3: complex] : zero_zero_complex
@ A2 )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_8592_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu3: nat] : zero_zero_real
@ A2 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_8593_sum_Oneutral__const,axiom,
! [A2: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [Uu3: int] : zero_zero_int
@ A2 )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_8594_of__nat__id,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N4: nat] : N4 ) ) ).
% of_nat_id
thf(fact_8595_int__sum,axiom,
! [F: int > nat,A2: set_int] :
( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
= ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A2 ) ) ).
% int_sum
thf(fact_8596_int__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3539618377306564664at_int
@ ^ [X3: nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A2 ) ) ).
% int_sum
thf(fact_8597_sum__subtractf__nat,axiom,
! [A2: set_complex,G: complex > nat,F: complex > nat] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ( groups5693394587270226106ex_nat
@ ^ [X3: complex] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_8598_sum__subtractf__nat,axiom,
! [A2: set_real,G: real > nat,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ( groups1935376822645274424al_nat
@ ^ [X3: real] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_8599_sum__subtractf__nat,axiom,
! [A2: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ( groups8294997508430121362at_nat
@ ^ [X3: set_nat] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( groups8294997508430121362at_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_8600_sum__subtractf__nat,axiom,
! [A2: set_int,G: int > nat,F: int > nat] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X3: int] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_8601_sum__subtractf__nat,axiom,
! [A2: set_nat,G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_8602_sum__SucD,axiom,
! [F: nat > nat,A2: set_nat,N2: nat] :
( ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( suc @ N2 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X2 ) ) ) ) ).
% sum_SucD
thf(fact_8603_sum__nth__roots,axiom,
! [N2: nat,C: complex] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( groups7754918857620584856omplex
@ ^ [X3: complex] : X3
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N2 )
= C ) ) )
= zero_zero_complex ) ) ).
% sum_nth_roots
thf(fact_8604_sum__roots__unity,axiom,
! [N2: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( groups7754918857620584856omplex
@ ^ [X3: complex] : X3
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N2 )
= one_one_complex ) ) )
= zero_zero_complex ) ) ).
% sum_roots_unity
thf(fact_8605_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > complex,A2: set_real] :
( ( ( groups5754745047067104278omplex @ G @ A2 )
!= zero_zero_complex )
=> ~ ! [A5: real] :
( ( member_real @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8606_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > complex,A2: set_nat] :
( ( ( groups2073611262835488442omplex @ G @ A2 )
!= zero_zero_complex )
=> ~ ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8607_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > complex,A2: set_int] :
( ( ( groups3049146728041665814omplex @ G @ A2 )
!= zero_zero_complex )
=> ~ ! [A5: int] :
( ( member_int @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8608_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > real,A2: set_complex] :
( ( ( groups5808333547571424918x_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A5: complex] :
( ( member_complex @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8609_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A2: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A5: real] :
( ( member_real @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8610_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > real,A2: set_int] :
( ( ( groups8778361861064173332t_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A5: int] :
( ( member_int @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8611_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > rat,A2: set_complex] :
( ( ( groups5058264527183730370ex_rat @ G @ A2 )
!= zero_zero_rat )
=> ~ ! [A5: complex] :
( ( member_complex @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8612_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > rat,A2: set_real] :
( ( ( groups1300246762558778688al_rat @ G @ A2 )
!= zero_zero_rat )
=> ~ ! [A5: real] :
( ( member_real @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8613_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > rat,A2: set_nat] :
( ( ( groups2906978787729119204at_rat @ G @ A2 )
!= zero_zero_rat )
=> ~ ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8614_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > rat,A2: set_int] :
( ( ( groups3906332499630173760nt_rat @ G @ A2 )
!= zero_zero_rat )
=> ~ ! [A5: int] :
( ( member_int @ A5 @ A2 )
=> ( ( G @ A5 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8615_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_8616_sum_Oneutral,axiom,
! [A2: set_complex,G: complex > complex] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( G @ X2 )
= zero_zero_complex ) )
=> ( ( groups7754918857620584856omplex @ G @ A2 )
= zero_zero_complex ) ) ).
% sum.neutral
thf(fact_8617_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_8618_sum_Oneutral,axiom,
! [A2: set_int,G: int > int] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( G @ X2 )
= zero_zero_int ) )
=> ( ( groups4538972089207619220nt_int @ G @ A2 )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_8619_sum__mono,axiom,
! [K5: set_complex,F: complex > rat,G: complex > rat] :
( ! [I2: complex] :
( ( member_complex @ I2 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_8620_sum__mono,axiom,
! [K5: set_real,F: real > rat,G: real > rat] :
( ! [I2: real] :
( ( member_real @ I2 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_8621_sum__mono,axiom,
! [K5: set_nat,F: nat > rat,G: nat > rat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_8622_sum__mono,axiom,
! [K5: set_int,F: int > rat,G: int > rat] :
( ! [I2: int] :
( ( member_int @ I2 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_8623_sum__mono,axiom,
! [K5: set_complex,F: complex > nat,G: complex > nat] :
( ! [I2: complex] :
( ( member_complex @ I2 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_8624_sum__mono,axiom,
! [K5: set_real,F: real > nat,G: real > nat] :
( ! [I2: real] :
( ( member_real @ I2 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_8625_sum__mono,axiom,
! [K5: set_int,F: int > nat,G: int > nat] :
( ! [I2: int] :
( ( member_int @ I2 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_8626_sum__mono,axiom,
! [K5: set_complex,F: complex > int,G: complex > int] :
( ! [I2: complex] :
( ( member_complex @ I2 @ K5 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_8627_sum__mono,axiom,
! [K5: set_real,F: real > int,G: real > int] :
( ! [I2: real] :
( ( member_real @ I2 @ K5 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_8628_sum__mono,axiom,
! [K5: set_nat,F: nat > int,G: nat > int] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K5 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_8629_sum_Odistrib,axiom,
! [G: nat > nat,H2: nat > nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( plus_plus_nat @ ( G @ X3 ) @ ( H2 @ X3 ) )
@ A2 )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).
% sum.distrib
thf(fact_8630_sum_Odistrib,axiom,
! [G: complex > complex,H2: complex > complex,A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X3: complex] : ( plus_plus_complex @ ( G @ X3 ) @ ( H2 @ X3 ) )
@ A2 )
= ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).
% sum.distrib
thf(fact_8631_sum_Odistrib,axiom,
! [G: nat > real,H2: nat > real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X3: nat] : ( plus_plus_real @ ( G @ X3 ) @ ( H2 @ X3 ) )
@ A2 )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).
% sum.distrib
thf(fact_8632_sum_Odistrib,axiom,
! [G: int > int,H2: int > int,A2: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( plus_plus_int @ ( G @ X3 ) @ ( H2 @ X3 ) )
@ A2 )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H2 @ A2 ) ) ) ).
% sum.distrib
thf(fact_8633_sum__divide__distrib,axiom,
! [F: complex > complex,A2: set_complex,R4: complex] :
( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R4 )
= ( groups7754918857620584856omplex
@ ^ [N4: complex] : ( divide1717551699836669952omplex @ ( F @ N4 ) @ R4 )
@ A2 ) ) ).
% sum_divide_distrib
thf(fact_8634_sum__divide__distrib,axiom,
! [F: nat > real,A2: set_nat,R4: real] :
( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R4 )
= ( groups6591440286371151544t_real
@ ^ [N4: nat] : ( divide_divide_real @ ( F @ N4 ) @ R4 )
@ A2 ) ) ).
% sum_divide_distrib
thf(fact_8635_sum__nonpos,axiom,
! [A2: set_complex,F: complex > rat] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X2 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_8636_sum__nonpos,axiom,
! [A2: set_real,F: real > rat] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X2 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_8637_sum__nonpos,axiom,
! [A2: set_nat,F: nat > rat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X2 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_8638_sum__nonpos,axiom,
! [A2: set_int,F: int > rat] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X2 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_8639_sum__nonpos,axiom,
! [A2: set_complex,F: complex > nat] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_8640_sum__nonpos,axiom,
! [A2: set_real,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_8641_sum__nonpos,axiom,
! [A2: set_int,F: int > nat] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_8642_sum__nonpos,axiom,
! [A2: set_complex,F: complex > int] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_8643_sum__nonpos,axiom,
! [A2: set_real,F: real > int] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_8644_sum__nonpos,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_8645_sum__nonneg,axiom,
! [A2: set_complex,F: complex > rat] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X2 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8646_sum__nonneg,axiom,
! [A2: set_real,F: real > rat] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X2 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8647_sum__nonneg,axiom,
! [A2: set_nat,F: nat > rat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X2 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8648_sum__nonneg,axiom,
! [A2: set_int,F: int > rat] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X2 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8649_sum__nonneg,axiom,
! [A2: set_complex,F: complex > nat] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8650_sum__nonneg,axiom,
! [A2: set_real,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8651_sum__nonneg,axiom,
! [A2: set_int,F: int > nat] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8652_sum__nonneg,axiom,
! [A2: set_complex,F: complex > int] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups5690904116761175830ex_int @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8653_sum__nonneg,axiom,
! [A2: set_real,F: real > int] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8654_sum__nonneg,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8655_sum__cong__Suc,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ! [X2: nat] :
( ( member_nat @ ( suc @ X2 ) @ A2 )
=> ( ( F @ ( suc @ X2 ) )
= ( G @ ( suc @ X2 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% sum_cong_Suc
thf(fact_8656_sum__cong__Suc,axiom,
! [A2: set_nat,F: nat > real,G: nat > real] :
( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ! [X2: nat] :
( ( member_nat @ ( suc @ X2 ) @ A2 )
=> ( ( F @ ( suc @ X2 ) )
= ( G @ ( suc @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ F @ A2 )
= ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).
% sum_cong_Suc
thf(fact_8657_Maclaurin__minus__cos__expansion,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ? [T: real] :
( ( ord_less_real @ X @ T )
& ( ord_less_real @ T @ 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_8658_Maclaurin__cos__expansion2,axiom,
! [X: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [T: real] :
( ( ord_less_real @ zero_zero_real @ T )
& ( ord_less_real @ T @ 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_8659_Maclaurin__sin__expansion3,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ? [T: real] :
( ( ord_less_real @ zero_zero_real @ T )
& ( ord_less_real @ T @ 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_8660_Maclaurin__sin__expansion4,axiom,
! [X: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [T: real] :
( ( ord_less_real @ zero_zero_real @ T )
& ( ord_less_eq_real @ T @ 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_8661_lessThan__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
= ( ord_less_set_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_8662_lessThan__iff,axiom,
! [I: rat,K: rat] :
( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
= ( ord_less_rat @ I @ K ) ) ).
% lessThan_iff
thf(fact_8663_lessThan__iff,axiom,
! [I: num,K: num] :
( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
= ( ord_less_num @ I @ K ) ) ).
% lessThan_iff
thf(fact_8664_lessThan__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I @ K ) ) ).
% lessThan_iff
thf(fact_8665_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_8666_lessThan__iff,axiom,
! [I: real,K: real] :
( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
= ( ord_less_real @ I @ K ) ) ).
% lessThan_iff
thf(fact_8667_lessThan__subset__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y ) )
= ( ord_less_eq_rat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_8668_lessThan__subset__iff,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y ) )
= ( ord_less_eq_num @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_8669_lessThan__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_8670_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_8671_lessThan__subset__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_8672_sum_OlessThan__Suc,axiom,
! [G: nat > rat,N2: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% sum.lessThan_Suc
thf(fact_8673_sum_OlessThan__Suc,axiom,
! [G: nat > int,N2: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% sum.lessThan_Suc
thf(fact_8674_sum_OlessThan__Suc,axiom,
! [G: nat > nat,N2: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% sum.lessThan_Suc
thf(fact_8675_sum_OlessThan__Suc,axiom,
! [G: nat > real,N2: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% sum.lessThan_Suc
thf(fact_8676_sumr__cos__zero__one,axiom,
! [N2: nat] :
( ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ zero_zero_real @ M2 ) )
@ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= one_one_real ) ).
% sumr_cos_zero_one
thf(fact_8677_lessThan__def,axiom,
( set_or890127255671739683et_nat
= ( ^ [U2: set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_set_nat @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_8678_lessThan__def,axiom,
( set_ord_lessThan_rat
= ( ^ [U2: rat] :
( collect_rat
@ ^ [X3: rat] : ( ord_less_rat @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_8679_lessThan__def,axiom,
( set_ord_lessThan_num
= ( ^ [U2: num] :
( collect_num
@ ^ [X3: num] : ( ord_less_num @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_8680_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X3: int] : ( ord_less_int @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_8681_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_8682_lessThan__def,axiom,
( set_or5984915006950818249n_real
= ( ^ [U2: real] :
( collect_real
@ ^ [X3: real] : ( ord_less_real @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_8683_lessThan__strict__subset__iff,axiom,
! [M: rat,N2: rat] :
( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N2 ) )
= ( ord_less_rat @ M @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_8684_lessThan__strict__subset__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N2 ) )
= ( ord_less_num @ M @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_8685_lessThan__strict__subset__iff,axiom,
! [M: int,N2: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N2 ) )
= ( ord_less_int @ M @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_8686_lessThan__strict__subset__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_8687_lessThan__strict__subset__iff,axiom,
! [M: real,N2: real] :
( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N2 ) )
= ( ord_less_real @ M @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_8688_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_8689_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_8690_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_8691_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_8692_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_8693_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_8694_sum_Onat__diff__reindex,axiom,
! [G: nat > nat,N2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N2 ) )
= ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% sum.nat_diff_reindex
thf(fact_8695_sum_Onat__diff__reindex,axiom,
! [G: nat > real,N2: nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N2 ) )
= ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% sum.nat_diff_reindex
thf(fact_8696_sum__diff__distrib,axiom,
! [Q: real > nat,P: real > nat,N2: real] :
( ! [X2: real] : ( ord_less_eq_nat @ ( Q @ X2 ) @ ( P @ X2 ) )
=> ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N2 ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N2 ) ) )
= ( groups1935376822645274424al_nat
@ ^ [X3: real] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q @ X3 ) )
@ ( set_or5984915006950818249n_real @ N2 ) ) ) ) ).
% sum_diff_distrib
thf(fact_8697_sum__diff__distrib,axiom,
! [Q: nat > nat,P: nat > nat,N2: nat] :
( ! [X2: nat] : ( ord_less_eq_nat @ ( Q @ X2 ) @ ( P @ X2 ) )
=> ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N2 ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q @ X3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% sum_diff_distrib
thf(fact_8698_suminf__le__const,axiom,
! [F: nat > int,X: int] :
( ( summable_int @ F )
=> ( ! [N: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
=> ( ord_less_eq_int @ ( suminf_int @ F ) @ X ) ) ) ).
% suminf_le_const
thf(fact_8699_suminf__le__const,axiom,
! [F: nat > nat,X: nat] :
( ( summable_nat @ F )
=> ( ! [N: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
=> ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X ) ) ) ).
% suminf_le_const
thf(fact_8700_suminf__le__const,axiom,
! [F: nat > real,X: real] :
( ( summable_real @ F )
=> ( ! [N: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
=> ( ord_less_eq_real @ ( suminf_real @ F ) @ X ) ) ) ).
% suminf_le_const
thf(fact_8701_sum_OlessThan__Suc__shift,axiom,
! [G: nat > rat,N2: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= ( plus_plus_rat @ ( G @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_8702_sum_OlessThan__Suc__shift,axiom,
! [G: nat > int,N2: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_8703_sum_OlessThan__Suc__shift,axiom,
! [G: nat > nat,N2: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_8704_sum_OlessThan__Suc__shift,axiom,
! [G: nat > real,N2: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_8705_sum__lessThan__telescope_H,axiom,
! [F: nat > rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [N4: nat] : ( minus_minus_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_8706_sum__lessThan__telescope_H,axiom,
! [F: nat > int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [N4: nat] : ( minus_minus_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_8707_sum__lessThan__telescope_H,axiom,
! [F: nat > real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_8708_sum__lessThan__telescope,axiom,
! [F: nat > rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [N4: nat] : ( minus_minus_rat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_8709_sum__lessThan__telescope,axiom,
! [F: nat > int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [N4: nat] : ( minus_minus_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_8710_sum__lessThan__telescope,axiom,
! [F: nat > real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [N4: nat] : ( minus_minus_real @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_8711_summableI__nonneg__bounded,axiom,
! [F: nat > int,X: int] :
( ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N ) )
=> ( ! [N: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
=> ( summable_int @ F ) ) ) ).
% summableI_nonneg_bounded
thf(fact_8712_summableI__nonneg__bounded,axiom,
! [F: nat > nat,X: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N ) )
=> ( ! [N: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
=> ( summable_nat @ F ) ) ) ).
% summableI_nonneg_bounded
thf(fact_8713_summableI__nonneg__bounded,axiom,
! [F: nat > real,X: real] :
( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N ) )
=> ( ! [N: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
=> ( summable_real @ F ) ) ) ).
% summableI_nonneg_bounded
thf(fact_8714_sums__iff__shift,axiom,
! [F: nat > real,N2: nat,S: real] :
( ( sums_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
@ S )
= ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% sums_iff_shift
thf(fact_8715_sums__iff__shift_H,axiom,
! [F: nat > real,N2: nat,S: real] :
( ( sums_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
@ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) )
= ( sums_real @ F @ S ) ) ).
% sums_iff_shift'
thf(fact_8716_sums__split__initial__segment,axiom,
! [F: nat > real,S: real,N2: nat] :
( ( sums_real @ F @ S )
=> ( sums_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
@ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% sums_split_initial_segment
thf(fact_8717_one__diff__power__eq,axiom,
! [X: complex,N2: nat] :
( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) )
= ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% one_diff_power_eq
thf(fact_8718_one__diff__power__eq,axiom,
! [X: rat,N2: nat] :
( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) )
= ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% one_diff_power_eq
thf(fact_8719_one__diff__power__eq,axiom,
! [X: int,N2: nat] :
( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N2 ) )
= ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% one_diff_power_eq
thf(fact_8720_one__diff__power__eq,axiom,
! [X: real,N2: nat] :
( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) )
= ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% one_diff_power_eq
thf(fact_8721_power__diff__1__eq,axiom,
! [X: complex,N2: nat] :
( ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ one_one_complex )
= ( times_times_complex @ ( minus_minus_complex @ X @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% power_diff_1_eq
thf(fact_8722_power__diff__1__eq,axiom,
! [X: rat,N2: nat] :
( ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ one_one_rat )
= ( times_times_rat @ ( minus_minus_rat @ X @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% power_diff_1_eq
thf(fact_8723_power__diff__1__eq,axiom,
! [X: int,N2: nat] :
( ( minus_minus_int @ ( power_power_int @ X @ N2 ) @ one_one_int )
= ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% power_diff_1_eq
thf(fact_8724_power__diff__1__eq,axiom,
! [X: real,N2: nat] :
( ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ one_one_real )
= ( times_times_real @ ( minus_minus_real @ X @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% power_diff_1_eq
thf(fact_8725_geometric__sum,axiom,
! [X: complex,N2: nat] :
( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ) ).
% geometric_sum
thf(fact_8726_geometric__sum,axiom,
! [X: rat,N2: nat] :
( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ) ).
% geometric_sum
thf(fact_8727_geometric__sum,axiom,
! [X: real,N2: nat] :
( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ) ).
% geometric_sum
thf(fact_8728_sum_OatMost__shift,axiom,
! [G: nat > rat,N2: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N2 ) )
= ( plus_plus_rat @ ( G @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% sum.atMost_shift
thf(fact_8729_sum_OatMost__shift,axiom,
! [G: nat > int,N2: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N2 ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% sum.atMost_shift
thf(fact_8730_sum_OatMost__shift,axiom,
! [G: nat > nat,N2: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N2 ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% sum.atMost_shift
thf(fact_8731_sum_OatMost__shift,axiom,
! [G: nat > real,N2: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N2 ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% sum.atMost_shift
thf(fact_8732_suminf__split__initial__segment,axiom,
! [F: nat > real,K: nat] :
( ( summable_real @ F )
=> ( ( suminf_real @ F )
= ( plus_plus_real
@ ( suminf_real
@ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ K ) ) )
@ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% suminf_split_initial_segment
thf(fact_8733_suminf__minus__initial__segment,axiom,
! [F: nat > real,K: nat] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ K ) ) )
= ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% suminf_minus_initial_segment
thf(fact_8734_sum__less__suminf,axiom,
! [F: nat > int,N2: nat] :
( ( summable_int @ F )
=> ( ! [M5: nat] :
( ( ord_less_eq_nat @ N2 @ M5 )
=> ( ord_less_int @ zero_zero_int @ ( F @ M5 ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ).
% sum_less_suminf
thf(fact_8735_sum__less__suminf,axiom,
! [F: nat > nat,N2: nat] :
( ( summable_nat @ F )
=> ( ! [M5: nat] :
( ( ord_less_eq_nat @ N2 @ M5 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ M5 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ).
% sum_less_suminf
thf(fact_8736_sum__less__suminf,axiom,
! [F: nat > real,N2: nat] :
( ( summable_real @ F )
=> ( ! [M5: nat] :
( ( ord_less_eq_nat @ N2 @ M5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ M5 ) ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ).
% sum_less_suminf
thf(fact_8737_sum__gp__strict,axiom,
! [X: complex,N2: nat] :
( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( semiri8010041392384452111omplex @ N2 ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% sum_gp_strict
thf(fact_8738_sum__gp__strict,axiom,
! [X: rat,N2: nat] :
( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( semiri681578069525770553at_rat @ N2 ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% sum_gp_strict
thf(fact_8739_sum__gp__strict,axiom,
! [X: real,N2: nat] :
( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( semiri5074537144036343181t_real @ N2 ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% sum_gp_strict
thf(fact_8740_lemma__termdiff1,axiom,
! [Z: complex,H2: complex,M: nat] :
( ( groups2073611262835488442omplex
@ ^ [P4: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_complex @ Z @ P4 ) ) @ ( power_power_complex @ Z @ M ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( groups2073611262835488442omplex
@ ^ [P4: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P4 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
@ ( set_ord_lessThan_nat @ M ) ) ) ).
% lemma_termdiff1
thf(fact_8741_lemma__termdiff1,axiom,
! [Z: rat,H2: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [P4: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_rat @ Z @ P4 ) ) @ ( power_power_rat @ Z @ M ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( groups2906978787729119204at_rat
@ ^ [P4: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P4 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
@ ( set_ord_lessThan_nat @ M ) ) ) ).
% lemma_termdiff1
thf(fact_8742_lemma__termdiff1,axiom,
! [Z: int,H2: int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [P4: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_int @ Z @ P4 ) ) @ ( power_power_int @ Z @ M ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( groups3539618377306564664at_int
@ ^ [P4: nat] : ( times_times_int @ ( power_power_int @ Z @ P4 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
@ ( set_ord_lessThan_nat @ M ) ) ) ).
% lemma_termdiff1
thf(fact_8743_lemma__termdiff1,axiom,
! [Z: real,H2: real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [P4: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_real @ Z @ P4 ) ) @ ( power_power_real @ Z @ M ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( groups6591440286371151544t_real
@ ^ [P4: nat] : ( times_times_real @ ( power_power_real @ Z @ P4 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
@ ( set_ord_lessThan_nat @ M ) ) ) ).
% lemma_termdiff1
thf(fact_8744_power__diff__sumr2,axiom,
! [X: complex,N2: nat,Y: complex] :
( ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_complex @ X @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% power_diff_sumr2
thf(fact_8745_power__diff__sumr2,axiom,
! [X: rat,N2: nat,Y: rat] :
( ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_rat @ X @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% power_diff_sumr2
thf(fact_8746_power__diff__sumr2,axiom,
! [X: int,N2: nat,Y: int] :
( ( minus_minus_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_int @ X @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% power_diff_sumr2
thf(fact_8747_power__diff__sumr2,axiom,
! [X: real,N2: nat,Y: real] :
( ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_real @ X @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% power_diff_sumr2
thf(fact_8748_diff__power__eq__sum,axiom,
! [X: complex,N2: nat,Y: complex] :
( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) @ ( power_power_complex @ Y @ ( suc @ N2 ) ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y )
@ ( groups2073611262835488442omplex
@ ^ [P4: nat] : ( times_times_complex @ ( power_power_complex @ X @ P4 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ P4 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_8749_diff__power__eq__sum,axiom,
! [X: rat,N2: nat,Y: rat] :
( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) @ ( power_power_rat @ Y @ ( suc @ N2 ) ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y )
@ ( groups2906978787729119204at_rat
@ ^ [P4: nat] : ( times_times_rat @ ( power_power_rat @ X @ P4 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N2 @ P4 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_8750_diff__power__eq__sum,axiom,
! [X: int,N2: nat,Y: int] :
( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N2 ) ) @ ( power_power_int @ Y @ ( suc @ N2 ) ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y )
@ ( groups3539618377306564664at_int
@ ^ [P4: nat] : ( times_times_int @ ( power_power_int @ X @ P4 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ P4 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_8751_diff__power__eq__sum,axiom,
! [X: real,N2: nat,Y: real] :
( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N2 ) ) @ ( power_power_real @ Y @ ( suc @ N2 ) ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y )
@ ( groups6591440286371151544t_real
@ ^ [P4: nat] : ( times_times_real @ ( power_power_real @ X @ P4 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ P4 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_8752_polyfun__linear__factor__root,axiom,
! [C: nat > complex,A: complex,N2: nat] :
( ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_complex )
=> ~ ! [B5: nat > complex] :
~ ! [Z3: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( times_times_complex @ ( minus_minus_complex @ Z3 @ A )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( B5 @ I3 ) @ ( power_power_complex @ Z3 @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8753_polyfun__linear__factor__root,axiom,
! [C: nat > rat,A: rat,N2: nat] :
( ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_rat )
=> ~ ! [B5: nat > rat] :
~ ! [Z3: rat] :
( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ Z3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( times_times_rat @ ( minus_minus_rat @ Z3 @ A )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( B5 @ I3 ) @ ( power_power_rat @ Z3 @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8754_polyfun__linear__factor__root,axiom,
! [C: nat > int,A: int,N2: nat] :
( ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_int )
=> ~ ! [B5: nat > int] :
~ ! [Z3: int] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ Z3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( times_times_int @ ( minus_minus_int @ Z3 @ A )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( B5 @ I3 ) @ ( power_power_int @ Z3 @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8755_polyfun__linear__factor__root,axiom,
! [C: nat > real,A: real,N2: nat] :
( ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= zero_zero_real )
=> ~ ! [B5: nat > real] :
~ ! [Z3: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( times_times_real @ ( minus_minus_real @ Z3 @ A )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( B5 @ I3 ) @ ( power_power_real @ Z3 @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8756_polyfun__linear__factor,axiom,
! [C: nat > complex,N2: nat,A: complex] :
? [B5: nat > complex] :
! [Z3: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( plus_plus_complex
@ ( times_times_complex @ ( minus_minus_complex @ Z3 @ A )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( B5 @ I3 ) @ ( power_power_complex @ Z3 @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% polyfun_linear_factor
thf(fact_8757_polyfun__linear__factor,axiom,
! [C: nat > rat,N2: nat,A: rat] :
? [B5: nat > rat] :
! [Z3: rat] :
( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ Z3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( plus_plus_rat
@ ( times_times_rat @ ( minus_minus_rat @ Z3 @ A )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( B5 @ I3 ) @ ( power_power_rat @ Z3 @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% polyfun_linear_factor
thf(fact_8758_polyfun__linear__factor,axiom,
! [C: nat > int,N2: nat,A: int] :
? [B5: nat > int] :
! [Z3: int] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ Z3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( plus_plus_int
@ ( times_times_int @ ( minus_minus_int @ Z3 @ A )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( B5 @ I3 ) @ ( power_power_int @ Z3 @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% polyfun_linear_factor
thf(fact_8759_polyfun__linear__factor,axiom,
! [C: nat > real,N2: nat,A: real] :
? [B5: nat > real] :
! [Z3: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( plus_plus_real
@ ( times_times_real @ ( minus_minus_real @ Z3 @ A )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( B5 @ I3 ) @ ( power_power_real @ Z3 @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% polyfun_linear_factor
thf(fact_8760_real__sum__nat__ivl__bounded2,axiom,
! [N2: nat,F: nat > rat,K5: rat,K: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N2 )
=> ( 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 @ N2 @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_8761_real__sum__nat__ivl__bounded2,axiom,
! [N2: nat,F: nat > int,K5: int,K: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N2 )
=> ( 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 @ N2 @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_8762_real__sum__nat__ivl__bounded2,axiom,
! [N2: nat,F: nat > nat,K5: nat,K: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N2 )
=> ( 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 @ N2 @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_8763_real__sum__nat__ivl__bounded2,axiom,
! [N2: nat,F: nat > real,K5: real,K: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N2 )
=> ( 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 @ N2 @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_8764_sum__less__suminf2,axiom,
! [F: nat > int,N2: nat,I: nat] :
( ( summable_int @ F )
=> ( ! [M5: nat] :
( ( ord_less_eq_nat @ N2 @ M5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ M5 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ I )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_8765_sum__less__suminf2,axiom,
! [F: nat > nat,N2: nat,I: nat] :
( ( summable_nat @ F )
=> ( ! [M5: nat] :
( ( ord_less_eq_nat @ N2 @ M5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M5 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ I )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_8766_sum__less__suminf2,axiom,
! [F: nat > real,N2: nat,I: nat] :
( ( summable_real @ F )
=> ( ! [M5: nat] :
( ( ord_less_eq_nat @ N2 @ M5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ M5 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ I )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_8767_one__diff__power__eq_H,axiom,
! [X: complex,N2: nat] :
( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) )
= ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% one_diff_power_eq'
thf(fact_8768_one__diff__power__eq_H,axiom,
! [X: rat,N2: nat] :
( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) )
= ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% one_diff_power_eq'
thf(fact_8769_one__diff__power__eq_H,axiom,
! [X: int,N2: nat] :
( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N2 ) )
= ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% one_diff_power_eq'
thf(fact_8770_one__diff__power__eq_H,axiom,
! [X: real,N2: nat] :
( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) )
= ( times_times_real @ ( minus_minus_real @ one_one_real @ X )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% one_diff_power_eq'
thf(fact_8771_Maclaurin__zero,axiom,
! [X: real,N2: nat,Diff: nat > complex > real] :
( ( X = zero_zero_real )
=> ( ( N2 != 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 @ N2 ) )
= ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).
% Maclaurin_zero
thf(fact_8772_Maclaurin__zero,axiom,
! [X: real,N2: nat,Diff: nat > real > real] :
( ( X = zero_zero_real )
=> ( ( N2 != 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 @ N2 ) )
= ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).
% Maclaurin_zero
thf(fact_8773_Maclaurin__zero,axiom,
! [X: real,N2: nat,Diff: nat > rat > real] :
( ( X = zero_zero_real )
=> ( ( N2 != 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 @ N2 ) )
= ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).
% Maclaurin_zero
thf(fact_8774_Maclaurin__zero,axiom,
! [X: real,N2: nat,Diff: nat > nat > real] :
( ( X = zero_zero_real )
=> ( ( N2 != 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 @ N2 ) )
= ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).
% Maclaurin_zero
thf(fact_8775_Maclaurin__zero,axiom,
! [X: real,N2: nat,Diff: nat > int > real] :
( ( X = zero_zero_real )
=> ( ( N2 != 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 @ N2 ) )
= ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).
% Maclaurin_zero
thf(fact_8776_Maclaurin__lemma,axiom,
! [H2: real,F: real > real,J: nat > real,N2: 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 @ N2 ) )
@ ( times_times_real @ B7 @ ( divide_divide_real @ ( power_power_real @ H2 @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_8777_sum__split__even__odd,axiom,
! [F: nat > real,G: nat > real,N2: nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( F @ I3 ) @ ( G @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% sum_split_even_odd
thf(fact_8778_Maclaurin__exp__le,axiom,
! [X: real,N2: nat] :
? [T: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T ) @ ( 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% Maclaurin_exp_le
thf(fact_8779_polyfun__diff__alt,axiom,
! [N2: nat,A: nat > complex,X: complex,Y: complex] :
( ( ord_less_eq_nat @ one_one_nat @ N2 )
=> ( ( minus_minus_complex
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y )
@ ( groups2073611262835488442omplex
@ ^ [J3: nat] :
( groups2073611262835488442omplex
@ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_complex @ Y @ K2 ) ) @ ( power_power_complex @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ J3 ) ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_8780_polyfun__diff__alt,axiom,
! [N2: nat,A: nat > rat,X: rat,Y: rat] :
( ( ord_less_eq_nat @ one_one_nat @ N2 )
=> ( ( minus_minus_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y )
@ ( groups2906978787729119204at_rat
@ ^ [J3: nat] :
( groups2906978787729119204at_rat
@ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_rat @ Y @ K2 ) ) @ ( power_power_rat @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ J3 ) ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_8781_polyfun__diff__alt,axiom,
! [N2: nat,A: nat > int,X: int,Y: int] :
( ( ord_less_eq_nat @ one_one_nat @ N2 )
=> ( ( minus_minus_int
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y )
@ ( groups3539618377306564664at_int
@ ^ [J3: nat] :
( groups3539618377306564664at_int
@ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_int @ Y @ K2 ) ) @ ( power_power_int @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ J3 ) ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_8782_polyfun__diff__alt,axiom,
! [N2: nat,A: nat > real,X: real,Y: real] :
( ( ord_less_eq_nat @ one_one_nat @ N2 )
=> ( ( minus_minus_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y )
@ ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( groups6591440286371151544t_real
@ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_real @ Y @ K2 ) ) @ ( power_power_real @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ J3 ) ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_8783_exp__first__terms,axiom,
! [K: nat] :
( exp_complex
= ( ^ [X3: complex] :
( plus_plus_complex
@ ( groups2073611262835488442omplex
@ ^ [N4: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N4 ) ) @ ( power_power_complex @ X3 @ N4 ) )
@ ( set_ord_lessThan_nat @ K ) )
@ ( suminf_complex
@ ^ [N4: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N4 @ K ) ) ) @ ( power_power_complex @ X3 @ ( plus_plus_nat @ N4 @ K ) ) ) ) ) ) ) ).
% exp_first_terms
thf(fact_8784_exp__first__terms,axiom,
! [K: nat] :
( exp_real
= ( ^ [X3: real] :
( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [N4: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N4 ) ) @ ( power_power_real @ X3 @ N4 ) )
@ ( set_ord_lessThan_nat @ K ) )
@ ( suminf_real
@ ^ [N4: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N4 @ K ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ N4 @ K ) ) ) ) ) ) ) ).
% exp_first_terms
thf(fact_8785_sum__pos__lt__pair,axiom,
! [F: nat > real,K: nat] :
( ( summable_real @ F )
=> ( ! [D4: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) @ one_one_nat ) ) ) ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_8786_Maclaurin__exp__lt,axiom,
! [X: real,N2: nat] :
( ( X != zero_zero_real )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [T: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T ) )
& ( ord_less_real @ ( abs_abs_real @ T ) @ ( 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_8787_lemma__termdiff2,axiom,
! [H2: complex,Z: complex,N2: nat] :
( ( H2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
= ( times_times_complex @ H2
@ ( groups2073611262835488442omplex
@ ^ [P4: nat] :
( groups2073611262835488442omplex
@ ^ [Q4: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q4 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_8788_lemma__termdiff2,axiom,
! [H2: rat,Z: rat,N2: nat] :
( ( H2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N2 ) @ ( power_power_rat @ Z @ N2 ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N2 @ ( 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 @ Z @ H2 ) @ Q4 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_8789_lemma__termdiff2,axiom,
! [H2: real,Z: real,N2: nat] :
( ( H2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N2 ) @ ( power_power_real @ Z @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N2 @ ( 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 @ Z @ H2 ) @ Q4 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_8790_Maclaurin__sin__expansion,axiom,
! [X: real,N2: nat] :
? [T: 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ).
% Maclaurin_sin_expansion
thf(fact_8791_Maclaurin__sin__expansion2,axiom,
! [X: real,N2: nat] :
? [T: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T ) @ ( 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% Maclaurin_sin_expansion2
thf(fact_8792_Maclaurin__cos__expansion,axiom,
! [X: real,N2: nat] :
? [T: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T ) @ ( 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% Maclaurin_cos_expansion
thf(fact_8793_bij__betw__roots__unity,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( bij_betw_nat_complex
@ ^ [K2: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
@ ( set_ord_lessThan_nat @ N2 )
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N2 )
= one_one_complex ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_8794_sum__gp,axiom,
! [N2: nat,M: nat,X: complex] :
( ( ( ord_less_nat @ N2 @ M )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= zero_zero_complex ) )
& ( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_8795_sum__gp,axiom,
! [N2: nat,M: nat,X: rat] :
( ( ( ord_less_nat @ N2 @ M )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= zero_zero_rat ) )
& ( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_8796_sum__gp,axiom,
! [N2: nat,M: nat,X: real] :
( ( ( ord_less_nat @ N2 @ M )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_8797_gchoose__row__sum__weighted,axiom,
! [R4: complex,M: nat] :
( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ R4 @ K2 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R4 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K2 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
= ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R4 @ ( suc @ M ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_8798_gchoose__row__sum__weighted,axiom,
! [R4: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K2: nat] : ( times_times_rat @ ( gbinomial_rat @ R4 @ K2 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R4 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K2 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
= ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R4 @ ( suc @ M ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_8799_gchoose__row__sum__weighted,axiom,
! [R4: real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ R4 @ K2 ) @ ( minus_minus_real @ ( divide_divide_real @ R4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
= ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R4 @ ( suc @ M ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_8800_gauss__sum__from__Suc__0,axiom,
! [N2: nat] :
( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
= ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_8801_gauss__sum__from__Suc__0,axiom,
! [N2: nat] :
( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
= ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_8802_Icc__eq__Icc,axiom,
! [L: set_nat,H2: set_nat,L4: set_nat,H3: set_nat] :
( ( ( set_or4548717258645045905et_nat @ L @ H2 )
= ( set_or4548717258645045905et_nat @ L4 @ H3 ) )
= ( ( ( L = L4 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_set_nat @ L @ H2 )
& ~ ( ord_less_eq_set_nat @ L4 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_8803_Icc__eq__Icc,axiom,
! [L: rat,H2: rat,L4: rat,H3: rat] :
( ( ( set_or633870826150836451st_rat @ L @ H2 )
= ( set_or633870826150836451st_rat @ L4 @ H3 ) )
= ( ( ( L = L4 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_rat @ L @ H2 )
& ~ ( ord_less_eq_rat @ L4 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_8804_Icc__eq__Icc,axiom,
! [L: num,H2: num,L4: num,H3: num] :
( ( ( set_or7049704709247886629st_num @ L @ H2 )
= ( set_or7049704709247886629st_num @ L4 @ H3 ) )
= ( ( ( L = L4 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_num @ L @ H2 )
& ~ ( ord_less_eq_num @ L4 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_8805_Icc__eq__Icc,axiom,
! [L: nat,H2: nat,L4: nat,H3: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H2 )
= ( set_or1269000886237332187st_nat @ L4 @ H3 ) )
= ( ( ( L = L4 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_nat @ L @ H2 )
& ~ ( ord_less_eq_nat @ L4 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_8806_Icc__eq__Icc,axiom,
! [L: int,H2: int,L4: int,H3: int] :
( ( ( set_or1266510415728281911st_int @ L @ H2 )
= ( set_or1266510415728281911st_int @ L4 @ H3 ) )
= ( ( ( L = L4 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_int @ L @ H2 )
& ~ ( ord_less_eq_int @ L4 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_8807_Icc__eq__Icc,axiom,
! [L: real,H2: real,L4: real,H3: real] :
( ( ( set_or1222579329274155063t_real @ L @ H2 )
= ( set_or1222579329274155063t_real @ L4 @ H3 ) )
= ( ( ( L = L4 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_real @ L @ H2 )
& ~ ( ord_less_eq_real @ L4 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_8808_atLeastAtMost__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_8809_atLeastAtMost__iff,axiom,
! [I: rat,L: rat,U: rat] :
( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
= ( ( ord_less_eq_rat @ L @ I )
& ( ord_less_eq_rat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_8810_atLeastAtMost__iff,axiom,
! [I: num,L: num,U: num] :
( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
= ( ( ord_less_eq_num @ L @ I )
& ( ord_less_eq_num @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_8811_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_8812_atLeastAtMost__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_eq_int @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_8813_atLeastAtMost__iff,axiom,
! [I: real,L: real,U: real] :
( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_eq_real @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_8814_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_8815_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_8816_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_8817_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_8818_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_8819_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_8820_Icc__subset__Iic__iff,axiom,
! [L: set_nat,H2: set_nat,H3: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L @ H2 ) @ ( set_or4236626031148496127et_nat @ H3 ) )
= ( ~ ( ord_less_eq_set_nat @ L @ H2 )
| ( ord_less_eq_set_nat @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8821_Icc__subset__Iic__iff,axiom,
! [L: rat,H2: rat,H3: rat] :
( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H2 ) @ ( set_ord_atMost_rat @ H3 ) )
= ( ~ ( ord_less_eq_rat @ L @ H2 )
| ( ord_less_eq_rat @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8822_Icc__subset__Iic__iff,axiom,
! [L: num,H2: num,H3: num] :
( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H2 ) @ ( set_ord_atMost_num @ H3 ) )
= ( ~ ( ord_less_eq_num @ L @ H2 )
| ( ord_less_eq_num @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8823_Icc__subset__Iic__iff,axiom,
! [L: nat,H2: nat,H3: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
= ( ~ ( ord_less_eq_nat @ L @ H2 )
| ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8824_Icc__subset__Iic__iff,axiom,
! [L: int,H2: int,H3: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H2 ) @ ( set_ord_atMost_int @ H3 ) )
= ( ~ ( ord_less_eq_int @ L @ H2 )
| ( ord_less_eq_int @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8825_Icc__subset__Iic__iff,axiom,
! [L: real,H2: real,H3: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H2 ) @ ( set_ord_atMost_real @ H3 ) )
= ( ~ ( ord_less_eq_real @ L @ H2 )
| ( ord_less_eq_real @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8826_sum_Ocl__ivl__Suc,axiom,
! [N2: nat,M: nat,G: nat > complex] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= zero_zero_complex ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_8827_sum_Ocl__ivl__Suc,axiom,
! [N2: nat,M: nat,G: nat > rat] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= zero_zero_rat ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_8828_sum_Ocl__ivl__Suc,axiom,
! [N2: nat,M: nat,G: nat > int] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= zero_zero_int ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_8829_sum_Ocl__ivl__Suc,axiom,
! [N2: nat,M: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_8830_sum_Ocl__ivl__Suc,axiom,
! [N2: nat,M: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_8831_all__nat__less,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( P @ M2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less
thf(fact_8832_ex__nat__less,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( P @ M2 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less
thf(fact_8833_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_8834_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > nat,M: nat,N2: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_8835_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > real,M: nat,N2: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_8836_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > nat,M: nat,K: nat,N2: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_8837_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > real,M: nat,K: nat,N2: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_8838_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_8839_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_8840_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_8841_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_8842_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_8843_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_8844_sum_OatLeastAtMost__rev,axiom,
! [G: nat > nat,N2: nat,M: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_8845_sum_OatLeastAtMost__rev,axiom,
! [G: nat > real,N2: nat,M: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_8846_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_8847_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_8848_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_8849_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_8850_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_8851_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > rat,N2: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_8852_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > int,N2: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_8853_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > nat,N2: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_8854_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > real,N2: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_8855_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N2: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_8856_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N2: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_8857_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N2: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_8858_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N2: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_8859_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N2: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( plus_plus_rat @ ( G @ ( suc @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_8860_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N2: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( plus_plus_int @ ( G @ ( suc @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_8861_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N2: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( plus_plus_nat @ ( G @ ( suc @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_8862_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N2: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( plus_plus_real @ ( G @ ( suc @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_8863_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N2: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
= ( plus_plus_rat @ ( G @ M )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_8864_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N2: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
= ( plus_plus_int @ ( G @ M )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_8865_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N2: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
= ( plus_plus_nat @ ( G @ M )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_8866_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N2: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
= ( plus_plus_real @ ( G @ M )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_8867_sum__Suc__diff,axiom,
! [M: nat,N2: nat,F: nat > rat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).
% sum_Suc_diff
thf(fact_8868_sum__Suc__diff,axiom,
! [M: nat,N2: nat,F: nat > int] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).
% sum_Suc_diff
thf(fact_8869_sum__Suc__diff,axiom,
! [M: nat,N2: nat,F: nat > real] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( minus_minus_real @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).
% sum_Suc_diff
thf(fact_8870_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > nat,N2: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
= ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_8871_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > real,N2: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
= ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_8872_sum__bounds__lt__plus1,axiom,
! [F: nat > nat,Mm: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
@ ( set_ord_lessThan_nat @ Mm ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_8873_sum__bounds__lt__plus1,axiom,
! [F: nat > real,Mm: nat] :
( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
@ ( set_ord_lessThan_nat @ Mm ) )
= ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_8874_sum_Onested__swap_H,axiom,
! [A: nat > nat > nat,N2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% sum.nested_swap'
thf(fact_8875_sum_Onested__swap_H,axiom,
! [A: nat > nat > real,N2: nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( groups6591440286371151544t_real @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% sum.nested_swap'
thf(fact_8876_sum__atLeastAtMost__code,axiom,
! [F: nat > complex,A: nat,B: nat] :
( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1517530859248394432omplex
@ ^ [A3: nat] : ( plus_plus_complex @ ( F @ A3 ) )
@ A
@ B
@ zero_zero_complex ) ) ).
% sum_atLeastAtMost_code
thf(fact_8877_sum__atLeastAtMost__code,axiom,
! [F: nat > rat,A: nat,B: nat] :
( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1949268297981939178at_rat
@ ^ [A3: nat] : ( plus_plus_rat @ ( F @ A3 ) )
@ A
@ B
@ zero_zero_rat ) ) ).
% sum_atLeastAtMost_code
thf(fact_8878_sum__atLeastAtMost__code,axiom,
! [F: nat > int,A: nat,B: nat] :
( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2581907887559384638at_int
@ ^ [A3: nat] : ( plus_plus_int @ ( F @ A3 ) )
@ A
@ B
@ zero_zero_int ) ) ).
% sum_atLeastAtMost_code
thf(fact_8879_sum__atLeastAtMost__code,axiom,
! [F: nat > nat,A: nat,B: nat] :
( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2584398358068434914at_nat
@ ^ [A3: nat] : ( plus_plus_nat @ ( F @ A3 ) )
@ A
@ B
@ zero_zero_nat ) ) ).
% sum_atLeastAtMost_code
thf(fact_8880_sum__atLeastAtMost__code,axiom,
! [F: nat > real,A: nat,B: nat] :
( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo3111899725591712190t_real
@ ^ [A3: nat] : ( plus_plus_real @ ( F @ A3 ) )
@ A
@ B
@ zero_zero_real ) ) ).
% sum_atLeastAtMost_code
thf(fact_8881_sum_Oub__add__nat,axiom,
! [M: nat,N2: nat,G: nat > rat,P3: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P3 ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_8882_sum_Oub__add__nat,axiom,
! [M: nat,N2: nat,G: nat > int,P3: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P3 ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_8883_sum_Oub__add__nat,axiom,
! [M: nat,N2: nat,G: nat > nat,P3: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P3 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_8884_sum_Oub__add__nat,axiom,
! [M: nat,N2: nat,G: nat > real,P3: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P3 ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_8885_sum__up__index__split,axiom,
! [F: nat > rat,M: nat,N2: nat] :
( ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).
% sum_up_index_split
thf(fact_8886_sum__up__index__split,axiom,
! [F: nat > int,M: nat,N2: nat] :
( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).
% sum_up_index_split
thf(fact_8887_sum__up__index__split,axiom,
! [F: nat > nat,M: nat,N2: nat] :
( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).
% sum_up_index_split
thf(fact_8888_sum__up__index__split,axiom,
! [F: nat > real,M: nat,N2: nat] :
( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).
% sum_up_index_split
thf(fact_8889_sum__natinterval__diff,axiom,
! [M: nat,N2: nat,F: nat > complex] :
( ( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= zero_zero_complex ) ) ) ).
% sum_natinterval_diff
thf(fact_8890_sum__natinterval__diff,axiom,
! [M: nat,N2: nat,F: nat > rat] :
( ( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= zero_zero_rat ) ) ) ).
% sum_natinterval_diff
thf(fact_8891_sum__natinterval__diff,axiom,
! [M: nat,N2: nat,F: nat > int] :
( ( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups3539618377306564664at_int
@ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups3539618377306564664at_int
@ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= zero_zero_int ) ) ) ).
% sum_natinterval_diff
thf(fact_8892_sum__natinterval__diff,axiom,
! [M: nat,N2: nat,F: nat > real] :
( ( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= zero_zero_real ) ) ) ).
% sum_natinterval_diff
thf(fact_8893_sum__telescope_H_H,axiom,
! [M: nat,N2: nat,F: nat > rat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
= ( minus_minus_rat @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).
% sum_telescope''
thf(fact_8894_sum__telescope_H_H,axiom,
! [M: nat,N2: nat,F: nat > int] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups3539618377306564664at_int
@ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
= ( minus_minus_int @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).
% sum_telescope''
thf(fact_8895_sum__telescope_H_H,axiom,
! [M: nat,N2: nat,F: nat > real] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
= ( minus_minus_real @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).
% sum_telescope''
thf(fact_8896_sum__power__shift,axiom,
! [M: nat,N2: nat,X: complex] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_8897_sum__power__shift,axiom,
! [M: nat,N2: nat,X: rat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_8898_sum__power__shift,axiom,
! [M: nat,N2: nat,X: int] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_8899_sum__power__shift,axiom,
! [M: nat,N2: nat,X: real] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_8900_summable__partial__sum__bound,axiom,
! [F: nat > complex,E2: real] :
( ( summable_complex @ F )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ~ ! [N7: nat] :
~ ! [M3: nat] :
( ( ord_less_eq_nat @ N7 @ M3 )
=> ! [N8: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M3 @ N8 ) ) ) @ E2 ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_8901_summable__partial__sum__bound,axiom,
! [F: nat > real,E2: real] :
( ( summable_real @ F )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ~ ! [N7: nat] :
~ ! [M3: nat] :
( ( ord_less_eq_nat @ N7 @ M3 )
=> ! [N8: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M3 @ N8 ) ) ) @ E2 ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_8902_sum__gp__multiplied,axiom,
! [M: nat,N2: nat,X: complex] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
= ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_8903_sum__gp__multiplied,axiom,
! [M: nat,N2: nat,X: rat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
= ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_8904_sum__gp__multiplied,axiom,
! [M: nat,N2: nat,X: int] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
= ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N2 ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_8905_sum__gp__multiplied,axiom,
! [M: nat,N2: nat,X: real] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
= ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_8906_sum_Oin__pairs,axiom,
! [G: nat > rat,M: nat,N2: 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 ) ) @ N2 ) ) ) )
= ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% sum.in_pairs
thf(fact_8907_sum_Oin__pairs,axiom,
! [G: nat > int,M: nat,N2: 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 ) ) @ N2 ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% sum.in_pairs
thf(fact_8908_sum_Oin__pairs,axiom,
! [G: nat > nat,M: nat,N2: 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 ) ) @ N2 ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% sum.in_pairs
thf(fact_8909_sum_Oin__pairs,axiom,
! [G: nat > real,M: nat,N2: 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 ) ) @ N2 ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% sum.in_pairs
thf(fact_8910_polyfun__eq__const,axiom,
! [C: nat > complex,N2: nat,K: complex] :
( ( ! [X3: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= K ) )
= ( ( ( C @ zero_zero_nat )
= K )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) )
=> ( ( C @ X3 )
= zero_zero_complex ) ) ) ) ).
% polyfun_eq_const
thf(fact_8911_polyfun__eq__const,axiom,
! [C: nat > real,N2: nat,K: real] :
( ( ! [X3: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= K ) )
= ( ( ( C @ zero_zero_nat )
= K )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) )
=> ( ( C @ X3 )
= zero_zero_real ) ) ) ) ).
% polyfun_eq_const
thf(fact_8912_gbinomial__sum__up__index,axiom,
! [K: nat,N2: nat] :
( ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
= ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% gbinomial_sum_up_index
thf(fact_8913_gbinomial__sum__up__index,axiom,
! [K: nat,N2: nat] :
( ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
= ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% gbinomial_sum_up_index
thf(fact_8914_gbinomial__sum__up__index,axiom,
! [K: nat,N2: nat] :
( ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
= ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% gbinomial_sum_up_index
thf(fact_8915_gauss__sum__nat,axiom,
! [N2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
= ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_nat
thf(fact_8916_double__arith__series,axiom,
! [A: int,D: int,N2: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
= ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_8917_double__arith__series,axiom,
! [A: complex,D: complex,N2: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
= ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_8918_double__arith__series,axiom,
! [A: rat,D: rat,N2: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
= ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_8919_double__arith__series,axiom,
! [A: nat,D: nat,N2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
= ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_8920_double__arith__series,axiom,
! [A: real,D: real,N2: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
= ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_8921_double__gauss__sum,axiom,
! [N2: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).
% double_gauss_sum
thf(fact_8922_double__gauss__sum,axiom,
! [N2: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).
% double_gauss_sum
thf(fact_8923_double__gauss__sum,axiom,
! [N2: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).
% double_gauss_sum
thf(fact_8924_double__gauss__sum,axiom,
! [N2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).
% double_gauss_sum
thf(fact_8925_double__gauss__sum,axiom,
! [N2: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).
% double_gauss_sum
thf(fact_8926_arith__series__nat,axiom,
! [A: nat,D: nat,N2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I3 @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
= ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N2 @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% arith_series_nat
thf(fact_8927_Sum__Icc__nat,axiom,
! [M: nat,N2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Sum_Icc_nat
thf(fact_8928_gauss__sum,axiom,
! [N2: nat] :
( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
= ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_8929_gauss__sum,axiom,
! [N2: nat] :
( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
= ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_8930_arith__series,axiom,
! [A: int,D: int,N2: nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
= ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_8931_arith__series,axiom,
! [A: nat,D: nat,N2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
= ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_8932_double__gauss__sum__from__Suc__0,axiom,
! [N2: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_8933_double__gauss__sum__from__Suc__0,axiom,
! [N2: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_8934_double__gauss__sum__from__Suc__0,axiom,
! [N2: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_8935_double__gauss__sum__from__Suc__0,axiom,
! [N2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_8936_double__gauss__sum__from__Suc__0,axiom,
! [N2: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_8937_sum__gp__offset,axiom,
! [X: complex,M: nat,N2: nat] :
( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
= ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% sum_gp_offset
thf(fact_8938_sum__gp__offset,axiom,
! [X: rat,M: nat,N2: nat] :
( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
= ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
= ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% sum_gp_offset
thf(fact_8939_sum__gp__offset,axiom,
! [X: real,M: nat,N2: nat] :
( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
= ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% sum_gp_offset
thf(fact_8940_polyfun__diff,axiom,
! [N2: nat,A: nat > complex,X: complex,Y: complex] :
( ( ord_less_eq_nat @ one_one_nat @ N2 )
=> ( ( minus_minus_complex
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y )
@ ( groups2073611262835488442omplex
@ ^ [J3: nat] :
( times_times_complex
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
@ ( power_power_complex @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% polyfun_diff
thf(fact_8941_polyfun__diff,axiom,
! [N2: nat,A: nat > rat,X: rat,Y: rat] :
( ( ord_less_eq_nat @ one_one_nat @ N2 )
=> ( ( minus_minus_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y )
@ ( groups2906978787729119204at_rat
@ ^ [J3: nat] :
( times_times_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
@ ( power_power_rat @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% polyfun_diff
thf(fact_8942_polyfun__diff,axiom,
! [N2: nat,A: nat > int,X: int,Y: int] :
( ( ord_less_eq_nat @ one_one_nat @ N2 )
=> ( ( minus_minus_int
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y )
@ ( groups3539618377306564664at_int
@ ^ [J3: nat] :
( times_times_int
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
@ ( power_power_int @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% polyfun_diff
thf(fact_8943_polyfun__diff,axiom,
! [N2: nat,A: nat > real,X: real,Y: real] :
( ( ord_less_eq_nat @ one_one_nat @ N2 )
=> ( ( minus_minus_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y )
@ ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
@ ( power_power_real @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% polyfun_diff
thf(fact_8944_pochhammer__times__pochhammer__half,axiom,
! [Z: real,N2: nat] :
( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
= ( groups129246275422532515t_real
@ ^ [K2: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_8945_pochhammer__times__pochhammer__half,axiom,
! [Z: complex,N2: nat] :
( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
= ( groups6464643781859351333omplex
@ ^ [K2: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_8946_pochhammer__times__pochhammer__half,axiom,
! [Z: rat,N2: nat] :
( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
= ( groups73079841787564623at_rat
@ ^ [K2: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K2 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_8947_vebt__buildup_Opelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
=> ( ( ( X = zero_zero_nat )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
=> ( ( ( X
= ( suc @ zero_zero_nat ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( 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 ) ) )
=> ( Y
= ( 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 ) ) ) ) ) ) ) ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.pelims
thf(fact_8948_divmod__algorithm__code_I6_J,axiom,
! [M: num,N2: num] :
( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( 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 @ N2 ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_8949_divmod__algorithm__code_I6_J,axiom,
! [M: num,N2: num] :
( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( 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 @ N2 ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_8950_divmod__algorithm__code_I6_J,axiom,
! [M: num,N2: num] :
( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( produc6916734918728496179nteger
@ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
@ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_8951_arctan__def,axiom,
( arctan
= ( ^ [Y4: real] :
( the_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 ) ) ) ) ) ).
% arctan_def
thf(fact_8952_Ints__prod,axiom,
! [A2: set_complex,F: complex > complex] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( member_complex @ ( F @ X2 ) @ ring_1_Ints_complex ) )
=> ( member_complex @ ( groups3708469109370488835omplex @ F @ A2 ) @ ring_1_Ints_complex ) ) ).
% Ints_prod
thf(fact_8953_Ints__prod,axiom,
! [A2: set_complex,F: complex > int] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( member_int @ ( F @ X2 ) @ ring_1_Ints_int ) )
=> ( member_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ring_1_Ints_int ) ) ).
% Ints_prod
thf(fact_8954_Ints__prod,axiom,
! [A2: set_real,F: real > complex] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( member_complex @ ( F @ X2 ) @ ring_1_Ints_complex ) )
=> ( member_complex @ ( groups713298508707869441omplex @ F @ A2 ) @ ring_1_Ints_complex ) ) ).
% Ints_prod
thf(fact_8955_Ints__prod,axiom,
! [A2: set_real,F: real > int] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( member_int @ ( F @ X2 ) @ ring_1_Ints_int ) )
=> ( member_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ring_1_Ints_int ) ) ).
% Ints_prod
thf(fact_8956_Ints__prod,axiom,
! [A2: set_nat,F: nat > complex] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_complex @ ( F @ X2 ) @ ring_1_Ints_complex ) )
=> ( member_complex @ ( groups6464643781859351333omplex @ F @ A2 ) @ ring_1_Ints_complex ) ) ).
% Ints_prod
thf(fact_8957_Ints__prod,axiom,
! [A2: set_int,F: int > complex] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( member_complex @ ( F @ X2 ) @ ring_1_Ints_complex ) )
=> ( member_complex @ ( groups7440179247065528705omplex @ F @ A2 ) @ ring_1_Ints_complex ) ) ).
% Ints_prod
thf(fact_8958_Ints__prod,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( member_real @ ( F @ X2 ) @ ring_1_Ints_real ) )
=> ( member_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ring_1_Ints_real ) ) ).
% Ints_prod
thf(fact_8959_Ints__prod,axiom,
! [A2: set_real,F: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( member_real @ ( F @ X2 ) @ ring_1_Ints_real ) )
=> ( member_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ring_1_Ints_real ) ) ).
% Ints_prod
thf(fact_8960_Ints__prod,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_real @ ( F @ X2 ) @ ring_1_Ints_real ) )
=> ( member_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ring_1_Ints_real ) ) ).
% Ints_prod
thf(fact_8961_Ints__prod,axiom,
! [A2: set_int,F: int > real] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( member_real @ ( F @ X2 ) @ ring_1_Ints_real ) )
=> ( member_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ring_1_Ints_real ) ) ).
% Ints_prod
thf(fact_8962_prod_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups708209901874060359at_nat
@ ^ [Uu3: nat] : one_one_nat
@ A2 )
= one_one_nat ) ).
% prod.neutral_const
thf(fact_8963_prod_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups705719431365010083at_int
@ ^ [Uu3: nat] : one_one_int
@ A2 )
= one_one_int ) ).
% prod.neutral_const
thf(fact_8964_prod_Oneutral__const,axiom,
! [A2: set_int] :
( ( groups1705073143266064639nt_int
@ ^ [Uu3: int] : one_one_int
@ A2 )
= one_one_int ) ).
% prod.neutral_const
thf(fact_8965_of__nat__prod,axiom,
! [F: int > nat,A2: set_int] :
( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
= ( groups1705073143266064639nt_int
@ ^ [X3: int] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A2 ) ) ).
% of_nat_prod
thf(fact_8966_of__nat__prod,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ F @ A2 ) )
= ( groups129246275422532515t_real
@ ^ [X3: nat] : ( semiri5074537144036343181t_real @ ( F @ X3 ) )
@ A2 ) ) ).
% of_nat_prod
thf(fact_8967_of__nat__prod,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri8010041392384452111omplex @ ( groups708209901874060359at_nat @ F @ A2 ) )
= ( groups6464643781859351333omplex
@ ^ [X3: nat] : ( semiri8010041392384452111omplex @ ( F @ X3 ) )
@ A2 ) ) ).
% of_nat_prod
thf(fact_8968_of__nat__prod,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri681578069525770553at_rat @ ( groups708209901874060359at_nat @ F @ A2 ) )
= ( groups73079841787564623at_rat
@ ^ [X3: nat] : ( semiri681578069525770553at_rat @ ( F @ X3 ) )
@ A2 ) ) ).
% of_nat_prod
thf(fact_8969_of__nat__prod,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ F @ A2 ) )
= ( groups708209901874060359at_nat
@ ^ [X3: nat] : ( semiri1316708129612266289at_nat @ ( F @ X3 ) )
@ A2 ) ) ).
% of_nat_prod
thf(fact_8970_of__nat__prod,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A2 ) )
= ( groups705719431365010083at_int
@ ^ [X3: nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A2 ) ) ).
% of_nat_prod
thf(fact_8971_of__int__prod,axiom,
! [F: nat > int,A2: set_nat] :
( ( ring_1_of_int_real @ ( groups705719431365010083at_int @ F @ A2 ) )
= ( groups129246275422532515t_real
@ ^ [X3: nat] : ( ring_1_of_int_real @ ( F @ X3 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_8972_of__int__prod,axiom,
! [F: nat > int,A2: set_nat] :
( ( ring_1_of_int_rat @ ( groups705719431365010083at_int @ F @ A2 ) )
= ( groups73079841787564623at_rat
@ ^ [X3: nat] : ( ring_1_of_int_rat @ ( F @ X3 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_8973_of__int__prod,axiom,
! [F: nat > int,A2: set_nat] :
( ( ring_1_of_int_int @ ( groups705719431365010083at_int @ F @ A2 ) )
= ( groups705719431365010083at_int
@ ^ [X3: nat] : ( ring_1_of_int_int @ ( F @ X3 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_8974_of__int__prod,axiom,
! [F: int > int,A2: set_int] :
( ( ring_1_of_int_real @ ( groups1705073143266064639nt_int @ F @ A2 ) )
= ( groups2316167850115554303t_real
@ ^ [X3: int] : ( ring_1_of_int_real @ ( F @ X3 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_8975_of__int__prod,axiom,
! [F: int > int,A2: set_int] :
( ( ring_1_of_int_rat @ ( groups1705073143266064639nt_int @ F @ A2 ) )
= ( groups1072433553688619179nt_rat
@ ^ [X3: int] : ( ring_1_of_int_rat @ ( F @ X3 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_8976_of__int__prod,axiom,
! [F: int > int,A2: set_int] :
( ( ring_1_of_int_int @ ( groups1705073143266064639nt_int @ F @ A2 ) )
= ( groups1705073143266064639nt_int
@ ^ [X3: int] : ( ring_1_of_int_int @ ( F @ X3 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_8977_prod_OlessThan__Suc,axiom,
! [G: nat > real,N2: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% prod.lessThan_Suc
thf(fact_8978_prod_OlessThan__Suc,axiom,
! [G: nat > rat,N2: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% prod.lessThan_Suc
thf(fact_8979_prod_OlessThan__Suc,axiom,
! [G: nat > nat,N2: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% prod.lessThan_Suc
thf(fact_8980_prod_OlessThan__Suc,axiom,
! [G: nat > int,N2: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% prod.lessThan_Suc
thf(fact_8981_prod_OatMost__Suc,axiom,
! [G: nat > real,N2: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% prod.atMost_Suc
thf(fact_8982_prod_OatMost__Suc,axiom,
! [G: nat > rat,N2: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% prod.atMost_Suc
thf(fact_8983_prod_OatMost__Suc,axiom,
! [G: nat > nat,N2: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% prod.atMost_Suc
thf(fact_8984_prod_OatMost__Suc,axiom,
! [G: nat > int,N2: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% prod.atMost_Suc
thf(fact_8985_prod_Ocl__ivl__Suc,axiom,
! [N2: nat,M: nat,G: nat > complex] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= one_one_complex ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_8986_prod_Ocl__ivl__Suc,axiom,
! [N2: nat,M: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= one_one_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_8987_prod_Ocl__ivl__Suc,axiom,
! [N2: nat,M: nat,G: nat > rat] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= one_one_rat ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_8988_prod_Ocl__ivl__Suc,axiom,
! [N2: nat,M: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= one_one_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_8989_prod_Ocl__ivl__Suc,axiom,
! [N2: nat,M: nat,G: nat > int] :
( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= one_one_int ) )
& ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_8990_divmod__algorithm__code_I5_J,axiom,
! [M: num,N2: num] :
( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( produc4245557441103728435nt_int
@ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
@ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_8991_divmod__algorithm__code_I5_J,axiom,
! [M: num,N2: num] :
( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( produc2626176000494625587at_nat
@ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
@ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_8992_divmod__algorithm__code_I5_J,axiom,
! [M: num,N2: num] :
( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( produc6916734918728496179nteger
@ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
@ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_8993_prod__power__distrib,axiom,
! [F: nat > nat,A2: set_nat,N2: nat] :
( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ N2 )
= ( groups708209901874060359at_nat
@ ^ [X3: nat] : ( power_power_nat @ ( F @ X3 ) @ N2 )
@ A2 ) ) ).
% prod_power_distrib
thf(fact_8994_prod__power__distrib,axiom,
! [F: nat > int,A2: set_nat,N2: nat] :
( ( power_power_int @ ( groups705719431365010083at_int @ F @ A2 ) @ N2 )
= ( groups705719431365010083at_int
@ ^ [X3: nat] : ( power_power_int @ ( F @ X3 ) @ N2 )
@ A2 ) ) ).
% prod_power_distrib
thf(fact_8995_prod__power__distrib,axiom,
! [F: int > int,A2: set_int,N2: nat] :
( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ N2 )
= ( groups1705073143266064639nt_int
@ ^ [X3: int] : ( power_power_int @ ( F @ X3 ) @ N2 )
@ A2 ) ) ).
% prod_power_distrib
thf(fact_8996_prod_Oneutral,axiom,
! [A2: set_nat,G: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( G @ X2 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ A2 )
= one_one_nat ) ) ).
% prod.neutral
thf(fact_8997_prod_Oneutral,axiom,
! [A2: set_nat,G: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( G @ X2 )
= one_one_int ) )
=> ( ( groups705719431365010083at_int @ G @ A2 )
= one_one_int ) ) ).
% prod.neutral
thf(fact_8998_prod_Oneutral,axiom,
! [A2: set_int,G: int > int] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( G @ X2 )
= one_one_int ) )
=> ( ( groups1705073143266064639nt_int @ G @ A2 )
= one_one_int ) ) ).
% prod.neutral
thf(fact_8999_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > complex,A2: set_complex] :
( ( ( groups3708469109370488835omplex @ G @ A2 )
!= one_one_complex )
=> ~ ! [A5: complex] :
( ( member_complex @ A5 @ A2 )
=> ( ( G @ A5 )
= one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_9000_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: real > complex,A2: set_real] :
( ( ( groups713298508707869441omplex @ G @ A2 )
!= one_one_complex )
=> ~ ! [A5: real] :
( ( member_real @ A5 @ A2 )
=> ( ( G @ A5 )
= one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_9001_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > complex,A2: set_nat] :
( ( ( groups6464643781859351333omplex @ G @ A2 )
!= one_one_complex )
=> ~ ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ( ( G @ A5 )
= one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_9002_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: int > complex,A2: set_int] :
( ( ( groups7440179247065528705omplex @ G @ A2 )
!= one_one_complex )
=> ~ ! [A5: int] :
( ( member_int @ A5 @ A2 )
=> ( ( G @ A5 )
= one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_9003_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > real,A2: set_complex] :
( ( ( groups766887009212190081x_real @ G @ A2 )
!= one_one_real )
=> ~ ! [A5: complex] :
( ( member_complex @ A5 @ A2 )
=> ( ( G @ A5 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_9004_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A2: set_real] :
( ( ( groups1681761925125756287l_real @ G @ A2 )
!= one_one_real )
=> ~ ! [A5: real] :
( ( member_real @ A5 @ A2 )
=> ( ( G @ A5 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_9005_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A2: set_nat] :
( ( ( groups129246275422532515t_real @ G @ A2 )
!= one_one_real )
=> ~ ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ( ( G @ A5 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_9006_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: int > real,A2: set_int] :
( ( ( groups2316167850115554303t_real @ G @ A2 )
!= one_one_real )
=> ~ ! [A5: int] :
( ( member_int @ A5 @ A2 )
=> ( ( G @ A5 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_9007_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > rat,A2: set_complex] :
( ( ( groups225925009352817453ex_rat @ G @ A2 )
!= one_one_rat )
=> ~ ! [A5: complex] :
( ( member_complex @ A5 @ A2 )
=> ( ( G @ A5 )
= one_one_rat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_9008_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: real > rat,A2: set_real] :
( ( ( groups4061424788464935467al_rat @ G @ A2 )
!= one_one_rat )
=> ~ ! [A5: real] :
( ( member_real @ A5 @ A2 )
=> ( ( G @ A5 )
= one_one_rat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_9009_mod__prod__eq,axiom,
! [F: nat > nat,A: nat,A2: set_nat] :
( ( modulo_modulo_nat
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ A ) ) ).
% mod_prod_eq
thf(fact_9010_mod__prod__eq,axiom,
! [F: nat > int,A: int,A2: set_nat] :
( ( modulo_modulo_int
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A2 ) @ A ) ) ).
% mod_prod_eq
thf(fact_9011_mod__prod__eq,axiom,
! [F: int > int,A: int,A2: set_int] :
( ( modulo_modulo_int
@ ( groups1705073143266064639nt_int
@ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ A ) ) ).
% mod_prod_eq
thf(fact_9012_prod__mono,axiom,
! [A2: set_complex,F: complex > rat,G: complex > rat] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_9013_prod__mono,axiom,
! [A2: set_real,F: real > rat,G: real > rat] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_9014_prod__mono,axiom,
! [A2: set_nat,F: nat > rat,G: nat > rat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_9015_prod__mono,axiom,
! [A2: set_int,F: int > rat,G: int > rat] :
( ! [I2: int] :
( ( member_int @ I2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_9016_prod__mono,axiom,
! [A2: set_complex,F: complex > nat,G: complex > nat] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
& ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_9017_prod__mono,axiom,
! [A2: set_real,F: real > nat,G: real > nat] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
& ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_9018_prod__mono,axiom,
! [A2: set_int,F: int > nat,G: int > nat] :
( ! [I2: int] :
( ( member_int @ I2 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
& ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_9019_prod__mono,axiom,
! [A2: set_complex,F: complex > int,G: complex > int] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I2 ) )
& ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_9020_prod__mono,axiom,
! [A2: set_real,F: real > int,G: real > int] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I2 ) )
& ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_9021_prod__mono,axiom,
! [A2: set_complex,F: complex > real,G: complex > real] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_9022_prod__nonneg,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% prod_nonneg
thf(fact_9023_prod__nonneg,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% prod_nonneg
thf(fact_9024_prod__nonneg,axiom,
! [A2: set_int,F: int > int] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).
% prod_nonneg
thf(fact_9025_prod__pos,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% prod_pos
thf(fact_9026_prod__pos,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% prod_pos
thf(fact_9027_prod__pos,axiom,
! [A2: set_int,F: int > int] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).
% prod_pos
thf(fact_9028_prod__ge__1,axiom,
! [A2: set_complex,F: complex > rat] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X2 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_9029_prod__ge__1,axiom,
! [A2: set_real,F: real > rat] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X2 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_9030_prod__ge__1,axiom,
! [A2: set_nat,F: nat > rat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X2 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_9031_prod__ge__1,axiom,
! [A2: set_int,F: int > rat] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X2 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_9032_prod__ge__1,axiom,
! [A2: set_complex,F: complex > nat] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_9033_prod__ge__1,axiom,
! [A2: set_real,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_9034_prod__ge__1,axiom,
! [A2: set_int,F: int > nat] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_9035_prod__ge__1,axiom,
! [A2: set_complex,F: complex > int] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_eq_int @ one_one_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_9036_prod__ge__1,axiom,
! [A2: set_real,F: real > int] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_int @ one_one_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_9037_prod__ge__1,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_9038_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > nat,M: nat,N2: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_9039_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > int,M: nat,N2: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_9040_power__sum,axiom,
! [C: real,F: nat > nat,A2: set_nat] :
( ( power_power_real @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups129246275422532515t_real
@ ^ [A3: nat] : ( power_power_real @ C @ ( F @ A3 ) )
@ A2 ) ) ).
% power_sum
thf(fact_9041_power__sum,axiom,
! [C: complex,F: nat > nat,A2: set_nat] :
( ( power_power_complex @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups6464643781859351333omplex
@ ^ [A3: nat] : ( power_power_complex @ C @ ( F @ A3 ) )
@ A2 ) ) ).
% power_sum
thf(fact_9042_power__sum,axiom,
! [C: rat,F: nat > nat,A2: set_nat] :
( ( power_power_rat @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups73079841787564623at_rat
@ ^ [A3: nat] : ( power_power_rat @ C @ ( F @ A3 ) )
@ A2 ) ) ).
% power_sum
thf(fact_9043_power__sum,axiom,
! [C: nat,F: nat > nat,A2: set_nat] :
( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups708209901874060359at_nat
@ ^ [A3: nat] : ( power_power_nat @ C @ ( F @ A3 ) )
@ A2 ) ) ).
% power_sum
thf(fact_9044_power__sum,axiom,
! [C: int,F: nat > nat,A2: set_nat] :
( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups705719431365010083at_int
@ ^ [A3: nat] : ( power_power_int @ C @ ( F @ A3 ) )
@ A2 ) ) ).
% power_sum
thf(fact_9045_power__sum,axiom,
! [C: int,F: int > nat,A2: set_int] :
( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
= ( groups1705073143266064639nt_int
@ ^ [A3: int] : ( power_power_int @ C @ ( F @ A3 ) )
@ A2 ) ) ).
% power_sum
thf(fact_9046_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > nat,M: nat,K: nat,N2: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_9047_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > int,M: nat,K: nat,N2: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_9048_prod__le__1,axiom,
! [A2: set_complex,F: complex > rat] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X2 ) )
& ( ord_less_eq_rat @ ( F @ X2 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_9049_prod__le__1,axiom,
! [A2: set_real,F: real > rat] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X2 ) )
& ( ord_less_eq_rat @ ( F @ X2 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_9050_prod__le__1,axiom,
! [A2: set_nat,F: nat > rat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X2 ) )
& ( ord_less_eq_rat @ ( F @ X2 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_9051_prod__le__1,axiom,
! [A2: set_int,F: int > rat] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X2 ) )
& ( ord_less_eq_rat @ ( F @ X2 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_9052_prod__le__1,axiom,
! [A2: set_complex,F: complex > nat] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) )
& ( ord_less_eq_nat @ ( F @ X2 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_9053_prod__le__1,axiom,
! [A2: set_real,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) )
& ( ord_less_eq_nat @ ( F @ X2 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_9054_prod__le__1,axiom,
! [A2: set_int,F: int > nat] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) )
& ( ord_less_eq_nat @ ( F @ X2 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_9055_prod__le__1,axiom,
! [A2: set_complex,F: complex > int] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) )
& ( ord_less_eq_int @ ( F @ X2 ) @ one_one_int ) ) )
=> ( ord_less_eq_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ one_one_int ) ) ).
% prod_le_1
thf(fact_9056_prod__le__1,axiom,
! [A2: set_real,F: real > int] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) )
& ( ord_less_eq_int @ ( F @ X2 ) @ one_one_int ) ) )
=> ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ one_one_int ) ) ).
% prod_le_1
thf(fact_9057_prod__le__1,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) )
& ( ord_less_eq_real @ ( F @ X2 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_9058_bset_I1_J,axiom,
! [D5: int,B3: set_int,P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B3 )
=> ( X2
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D5 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B3 )
=> ( X2
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X2 )
=> ( Q @ ( minus_minus_int @ X2 @ D5 ) ) ) )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B3 )
=> ( X4
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
=> ( ( P @ ( minus_minus_int @ X4 @ D5 ) )
& ( Q @ ( minus_minus_int @ X4 @ D5 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_9059_bset_I2_J,axiom,
! [D5: int,B3: set_int,P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B3 )
=> ( X2
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D5 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B3 )
=> ( X2
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X2 )
=> ( Q @ ( minus_minus_int @ X2 @ D5 ) ) ) )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B3 )
=> ( X4
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
=> ( ( P @ ( minus_minus_int @ X4 @ D5 ) )
| ( Q @ ( minus_minus_int @ X4 @ D5 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_9060_aset_I1_J,axiom,
! [D5: int,A2: set_int,P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X2
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D5 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X2
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X2 )
=> ( Q @ ( plus_plus_int @ X2 @ D5 ) ) ) )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X4
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
=> ( ( P @ ( plus_plus_int @ X4 @ D5 ) )
& ( Q @ ( plus_plus_int @ X4 @ D5 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_9061_aset_I2_J,axiom,
! [D5: int,A2: set_int,P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X2
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D5 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X2
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X2 )
=> ( Q @ ( plus_plus_int @ X2 @ D5 ) ) ) )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X4
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
=> ( ( P @ ( plus_plus_int @ X4 @ D5 ) )
| ( Q @ ( plus_plus_int @ X4 @ D5 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_9062_prod_Onat__diff__reindex,axiom,
! [G: nat > nat,N2: nat] :
( ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N2 ) )
= ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% prod.nat_diff_reindex
thf(fact_9063_prod_Onat__diff__reindex,axiom,
! [G: nat > int,N2: nat] :
( ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N2 ) )
= ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% prod.nat_diff_reindex
thf(fact_9064_prod_OatLeastAtMost__rev,axiom,
! [G: nat > nat,N2: nat,M: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_9065_prod_OatLeastAtMost__rev,axiom,
! [G: nat > int,N2: nat,M: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_9066_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > real,N2: nat] :
( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_9067_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > rat,N2: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_9068_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > nat,N2: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_9069_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > int,N2: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_9070_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N2: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_9071_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N2: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_9072_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N2: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_9073_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N2: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
= ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_9074_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N2: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( times_times_real @ ( G @ ( suc @ N2 ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_9075_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N2: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( times_times_rat @ ( G @ ( suc @ N2 ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_9076_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N2: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( times_times_nat @ ( G @ ( suc @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_9077_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N2: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
= ( times_times_int @ ( G @ ( suc @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_9078_bset_I9_J,axiom,
! [D: int,D5: int,B3: set_int,T2: int] :
( ( dvd_dvd_int @ D @ D5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B3 )
=> ( X4
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T2 ) )
=> ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X4 @ D5 ) @ T2 ) ) ) ) ) ).
% bset(9)
thf(fact_9079_bset_I10_J,axiom,
! [D: int,D5: int,B3: set_int,T2: int] :
( ( dvd_dvd_int @ D @ D5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B3 )
=> ( X4
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T2 ) )
=> ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X4 @ D5 ) @ T2 ) ) ) ) ) ).
% bset(10)
thf(fact_9080_aset_I9_J,axiom,
! [D: int,D5: int,A2: set_int,T2: int] :
( ( dvd_dvd_int @ D @ D5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X4
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T2 ) )
=> ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X4 @ D5 ) @ T2 ) ) ) ) ) ).
% aset(9)
thf(fact_9081_aset_I10_J,axiom,
! [D: int,D5: int,A2: set_int,T2: int] :
( ( dvd_dvd_int @ D @ D5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X4
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T2 ) )
=> ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X4 @ D5 ) @ T2 ) ) ) ) ) ).
% aset(10)
thf(fact_9082_prod_OlessThan__Suc__shift,axiom,
! [G: nat > real,N2: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= ( times_times_real @ ( G @ zero_zero_nat )
@ ( groups129246275422532515t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_9083_prod_OlessThan__Suc__shift,axiom,
! [G: nat > rat,N2: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= ( times_times_rat @ ( G @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_9084_prod_OlessThan__Suc__shift,axiom,
! [G: nat > nat,N2: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= ( times_times_nat @ ( G @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_9085_prod_OlessThan__Suc__shift,axiom,
! [G: nat > int,N2: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
= ( times_times_int @ ( G @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_9086_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N2: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
= ( times_times_real @ ( G @ M )
@ ( groups129246275422532515t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_9087_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N2: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
= ( times_times_rat @ ( G @ M )
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_9088_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N2: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
= ( times_times_nat @ ( G @ M )
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_9089_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N2: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
= ( times_times_int @ ( G @ M )
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_9090_prod_OatMost__Suc__shift,axiom,
! [G: nat > real,N2: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( times_times_real @ ( G @ zero_zero_nat )
@ ( groups129246275422532515t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_9091_prod_OatMost__Suc__shift,axiom,
! [G: nat > rat,N2: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( times_times_rat @ ( G @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_9092_prod_OatMost__Suc__shift,axiom,
! [G: nat > nat,N2: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( times_times_nat @ ( G @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_9093_prod_OatMost__Suc__shift,axiom,
! [G: nat > int,N2: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
= ( times_times_int @ ( G @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_9094_prod_OatLeast1__atMost__eq,axiom,
! [G: nat > nat,N2: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
= ( groups708209901874060359at_nat
@ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_9095_prod_OatLeast1__atMost__eq,axiom,
! [G: nat > int,N2: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
= ( groups705719431365010083at_int
@ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_9096_fact__prod,axiom,
( semiri1406184849735516958ct_int
= ( ^ [N4: nat] :
( semiri1314217659103216013at_int
@ ( groups708209901874060359at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).
% fact_prod
thf(fact_9097_fact__prod,axiom,
( semiri5044797733671781792omplex
= ( ^ [N4: nat] :
( semiri8010041392384452111omplex
@ ( groups708209901874060359at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).
% fact_prod
thf(fact_9098_fact__prod,axiom,
( semiri773545260158071498ct_rat
= ( ^ [N4: nat] :
( semiri681578069525770553at_rat
@ ( groups708209901874060359at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).
% fact_prod
thf(fact_9099_fact__prod,axiom,
( semiri1408675320244567234ct_nat
= ( ^ [N4: nat] :
( semiri1316708129612266289at_nat
@ ( groups708209901874060359at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).
% fact_prod
thf(fact_9100_fact__prod,axiom,
( semiri2265585572941072030t_real
= ( ^ [N4: nat] :
( semiri5074537144036343181t_real
@ ( groups708209901874060359at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).
% fact_prod
thf(fact_9101_prod_Onested__swap_H,axiom,
! [A: nat > nat > nat,N2: nat] :
( ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( groups708209901874060359at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( groups708209901874060359at_nat
@ ^ [J3: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% prod.nested_swap'
thf(fact_9102_prod_Onested__swap_H,axiom,
! [A: nat > nat > int,N2: nat] :
( ( groups705719431365010083at_int
@ ^ [I3: nat] : ( groups705719431365010083at_int @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( groups705719431365010083at_int
@ ^ [J3: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% prod.nested_swap'
thf(fact_9103_prod__atLeastAtMost__code,axiom,
! [F: nat > complex,A: nat,B: nat] :
( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1517530859248394432omplex
@ ^ [A3: nat] : ( times_times_complex @ ( F @ A3 ) )
@ A
@ B
@ one_one_complex ) ) ).
% prod_atLeastAtMost_code
thf(fact_9104_prod__atLeastAtMost__code,axiom,
! [F: nat > real,A: nat,B: nat] :
( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo3111899725591712190t_real
@ ^ [A3: nat] : ( times_times_real @ ( F @ A3 ) )
@ A
@ B
@ one_one_real ) ) ).
% prod_atLeastAtMost_code
thf(fact_9105_prod__atLeastAtMost__code,axiom,
! [F: nat > rat,A: nat,B: nat] :
( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1949268297981939178at_rat
@ ^ [A3: nat] : ( times_times_rat @ ( F @ A3 ) )
@ A
@ B
@ one_one_rat ) ) ).
% prod_atLeastAtMost_code
thf(fact_9106_prod__atLeastAtMost__code,axiom,
! [F: nat > nat,A: nat,B: nat] :
( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2584398358068434914at_nat
@ ^ [A3: nat] : ( times_times_nat @ ( F @ A3 ) )
@ A
@ B
@ one_one_nat ) ) ).
% prod_atLeastAtMost_code
thf(fact_9107_prod__atLeastAtMost__code,axiom,
! [F: nat > int,A: nat,B: nat] :
( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2581907887559384638at_int
@ ^ [A3: nat] : ( times_times_int @ ( F @ A3 ) )
@ A
@ B
@ one_one_int ) ) ).
% prod_atLeastAtMost_code
thf(fact_9108_prod_Oub__add__nat,axiom,
! [M: nat,N2: nat,G: nat > real,P3: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P3 ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_9109_prod_Oub__add__nat,axiom,
! [M: nat,N2: nat,G: nat > rat,P3: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P3 ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_9110_prod_Oub__add__nat,axiom,
! [M: nat,N2: nat,G: nat > nat,P3: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P3 ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_9111_prod_Oub__add__nat,axiom,
! [M: nat,N2: nat,G: nat > int,P3: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P3 ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_9112_periodic__finite__ex,axiom,
! [D: int,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int,K3: int] :
( ( P @ X2 )
= ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
& ( P @ X3 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_9113_aset_I7_J,axiom,
! [D5: int,A2: set_int,T2: int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X4
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ T2 @ X4 )
=> ( ord_less_int @ T2 @ ( plus_plus_int @ X4 @ D5 ) ) ) ) ) ).
% aset(7)
thf(fact_9114_aset_I5_J,axiom,
! [D5: int,T2: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ( member_int @ T2 @ A2 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X4
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ X4 @ T2 )
=> ( ord_less_int @ ( plus_plus_int @ X4 @ D5 ) @ T2 ) ) ) ) ) ).
% aset(5)
thf(fact_9115_aset_I4_J,axiom,
! [D5: int,T2: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ( member_int @ T2 @ A2 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X4
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X4 != T2 )
=> ( ( plus_plus_int @ X4 @ D5 )
!= T2 ) ) ) ) ) ).
% aset(4)
thf(fact_9116_aset_I3_J,axiom,
! [D5: int,T2: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ( member_int @ ( plus_plus_int @ T2 @ one_one_int ) @ A2 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X4
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X4 = T2 )
=> ( ( plus_plus_int @ X4 @ D5 )
= T2 ) ) ) ) ) ).
% aset(3)
thf(fact_9117_bset_I7_J,axiom,
! [D5: int,T2: int,B3: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ( member_int @ T2 @ B3 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B3 )
=> ( X4
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ T2 @ X4 )
=> ( ord_less_int @ T2 @ ( minus_minus_int @ X4 @ D5 ) ) ) ) ) ) ).
% bset(7)
thf(fact_9118_bset_I5_J,axiom,
! [D5: int,B3: set_int,T2: int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B3 )
=> ( X4
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ X4 @ T2 )
=> ( ord_less_int @ ( minus_minus_int @ X4 @ D5 ) @ T2 ) ) ) ) ).
% bset(5)
thf(fact_9119_bset_I4_J,axiom,
! [D5: int,T2: int,B3: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ( member_int @ T2 @ B3 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B3 )
=> ( X4
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X4 != T2 )
=> ( ( minus_minus_int @ X4 @ D5 )
!= T2 ) ) ) ) ) ).
% bset(4)
thf(fact_9120_bset_I3_J,axiom,
! [D5: int,T2: int,B3: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ( member_int @ ( minus_minus_int @ T2 @ one_one_int ) @ B3 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B3 )
=> ( X4
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X4 = T2 )
=> ( ( minus_minus_int @ X4 @ D5 )
= T2 ) ) ) ) ) ).
% bset(3)
thf(fact_9121_norm__prod__diff,axiom,
! [I5: set_complex,Z: complex > real,W: complex > real] :
( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups766887009212190081x_real @ Z @ I5 ) @ ( groups766887009212190081x_real @ W @ I5 ) ) )
@ ( groups5808333547571424918x_real
@ ^ [I3: complex] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_9122_norm__prod__diff,axiom,
! [I5: set_real,Z: real > real,W: real > real] :
( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups1681761925125756287l_real @ Z @ I5 ) @ ( groups1681761925125756287l_real @ W @ I5 ) ) )
@ ( groups8097168146408367636l_real
@ ^ [I3: real] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_9123_norm__prod__diff,axiom,
! [I5: set_set_nat,Z: set_nat > real,W: set_nat > real] :
( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups3619160379726066777t_real @ Z @ I5 ) @ ( groups3619160379726066777t_real @ W @ I5 ) ) )
@ ( groups5107569545109728110t_real
@ ^ [I3: set_nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_9124_norm__prod__diff,axiom,
! [I5: set_int,Z: int > real,W: int > real] :
( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups2316167850115554303t_real @ Z @ I5 ) @ ( groups2316167850115554303t_real @ W @ I5 ) ) )
@ ( groups8778361861064173332t_real
@ ^ [I3: int] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_9125_norm__prod__diff,axiom,
! [I5: set_complex,Z: complex > complex,W: complex > complex] :
( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups3708469109370488835omplex @ Z @ I5 ) @ ( groups3708469109370488835omplex @ W @ I5 ) ) )
@ ( groups5808333547571424918x_real
@ ^ [I3: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_9126_norm__prod__diff,axiom,
! [I5: set_real,Z: real > complex,W: real > complex] :
( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups713298508707869441omplex @ Z @ I5 ) @ ( groups713298508707869441omplex @ W @ I5 ) ) )
@ ( groups8097168146408367636l_real
@ ^ [I3: real] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_9127_norm__prod__diff,axiom,
! [I5: set_set_nat,Z: set_nat > complex,W: set_nat > complex] :
( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups1092910753850256091omplex @ Z @ I5 ) @ ( groups1092910753850256091omplex @ W @ I5 ) ) )
@ ( groups5107569545109728110t_real
@ ^ [I3: set_nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_9128_norm__prod__diff,axiom,
! [I5: set_int,Z: int > complex,W: int > complex] :
( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups7440179247065528705omplex @ Z @ I5 ) @ ( groups7440179247065528705omplex @ W @ I5 ) ) )
@ ( groups8778361861064173332t_real
@ ^ [I3: int] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_9129_norm__prod__diff,axiom,
! [I5: set_nat,Z: nat > real,W: nat > real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups129246275422532515t_real @ Z @ I5 ) @ ( groups129246275422532515t_real @ W @ I5 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_9130_norm__prod__diff,axiom,
! [I5: set_nat,Z: nat > complex,W: nat > complex] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups6464643781859351333omplex @ Z @ I5 ) @ ( groups6464643781859351333omplex @ W @ I5 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_9131_prod_OatMost__shift,axiom,
! [G: nat > real,N2: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N2 ) )
= ( times_times_real @ ( G @ zero_zero_nat )
@ ( groups129246275422532515t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% prod.atMost_shift
thf(fact_9132_prod_OatMost__shift,axiom,
! [G: nat > rat,N2: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N2 ) )
= ( times_times_rat @ ( G @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% prod.atMost_shift
thf(fact_9133_prod_OatMost__shift,axiom,
! [G: nat > nat,N2: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N2 ) )
= ( times_times_nat @ ( G @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% prod.atMost_shift
thf(fact_9134_prod_OatMost__shift,axiom,
! [G: nat > int,N2: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N2 ) )
= ( times_times_int @ ( G @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% prod.atMost_shift
thf(fact_9135_fact__eq__fact__times,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( semiri1408675320244567234ct_nat @ M )
= ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N2 )
@ ( groups708209901874060359at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_9136_bset_I6_J,axiom,
! [D5: int,B3: set_int,T2: int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B3 )
=> ( X4
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ X4 @ T2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ X4 @ D5 ) @ T2 ) ) ) ) ).
% bset(6)
thf(fact_9137_bset_I8_J,axiom,
! [D5: int,T2: int,B3: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ( member_int @ ( minus_minus_int @ T2 @ one_one_int ) @ B3 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B3 )
=> ( X4
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ T2 @ X4 )
=> ( ord_less_eq_int @ T2 @ ( minus_minus_int @ X4 @ D5 ) ) ) ) ) ) ).
% bset(8)
thf(fact_9138_aset_I6_J,axiom,
! [D5: int,T2: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ( member_int @ ( plus_plus_int @ T2 @ one_one_int ) @ A2 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X4
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ X4 @ T2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ X4 @ D5 ) @ T2 ) ) ) ) ) ).
% aset(6)
thf(fact_9139_aset_I8_J,axiom,
! [D5: int,A2: set_int,T2: int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X4
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ T2 @ X4 )
=> ( ord_less_eq_int @ T2 @ ( plus_plus_int @ X4 @ D5 ) ) ) ) ) ).
% aset(8)
thf(fact_9140_cpmi,axiom,
! [D5: int,P: int > $o,P2: int > $o,B3: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z3 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B3 )
=> ( X2
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D5 ) ) ) )
=> ( ! [X2: int,K3: int] :
( ( P2 @ X2 )
= ( P2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D5 ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
& ( P2 @ X3 ) )
| ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
& ? [Y4: int] :
( ( member_int @ Y4 @ B3 )
& ( P @ ( plus_plus_int @ Y4 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_9141_cppi,axiom,
! [D5: int,P: int > $o,P2: int > $o,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ? [Z3: int] :
! [X2: int] :
( ( ord_less_int @ Z3 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X2
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D5 ) ) ) )
=> ( ! [X2: int,K3: int] :
( ( P2 @ X2 )
= ( P2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D5 ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
& ( P2 @ X3 ) )
| ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
& ? [Y4: int] :
( ( member_int @ Y4 @ A2 )
& ( P @ ( minus_minus_int @ Y4 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_9142_pochhammer__Suc__prod,axiom,
! [A: real,N2: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
= ( groups129246275422532515t_real
@ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).
% pochhammer_Suc_prod
thf(fact_9143_pochhammer__Suc__prod,axiom,
! [A: complex,N2: nat] :
( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
= ( groups6464643781859351333omplex
@ ^ [I3: nat] : ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).
% pochhammer_Suc_prod
thf(fact_9144_pochhammer__Suc__prod,axiom,
! [A: rat,N2: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).
% pochhammer_Suc_prod
thf(fact_9145_pochhammer__Suc__prod,axiom,
! [A: nat,N2: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).
% pochhammer_Suc_prod
thf(fact_9146_pochhammer__Suc__prod,axiom,
! [A: int,N2: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).
% pochhammer_Suc_prod
thf(fact_9147_pochhammer__prod__rev,axiom,
( comm_s7457072308508201937r_real
= ( ^ [A3: real,N4: nat] :
( groups129246275422532515t_real
@ ^ [I3: nat] : ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N4 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_9148_pochhammer__prod__rev,axiom,
( comm_s2602460028002588243omplex
= ( ^ [A3: complex,N4: nat] :
( groups6464643781859351333omplex
@ ^ [I3: nat] : ( plus_plus_complex @ A3 @ ( semiri8010041392384452111omplex @ ( minus_minus_nat @ N4 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_9149_pochhammer__prod__rev,axiom,
( comm_s4028243227959126397er_rat
= ( ^ [A3: rat,N4: nat] :
( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A3 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N4 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_9150_pochhammer__prod__rev,axiom,
( comm_s4663373288045622133er_nat
= ( ^ [A3: nat,N4: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N4 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_9151_pochhammer__prod__rev,axiom,
( comm_s4660882817536571857er_int
= ( ^ [A3: int,N4: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N4 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_9152_fact__div__fact,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) )
= ( groups708209901874060359at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_9153_prod_Oin__pairs,axiom,
! [G: nat > real,M: nat,N2: 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 ) ) @ N2 ) ) ) )
= ( groups129246275422532515t_real
@ ^ [I3: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% prod.in_pairs
thf(fact_9154_prod_Oin__pairs,axiom,
! [G: nat > rat,M: nat,N2: 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 ) ) @ N2 ) ) ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% prod.in_pairs
thf(fact_9155_prod_Oin__pairs,axiom,
! [G: nat > nat,M: nat,N2: 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 ) ) @ N2 ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% prod.in_pairs
thf(fact_9156_prod_Oin__pairs,axiom,
! [G: nat > int,M: nat,N2: 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 ) ) @ N2 ) ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% prod.in_pairs
thf(fact_9157_prod_Oin__pairs__0,axiom,
! [G: nat > real,N2: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
= ( groups129246275422532515t_real
@ ^ [I3: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% prod.in_pairs_0
thf(fact_9158_prod_Oin__pairs__0,axiom,
! [G: nat > rat,N2: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% prod.in_pairs_0
thf(fact_9159_prod_Oin__pairs__0,axiom,
! [G: nat > nat,N2: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% prod.in_pairs_0
thf(fact_9160_prod_Oin__pairs__0,axiom,
! [G: nat > int,N2: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% prod.in_pairs_0
thf(fact_9161_pochhammer__Suc__prod__rev,axiom,
! [A: real,N2: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
= ( groups129246275422532515t_real
@ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_9162_pochhammer__Suc__prod__rev,axiom,
! [A: complex,N2: nat] :
( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
= ( groups6464643781859351333omplex
@ ^ [I3: nat] : ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ ( minus_minus_nat @ N2 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_9163_pochhammer__Suc__prod__rev,axiom,
! [A: rat,N2: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_9164_pochhammer__Suc__prod__rev,axiom,
! [A: nat,N2: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_9165_pochhammer__Suc__prod__rev,axiom,
! [A: int,N2: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_9166_prod_Ozero__middle,axiom,
! [P3: nat,K: nat,G: nat > complex,H2: nat > complex] :
( ( ord_less_eq_nat @ one_one_nat @ P3 )
=> ( ( ord_less_eq_nat @ K @ P3 )
=> ( ( 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 @ P3 ) )
= ( groups6464643781859351333omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_9167_prod_Ozero__middle,axiom,
! [P3: nat,K: nat,G: nat > real,H2: nat > real] :
( ( ord_less_eq_nat @ one_one_nat @ P3 )
=> ( ( ord_less_eq_nat @ K @ P3 )
=> ( ( 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 @ P3 ) )
= ( groups129246275422532515t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_9168_prod_Ozero__middle,axiom,
! [P3: nat,K: nat,G: nat > rat,H2: nat > rat] :
( ( ord_less_eq_nat @ one_one_nat @ P3 )
=> ( ( ord_less_eq_nat @ K @ P3 )
=> ( ( 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 @ P3 ) )
= ( groups73079841787564623at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_9169_prod_Ozero__middle,axiom,
! [P3: nat,K: nat,G: nat > nat,H2: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P3 )
=> ( ( ord_less_eq_nat @ K @ P3 )
=> ( ( 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 @ P3 ) )
= ( groups708209901874060359at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_9170_prod_Ozero__middle,axiom,
! [P3: nat,K: nat,G: nat > int,H2: nat > int] :
( ( ord_less_eq_nat @ one_one_nat @ P3 )
=> ( ( ord_less_eq_nat @ K @ P3 )
=> ( ( 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 @ P3 ) )
= ( groups705719431365010083at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_9171_gbinomial__Suc,axiom,
! [A: complex,K: nat] :
( ( gbinomial_complex @ A @ ( suc @ K ) )
= ( divide1717551699836669952omplex
@ ( groups6464643781859351333omplex
@ ^ [I3: nat] : ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri5044797733671781792omplex @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_9172_gbinomial__Suc,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ A @ ( suc @ K ) )
= ( divide_divide_rat
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri773545260158071498ct_rat @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_9173_gbinomial__Suc,axiom,
! [A: real,K: nat] :
( ( gbinomial_real @ A @ ( suc @ K ) )
= ( divide_divide_real
@ ( groups129246275422532515t_real
@ ^ [I3: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri2265585572941072030t_real @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_9174_gbinomial__Suc,axiom,
! [A: nat,K: nat] :
( ( gbinomial_nat @ A @ ( suc @ K ) )
= ( divide_divide_nat
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri1408675320244567234ct_nat @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_9175_gbinomial__Suc,axiom,
! [A: int,K: nat] :
( ( gbinomial_int @ A @ ( suc @ K ) )
= ( divide_divide_int
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri1406184849735516958ct_int @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_9176_divmod__step__nat__def,axiom,
( unique5026877609467782581ep_nat
= ( ^ [L2: num] :
( produc2626176000494625587at_nat
@ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ 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 @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_9177_divmod__step__int__def,axiom,
( unique5024387138958732305ep_int
= ( ^ [L2: num] :
( produc4245557441103728435nt_int
@ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ 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 @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_9178_pi__half,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( the_real
@ ^ [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 ) ) ) ) ).
% pi_half
thf(fact_9179_pi__def,axiom,
( pi
= ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( the_real
@ ^ [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 ) ) ) ) ) ).
% pi_def
thf(fact_9180_Sum__Icc__int,axiom,
! [M: int,N2: int] :
( ( ord_less_eq_int @ M @ N2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [X3: int] : X3
@ ( set_or1266510415728281911st_int @ M @ N2 ) )
= ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N2 @ ( plus_plus_int @ N2 @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_9181_divmod__step__def,axiom,
( unique5026877609467782581ep_nat
= ( ^ [L2: num] :
( produc2626176000494625587at_nat
@ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ 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 @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_def
thf(fact_9182_divmod__step__def,axiom,
( unique5024387138958732305ep_int
= ( ^ [L2: num] :
( produc4245557441103728435nt_int
@ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ 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 @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_def
thf(fact_9183_divmod__step__def,axiom,
( unique4921790084139445826nteger
= ( ^ [L2: num] :
( produc6916734918728496179nteger
@ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_def
thf(fact_9184_arcsin__def,axiom,
( arcsin
= ( ^ [Y4: real] :
( the_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 ) ) ) ) ) ).
% arcsin_def
thf(fact_9185_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M2: nat,N4: nat] :
( if_Pro6206227464963214023at_nat
@ ( ( N4 = zero_zero_nat )
| ( ord_less_nat @ M2 @ N4 ) )
@ ( product_Pair_nat_nat @ zero_zero_nat @ M2 )
@ ( produc2626176000494625587at_nat
@ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
@ ( divmod_nat @ ( minus_minus_nat @ M2 @ N4 ) @ N4 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_9186_set__encode__def,axiom,
( nat_set_encode
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% set_encode_def
thf(fact_9187_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_9188_valid__eq1,axiom,
! [T2: vEBT_VEBT,D: nat] :
( ( vEBT_invar_vebt @ T2 @ D )
=> ( vEBT_VEBT_valid @ T2 @ D ) ) ).
% valid_eq1
thf(fact_9189_valid__eq2,axiom,
! [T2: vEBT_VEBT,D: nat] :
( ( vEBT_VEBT_valid @ T2 @ D )
=> ( vEBT_invar_vebt @ T2 @ D ) ) ).
% valid_eq2
thf(fact_9190_int__prod,axiom,
! [F: int > nat,A2: set_int] :
( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
= ( groups1705073143266064639nt_int
@ ^ [X3: int] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A2 ) ) ).
% int_prod
thf(fact_9191_int__prod,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A2 ) )
= ( groups705719431365010083at_int
@ ^ [X3: nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A2 ) ) ).
% int_prod
thf(fact_9192_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_9193_prod__int__plus__eq,axiom,
! [I: nat,J: nat] :
( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
= ( groups1705073143266064639nt_int
@ ^ [X3: int] : X3
@ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_9194_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M2: nat,N4: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M2 @ N4 ) @ ( modulo_modulo_nat @ M2 @ N4 ) ) ) ) ).
% divmod_nat_def
thf(fact_9195_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_9196_Sum__Ico__nat,axiom,
! [M: nat,N2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ ( set_or4665077453230672383an_nat @ M @ N2 ) )
= ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Sum_Ico_nat
thf(fact_9197_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_9198_ex__nat__less__eq,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ( P @ M2 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_9199_all__nat__less__eq,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less_eq
thf(fact_9200_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_9201_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_9202_prod__Suc__Suc__fact,axiom,
! [N2: nat] :
( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
= ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% prod_Suc_Suc_fact
thf(fact_9203_prod__Suc__fact,axiom,
! [N2: nat] :
( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
= ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% prod_Suc_fact
thf(fact_9204_Chebyshev__sum__upper__nat,axiom,
! [N2: nat,A: nat > nat,B: nat > nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I2 ) ) ) )
=> ( ord_less_eq_nat
@ ( times_times_nat @ N2
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( B @ I3 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
@ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_9205_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_9206_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
= ( set_or1266510415728281911st_int @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_9207_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_9208_Cauchy__iff2,axiom,
( topolo4055970368930404560y_real
= ( ^ [X6: nat > real] :
! [J3: nat] :
? [M9: nat] :
! [M2: nat] :
( ( ord_less_eq_nat @ M9 @ M2 )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M9 @ N4 )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X6 @ M2 ) @ ( X6 @ N4 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_9209_Code__Target__Int_Opositive__def,axiom,
code_Target_positive = numeral_numeral_int ).
% Code_Target_Int.positive_def
thf(fact_9210_divmod__step__integer__def,axiom,
( unique4921790084139445826nteger
= ( ^ [L2: num] :
( produc6916734918728496179nteger
@ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_9211_csqrt_Osimps_I1_J,axiom,
! [Z: complex] :
( ( re @ ( csqrt @ Z ) )
= ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% csqrt.simps(1)
thf(fact_9212_Re__divide__of__nat,axiom,
! [Z: complex,N2: nat] :
( ( re @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) )
= ( divide_divide_real @ ( re @ Z ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% Re_divide_of_nat
thf(fact_9213_Re__divide__of__real,axiom,
! [Z: complex,R4: real] :
( ( re @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R4 ) ) )
= ( divide_divide_real @ ( re @ Z ) @ R4 ) ) ).
% Re_divide_of_real
thf(fact_9214_Re__sgn,axiom,
! [Z: complex] :
( ( re @ ( sgn_sgn_complex @ Z ) )
= ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% Re_sgn
thf(fact_9215_Re__divide__numeral,axiom,
! [Z: complex,W: num] :
( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
= ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% Re_divide_numeral
thf(fact_9216_plus__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
= K ) ).
% plus_integer_code(1)
thf(fact_9217_plus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
= L ) ).
% plus_integer_code(2)
thf(fact_9218_divmod__integer_H__def,axiom,
( unique3479559517661332726nteger
= ( ^ [M2: num,N4: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N4 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N4 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_9219_sgn__integer__code,axiom,
( sgn_sgn_Code_integer
= ( ^ [K2: code_integer] : ( if_Code_integer @ ( K2 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% sgn_integer_code
thf(fact_9220_one__complex_Osimps_I1_J,axiom,
( ( re @ one_one_complex )
= one_one_real ) ).
% one_complex.simps(1)
thf(fact_9221_plus__complex_Osimps_I1_J,axiom,
! [X: complex,Y: complex] :
( ( re @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).
% plus_complex.simps(1)
thf(fact_9222_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_9223_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_9224_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_9225_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_9226_cmod__plus__Re__le__0__iff,axiom,
! [Z: complex] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
= ( ( re @ Z )
= ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% cmod_plus_Re_le_0_iff
thf(fact_9227_csqrt_Ocode,axiom,
( csqrt
= ( ^ [Z5: complex] :
( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
@ ( times_times_real
@ ( if_real
@ ( ( im @ Z5 )
= zero_zero_real )
@ one_one_real
@ ( sgn_sgn_real @ ( im @ Z5 ) ) )
@ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% csqrt.code
thf(fact_9228_csqrt_Osimps_I2_J,axiom,
! [Z: complex] :
( ( im @ ( csqrt @ Z ) )
= ( times_times_real
@ ( if_real
@ ( ( im @ Z )
= zero_zero_real )
@ one_one_real
@ ( sgn_sgn_real @ ( im @ Z ) ) )
@ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% csqrt.simps(2)
thf(fact_9229_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K2: int] :
( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K2 ) ) )
@ ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
@ ( if_Code_integer
@ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_9230_Complex__divide,axiom,
( divide1717551699836669952omplex
= ( ^ [X3: complex,Y4: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y4 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( 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 ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y4 ) ) @ ( times_times_real @ ( re @ X3 ) @ ( 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 ) ) ) ) ) ) ) ) ).
% Complex_divide
thf(fact_9231_Im__divide__of__real,axiom,
! [Z: complex,R4: real] :
( ( im @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R4 ) ) )
= ( divide_divide_real @ ( im @ Z ) @ R4 ) ) ).
% Im_divide_of_real
thf(fact_9232_Im__sgn,axiom,
! [Z: complex] :
( ( im @ ( sgn_sgn_complex @ Z ) )
= ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% Im_sgn
thf(fact_9233_Im__divide__numeral,axiom,
! [Z: complex,W: num] :
( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
= ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% Im_divide_numeral
thf(fact_9234_Im__divide__of__nat,axiom,
! [Z: complex,N2: nat] :
( ( im @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) )
= ( divide_divide_real @ ( im @ Z ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% Im_divide_of_nat
thf(fact_9235_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_9236_zero__integer__def,axiom,
( zero_z3403309356797280102nteger
= ( code_integer_of_int @ zero_zero_int ) ) ).
% zero_integer_def
thf(fact_9237_divide__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( divide_divide_int @ Xa @ X ) ) ) ).
% divide_integer.abs_eq
thf(fact_9238_less__integer__code_I1_J,axiom,
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% less_integer_code(1)
thf(fact_9239_abs__integer__code,axiom,
( abs_abs_Code_integer
= ( ^ [K2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K2 ) @ K2 ) ) ) ).
% abs_integer_code
thf(fact_9240_less__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( ord_less_int @ Xa @ X ) ) ).
% less_integer.abs_eq
thf(fact_9241_imaginary__unit_Osimps_I2_J,axiom,
( ( im @ imaginary_unit )
= one_one_real ) ).
% imaginary_unit.simps(2)
thf(fact_9242_one__complex_Osimps_I2_J,axiom,
( ( im @ one_one_complex )
= zero_zero_real ) ).
% one_complex.simps(2)
thf(fact_9243_plus__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( plus_plus_int @ Xa @ X ) ) ) ).
% plus_integer.abs_eq
thf(fact_9244_one__integer__def,axiom,
( one_one_Code_integer
= ( code_integer_of_int @ one_one_int ) ) ).
% one_integer_def
thf(fact_9245_less__eq__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( ord_less_eq_int @ Xa @ X ) ) ).
% less_eq_integer.abs_eq
thf(fact_9246_plus__complex_Osimps_I2_J,axiom,
! [X: complex,Y: complex] :
( ( im @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).
% plus_complex.simps(2)
thf(fact_9247_times__complex_Osimps_I2_J,axiom,
! [X: complex,Y: complex] :
( ( im @ ( times_times_complex @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ).
% times_complex.simps(2)
thf(fact_9248_plus__complex_Ocode,axiom,
( plus_plus_complex
= ( ^ [X3: complex,Y4: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X3 ) @ ( re @ Y4 ) ) @ ( plus_plus_real @ ( im @ X3 ) @ ( im @ Y4 ) ) ) ) ) ).
% plus_complex.code
thf(fact_9249_csqrt__principal,axiom,
! [Z: complex] :
( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
| ( ( ( re @ ( csqrt @ Z ) )
= zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% csqrt_principal
thf(fact_9250_cmod__le,axiom,
! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% cmod_le
thf(fact_9251_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_9252_times__complex_Ocode,axiom,
( times_times_complex
= ( ^ [X3: complex,Y4: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y4 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y4 ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y4 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y4 ) ) ) ) ) ) ).
% times_complex.code
thf(fact_9253_cmod__power2,axiom,
! [Z: complex] :
( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% cmod_power2
thf(fact_9254_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_9255_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_9256_complex__eq__0,axiom,
! [Z: complex] :
( ( Z = zero_zero_complex )
= ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_real ) ) ).
% complex_eq_0
thf(fact_9257_norm__complex__def,axiom,
( real_V1022390504157884413omplex
= ( ^ [Z5: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% norm_complex_def
thf(fact_9258_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_9259_complex__neq__0,axiom,
! [Z: complex] :
( ( Z != zero_zero_complex )
= ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% complex_neq_0
thf(fact_9260_Re__divide,axiom,
! [X: complex,Y: complex] :
( ( re @ ( divide1717551699836669952omplex @ X @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( 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 ) ) ) ) ) ) ).
% Re_divide
thf(fact_9261_csqrt__unique,axiom,
! [W: complex,Z: complex] :
( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= Z )
=> ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
| ( ( ( re @ W )
= zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
=> ( ( csqrt @ Z )
= W ) ) ) ).
% csqrt_unique
thf(fact_9262_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_9263_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_9264_Im__divide,axiom,
! [X: complex,Y: complex] :
( ( im @ ( divide1717551699836669952omplex @ X @ Y ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X ) @ ( 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 ) ) ) ) ) ) ).
% Im_divide
thf(fact_9265_complex__abs__le__norm,axiom,
! [Z: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% complex_abs_le_norm
thf(fact_9266_complex__unit__circle,axiom,
! [Z: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real ) ) ).
% complex_unit_circle
thf(fact_9267_inverse__complex_Ocode,axiom,
( invers8013647133539491842omplex
= ( ^ [X3: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X3 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X3 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% inverse_complex.code
thf(fact_9268_Im__Reals__divide,axiom,
! [R4: complex,Z: complex] :
( ( member_complex @ R4 @ real_V2521375963428798218omplex )
=> ( ( im @ ( divide1717551699836669952omplex @ R4 @ Z ) )
= ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R4 ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% Im_Reals_divide
thf(fact_9269_Re__Reals__divide,axiom,
! [R4: complex,Z: complex] :
( ( member_complex @ R4 @ real_V2521375963428798218omplex )
=> ( ( re @ ( divide1717551699836669952omplex @ R4 @ Z ) )
= ( divide_divide_real @ ( times_times_real @ ( re @ R4 ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% Re_Reals_divide
thf(fact_9270_complex__mult__cnj,axiom,
! [Z: complex] :
( ( times_times_complex @ Z @ ( cnj @ Z ) )
= ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% complex_mult_cnj
thf(fact_9271_complex__cnj__divide,axiom,
! [X: complex,Y: complex] :
( ( cnj @ ( divide1717551699836669952omplex @ X @ Y ) )
= ( divide1717551699836669952omplex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
% complex_cnj_divide
thf(fact_9272_complex__cnj__one__iff,axiom,
! [Z: complex] :
( ( ( cnj @ Z )
= one_one_complex )
= ( Z = one_one_complex ) ) ).
% complex_cnj_one_iff
thf(fact_9273_complex__cnj__one,axiom,
( ( cnj @ one_one_complex )
= one_one_complex ) ).
% complex_cnj_one
thf(fact_9274_complex__cnj__add,axiom,
! [X: complex,Y: complex] :
( ( cnj @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
% complex_cnj_add
thf(fact_9275_Re__divide__Reals,axiom,
! [R4: complex,Z: complex] :
( ( member_complex @ R4 @ real_V2521375963428798218omplex )
=> ( ( re @ ( divide1717551699836669952omplex @ Z @ R4 ) )
= ( divide_divide_real @ ( re @ Z ) @ ( re @ R4 ) ) ) ) ).
% Re_divide_Reals
thf(fact_9276_Im__divide__Reals,axiom,
! [R4: complex,Z: complex] :
( ( member_complex @ R4 @ real_V2521375963428798218omplex )
=> ( ( im @ ( divide1717551699836669952omplex @ Z @ R4 ) )
= ( divide_divide_real @ ( im @ Z ) @ ( re @ R4 ) ) ) ) ).
% Im_divide_Reals
thf(fact_9277_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_9278_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_9279_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_9280_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_9281_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_9282_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_9283_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_9284_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_9285_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_9286_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_9287_complex__mod__mult__cnj,axiom,
! [Z: complex] :
( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
= ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% complex_mod_mult_cnj
thf(fact_9288_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_9289_complex__norm__square,axiom,
! [Z: complex] :
( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).
% complex_norm_square
thf(fact_9290_complex__add__cnj,axiom,
! [Z: complex] :
( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
= ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).
% complex_add_cnj
thf(fact_9291_complex__diff__cnj,axiom,
! [Z: complex] :
( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
= ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% complex_diff_cnj
thf(fact_9292_complex__div__cnj,axiom,
( divide1717551699836669952omplex
= ( ^ [A3: complex,B2: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ ( cnj @ B2 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% complex_div_cnj
thf(fact_9293_cnj__add__mult__eq__Re,axiom,
! [Z: complex,W: complex] :
( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
= ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% cnj_add_mult_eq_Re
thf(fact_9294_integer__of__num_I3_J,axiom,
! [N2: num] :
( ( code_integer_of_num @ ( bit1 @ N2 ) )
= ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N2 ) @ ( code_integer_of_num @ N2 ) ) @ one_one_Code_integer ) ) ).
% integer_of_num(3)
thf(fact_9295_bit__cut__integer__def,axiom,
( code_bit_cut_integer
= ( ^ [K2: code_integer] :
( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
@ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K2 ) ) ) ) ).
% bit_cut_integer_def
thf(fact_9296_divmod__integer__def,axiom,
( code_divmod_integer
= ( ^ [K2: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K2 @ L2 ) @ ( modulo364778990260209775nteger @ K2 @ L2 ) ) ) ) ).
% divmod_integer_def
thf(fact_9297_divmod__abs__def,axiom,
( code_divmod_abs
= ( ^ [K2: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L2 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ).
% divmod_abs_def
thf(fact_9298_integer__of__num__triv_I1_J,axiom,
( ( code_integer_of_num @ one )
= one_one_Code_integer ) ).
% integer_of_num_triv(1)
thf(fact_9299_integer__of__num_I2_J,axiom,
! [N2: num] :
( ( code_integer_of_num @ ( bit0 @ N2 ) )
= ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N2 ) @ ( code_integer_of_num @ N2 ) ) ) ).
% integer_of_num(2)
thf(fact_9300_integer__of__num__triv_I2_J,axiom,
( ( code_integer_of_num @ ( bit0 @ one ) )
= ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% integer_of_num_triv(2)
thf(fact_9301_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K2: code_integer] :
( if_Pro5737122678794959658eger_o @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
@ ( produc9125791028180074456eger_o
@ ^ [R5: code_integer,S4: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S4 ) ) @ ( S4 = one_one_Code_integer ) )
@ ( code_divmod_abs @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_9302_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K2: code_integer,L2: code_integer] :
( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ ( code_divmod_abs @ K2 @ L2 )
@ ( produc6916734918728496179nteger
@ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S4 ) ) )
@ ( code_divmod_abs @ K2 @ L2 ) ) )
@ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
@ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K2 @ L2 )
@ ( produc6916734918728496179nteger
@ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S4 ) ) )
@ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_9303_or__int__rec,axiom,
( bit_se1409905431419307370or_int
= ( ^ [K2: int,L2: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
| ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% or_int_rec
thf(fact_9304_or__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
= ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% or_nonnegative_int_iff
thf(fact_9305_or__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
= ( ( ord_less_int @ K @ zero_zero_int )
| ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% or_negative_int_iff
thf(fact_9306_or__minus__numerals_I6_J,axiom,
! [N2: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ) ).
% or_minus_numerals(6)
thf(fact_9307_or__minus__numerals_I2_J,axiom,
! [N2: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ) ).
% or_minus_numerals(2)
thf(fact_9308_and__minus__minus__numerals,axiom,
! [M: num,N2: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ) ) ) ).
% and_minus_minus_numerals
thf(fact_9309_or__minus__minus__numerals,axiom,
! [M: num,N2: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ) ) ) ).
% or_minus_minus_numerals
thf(fact_9310_bit__or__int__iff,axiom,
! [K: int,L: int,N2: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ N2 )
= ( ( bit_se1146084159140164899it_int @ K @ N2 )
| ( bit_se1146084159140164899it_int @ L @ N2 ) ) ) ).
% bit_or_int_iff
thf(fact_9311_OR__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y ) ) ) ) ).
% OR_lower
thf(fact_9312_or__greater__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).
% or_greater_eq
thf(fact_9313_plus__and__or,axiom,
! [X: int,Y: int] :
( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Y ) )
= ( plus_plus_int @ X @ Y ) ) ).
% plus_and_or
thf(fact_9314_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat != zero_zero_nat )
= ( case_nat_o @ $false
@ ^ [Uu3: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_9315_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat = zero_zero_nat )
= ( case_nat_o @ $true
@ ^ [Uu3: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
thf(fact_9316_or__int__def,axiom,
( bit_se1409905431419307370or_int
= ( ^ [K2: int,L2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ ( bit_ri7919022796975470100ot_int @ L2 ) ) ) ) ) ).
% or_int_def
thf(fact_9317_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_9318_xor__int__def,axiom,
( bit_se6526347334894502574or_int
= ( ^ [K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ L2 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ L2 ) ) ) ) ).
% xor_int_def
thf(fact_9319_concat__bit__def,axiom,
( bit_concat_bit
= ( ^ [N4: nat,K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N4 @ K2 ) @ ( bit_se545348938243370406it_int @ N4 @ L2 ) ) ) ) ).
% concat_bit_def
thf(fact_9320_set__bit__int__def,axiom,
( bit_se7879613467334960850it_int
= ( ^ [N4: nat,K2: int] : ( bit_se1409905431419307370or_int @ K2 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).
% set_bit_int_def
thf(fact_9321_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
= ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N2 ) ) ).
% less_eq_nat.simps(2)
thf(fact_9322_or__not__numerals_I2_J,axiom,
! [N2: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
= ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ) ).
% or_not_numerals(2)
thf(fact_9323_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_9324_diff__Suc,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ M @ ( suc @ N2 ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [K2: nat] : K2
@ ( minus_minus_nat @ M @ N2 ) ) ) ).
% diff_Suc
thf(fact_9325_or__not__numerals_I3_J,axiom,
! [N2: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
= ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ) ).
% or_not_numerals(3)
thf(fact_9326_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_9327_or__not__numerals_I6_J,axiom,
! [M: num,N2: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% or_not_numerals(6)
thf(fact_9328_OR__upper,axiom,
! [X: int,N2: nat,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
=> ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
=> ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% OR_upper
thf(fact_9329_or__not__numerals_I5_J,axiom,
! [M: num,N2: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
= ( 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 @ N2 ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_9330_or__not__numerals_I9_J,axiom,
! [M: num,N2: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
= ( 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 @ N2 ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_9331_or__not__numerals_I8_J,axiom,
! [M: num,N2: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
= ( 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 @ N2 ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_9332_or__minus__numerals_I1_J,axiom,
! [N2: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N2 ) ) ) ) ) ).
% or_minus_numerals(1)
thf(fact_9333_or__minus__numerals_I5_J,axiom,
! [N2: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N2 ) ) ) ) ) ).
% or_minus_numerals(5)
thf(fact_9334_or__int__unfold,axiom,
( bit_se1409905431419307370or_int
= ( ^ [K2: int,L2: int] :
( if_int
@ ( ( K2
= ( uminus_uminus_int @ one_one_int ) )
| ( L2
= ( uminus_uminus_int @ one_one_int ) ) )
@ ( uminus_uminus_int @ one_one_int )
@ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_9335_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_9336_or__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(2)
thf(fact_9337_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_9338_or__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(1)
thf(fact_9339_or__minus__numerals_I8_J,axiom,
! [N2: num,M: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N2 ) ) ) ) ) ).
% or_minus_numerals(8)
thf(fact_9340_or__minus__numerals_I4_J,axiom,
! [M: num,N2: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N2 ) ) ) ) ) ).
% or_minus_numerals(4)
thf(fact_9341_or__minus__numerals_I3_J,axiom,
! [M: num,N2: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N2 ) ) ) ) ) ).
% or_minus_numerals(3)
thf(fact_9342_or__minus__numerals_I7_J,axiom,
! [N2: num,M: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N2 ) ) ) ) ) ).
% or_minus_numerals(7)
thf(fact_9343_or__not__num__neg_Osimps_I1_J,axiom,
( ( bit_or_not_num_neg @ one @ one )
= one ) ).
% or_not_num_neg.simps(1)
thf(fact_9344_set__bit__nat__def,axiom,
( bit_se7882103937844011126it_nat
= ( ^ [M2: nat,N4: nat] : ( bit_se1412395901928357646or_nat @ N4 @ ( bit_se547839408752420682it_nat @ M2 @ one_one_nat ) ) ) ) ).
% set_bit_nat_def
thf(fact_9345_or__not__num__neg_Osimps_I4_J,axiom,
! [N2: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ one )
= ( bit0 @ one ) ) ).
% or_not_num_neg.simps(4)
thf(fact_9346_or__not__num__neg_Osimps_I6_J,axiom,
! [N2: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ ( bit1 @ M ) )
= ( bit0 @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).
% or_not_num_neg.simps(6)
thf(fact_9347_or__not__num__neg_Osimps_I7_J,axiom,
! [N2: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ one )
= one ) ).
% or_not_num_neg.simps(7)
thf(fact_9348_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_9349_or__not__num__neg_Osimps_I5_J,axiom,
! [N2: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).
% or_not_num_neg.simps(5)
thf(fact_9350_or__not__num__neg_Osimps_I9_J,axiom,
! [N2: num,M: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ ( bit1 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).
% or_not_num_neg.simps(9)
thf(fact_9351_or__nat__def,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M2: nat,N4: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% or_nat_def
thf(fact_9352_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_9353_or__not__num__neg_Osimps_I8_J,axiom,
! [N2: num,M: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).
% or_not_num_neg.simps(8)
thf(fact_9354_or__not__num__neg_Oelims,axiom,
! [X: num,Xa: num,Y: num] :
( ( ( bit_or_not_num_neg @ X @ Xa )
= Y )
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( Y != one ) ) )
=> ( ( ( X = one )
=> ! [M5: num] :
( ( Xa
= ( bit0 @ M5 ) )
=> ( Y
!= ( bit1 @ M5 ) ) ) )
=> ( ( ( X = one )
=> ! [M5: num] :
( ( Xa
= ( bit1 @ M5 ) )
=> ( Y
!= ( bit1 @ M5 ) ) ) )
=> ( ( ? [N: num] :
( X
= ( bit0 @ N ) )
=> ( ( Xa = one )
=> ( Y
!= ( bit0 @ one ) ) ) )
=> ( ! [N: num] :
( ( X
= ( bit0 @ N ) )
=> ! [M5: num] :
( ( Xa
= ( bit0 @ M5 ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N @ M5 ) ) ) ) )
=> ( ! [N: num] :
( ( X
= ( bit0 @ N ) )
=> ! [M5: num] :
( ( Xa
= ( bit1 @ M5 ) )
=> ( Y
!= ( bit0 @ ( bit_or_not_num_neg @ N @ M5 ) ) ) ) )
=> ( ( ? [N: num] :
( X
= ( bit1 @ N ) )
=> ( ( Xa = one )
=> ( Y != one ) ) )
=> ( ! [N: num] :
( ( X
= ( bit1 @ N ) )
=> ! [M5: num] :
( ( Xa
= ( bit0 @ M5 ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N @ M5 ) ) ) ) )
=> ~ ! [N: num] :
( ( X
= ( bit1 @ N ) )
=> ! [M5: num] :
( ( Xa
= ( bit1 @ M5 ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N @ M5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.elims
thf(fact_9355_numeral__or__not__num__eq,axiom,
! [M: num,N2: num] :
( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N2 ) )
= ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% numeral_or_not_num_eq
thf(fact_9356_int__numeral__not__or__num__neg,axiom,
! [M: num,N2: num] :
( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) ).
% int_numeral_not_or_num_neg
thf(fact_9357_int__numeral__or__not__num__neg,axiom,
! [M: num,N2: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N2 ) ) ) ) ).
% int_numeral_or_not_num_neg
thf(fact_9358_floor__real__def,axiom,
( archim6058952711729229775r_real
= ( ^ [X3: real] :
( the_int
@ ^ [Z5: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z5 ) @ X3 )
& ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_9359_Suc__0__or__eq,axiom,
! [N2: nat] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% Suc_0_or_eq
thf(fact_9360_or__Suc__0__eq,axiom,
! [N2: nat] :
( ( bit_se1412395901928357646or_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% or_Suc_0_eq
thf(fact_9361_or__nat__rec,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M2: nat,N4: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
| ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_9362_max__enat__simps_I2_J,axiom,
! [Q3: extended_enat] :
( ( ord_ma741700101516333627d_enat @ Q3 @ zero_z5237406670263579293d_enat )
= Q3 ) ).
% max_enat_simps(2)
thf(fact_9363_max__enat__simps_I3_J,axiom,
! [Q3: extended_enat] :
( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q3 )
= Q3 ) ).
% max_enat_simps(3)
thf(fact_9364_max__Suc__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( suc @ ( ord_max_nat @ M @ N2 ) ) ) ).
% max_Suc_Suc
thf(fact_9365_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_max_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_9366_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_9367_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_9368_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_9369_max__0L,axiom,
! [N2: nat] :
( ( ord_max_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% max_0L
thf(fact_9370_max__0R,axiom,
! [N2: nat] :
( ( ord_max_nat @ N2 @ zero_zero_nat )
= N2 ) ).
% max_0R
thf(fact_9371_max__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
= ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).
% max_numeral_Suc
thf(fact_9372_max__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( ord_max_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
= ( suc @ ( ord_max_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).
% max_Suc_numeral
thf(fact_9373_nat__mult__max__right,axiom,
! [M: nat,N2: nat,Q3: nat] :
( ( times_times_nat @ M @ ( ord_max_nat @ N2 @ Q3 ) )
= ( ord_max_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q3 ) ) ) ).
% nat_mult_max_right
thf(fact_9374_nat__mult__max__left,axiom,
! [M: nat,N2: nat,Q3: nat] :
( ( times_times_nat @ ( ord_max_nat @ M @ N2 ) @ Q3 )
= ( ord_max_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N2 @ Q3 ) ) ) ).
% nat_mult_max_left
thf(fact_9375_nat__add__max__left,axiom,
! [M: nat,N2: nat,Q3: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M @ N2 ) @ Q3 )
= ( ord_max_nat @ ( plus_plus_nat @ M @ Q3 ) @ ( plus_plus_nat @ N2 @ Q3 ) ) ) ).
% nat_add_max_left
thf(fact_9376_nat__add__max__right,axiom,
! [M: nat,N2: nat,Q3: nat] :
( ( plus_plus_nat @ M @ ( ord_max_nat @ N2 @ Q3 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M @ N2 ) @ ( plus_plus_nat @ M @ Q3 ) ) ) ).
% nat_add_max_right
thf(fact_9377_nat__minus__add__max,axiom,
! [N2: nat,M: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N2 @ M ) @ M )
= ( ord_max_nat @ N2 @ M ) ) ).
% nat_minus_add_max
thf(fact_9378_max__Suc1,axiom,
! [N2: nat,M: nat] :
( ( ord_max_nat @ ( suc @ N2 ) @ M )
= ( case_nat_nat @ ( suc @ N2 )
@ ^ [M4: nat] : ( suc @ ( ord_max_nat @ N2 @ M4 ) )
@ M ) ) ).
% max_Suc1
thf(fact_9379_max__Suc2,axiom,
! [M: nat,N2: nat] :
( ( ord_max_nat @ M @ ( suc @ N2 ) )
= ( case_nat_nat @ ( suc @ N2 )
@ ^ [M4: nat] : ( suc @ ( ord_max_nat @ M4 @ N2 ) )
@ M ) ) ).
% max_Suc2
thf(fact_9380_or__nat__unfold,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M2: nat,N4: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( 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 @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_9381_floor__rat__def,axiom,
( archim3151403230148437115or_rat
= ( ^ [X3: rat] :
( the_int
@ ^ [Z5: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z5 ) @ X3 )
& ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_9382_abs__rat__def,axiom,
( abs_abs_rat
= ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% abs_rat_def
thf(fact_9383_less__eq__rat__def,axiom,
( ord_less_eq_rat
= ( ^ [X3: rat,Y4: rat] :
( ( ord_less_rat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% less_eq_rat_def
thf(fact_9384_obtain__pos__sum,axiom,
! [R4: rat] :
( ( ord_less_rat @ zero_zero_rat @ R4 )
=> ~ ! [S3: rat] :
( ( ord_less_rat @ zero_zero_rat @ S3 )
=> ! [T: rat] :
( ( ord_less_rat @ zero_zero_rat @ T )
=> ( R4
!= ( plus_plus_rat @ S3 @ T ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_9385_sgn__rat__def,axiom,
( sgn_sgn_rat
= ( ^ [A3: rat] : ( if_rat @ ( A3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% sgn_rat_def
thf(fact_9386_rat__inverse__code,axiom,
! [P3: rat] :
( ( quotient_of @ ( inverse_inverse_rat @ P3 ) )
= ( produc4245557441103728435nt_int
@ ^ [A3: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B2 ) @ ( abs_abs_int @ A3 ) ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_inverse_code
thf(fact_9387_normalize__negative,axiom,
! [Q3: int,P3: int] :
( ( ord_less_int @ Q3 @ zero_zero_int )
=> ( ( normalize @ ( product_Pair_int_int @ P3 @ Q3 ) )
= ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P3 ) @ ( uminus_uminus_int @ Q3 ) ) ) ) ) ).
% normalize_negative
thf(fact_9388_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_9389_normalize__denom__zero,axiom,
! [P3: int] :
( ( normalize @ ( product_Pair_int_int @ P3 @ zero_zero_int ) )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% normalize_denom_zero
thf(fact_9390_rat__one__code,axiom,
( ( quotient_of @ one_one_rat )
= ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% rat_one_code
thf(fact_9391_rat__zero__code,axiom,
( ( quotient_of @ zero_zero_rat )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% rat_zero_code
thf(fact_9392_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_9393_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_9394_divide__rat__def,axiom,
( divide_divide_rat
= ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% divide_rat_def
thf(fact_9395_diff__rat__def,axiom,
( minus_minus_rat
= ( ^ [Q4: rat,R5: rat] : ( plus_plus_rat @ Q4 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).
% diff_rat_def
thf(fact_9396_rat__divide__code,axiom,
! [P3: rat,Q3: rat] :
( ( quotient_of @ ( divide_divide_rat @ P3 @ Q3 ) )
= ( produc4245557441103728435nt_int
@ ^ [A3: int,C2: int] :
( produc4245557441103728435nt_int
@ ^ [B2: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D3 ) @ ( times_times_int @ C2 @ B2 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_divide_code
thf(fact_9397_quotient__of__div,axiom,
! [R4: rat,N2: int,D: int] :
( ( ( quotient_of @ R4 )
= ( product_Pair_int_int @ N2 @ D ) )
=> ( R4
= ( divide_divide_rat @ ( ring_1_of_int_rat @ N2 ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% quotient_of_div
thf(fact_9398_rat__plus__code,axiom,
! [P3: rat,Q3: rat] :
( ( quotient_of @ ( plus_plus_rat @ P3 @ Q3 ) )
= ( produc4245557441103728435nt_int
@ ^ [A3: int,C2: int] :
( produc4245557441103728435nt_int
@ ^ [B2: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D3 ) @ ( times_times_int @ B2 @ C2 ) ) @ ( times_times_int @ C2 @ D3 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_plus_code
thf(fact_9399_quotient__of__denom__pos,axiom,
! [R4: rat,P3: int,Q3: int] :
( ( ( quotient_of @ R4 )
= ( product_Pair_int_int @ P3 @ Q3 ) )
=> ( ord_less_int @ zero_zero_int @ Q3 ) ) ).
% quotient_of_denom_pos
thf(fact_9400_normalize__denom__pos,axiom,
! [R4: product_prod_int_int,P3: int,Q3: int] :
( ( ( normalize @ R4 )
= ( product_Pair_int_int @ P3 @ Q3 ) )
=> ( ord_less_int @ zero_zero_int @ Q3 ) ) ).
% normalize_denom_pos
thf(fact_9401_normalize__crossproduct,axiom,
! [Q3: int,S: int,P3: int,R4: int] :
( ( Q3 != zero_zero_int )
=> ( ( S != zero_zero_int )
=> ( ( ( normalize @ ( product_Pair_int_int @ P3 @ Q3 ) )
= ( normalize @ ( product_Pair_int_int @ R4 @ S ) ) )
=> ( ( times_times_int @ P3 @ S )
= ( times_times_int @ R4 @ Q3 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_9402_pred__def,axiom,
( pred
= ( case_nat_nat @ zero_zero_nat
@ ^ [X24: nat] : X24 ) ) ).
% pred_def
thf(fact_9403_quotient__of__int,axiom,
! [A: int] :
( ( quotient_of @ ( of_int @ A ) )
= ( product_Pair_int_int @ A @ one_one_int ) ) ).
% quotient_of_int
thf(fact_9404_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% bezw_0
thf(fact_9405_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq_nat @ Xa @ X )
=> ( Y
= ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_9406_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K2: nat,M2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M2 @ K2 ) @ ( product_Pair_nat_nat @ M2 @ ( minus_minus_nat @ K2 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M2 @ ( suc @ K2 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_9407_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_9408_Frct__code__post_I6_J,axiom,
! [K: num,L: num] :
( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
= ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).
% Frct_code_post(6)
thf(fact_9409_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N9: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N9 )
=> ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_9410_bounded__nat__set__is__finite,axiom,
! [N3: set_nat,N2: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N3 )
=> ( ord_less_nat @ X2 @ N2 ) )
=> ( finite_finite_nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_9411_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N9: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N9 )
=> ( ord_less_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_9412_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less_nat @ K2 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_9413_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ N @ ( F @ N ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_9414_set__encode__inf,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( nat_set_encode @ A2 )
= zero_zero_nat ) ) ).
% set_encode_inf
thf(fact_9415_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_9416_subset__eq__atLeast0__atMost__finite,axiom,
! [N3: set_nat,N2: nat] :
( ( ord_less_eq_set_nat @ N3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
=> ( finite_finite_nat @ N3 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_9417_subset__eq__atLeast0__lessThan__finite,axiom,
! [N3: set_nat,N2: nat] :
( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
=> ( finite_finite_nat @ N3 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_9418_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_9419_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_9420_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_9421_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_9422_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_9423_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_9424_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_9425_finite__interval__int1,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_eq_int @ A @ I3 )
& ( ord_less_eq_int @ I3 @ B ) ) ) ) ).
% finite_interval_int1
thf(fact_9426_finite__interval__int4,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_int @ A @ I3 )
& ( ord_less_int @ I3 @ B ) ) ) ) ).
% finite_interval_int4
thf(fact_9427_finite__interval__int2,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_eq_int @ A @ I3 )
& ( ord_less_int @ I3 @ B ) ) ) ) ).
% finite_interval_int2
thf(fact_9428_finite__interval__int3,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_int @ A @ I3 )
& ( ord_less_eq_int @ I3 @ B ) ) ) ) ).
% finite_interval_int3
thf(fact_9429_finite__nth__roots,axiom,
! [N2: nat,C: complex] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N2 )
= C ) ) ) ) ).
% finite_nth_roots
thf(fact_9430_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_9431_finite__divisors__int,axiom,
! [I: int] :
( ( I != zero_zero_int )
=> ( finite_finite_int
@ ( collect_int
@ ^ [D3: int] : ( dvd_dvd_int @ D3 @ I ) ) ) ) ).
% finite_divisors_int
thf(fact_9432_bij__betw__nth__root__unity,axiom,
! [C: complex,N2: nat] :
( ( C != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N2 @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) )
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N2 )
= one_one_complex ) )
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N2 )
= C ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_9433_set__encode__insert,axiom,
! [A2: set_nat,N2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ N2 @ A2 )
=> ( ( nat_set_encode @ ( insert_nat @ N2 @ A2 ) )
= ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% set_encode_insert
thf(fact_9434_real__root__zero,axiom,
! [N2: nat] :
( ( root @ N2 @ zero_zero_real )
= zero_zero_real ) ).
% real_root_zero
thf(fact_9435_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ zero_zero_nat ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_9436_root__0,axiom,
! [X: real] :
( ( root @ zero_zero_nat @ X )
= zero_zero_real ) ).
% root_0
thf(fact_9437_real__root__eq__iff,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( root @ N2 @ X )
= ( root @ N2 @ Y ) )
= ( X = Y ) ) ) ).
% real_root_eq_iff
thf(fact_9438_real__root__eq__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( root @ N2 @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ) ).
% real_root_eq_0_iff
thf(fact_9439_real__root__less__iff,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% real_root_less_iff
thf(fact_9440_real__root__le__iff,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% real_root_le_iff
thf(fact_9441_real__root__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( root @ N2 @ one_one_real )
= one_one_real ) ) ).
% real_root_one
thf(fact_9442_real__root__eq__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( root @ N2 @ X )
= one_one_real )
= ( X = one_one_real ) ) ) ).
% real_root_eq_1_iff
thf(fact_9443_real__root__lt__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ ( root @ N2 @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% real_root_lt_0_iff
thf(fact_9444_real__root__gt__0__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ ( root @ N2 @ Y ) )
= ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_9445_real__root__ge__0__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ Y ) )
= ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_9446_real__root__le__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% real_root_le_0_iff
thf(fact_9447_real__root__gt__1__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ one_one_real @ ( root @ N2 @ Y ) )
= ( ord_less_real @ one_one_real @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_9448_real__root__lt__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ ( root @ N2 @ X ) @ one_one_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% real_root_lt_1_iff
thf(fact_9449_real__root__ge__1__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ one_one_real @ ( root @ N2 @ Y ) )
= ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_9450_real__root__le__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% real_root_le_1_iff
thf(fact_9451_real__root__pow__pos2,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_9452_real__root__minus,axiom,
! [N2: nat,X: real] :
( ( root @ N2 @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( root @ N2 @ X ) ) ) ).
% real_root_minus
thf(fact_9453_real__root__mult__exp,axiom,
! [M: nat,N2: nat,X: real] :
( ( root @ ( times_times_nat @ M @ N2 ) @ X )
= ( root @ M @ ( root @ N2 @ X ) ) ) ).
% real_root_mult_exp
thf(fact_9454_real__root__mult,axiom,
! [N2: nat,X: real,Y: real] :
( ( root @ N2 @ ( times_times_real @ X @ Y ) )
= ( times_times_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ).
% real_root_mult
thf(fact_9455_real__root__divide,axiom,
! [N2: nat,X: real,Y: real] :
( ( root @ N2 @ ( divide_divide_real @ X @ Y ) )
= ( divide_divide_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ).
% real_root_divide
thf(fact_9456_real__root__commute,axiom,
! [M: nat,N2: nat,X: real] :
( ( root @ M @ ( root @ N2 @ X ) )
= ( root @ N2 @ ( root @ M @ X ) ) ) ).
% real_root_commute
thf(fact_9457_real__root__inverse,axiom,
! [N2: nat,X: real] :
( ( root @ N2 @ ( inverse_inverse_real @ X ) )
= ( inverse_inverse_real @ ( root @ N2 @ X ) ) ) ).
% real_root_inverse
thf(fact_9458_real__root__pos__pos__le,axiom,
! [X: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ).
% real_root_pos_pos_le
thf(fact_9459_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_9460_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost_nat @ ( suc @ K ) )
= ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% atMost_Suc
thf(fact_9461_real__root__less__mono,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_9462_real__root__le__mono,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_9463_real__root__power,axiom,
! [N2: nat,X: real,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( root @ N2 @ ( power_power_real @ X @ K ) )
= ( power_power_real @ ( root @ N2 @ X ) @ K ) ) ) ).
% real_root_power
thf(fact_9464_real__root__abs,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( root @ N2 @ ( abs_abs_real @ X ) )
= ( abs_abs_real @ ( root @ N2 @ X ) ) ) ) ).
% real_root_abs
thf(fact_9465_sgn__root,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( sgn_sgn_real @ ( root @ N2 @ X ) )
= ( sgn_sgn_real @ X ) ) ) ).
% sgn_root
thf(fact_9466_atLeast0__atMost__Suc,axiom,
! [N2: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
= ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_9467_atLeast0__lessThan__Suc,axiom,
! [N2: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
= ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_9468_atLeastAtMost__insertL,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
= ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% atLeastAtMost_insertL
thf(fact_9469_atLeastAtMostSuc__conv,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) )
= ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_9470_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( set_or1269000886237332187st_nat @ M @ N2 )
= ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_9471_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_9472_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_9473_real__root__gt__zero,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_9474_real__root__strict__decreasing,axiom,
! [N2: nat,N3: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( root @ N3 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_9475_sqrt__def,axiom,
( sqrt
= ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% sqrt_def
thf(fact_9476_root__abs__power,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( abs_abs_real @ ( root @ N2 @ ( power_power_real @ Y @ N2 ) ) )
= ( abs_abs_real @ Y ) ) ) ).
% root_abs_power
thf(fact_9477_real__root__pos__pos,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_9478_real__root__strict__increasing,axiom,
! [N2: nat,N3: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N3 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_9479_real__root__decreasing,axiom,
! [N2: nat,N3: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( root @ N3 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_9480_real__root__pow__pos,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
= X ) ) ) ).
% real_root_pow_pos
thf(fact_9481_real__root__pos__unique,axiom,
! [N2: nat,Y: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( power_power_real @ Y @ N2 )
= X )
=> ( ( root @ N2 @ X )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_9482_real__root__power__cancel,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( root @ N2 @ ( power_power_real @ X @ N2 ) )
= X ) ) ) ).
% real_root_power_cancel
thf(fact_9483_odd__real__root__power__cancel,axiom,
! [N2: nat,X: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( root @ N2 @ ( power_power_real @ X @ N2 ) )
= X ) ) ).
% odd_real_root_power_cancel
thf(fact_9484_odd__real__root__unique,axiom,
! [N2: nat,Y: real,X: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( ( power_power_real @ Y @ N2 )
= X )
=> ( ( root @ N2 @ X )
= Y ) ) ) ).
% odd_real_root_unique
thf(fact_9485_odd__real__root__pow,axiom,
! [N2: nat,X: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
= X ) ) ).
% odd_real_root_pow
thf(fact_9486_real__root__increasing,axiom,
! [N2: nat,N3: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N3 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_9487_root__sgn__power,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( root @ N2 @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) ) )
= Y ) ) ).
% root_sgn_power
thf(fact_9488_sgn__power__root,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N2 @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N2 @ X ) ) @ N2 ) )
= X ) ) ).
% sgn_power_root
thf(fact_9489_set__decode__plus__power__2,axiom,
! [N2: nat,Z: nat] :
( ~ ( member_nat @ N2 @ ( nat_set_decode @ Z ) )
=> ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ Z ) )
= ( insert_nat @ N2 @ ( nat_set_decode @ Z ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_9490_ln__root,axiom,
! [N2: nat,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ln_ln_real @ ( root @ N2 @ B ) )
= ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% ln_root
thf(fact_9491_log__root,axiom,
! [N2: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( log @ B @ ( root @ N2 @ A ) )
= ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% log_root
thf(fact_9492_log__base__root,axiom,
! [N2: nat,B: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( log @ ( root @ N2 @ B ) @ X )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X ) ) ) ) ) ).
% log_base_root
thf(fact_9493_split__root,axiom,
! [P: real > $o,N2: nat,X: real] :
( ( P @ ( root @ N2 @ X ) )
= ( ( ( N2 = zero_zero_nat )
=> ( P @ zero_zero_real ) )
& ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ! [Y4: real] :
( ( ( times_times_real @ ( sgn_sgn_real @ Y4 ) @ ( power_power_real @ ( abs_abs_real @ Y4 ) @ N2 ) )
= X )
=> ( P @ Y4 ) ) ) ) ) ).
% split_root
thf(fact_9494_root__powr__inverse,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( root @ N2 @ X )
= ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_9495_infinite__int__iff__unbounded,axiom,
! [S2: set_int] :
( ( ~ ( finite_finite_int @ S2 ) )
= ( ! [M2: int] :
? [N4: int] :
( ( ord_less_int @ M2 @ ( abs_abs_int @ N4 ) )
& ( member_int @ N4 @ S2 ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_9496_infinite__int__iff__unbounded__le,axiom,
! [S2: set_int] :
( ( ~ ( finite_finite_int @ S2 ) )
= ( ! [M2: int] :
? [N4: int] :
( ( ord_less_eq_int @ M2 @ ( abs_abs_int @ N4 ) )
& ( member_int @ N4 @ S2 ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_9497_card__Collect__less__nat,axiom,
! [N2: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_nat @ I3 @ N2 ) ) )
= N2 ) ).
% card_Collect_less_nat
thf(fact_9498_card__atMost,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_9499_card__Collect__le__nat,axiom,
! [N2: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N2 ) ) )
= ( suc @ N2 ) ) ).
% card_Collect_le_nat
thf(fact_9500_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% card_atLeastAtMost
thf(fact_9501_card__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
= ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).
% card_atLeastAtMost_int
thf(fact_9502_card__less,axiom,
! [M7: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ K2 @ M7 )
& ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_9503_card__less__Suc,axiom,
! [M7: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ ( suc @ K2 ) @ M7 )
& ( ord_less_nat @ K2 @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ K2 @ M7 )
& ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_9504_card__less__Suc2,axiom,
! [M7: set_nat,I: nat] :
( ~ ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ ( suc @ K2 ) @ M7 )
& ( ord_less_nat @ K2 @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ K2 @ M7 )
& ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_9505_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
= ( nat2 @ U ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_9506_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_9507_atLeastAtMostPlus1__int__conv,axiom,
! [M: int,N2: int] :
( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N2 ) )
=> ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N2 ) )
= ( insert_int @ ( plus_plus_int @ one_one_int @ N2 ) @ ( set_or1266510415728281911st_int @ M @ N2 ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_9508_subset__eq__atLeast0__lessThan__card,axiom,
! [N3: set_nat,N2: nat] :
( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N3 ) @ N2 ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_9509_card__sum__le__nat__sum,axiom,
! [S2: set_nat] :
( ord_less_eq_nat
@ ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
@ ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ S2 ) ) ).
% card_sum_le_nat_sum
thf(fact_9510_card__nth__roots,axiom,
! [C: complex,N2: nat] :
( ( C != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N2 )
= C ) ) )
= N2 ) ) ) ).
% card_nth_roots
thf(fact_9511_card__roots__unity__eq,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N2 )
= one_one_complex ) ) )
= N2 ) ) ).
% card_roots_unity_eq
thf(fact_9512_infinite__nat__iff__unbounded,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M2: nat] :
? [N4: nat] :
( ( ord_less_nat @ M2 @ N4 )
& ( member_nat @ N4 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_9513_unbounded__k__infinite,axiom,
! [K: nat,S2: set_nat] :
( ! [M5: nat] :
( ( ord_less_nat @ K @ M5 )
=> ? [N8: nat] :
( ( ord_less_nat @ M5 @ N8 )
& ( member_nat @ N8 @ S2 ) ) )
=> ~ ( finite_finite_nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_9514_infinite__nat__iff__unbounded__le,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M2: nat] :
? [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
& ( member_nat @ N4 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_9515_and__int_Oelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se725231765392027082nd_int @ X @ Xa )
= Y )
=> ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_9516_and__int_Osimps,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K2: int,L2: int] :
( if_int
@ ( ( member_int @ K2 @ ( 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 ) ) ) )
@ ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
@ ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_9517_simp__from__to,axiom,
( set_or1266510415728281911st_int
= ( ^ [I3: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I3 ) @ bot_bot_set_int @ ( insert_int @ I3 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_9518_and__int_Opsimps,axiom,
! [K: int,L: int] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
=> ( ( ( ( member_int @ K @ ( 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 ) ) ) )
=> ( ( bit_se725231765392027082nd_int @ K @ L )
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member_int @ K @ ( 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 ) ) ) )
=> ( ( bit_se725231765392027082nd_int @ K @ L )
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_9519_and__int_Opelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se725231765392027082nd_int @ X @ Xa )
= Y )
=> ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) )
=> ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% and_int.pelims
thf(fact_9520_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_9521_set__decode__zero,axiom,
( ( nat_set_decode @ zero_zero_nat )
= bot_bot_set_nat ) ).
% set_decode_zero
thf(fact_9522_set__encode__empty,axiom,
( ( nat_set_encode @ bot_bot_set_nat )
= zero_zero_nat ) ).
% set_encode_empty
thf(fact_9523_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
= ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% atLeastLessThan_singleton
thf(fact_9524_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_9525_bot__enat__def,axiom,
bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% bot_enat_def
thf(fact_9526_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_9527_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_9528_lessThan__empty__iff,axiom,
! [N2: nat] :
( ( ( set_ord_lessThan_nat @ N2 )
= bot_bot_set_nat )
= ( N2 = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_9529_atLeastLessThanSuc,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_eq_nat @ M @ N2 )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
= ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N2 )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThanSuc
thf(fact_9530_atLeast1__lessThan__eq__remove0,axiom,
! [N2: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_9531_atLeast1__atMost__eq__remove0,axiom,
! [N2: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_9532_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_9533_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [K3: int,L3: int] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L3 ) )
=> ( ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( P @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
=> ( P @ K3 @ L3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_9534_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [I2: int,J2: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
=> ( ( ( ord_less_eq_int @ I2 @ J2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
=> ( P @ I2 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_9535_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_9536_drop__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_numeral_minus_bit1
thf(fact_9537_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_9538_drop__bit__nonnegative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N2 @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% drop_bit_nonnegative_int_iff
thf(fact_9539_drop__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N2 @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% drop_bit_negative_int_iff
thf(fact_9540_drop__bit__minus__one,axiom,
! [N2: nat] :
( ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% drop_bit_minus_one
thf(fact_9541_fst__divmod__nat,axiom,
! [M: nat,N2: nat] :
( ( product_fst_nat_nat @ ( divmod_nat @ M @ N2 ) )
= ( divide_divide_nat @ M @ N2 ) ) ).
% fst_divmod_nat
thf(fact_9542_snd__divmod__nat,axiom,
! [M: nat,N2: nat] :
( ( product_snd_nat_nat @ ( divmod_nat @ M @ N2 ) )
= ( modulo_modulo_nat @ M @ N2 ) ) ).
% snd_divmod_nat
thf(fact_9543_drop__bit__Suc__minus__bit0,axiom,
! [N2: nat,K: num] :
( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% drop_bit_Suc_minus_bit0
thf(fact_9544_drop__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% drop_bit_numeral_minus_bit0
thf(fact_9545_drop__bit__Suc__minus__bit1,axiom,
! [N2: nat,K: num] :
( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_Suc_minus_bit1
thf(fact_9546_drop__bit__push__bit__int,axiom,
! [M: nat,N2: nat,K: int] :
( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N2 @ K ) )
= ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N2 ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N2 @ M ) @ K ) ) ) ).
% drop_bit_push_bit_int
thf(fact_9547_drop__bit__int__def,axiom,
( bit_se8568078237143864401it_int
= ( ^ [N4: nat,K2: int] : ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% drop_bit_int_def
thf(fact_9548_finite__enumerate,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ? [R2: nat > nat] :
( ( strict1292158309912662752at_nat @ R2 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
& ! [N8: nat] :
( ( ord_less_nat @ N8 @ ( finite_card_nat @ S2 ) )
=> ( member_nat @ ( R2 @ N8 ) @ S2 ) ) ) ) ).
% finite_enumerate
thf(fact_9549_fst__divmod__integer,axiom,
! [K: code_integer,L: code_integer] :
( ( produc8508995932063986495nteger @ ( code_divmod_integer @ K @ L ) )
= ( divide6298287555418463151nteger @ K @ L ) ) ).
% fst_divmod_integer
thf(fact_9550_fst__divmod__abs,axiom,
! [K: code_integer,L: code_integer] :
( ( produc8508995932063986495nteger @ ( code_divmod_abs @ K @ L ) )
= ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L ) ) ) ).
% fst_divmod_abs
thf(fact_9551_drop__bit__of__Suc__0,axiom,
! [N2: nat] :
( ( bit_se8570568707652914677it_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).
% drop_bit_of_Suc_0
thf(fact_9552_drop__bit__nat__eq,axiom,
! [N2: nat,K: int] :
( ( bit_se8570568707652914677it_nat @ N2 @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se8568078237143864401it_int @ N2 @ K ) ) ) ).
% drop_bit_nat_eq
thf(fact_9553_quotient__of__denom__pos_H,axiom,
! [R4: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R4 ) ) ) ).
% quotient_of_denom_pos'
thf(fact_9554_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ Y )
=> ( ( bezw @ X @ Y )
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_9555_bezw_Osimps,axiom,
( bezw
= ( ^ [X3: nat,Y4: nat] : ( if_Pro3027730157355071871nt_int @ ( Y4 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X3 @ Y4 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X3 @ Y4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X3 @ Y4 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y4 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_9556_bezw_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_int_int] :
( ( ( bezw @ X @ Xa )
= Y )
=> ( ( ( Xa = zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa != zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_9557_drop__bit__nat__def,axiom,
( bit_se8570568707652914677it_nat
= ( ^ [N4: nat,M2: nat] : ( divide_divide_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% drop_bit_nat_def
thf(fact_9558_rat__sgn__code,axiom,
! [P3: rat] :
( ( quotient_of @ ( sgn_sgn_rat @ P3 ) )
= ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P3 ) ) ) @ one_one_int ) ) ).
% rat_sgn_code
thf(fact_9559_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N2 @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_9560_one__mod__minus__numeral,axiom,
! [N2: num] :
( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ) ).
% one_mod_minus_numeral
thf(fact_9561_minus__one__mod__numeral,axiom,
! [N2: num] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
= ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).
% minus_one_mod_numeral
thf(fact_9562_minus__numeral__mod__numeral,axiom,
! [M: num,N2: num] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
= ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).
% minus_numeral_mod_numeral
thf(fact_9563_numeral__mod__minus__numeral,axiom,
! [M: num,N2: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ) ).
% numeral_mod_minus_numeral
thf(fact_9564_Divides_Oadjust__mod__def,axiom,
( adjust_mod
= ( ^ [L2: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L2 @ R5 ) ) ) ) ).
% Divides.adjust_mod_def
thf(fact_9565_bezw_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_int_int] :
( ( ( bezw @ X @ Xa )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
=> ~ ( ( ( ( Xa = zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa != zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% bezw.pelims
thf(fact_9566_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_9567_gcd__1__int,axiom,
! [M: int] :
( ( gcd_gcd_int @ M @ one_one_int )
= one_one_int ) ).
% gcd_1_int
thf(fact_9568_gcd__pos__int,axiom,
! [M: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N2 ) )
= ( ( M != zero_zero_int )
| ( N2 != zero_zero_int ) ) ) ).
% gcd_pos_int
thf(fact_9569_gcd__neg__numeral__1__int,axiom,
! [N2: num,X: int] :
( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ X )
= ( gcd_gcd_int @ ( numeral_numeral_int @ N2 ) @ X ) ) ).
% gcd_neg_numeral_1_int
thf(fact_9570_gcd__neg__numeral__2__int,axiom,
! [X: int,N2: num] :
( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( gcd_gcd_int @ X @ ( numeral_numeral_int @ N2 ) ) ) ).
% gcd_neg_numeral_2_int
thf(fact_9571_gcd__0__int,axiom,
! [X: int] :
( ( gcd_gcd_int @ X @ zero_zero_int )
= ( abs_abs_int @ X ) ) ).
% gcd_0_int
thf(fact_9572_gcd__0__left__int,axiom,
! [X: int] :
( ( gcd_gcd_int @ zero_zero_int @ X )
= ( abs_abs_int @ X ) ) ).
% gcd_0_left_int
thf(fact_9573_gcd__ge__0__int,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ).
% gcd_ge_0_int
thf(fact_9574_bezout__int,axiom,
! [X: int,Y: int] :
? [U3: int,V2: int] :
( ( plus_plus_int @ ( times_times_int @ U3 @ X ) @ ( times_times_int @ V2 @ Y ) )
= ( gcd_gcd_int @ X @ Y ) ) ).
% bezout_int
thf(fact_9575_gcd__le1__int,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% gcd_le1_int
thf(fact_9576_gcd__le2__int,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% gcd_le2_int
thf(fact_9577_gcd__cases__int,axiom,
! [X: int,Y: int,P: int > $o] :
( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( P @ ( gcd_gcd_int @ X @ Y ) ) ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
=> ( ( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
=> ( ( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
=> ( P @ ( gcd_gcd_int @ X @ Y ) ) ) ) ) ) ).
% gcd_cases_int
thf(fact_9578_gcd__unique__int,axiom,
! [D: int,A: int,B: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ D )
& ( dvd_dvd_int @ D @ A )
& ( dvd_dvd_int @ D @ B )
& ! [E3: int] :
( ( ( dvd_dvd_int @ E3 @ A )
& ( dvd_dvd_int @ E3 @ B ) )
=> ( dvd_dvd_int @ E3 @ D ) ) )
= ( D
= ( gcd_gcd_int @ A @ B ) ) ) ).
% gcd_unique_int
thf(fact_9579_gcd__non__0__int,axiom,
! [Y: int,X: int] :
( ( ord_less_int @ zero_zero_int @ Y )
=> ( ( gcd_gcd_int @ X @ Y )
= ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).
% gcd_non_0_int
thf(fact_9580_gcd__code__int,axiom,
( gcd_gcd_int
= ( ^ [K2: int,L2: int] : ( abs_abs_int @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( gcd_gcd_int @ L2 @ ( modulo_modulo_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ).
% gcd_code_int
thf(fact_9581_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq_nat @ Xa @ X )
=> ( Y
= ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_9582_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K2: code_integer,L2: code_integer] :
( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
@ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
@ ( if_Pro6119634080678213985nteger
@ ( ( sgn_sgn_Code_integer @ K2 )
= ( sgn_sgn_Code_integer @ L2 ) )
@ ( code_divmod_abs @ K2 @ L2 )
@ ( produc6916734918728496179nteger
@ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S4 ) ) )
@ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_9583_gcd__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( gcd_gcd_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% gcd_nat.eq_neutr_iff
thf(fact_9584_gcd__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( gcd_gcd_nat @ zero_zero_nat @ A )
= A ) ).
% gcd_nat.left_neutral
thf(fact_9585_gcd__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( gcd_gcd_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% gcd_nat.neutr_eq_iff
thf(fact_9586_gcd__nat_Oright__neutral,axiom,
! [A: nat] :
( ( gcd_gcd_nat @ A @ zero_zero_nat )
= A ) ).
% gcd_nat.right_neutral
thf(fact_9587_gcd__0__nat,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ X @ zero_zero_nat )
= X ) ).
% gcd_0_nat
thf(fact_9588_gcd__0__left__nat,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ zero_zero_nat @ X )
= X ) ).
% gcd_0_left_nat
thf(fact_9589_gcd__1__nat,axiom,
! [M: nat] :
( ( gcd_gcd_nat @ M @ one_one_nat )
= one_one_nat ) ).
% gcd_1_nat
thf(fact_9590_gcd__Suc__0,axiom,
! [M: nat] :
( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( suc @ zero_zero_nat ) ) ).
% gcd_Suc_0
thf(fact_9591_gcd__pos__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N2 ) )
= ( ( M != zero_zero_nat )
| ( N2 != zero_zero_nat ) ) ) ).
% gcd_pos_nat
thf(fact_9592_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_9593_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_9594_gcd__diff1__nat,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 )
= ( gcd_gcd_nat @ M @ N2 ) ) ) ).
% gcd_diff1_nat
thf(fact_9595_gcd__diff2__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( gcd_gcd_nat @ ( minus_minus_nat @ N2 @ M ) @ N2 )
= ( gcd_gcd_nat @ M @ N2 ) ) ) ).
% gcd_diff2_nat
thf(fact_9596_gcd__nat_Oelims,axiom,
! [X: nat,Xa: nat,Y: nat] :
( ( ( gcd_gcd_nat @ X @ Xa )
= Y )
=> ( ( ( Xa = zero_zero_nat )
=> ( Y = X ) )
& ( ( Xa != zero_zero_nat )
=> ( Y
= ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) ) ) ).
% gcd_nat.elims
thf(fact_9597_gcd__nat_Osimps,axiom,
( gcd_gcd_nat
= ( ^ [X3: nat,Y4: nat] : ( if_nat @ ( Y4 = zero_zero_nat ) @ X3 @ ( gcd_gcd_nat @ Y4 @ ( modulo_modulo_nat @ X3 @ Y4 ) ) ) ) ) ).
% gcd_nat.simps
thf(fact_9598_gcd__non__0__nat,axiom,
! [Y: nat,X: nat] :
( ( Y != zero_zero_nat )
=> ( ( gcd_gcd_nat @ X @ Y )
= ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).
% gcd_non_0_nat
thf(fact_9599_bezout__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [X2: nat,Y2: nat] :
( ( times_times_nat @ A @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% bezout_nat
thf(fact_9600_bezout__gcd__nat_H,axiom,
! [B: nat,A: nat] :
? [X2: nat,Y2: nat] :
( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y2 ) @ ( times_times_nat @ A @ X2 ) )
& ( ( minus_minus_nat @ ( times_times_nat @ A @ X2 ) @ ( times_times_nat @ B @ Y2 ) )
= ( gcd_gcd_nat @ A @ B ) ) )
| ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y2 ) @ ( times_times_nat @ B @ X2 ) )
& ( ( minus_minus_nat @ ( times_times_nat @ B @ X2 ) @ ( times_times_nat @ A @ Y2 ) )
= ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_9601_bezw__aux,axiom,
! [X: nat,Y: nat] :
( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y ) )
= ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).
% bezw_aux
thf(fact_9602_gcd__nat_Opelims,axiom,
! [X: nat,Xa: nat,Y: nat] :
( ( ( gcd_gcd_nat @ X @ Xa )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
=> ~ ( ( ( ( Xa = zero_zero_nat )
=> ( Y = X ) )
& ( ( Xa != zero_zero_nat )
=> ( Y
= ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_9603_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
= ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_9604_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_9605_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
= ( set_or5832277885323065728an_int @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_9606_Code__Target__Int_Onegative__def,axiom,
( code_Target_negative
= ( comp_int_int_num @ uminus_uminus_int @ numeral_numeral_int ) ) ).
% Code_Target_Int.negative_def
thf(fact_9607_xor__minus__numerals_I2_J,axiom,
! [K: int,N2: num] :
( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N2 @ one ) ) ) ) ).
% xor_minus_numerals(2)
thf(fact_9608_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% card_greaterThanLessThan
thf(fact_9609_xor__minus__numerals_I1_J,axiom,
! [N2: num,K: int] :
( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ K )
= ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N2 @ one ) @ K ) ) ) ).
% xor_minus_numerals(1)
thf(fact_9610_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_9611_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_9612_sub__BitM__One__eq,axiom,
! [N2: num] :
( ( neg_numeral_sub_int @ ( bitM @ N2 ) @ one )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N2 @ one ) ) ) ).
% sub_BitM_One_eq
thf(fact_9613_nat__of__integer__non__positive,axiom,
! [K: code_integer] :
( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
=> ( ( code_nat_of_integer @ K )
= zero_zero_nat ) ) ).
% nat_of_integer_non_positive
thf(fact_9614_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
@ ^ [X3: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X3 )
@ ^ [X3: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X3 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_9615_Suc__funpow,axiom,
! [N2: nat] :
( ( compow_nat_nat @ N2 @ suc )
= ( plus_plus_nat @ N2 ) ) ).
% Suc_funpow
thf(fact_9616_nat__of__integer__code__post_I1_J,axiom,
( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
= zero_zero_nat ) ).
% nat_of_integer_code_post(1)
thf(fact_9617_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_9618_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_9619_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
@ ^ [M2: nat,N4: nat] :
( ( dvd_dvd_nat @ M2 @ N4 )
& ( M2 != N4 ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_9620_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K2: code_integer] :
( if_nat @ ( ord_le3102999989581377725nteger @ K2 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
@ ( produc1555791787009142072er_nat
@ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
@ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_9621_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K2: code_integer] :
( if_int @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K2 ) ) )
@ ( if_int @ ( K2 = zero_z3403309356797280102nteger ) @ zero_zero_int
@ ( produc1553301316500091796er_int
@ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
@ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_9622_times__int_Oabs__eq,axiom,
! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y4 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y4 @ U2 ) ) ) )
@ Xa
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_9623_zero__integer_Orep__eq,axiom,
( ( code_int_of_integer @ zero_z3403309356797280102nteger )
= zero_zero_int ) ).
% zero_integer.rep_eq
thf(fact_9624_int__of__integer__numeral,axiom,
! [K: num] :
( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
= ( numeral_numeral_int @ K ) ) ).
% int_of_integer_numeral
thf(fact_9625_plus__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X @ Xa ) )
= ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% plus_integer.rep_eq
thf(fact_9626_one__integer_Orep__eq,axiom,
( ( code_int_of_integer @ one_one_Code_integer )
= one_one_int ) ).
% one_integer.rep_eq
thf(fact_9627_divide__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa ) )
= ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% divide_integer.rep_eq
thf(fact_9628_eq__Abs__Integ,axiom,
! [Z: int] :
~ ! [X2: nat,Y2: nat] :
( Z
!= ( abs_Integ @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) ) ).
% eq_Abs_Integ
thf(fact_9629_int_Oabs__induct,axiom,
! [P: int > $o,X: int] :
( ! [Y2: product_prod_nat_nat] : ( P @ ( abs_Integ @ Y2 ) )
=> ( P @ X ) ) ).
% int.abs_induct
thf(fact_9630_less__integer_Orep__eq,axiom,
( ord_le6747313008572928689nteger
= ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_integer.rep_eq
thf(fact_9631_integer__less__iff,axiom,
( ord_le6747313008572928689nteger
= ( ^ [K2: code_integer,L2: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% integer_less_iff
thf(fact_9632_less__eq__integer_Orep__eq,axiom,
( ord_le3102999989581377725nteger
= ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_eq_integer.rep_eq
thf(fact_9633_integer__less__eq__iff,axiom,
( ord_le3102999989581377725nteger
= ( ^ [K2: code_integer,L2: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% integer_less_eq_iff
thf(fact_9634_nat_Oabs__eq,axiom,
! [X: product_prod_nat_nat] :
( ( nat2 @ ( abs_Integ @ X ) )
= ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).
% nat.abs_eq
thf(fact_9635_zero__int__def,axiom,
( zero_zero_int
= ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% zero_int_def
thf(fact_9636_int__def,axiom,
( semiri1314217659103216013at_int
= ( ^ [N4: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N4 @ zero_zero_nat ) ) ) ) ).
% int_def
thf(fact_9637_uminus__int_Oabs__eq,axiom,
! [X: product_prod_nat_nat] :
( ( uminus_uminus_int @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X3: nat,Y4: nat] : ( product_Pair_nat_nat @ Y4 @ X3 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_9638_one__int__def,axiom,
( one_one_int
= ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% one_int_def
thf(fact_9639_less__int_Oabs__eq,axiom,
! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( ord_less_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y4: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) )
@ Xa
@ X ) ) ).
% less_int.abs_eq
thf(fact_9640_less__eq__int_Oabs__eq,axiom,
! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( ord_less_eq_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y4: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) )
@ Xa
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_9641_plus__int_Oabs__eq,axiom,
! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( plus_plus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y4 @ V4 ) ) )
@ Xa
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_9642_minus__int_Oabs__eq,axiom,
! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( minus_minus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y4 @ U2 ) ) )
@ Xa
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_9643_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_9644_num__of__nat_Osimps_I2_J,axiom,
! [N2: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( num_of_nat @ ( suc @ N2 ) )
= ( inc @ ( num_of_nat @ N2 ) ) ) )
& ( ~ ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( num_of_nat @ ( suc @ N2 ) )
= one ) ) ) ).
% num_of_nat.simps(2)
thf(fact_9645_num__of__nat__numeral__eq,axiom,
! [Q3: num] :
( ( num_of_nat @ ( numeral_numeral_nat @ Q3 ) )
= Q3 ) ).
% num_of_nat_numeral_eq
thf(fact_9646_Gcd__nat__eq__one,axiom,
! [N3: set_nat] :
( ( member_nat @ one_one_nat @ N3 )
=> ( ( gcd_Gcd_nat @ N3 )
= one_one_nat ) ) ).
% Gcd_nat_eq_one
thf(fact_9647_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ zero_zero_nat )
= one ) ).
% num_of_nat.simps(1)
thf(fact_9648_numeral__num__of__nat,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( numeral_numeral_nat @ ( num_of_nat @ N2 ) )
= N2 ) ) ).
% numeral_num_of_nat
thf(fact_9649_num__of__nat__One,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ one_one_nat )
=> ( ( num_of_nat @ N2 )
= one ) ) ).
% num_of_nat_One
thf(fact_9650_num__of__nat__double,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( num_of_nat @ ( plus_plus_nat @ N2 @ N2 ) )
= ( bit0 @ ( num_of_nat @ N2 ) ) ) ) ).
% num_of_nat_double
thf(fact_9651_num__of__nat__plus__distrib,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( num_of_nat @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N2 ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_9652_less__eq__int_Orep__eq,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Xa4: int] :
( produc8739625826339149834_nat_o
@ ^ [Y4: nat,Z5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y4 @ V4 ) @ ( plus_plus_nat @ U2 @ Z5 ) ) )
@ ( rep_Integ @ X3 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_9653_less__int_Orep__eq,axiom,
( ord_less_int
= ( ^ [X3: int,Xa4: int] :
( produc8739625826339149834_nat_o
@ ^ [Y4: nat,Z5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y4 @ V4 ) @ ( plus_plus_nat @ U2 @ Z5 ) ) )
@ ( rep_Integ @ X3 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_9654_Gcd__int__greater__eq__0,axiom,
! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).
% Gcd_int_greater_eq_0
thf(fact_9655_nat_Orep__eq,axiom,
( nat2
= ( ^ [X3: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X3 ) ) ) ) ).
% nat.rep_eq
thf(fact_9656_uminus__int__def,axiom,
( uminus_uminus_int
= ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X3: nat,Y4: nat] : ( product_Pair_nat_nat @ Y4 @ X3 ) ) ) ) ).
% uminus_int_def
thf(fact_9657_prod__encode__def,axiom,
( nat_prod_encode
= ( produc6842872674320459806at_nat
@ ^ [M2: nat,N4: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M2 @ N4 ) ) @ M2 ) ) ) ).
% prod_encode_def
thf(fact_9658_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_9659_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_9660_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_9661_times__int__def,axiom,
( times_times_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y4 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y4 @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_9662_minus__int__def,axiom,
( minus_minus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y4 @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_9663_plus__int__def,axiom,
( plus_plus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y4 @ V4 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_9664_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Y
= ( Xa != one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Deg2 = Xa )
& ! [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 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi2: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi2 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( 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 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
& ( ( Mi2 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X3: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
=> ( ( ord_less_nat @ Mi2 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_9665_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_valid @ X @ Xa )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa != one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ~ ( ( Deg2 = Xa )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( 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 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi2: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi2 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( 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 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
& ( ( Mi2 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X3: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
=> ( ( ord_less_nat @ Mi2 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_9666_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
= ( ( Deg = Deg4 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_VEBT_valid @ X3 @ ( 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 @ TreeList )
= ( 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 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi2: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi2 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( 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 @ TreeList @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
& ( ( Mi2 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
& ! [X3: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X3 )
=> ( ( ord_less_nat @ Mi2 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_9667_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N2: nat,J: nat,I: nat] :
( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N2 )
= ( suc @ ( plus_plus_nat @ I @ N2 ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_9668_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa = one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( Deg2 = Xa )
& ! [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 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi2: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi2 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( 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 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
& ( ( Mi2 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X3: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
=> ( ( ord_less_nat @ Mi2 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_9669_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Y
= ( Xa = one_one_nat ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( Y
= ( ( Deg2 = Xa )
& ! [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 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi2: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi2 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( 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 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
& ( ( Mi2 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X3: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
=> ( ( ord_less_nat @ Mi2 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_9670_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_valid @ X @ Xa )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
=> ( Xa != 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 ) @ Xa ) )
=> ~ ( ( Deg2 = Xa )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( 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 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi2: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi2 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( 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 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
& ( ( Mi2 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X3: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
=> ( ( ord_less_nat @ Mi2 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_9671_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
=> ( Xa = 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 ) @ Xa ) )
=> ( ( Deg2 = Xa )
& ! [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 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi2: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi2 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( 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 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi2 = Ma3 )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X6 ) ) )
& ( ( Mi2 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X3: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
=> ( ( ord_less_nat @ Mi2 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_9672_take__bit__numeral__minus__numeral__int,axiom,
! [M: num,N2: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( 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 ) @ N2 ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_9673_and__minus__numerals_I3_J,axiom,
! [M: num,N2: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).
% and_minus_numerals(3)
thf(fact_9674_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_9675_take__bit__num__simps_I2_J,axiom,
! [N2: nat] :
( ( bit_take_bit_num @ ( suc @ N2 ) @ one )
= ( some_num @ one ) ) ).
% take_bit_num_simps(2)
thf(fact_9676_take__bit__num__simps_I5_J,axiom,
! [R4: num] :
( ( bit_take_bit_num @ ( numeral_numeral_nat @ R4 ) @ one )
= ( some_num @ one ) ) ).
% take_bit_num_simps(5)
thf(fact_9677_take__bit__num__simps_I3_J,axiom,
! [N2: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) )
= ( case_o6005452278849405969um_num @ none_num
@ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ N2 @ M ) ) ) ).
% take_bit_num_simps(3)
thf(fact_9678_take__bit__num__simps_I4_J,axiom,
! [N2: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M ) )
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) ) ) ).
% take_bit_num_simps(4)
thf(fact_9679_take__bit__num__simps_I6_J,axiom,
! [R4: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral_nat @ R4 ) @ ( bit0 @ M ) )
= ( case_o6005452278849405969um_num @ none_num
@ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R4 ) @ M ) ) ) ).
% take_bit_num_simps(6)
thf(fact_9680_take__bit__num__simps_I7_J,axiom,
! [R4: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral_nat @ R4 ) @ ( bit1 @ M ) )
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R4 ) @ M ) ) ) ) ).
% take_bit_num_simps(7)
thf(fact_9681_and__minus__numerals_I8_J,axiom,
! [N2: num,M: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).
% and_minus_numerals(8)
thf(fact_9682_and__minus__numerals_I4_J,axiom,
! [M: num,N2: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).
% and_minus_numerals(4)
thf(fact_9683_and__minus__numerals_I7_J,axiom,
! [N2: num,M: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).
% and_minus_numerals(7)
thf(fact_9684_and__not__num_Osimps_I1_J,axiom,
( ( bit_and_not_num @ one @ one )
= none_num ) ).
% and_not_num.simps(1)
thf(fact_9685_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
! [N2: nat,M: num] :
( ( bit_take_bit_num @ N2 @ ( bit0 @ M ) )
= ( case_nat_option_num @ none_num
@ ^ [N4: nat] :
( case_o6005452278849405969um_num @ none_num
@ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ N4 @ M ) )
@ N2 ) ) ).
% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_9686_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
! [N2: nat] :
( ( bit_take_bit_num @ N2 @ one )
= ( case_nat_option_num @ none_num
@ ^ [N4: nat] : ( some_num @ one )
@ N2 ) ) ).
% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_9687_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_9688_and__not__num_Osimps_I2_J,axiom,
! [N2: num] :
( ( bit_and_not_num @ one @ ( bit0 @ N2 ) )
= ( some_num @ one ) ) ).
% and_not_num.simps(2)
thf(fact_9689_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_9690_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_9691_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_9692_and__not__num_Osimps_I3_J,axiom,
! [N2: num] :
( ( bit_and_not_num @ one @ ( bit1 @ N2 ) )
= none_num ) ).
% and_not_num.simps(3)
thf(fact_9693_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
! [N2: nat,M: num] :
( ( bit_take_bit_num @ N2 @ ( bit1 @ M ) )
= ( case_nat_option_num @ none_num
@ ^ [N4: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N4 @ M ) ) )
@ N2 ) ) ).
% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_9694_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_9695_and__not__num__eq__Some__iff,axiom,
! [M: num,N2: num,Q3: num] :
( ( ( bit_and_not_num @ M @ N2 )
= ( some_num @ Q3 ) )
= ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
= ( numeral_numeral_int @ Q3 ) ) ) ).
% and_not_num_eq_Some_iff
thf(fact_9696_and__not__num_Osimps_I8_J,axiom,
! [M: num,N2: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M @ N2 ) ) ) ).
% and_not_num.simps(8)
thf(fact_9697_and__not__num__eq__None__iff,axiom,
! [M: num,N2: num] :
( ( ( bit_and_not_num @ M @ N2 )
= none_num )
= ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
= zero_zero_int ) ) ).
% and_not_num_eq_None_iff
thf(fact_9698_int__numeral__not__and__num,axiom,
! [M: num,N2: num] :
( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N2 @ M ) ) ) ).
% int_numeral_not_and_num
thf(fact_9699_int__numeral__and__not__num,axiom,
! [M: num,N2: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N2 ) ) ) ).
% int_numeral_and_not_num
thf(fact_9700_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N4: nat,M2: num] :
( if_option_num
@ ( ( bit_se2925701944663578781it_nat @ N4 @ ( numeral_numeral_nat @ M2 ) )
= zero_zero_nat )
@ none_num
@ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N4 @ ( numeral_numeral_nat @ M2 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_9701_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N2: nat,J: nat,I: nat] :
( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ I ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N2 )
= ( suc @ ( plus_plus_nat @ I @ N2 ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_9702_pow_Osimps_I3_J,axiom,
! [X: num,Y: num] :
( ( pow @ X @ ( bit1 @ Y ) )
= ( times_times_num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).
% pow.simps(3)
thf(fact_9703_sqr_Osimps_I2_J,axiom,
! [N2: num] :
( ( sqr @ ( bit0 @ N2 ) )
= ( bit0 @ ( bit0 @ ( sqr @ N2 ) ) ) ) ).
% sqr.simps(2)
thf(fact_9704_sqr_Osimps_I1_J,axiom,
( ( sqr @ one )
= one ) ).
% sqr.simps(1)
thf(fact_9705_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_9706_sqr__conv__mult,axiom,
( sqr
= ( ^ [X3: num] : ( times_times_num @ X3 @ X3 ) ) ) ).
% sqr_conv_mult
thf(fact_9707_pow_Osimps_I2_J,axiom,
! [X: num,Y: num] :
( ( pow @ X @ ( bit0 @ Y ) )
= ( sqr @ ( pow @ X @ Y ) ) ) ).
% pow.simps(2)
thf(fact_9708_sqr_Osimps_I3_J,axiom,
! [N2: num] :
( ( sqr @ ( bit1 @ N2 ) )
= ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N2 ) @ N2 ) ) ) ) ).
% sqr.simps(3)
thf(fact_9709_and__not__num_Oelims,axiom,
! [X: num,Xa: num,Y: option_num] :
( ( ( bit_and_not_num @ X @ Xa )
= Y )
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( Y != none_num ) ) )
=> ( ( ( X = one )
=> ( ? [N: num] :
( Xa
= ( bit0 @ N ) )
=> ( Y
!= ( some_num @ one ) ) ) )
=> ( ( ( X = one )
=> ( ? [N: num] :
( Xa
= ( bit1 @ N ) )
=> ( Y != none_num ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one )
=> ( Y
!= ( some_num @ ( bit0 @ M5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit0 @ N ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit1 @ N ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one )
=> ( Y
!= ( some_num @ ( bit0 @ M5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit0 @ N ) )
=> ( Y
!= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M5 @ N ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit1 @ N ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
thf(fact_9710_Bit__Operations_Otake__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N4: nat,M2: num] :
( produc478579273971653890on_num
@ ^ [A3: nat,X3: 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 ) ) )
@ X3 )
@ A3 )
@ ( product_Pair_nat_num @ N4 @ M2 ) ) ) ) ).
% Bit_Operations.take_bit_num_code
thf(fact_9711_xor__num_Osimps_I6_J,axiom,
! [M: num,N2: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ).
% xor_num.simps(6)
thf(fact_9712_xor__num_Osimps_I5_J,axiom,
! [M: num,N2: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ).
% xor_num.simps(5)
thf(fact_9713_xor__num_Osimps_I9_J,axiom,
! [M: num,N2: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ).
% xor_num.simps(9)
thf(fact_9714_and__not__num_Osimps_I5_J,axiom,
! [M: num,N2: num] :
( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ).
% and_not_num.simps(5)
thf(fact_9715_and__not__num_Osimps_I6_J,axiom,
! [M: num,N2: num] :
( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ).
% and_not_num.simps(6)
thf(fact_9716_and__not__num_Osimps_I9_J,axiom,
! [M: num,N2: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ).
% and_not_num.simps(9)
thf(fact_9717_xor__num_Osimps_I1_J,axiom,
( ( bit_un2480387367778600638or_num @ one @ one )
= none_num ) ).
% xor_num.simps(1)
thf(fact_9718_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
= ( set_or6656581121297822940st_int @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_9719_xor__num_Oelims,axiom,
! [X: num,Xa: num,Y: option_num] :
( ( ( bit_un2480387367778600638or_num @ X @ Xa )
= Y )
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( Y != none_num ) ) )
=> ( ( ( X = one )
=> ! [N: num] :
( ( Xa
= ( bit0 @ N ) )
=> ( Y
!= ( some_num @ ( bit1 @ N ) ) ) ) )
=> ( ( ( X = one )
=> ! [N: num] :
( ( Xa
= ( bit1 @ N ) )
=> ( Y
!= ( some_num @ ( bit0 @ N ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one )
=> ( Y
!= ( some_num @ ( bit1 @ M5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit0 @ N ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit1 @ N ) )
=> ( Y
!= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one )
=> ( Y
!= ( some_num @ ( bit0 @ M5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit0 @ N ) )
=> ( Y
!= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N ) ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit1 @ N ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.elims
thf(fact_9720_xor__num_Osimps_I2_J,axiom,
! [N2: num] :
( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N2 ) )
= ( some_num @ ( bit1 @ N2 ) ) ) ).
% xor_num.simps(2)
thf(fact_9721_xor__num_Osimps_I3_J,axiom,
! [N2: num] :
( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N2 ) )
= ( some_num @ ( bit0 @ N2 ) ) ) ).
% xor_num.simps(3)
thf(fact_9722_xor__num_Osimps_I4_J,axiom,
! [M: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
= ( some_num @ ( bit1 @ M ) ) ) ).
% xor_num.simps(4)
thf(fact_9723_xor__num_Osimps_I7_J,axiom,
! [M: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
= ( some_num @ ( bit0 @ M ) ) ) ).
% xor_num.simps(7)
thf(fact_9724_xor__num_Osimps_I8_J,axiom,
! [M: num,N2: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ).
% xor_num.simps(8)
thf(fact_9725_and__num_Oelims,axiom,
! [X: num,Xa: num,Y: option_num] :
( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
= Y )
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( Y
!= ( some_num @ one ) ) ) )
=> ( ( ( X = one )
=> ( ? [N: num] :
( Xa
= ( bit0 @ N ) )
=> ( Y != none_num ) ) )
=> ( ( ( X = one )
=> ( ? [N: num] :
( Xa
= ( bit1 @ N ) )
=> ( Y
!= ( some_num @ one ) ) ) )
=> ( ( ? [M5: num] :
( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one )
=> ( Y != none_num ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit0 @ N ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit1 @ N ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N ) ) ) ) )
=> ( ( ? [M5: num] :
( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one )
=> ( Y
!= ( some_num @ one ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit0 @ N ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit1 @ N ) )
=> ( Y
!= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M5 @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.elims
thf(fact_9726_xor__num__dict,axiom,
bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% xor_num_dict
thf(fact_9727_and__num_Osimps_I1_J,axiom,
( ( bit_un7362597486090784418nd_num @ one @ one )
= ( some_num @ one ) ) ).
% and_num.simps(1)
thf(fact_9728_and__num_Osimps_I5_J,axiom,
! [M: num,N2: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).
% and_num.simps(5)
thf(fact_9729_and__num_Osimps_I3_J,axiom,
! [N2: num] :
( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N2 ) )
= ( some_num @ one ) ) ).
% and_num.simps(3)
thf(fact_9730_and__num_Osimps_I7_J,axiom,
! [M: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
= ( some_num @ one ) ) ).
% and_num.simps(7)
thf(fact_9731_and__num_Osimps_I2_J,axiom,
! [N2: num] :
( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N2 ) )
= none_num ) ).
% and_num.simps(2)
thf(fact_9732_and__num_Osimps_I4_J,axiom,
! [M: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
= none_num ) ).
% and_num.simps(4)
thf(fact_9733_and__num_Osimps_I8_J,axiom,
! [M: num,N2: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).
% and_num.simps(8)
thf(fact_9734_and__num_Osimps_I6_J,axiom,
! [M: num,N2: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).
% and_num.simps(6)
thf(fact_9735_and__num_Osimps_I9_J,axiom,
! [M: num,N2: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).
% and_num.simps(9)
thf(fact_9736_and__num__dict,axiom,
bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% and_num_dict
thf(fact_9737_Rats__eq__int__div__nat,axiom,
( field_5140801741446780682s_real
= ( collect_real
@ ^ [Uu3: real] :
? [I3: int,N4: nat] :
( ( Uu3
= ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( semiri5074537144036343181t_real @ N4 ) ) )
& ( N4 != zero_zero_nat ) ) ) ) ).
% Rats_eq_int_div_nat
thf(fact_9738_image__minus__const__atLeastLessThan__nat,axiom,
! [C: nat,Y: nat,X: nat] :
( ( ( ord_less_nat @ C @ Y )
=> ( ( image_nat_nat
@ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
& ( ~ ( ord_less_nat @ C @ Y )
=> ( ( ( ord_less_nat @ X @ Y )
=> ( ( image_nat_nat
@ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
& ( ~ ( ord_less_nat @ X @ Y )
=> ( ( image_nat_nat
@ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= bot_bot_set_nat ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_9739_bij__betw__Suc,axiom,
! [M7: set_nat,N3: set_nat] :
( ( bij_betw_nat_nat @ suc @ M7 @ N3 )
= ( ( image_nat_nat @ suc @ M7 )
= N3 ) ) ).
% bij_betw_Suc
thf(fact_9740_Rats__abs__iff,axiom,
! [X: real] :
( ( member_real @ ( abs_abs_real @ X ) @ field_5140801741446780682s_real )
= ( member_real @ X @ field_5140801741446780682s_real ) ) ).
% Rats_abs_iff
thf(fact_9741_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
= ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_9742_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_9743_Rats__no__top__le,axiom,
! [X: real] :
? [X2: real] :
( ( member_real @ X2 @ field_5140801741446780682s_real )
& ( ord_less_eq_real @ X @ X2 ) ) ).
% Rats_no_top_le
thf(fact_9744_Rats__dense__in__real,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [X2: real] :
( ( member_real @ X2 @ field_5140801741446780682s_real )
& ( ord_less_real @ X @ X2 )
& ( ord_less_real @ X2 @ Y ) ) ) ).
% Rats_dense_in_real
thf(fact_9745_Rats__no__bot__less,axiom,
! [X: real] :
? [X2: real] :
( ( member_real @ X2 @ field_5140801741446780682s_real )
& ( ord_less_real @ X2 @ X ) ) ).
% Rats_no_bot_less
thf(fact_9746_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_9747_image__Suc__lessThan,axiom,
! [N2: nat] :
( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ).
% image_Suc_lessThan
thf(fact_9748_image__Suc__atMost,axiom,
! [N2: nat] :
( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N2 ) ) ) ).
% image_Suc_atMost
thf(fact_9749_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N2: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_9750_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N2: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_9751_lessThan__Suc__eq__insert__0,axiom,
! [N2: nat] :
( ( set_ord_lessThan_nat @ ( suc @ N2 ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_9752_atMost__Suc__eq__insert__0,axiom,
! [N2: nat] :
( ( set_ord_atMost_nat @ ( suc @ N2 ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_9753_Rats__eq__int__div__int,axiom,
( field_5140801741446780682s_real
= ( collect_real
@ ^ [Uu3: real] :
? [I3: int,J3: int] :
( ( Uu3
= ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( ring_1_of_int_real @ J3 ) ) )
& ( J3 != zero_zero_int ) ) ) ) ).
% Rats_eq_int_div_int
thf(fact_9754_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_9755_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_9756_floor__Fract,axiom,
! [A: int,B: int] :
( ( archim3151403230148437115or_rat @ ( fract @ A @ B ) )
= ( divide_divide_int @ A @ B ) ) ).
% floor_Fract
thf(fact_9757_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_9758_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_9759_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_9760_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_9761_eq__rat_I3_J,axiom,
! [A: int,C: int] :
( ( fract @ zero_zero_int @ A )
= ( fract @ zero_zero_int @ C ) ) ).
% eq_rat(3)
thf(fact_9762_eq__rat_I2_J,axiom,
! [A: int] :
( ( fract @ A @ zero_zero_int )
= ( fract @ zero_zero_int @ one_one_int ) ) ).
% eq_rat(2)
thf(fact_9763_Rat__induct__pos,axiom,
! [P: rat > $o,Q3: rat] :
( ! [A5: int,B5: int] :
( ( ord_less_int @ zero_zero_int @ B5 )
=> ( P @ ( fract @ A5 @ B5 ) ) )
=> ( P @ Q3 ) ) ).
% Rat_induct_pos
thf(fact_9764_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_9765_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_9766_Fract__of__nat__eq,axiom,
! [K: nat] :
( ( fract @ ( semiri1314217659103216013at_int @ K ) @ one_one_int )
= ( semiri681578069525770553at_rat @ K ) ) ).
% Fract_of_nat_eq
thf(fact_9767_rat__number__collapse_I6_J,axiom,
! [K: int] :
( ( fract @ K @ zero_zero_int )
= zero_zero_rat ) ).
% rat_number_collapse(6)
thf(fact_9768_rat__number__collapse_I1_J,axiom,
! [K: int] :
( ( fract @ zero_zero_int @ K )
= zero_zero_rat ) ).
% rat_number_collapse(1)
thf(fact_9769_Fract__coprime,axiom,
! [A: int,B: int] :
( ( fract @ ( divide_divide_int @ A @ ( gcd_gcd_int @ A @ B ) ) @ ( divide_divide_int @ B @ ( gcd_gcd_int @ A @ B ) ) )
= ( fract @ A @ B ) ) ).
% Fract_coprime
thf(fact_9770_One__rat__def,axiom,
( one_one_rat
= ( fract @ one_one_int @ one_one_int ) ) ).
% One_rat_def
thf(fact_9771_Fract__of__int__eq,axiom,
! [K: int] :
( ( fract @ K @ one_one_int )
= ( ring_1_of_int_rat @ K ) ) ).
% Fract_of_int_eq
thf(fact_9772_Fract__of__int__quotient,axiom,
( fract
= ( ^ [K2: int,L2: int] : ( divide_divide_rat @ ( ring_1_of_int_rat @ K2 ) @ ( ring_1_of_int_rat @ L2 ) ) ) ) ).
% Fract_of_int_quotient
thf(fact_9773_Zero__rat__def,axiom,
( zero_zero_rat
= ( fract @ zero_zero_int @ one_one_int ) ) ).
% Zero_rat_def
thf(fact_9774_rat__number__collapse_I3_J,axiom,
! [W: num] :
( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
= ( numeral_numeral_rat @ W ) ) ).
% rat_number_collapse(3)
thf(fact_9775_rat__number__expand_I3_J,axiom,
( numeral_numeral_rat
= ( ^ [K2: num] : ( fract @ ( numeral_numeral_int @ K2 ) @ one_one_int ) ) ) ).
% rat_number_expand(3)
thf(fact_9776_Fract__less__zero__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% Fract_less_zero_iff
thf(fact_9777_zero__less__Fract__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% zero_less_Fract_iff
thf(fact_9778_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_9779_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_9780_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_9781_Fract__add__one,axiom,
! [N2: int,M: int] :
( ( N2 != zero_zero_int )
=> ( ( fract @ ( plus_plus_int @ M @ N2 ) @ N2 )
= ( plus_plus_rat @ ( fract @ M @ N2 ) @ one_one_rat ) ) ) ).
% Fract_add_one
thf(fact_9782_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image_int_int
@ ^ [X3: int] : ( plus_plus_int @ X3 @ L )
@ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
= ( set_or4662586982721622107an_int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_9783_zero__le__Fract__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% zero_le_Fract_iff
thf(fact_9784_Fract__le__zero__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% Fract_le_zero_iff
thf(fact_9785_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_9786_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_9787_rat__number__collapse_I4_J,axiom,
! [W: num] :
( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% rat_number_collapse(4)
thf(fact_9788_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_9789_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
= ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_9790_of__nat__eq__id,axiom,
semiri1316708129612266289at_nat = id_nat ).
% of_nat_eq_id
thf(fact_9791_less__int__def,axiom,
( ord_less_int
= ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y4: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) ) ) ) ).
% less_int_def
thf(fact_9792_less__eq__int__def,axiom,
( ord_less_eq_int
= ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y4: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_9793_nat__def,axiom,
( nat2
= ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% nat_def
thf(fact_9794_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_9795_Rat_Opositive__add,axiom,
! [X: rat,Y: rat] :
( ( positive @ X )
=> ( ( positive @ Y )
=> ( positive @ ( plus_plus_rat @ X @ Y ) ) ) ) ).
% Rat.positive_add
thf(fact_9796_less__rat__def,axiom,
( ord_less_rat
= ( ^ [X3: rat,Y4: rat] : ( positive @ ( minus_minus_rat @ Y4 @ X3 ) ) ) ) ).
% less_rat_def
thf(fact_9797_Rat_Opositive_Orep__eq,axiom,
( positive
= ( ^ [X3: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X3 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X3 ) ) ) ) ) ) ).
% Rat.positive.rep_eq
thf(fact_9798_Rat_Opositive__def,axiom,
( positive
= ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
@ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ).
% Rat.positive_def
thf(fact_9799_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K2: code_integer] :
( if_num @ ( ord_le3102999989581377725nteger @ K2 @ one_one_Code_integer ) @ one
@ ( produc7336495610019696514er_num
@ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
@ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_9800_of__rat__dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Q2: rat] :
( ( ord_less_real @ X @ ( field_7254667332652039916t_real @ Q2 ) )
& ( ord_less_real @ ( field_7254667332652039916t_real @ Q2 ) @ Y ) ) ) ).
% of_rat_dense
thf(fact_9801_plus__rat__def,axiom,
( plus_plus_rat
= ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
@ ^ [X3: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y4 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y4 ) ) ) ) ) ).
% plus_rat_def
thf(fact_9802_inverse__rat__def,axiom,
( inverse_inverse_rat
= ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
@ ^ [X3: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X3 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ) ).
% inverse_rat_def
thf(fact_9803_one__rat__def,axiom,
( one_one_rat
= ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).
% one_rat_def
thf(fact_9804_Fract_Oabs__eq,axiom,
( fract
= ( ^ [Xa4: int,X3: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X3 ) ) ) ) ) ).
% Fract.abs_eq
thf(fact_9805_zero__rat__def,axiom,
( zero_zero_rat
= ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).
% zero_rat_def
thf(fact_9806_plus__rat_Oabs__eq,axiom,
! [Xa: product_prod_int_int,X: product_prod_int_int] :
( ( ratrel @ Xa @ Xa )
=> ( ( ratrel @ X @ X )
=> ( ( plus_plus_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% plus_rat.abs_eq
thf(fact_9807_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_9808_ratrel__iff,axiom,
( ratrel
= ( ^ [X3: product_prod_int_int,Y4: product_prod_int_int] :
( ( ( product_snd_int_int @ X3 )
!= zero_zero_int )
& ( ( product_snd_int_int @ Y4 )
!= zero_zero_int )
& ( ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y4 ) )
= ( times_times_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ).
% ratrel_iff
thf(fact_9809_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_9810_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_9811_rat__less__code,axiom,
( ord_less_rat
= ( ^ [P4: rat,Q4: rat] :
( produc4947309494688390418_int_o
@ ^ [A3: int,C2: int] :
( produc4947309494688390418_int_o
@ ^ [B2: int,D3: int] : ( ord_less_int @ ( times_times_int @ A3 @ D3 ) @ ( times_times_int @ C2 @ B2 ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ) ).
% rat_less_code
thf(fact_9812_rat__less__eq__code,axiom,
( ord_less_eq_rat
= ( ^ [P4: rat,Q4: rat] :
( produc4947309494688390418_int_o
@ ^ [A3: int,C2: int] :
( produc4947309494688390418_int_o
@ ^ [B2: int,D3: int] : ( ord_less_eq_int @ ( times_times_int @ A3 @ D3 ) @ ( times_times_int @ C2 @ B2 ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ) ).
% rat_less_eq_code
thf(fact_9813_ratrel__def,axiom,
( ratrel
= ( ^ [X3: product_prod_int_int,Y4: product_prod_int_int] :
( ( ( product_snd_int_int @ X3 )
!= zero_zero_int )
& ( ( product_snd_int_int @ Y4 )
!= zero_zero_int )
& ( ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y4 ) )
= ( times_times_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ).
% ratrel_def
thf(fact_9814_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_9815_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D3: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z7: int,Z5: int] :
( ( ord_less_eq_int @ D3 @ Z7 )
& ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_9816_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D3: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z7: int,Z5: int] :
( ( ord_less_eq_int @ D3 @ Z5 )
& ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_9817_min__Suc__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( suc @ ( ord_min_nat @ M @ N2 ) ) ) ).
% min_Suc_Suc
thf(fact_9818_min__0L,axiom,
! [N2: nat] :
( ( ord_min_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% min_0L
thf(fact_9819_min__0R,axiom,
! [N2: nat] :
( ( ord_min_nat @ N2 @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_9820_Max__divisors__self__nat,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ N2 ) ) )
= N2 ) ) ).
% Max_divisors_self_nat
thf(fact_9821_min__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
= ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).
% min_numeral_Suc
thf(fact_9822_min__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( ord_min_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
= ( suc @ ( ord_min_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).
% min_Suc_numeral
thf(fact_9823_nat__mult__min__right,axiom,
! [M: nat,N2: nat,Q3: nat] :
( ( times_times_nat @ M @ ( ord_min_nat @ N2 @ Q3 ) )
= ( ord_min_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q3 ) ) ) ).
% nat_mult_min_right
thf(fact_9824_nat__mult__min__left,axiom,
! [M: nat,N2: nat,Q3: nat] :
( ( times_times_nat @ ( ord_min_nat @ M @ N2 ) @ Q3 )
= ( ord_min_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N2 @ Q3 ) ) ) ).
% nat_mult_min_left
thf(fact_9825_min__diff,axiom,
! [M: nat,I: nat,N2: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N2 @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N2 ) @ I ) ) ).
% min_diff
thf(fact_9826_inf__nat__def,axiom,
inf_inf_nat = ord_min_nat ).
% inf_nat_def
thf(fact_9827_concat__bit__assoc__sym,axiom,
! [M: nat,N2: nat,K: int,L: int,R4: int] :
( ( bit_concat_bit @ M @ ( bit_concat_bit @ N2 @ K @ L ) @ R4 )
= ( bit_concat_bit @ ( ord_min_nat @ M @ N2 ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N2 ) @ L @ R4 ) ) ) ).
% concat_bit_assoc_sym
thf(fact_9828_take__bit__concat__bit__eq,axiom,
! [M: nat,N2: nat,K: int,L: int] :
( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N2 @ K @ L ) )
= ( bit_concat_bit @ ( ord_min_nat @ M @ N2 ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N2 ) @ L ) ) ) ).
% take_bit_concat_bit_eq
thf(fact_9829_min__Suc1,axiom,
! [N2: nat,M: nat] :
( ( ord_min_nat @ ( suc @ N2 ) @ M )
= ( case_nat_nat @ zero_zero_nat
@ ^ [M4: nat] : ( suc @ ( ord_min_nat @ N2 @ M4 ) )
@ M ) ) ).
% min_Suc1
thf(fact_9830_min__Suc2,axiom,
! [M: nat,N2: nat] :
( ( ord_min_nat @ M @ ( suc @ N2 ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [M4: nat] : ( suc @ ( ord_min_nat @ M4 @ N2 ) )
@ M ) ) ).
% min_Suc2
thf(fact_9831_card__le__Suc__Max,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_9832_divide__nat__def,axiom,
( divide_divide_nat
= ( ^ [M2: nat,N4: nat] :
( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat
@ ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [K2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K2 @ N4 ) @ M2 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_9833_gcd__is__Max__divisors__nat,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( gcd_gcd_nat @ M @ N2 )
= ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [D3: nat] :
( ( dvd_dvd_nat @ D3 @ M )
& ( dvd_dvd_nat @ D3 @ N2 ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_9834_min__enat__simps_I2_J,axiom,
! [Q3: extended_enat] :
( ( ord_mi8085742599997312461d_enat @ Q3 @ zero_z5237406670263579293d_enat )
= zero_z5237406670263579293d_enat ) ).
% min_enat_simps(2)
thf(fact_9835_min__enat__simps_I3_J,axiom,
! [Q3: extended_enat] :
( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q3 )
= zero_z5237406670263579293d_enat ) ).
% min_enat_simps(3)
thf(fact_9836_Max__divisors__self__int,axiom,
! [N2: int] :
( ( N2 != zero_zero_int )
=> ( ( lattic8263393255366662781ax_int
@ ( collect_int
@ ^ [D3: int] : ( dvd_dvd_int @ D3 @ N2 ) ) )
= ( abs_abs_int @ N2 ) ) ) ).
% Max_divisors_self_int
thf(fact_9837_inf__int__def,axiom,
inf_inf_int = ord_min_int ).
% inf_int_def
thf(fact_9838_gcd__is__Max__divisors__int,axiom,
! [N2: int,M: int] :
( ( N2 != zero_zero_int )
=> ( ( gcd_gcd_int @ M @ N2 )
= ( lattic8263393255366662781ax_int
@ ( collect_int
@ ^ [D3: int] :
( ( dvd_dvd_int @ D3 @ M )
& ( dvd_dvd_int @ D3 @ N2 ) ) ) ) ) ) ).
% gcd_is_Max_divisors_int
thf(fact_9839_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
@ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ M2 ) )
@ M7 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_9840_inf__enat__def,axiom,
inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).
% inf_enat_def
thf(fact_9841_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_9842_Inf__real__def,axiom,
( comple4887499456419720421f_real
= ( ^ [X6: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X6 ) ) ) ) ) ).
% Inf_real_def
thf(fact_9843_Sup__int__def,axiom,
( complete_Sup_Sup_int
= ( ^ [X6: set_int] :
( the_int
@ ^ [X3: int] :
( ( member_int @ X3 @ X6 )
& ! [Y4: int] :
( ( member_int @ Y4 @ X6 )
=> ( ord_less_eq_int @ Y4 @ X3 ) ) ) ) ) ) ).
% Sup_int_def
thf(fact_9844_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_9845_range__mod,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( image_nat_nat
@ ^ [M2: nat] : ( modulo_modulo_nat @ M2 @ N2 )
@ top_top_set_nat )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).
% range_mod
thf(fact_9846_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_9847_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_9848_range__abs__Nats,axiom,
( ( image_int_int @ abs_abs_int @ top_top_set_int )
= semiring_1_Nats_int ) ).
% range_abs_Nats
thf(fact_9849_card__UNIV__bool,axiom,
( ( finite_card_o @ top_top_set_o )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% card_UNIV_bool
thf(fact_9850_infinite__UNIV__int,axiom,
~ ( finite_finite_int @ top_top_set_int ) ).
% infinite_UNIV_int
thf(fact_9851_int__in__range__abs,axiom,
! [N2: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).
% int_in_range_abs
thf(fact_9852_root__def,axiom,
( root
= ( ^ [N4: nat,X3: real] :
( if_real @ ( N4 = zero_zero_nat ) @ zero_zero_real
@ ( the_in5290026491893676941l_real @ top_top_set_real
@ ^ [Y4: real] : ( times_times_real @ ( sgn_sgn_real @ Y4 ) @ ( power_power_real @ ( abs_abs_real @ Y4 ) @ N4 ) )
@ X3 ) ) ) ) ).
% root_def
thf(fact_9853_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_9854_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_9855_char_Osize_I2_J,axiom,
! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= zero_zero_nat ) ).
% char.size(2)
thf(fact_9856_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_9857_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_9858_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_9859_char_Osize__gen,axiom,
! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= zero_zero_nat ) ).
% char.size_gen
thf(fact_9860_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_9861_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_9862_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_9863_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ ( suc @ I ) @ J )
=> ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
= ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_9864_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ ( suc @ I ) @ J )
=> ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
= ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_9865_list__encode_Oelims,axiom,
! [X: list_nat,Y: nat] :
( ( ( nat_list_encode @ X )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != zero_zero_nat ) )
=> ~ ! [X2: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs2 ) )
=> ( Y
!= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X2 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_9866_list__encode_Osimps_I1_J,axiom,
( ( nat_list_encode @ nil_nat )
= zero_zero_nat ) ).
% list_encode.simps(1)
thf(fact_9867_list__encode_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( nat_list_encode @ ( cons_nat @ X @ Xs ) )
= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% list_encode.simps(2)
thf(fact_9868_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_9869_upto_Opelims,axiom,
! [X: int,Xa: int,Y: list_int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq_int @ X @ Xa )
=> ( Y
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa )
=> ( Y = nil_int ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_9870_upto_Opsimps,axiom,
! [I: int,J: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
=> ( ( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
& ( ~ ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= nil_int ) ) ) ) ).
% upto.psimps
thf(fact_9871_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less_int @ J @ I )
=> ( ( upto @ I @ J )
= nil_int ) ) ).
% upto_empty
thf(fact_9872_upto__Nil2,axiom,
! [I: int,J: int] :
( ( nil_int
= ( upto @ I @ J ) )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil2
thf(fact_9873_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= nil_int )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil
thf(fact_9874_upto__single,axiom,
! [I: int] :
( ( upto @ I @ I )
= ( cons_int @ I @ nil_int ) ) ).
% upto_single
thf(fact_9875_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
=> ( ( nth_int @ ( upto @ I @ J ) @ K )
= ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% nth_upto
thf(fact_9876_length__upto,axiom,
! [I: int,J: int] :
( ( size_size_list_int @ ( upto @ I @ J ) )
= ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% length_upto
thf(fact_9877_upto__rec__numeral_I1_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
= ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N2 ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
= nil_int ) ) ) ).
% upto_rec_numeral(1)
thf(fact_9878_upto__rec__numeral_I2_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(2)
thf(fact_9879_upto__rec__numeral_I3_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
= ( 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 @ N2 ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
= nil_int ) ) ) ).
% upto_rec_numeral(3)
thf(fact_9880_upto__rec__numeral_I4_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( 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 @ N2 ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(4)
thf(fact_9881_upto__code,axiom,
( upto
= ( ^ [I3: int,J3: int] : ( upto_aux @ I3 @ J3 @ nil_int ) ) ) ).
% upto_code
thf(fact_9882_upto__aux__def,axiom,
( upto_aux
= ( ^ [I3: int,J3: int] : ( append_int @ ( upto @ I3 @ J3 ) ) ) ) ).
% upto_aux_def
thf(fact_9883_distinct__upto,axiom,
! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).
% distinct_upto
thf(fact_9884_atLeastAtMost__upto,axiom,
( set_or1266510415728281911st_int
= ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ J3 ) ) ) ) ).
% atLeastAtMost_upto
thf(fact_9885_upto__split2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_9886_upto__split1,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_9887_atLeastLessThan__upto,axiom,
( set_or4662586982721622107an_int
= ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_9888_greaterThanAtMost__upto,axiom,
( set_or6656581121297822940st_int
= ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_9889_upto_Oelims,axiom,
! [X: int,Xa: int,Y: list_int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq_int @ X @ Xa )
=> ( Y
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa )
=> ( Y = nil_int ) ) ) ) ).
% upto.elims
thf(fact_9890_upto_Osimps,axiom,
( upto
= ( ^ [I3: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I3 @ J3 ) @ ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% upto.simps
thf(fact_9891_upto__rec1,axiom,
! [I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_9892_upto__rec2,axiom,
! [I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% upto_rec2
thf(fact_9893_greaterThanLessThan__upto,axiom,
( set_or5832277885323065728an_int
= ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_9894_upto__split3,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_9895_DERIV__even__real__root,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_9896_DERIV__real__root__generic,axiom,
! [N2: nat,X: real,D5: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( X != zero_zero_real )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( D5
= ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( D5
= ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( D5
= ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
=> ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ D5 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_9897_MVT2,axiom,
! [A: real,B: real,F: real > real,F3: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X2: real] :
( ( ord_less_eq_real @ A @ X2 )
=> ( ( ord_less_eq_real @ X2 @ B )
=> ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) )
=> ? [Z2: real] :
( ( ord_less_real @ A @ Z2 )
& ( ord_less_real @ Z2 @ B )
& ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
= ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F3 @ Z2 ) ) ) ) ) ) ).
% MVT2
thf(fact_9898_DERIV__local__const,axiom,
! [F: real > real,L: real,X: real,D: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y2: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y2 ) ) @ D )
=> ( ( F @ X )
= ( F @ Y2 ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_const
thf(fact_9899_DERIV__ln,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_ln
thf(fact_9900_DERIV__pos__inc__right,axiom,
! [F: real > real,L: real,X: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_9901_DERIV__neg__dec__right,axiom,
! [F: real > real,L: real,X: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_9902_DERIV__const__ratio__const2,axiom,
! [A: real,B: real,F: real > real,K: real] :
( ( A != B )
=> ( ! [X2: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
= K ) ) ) ).
% DERIV_const_ratio_const2
thf(fact_9903_DERIV__neg__dec__left,axiom,
! [F: real > real,L: real,X: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_9904_DERIV__pos__inc__left,axiom,
! [F: real > real,L: real,X: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_9905_DERIV__isconst3,axiom,
! [A: real,B: real,X: real,Y: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) )
=> ( ( F @ X )
= ( F @ Y ) ) ) ) ) ) ).
% DERIV_isconst3
thf(fact_9906_has__real__derivative__neg__dec__right,axiom,
! [F: real > real,L: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S2 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_9907_has__real__derivative__pos__inc__right,axiom,
! [F: real > real,L: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S2 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_9908_has__real__derivative__pos__inc__left,axiom,
! [F: real > real,L: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S2 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_9909_has__real__derivative__neg__dec__left,axiom,
! [F: real > real,L: real,X: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S2 )
=> ( ( ord_less_real @ H4 @ D4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_9910_DERIV__neg__imp__decreasing,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X2: real] :
( ( ord_less_eq_real @ A @ X2 )
=> ( ( ord_less_eq_real @ X2 @ B )
=> ? [Y3: real] :
( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
& ( ord_less_real @ Y3 @ zero_zero_real ) ) ) )
=> ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% DERIV_neg_imp_decreasing
thf(fact_9911_DERIV__pos__imp__increasing,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X2: real] :
( ( ord_less_eq_real @ A @ X2 )
=> ( ( ord_less_eq_real @ X2 @ B )
=> ? [Y3: real] :
( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
& ( ord_less_real @ zero_zero_real @ Y3 ) ) ) )
=> ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% DERIV_pos_imp_increasing
thf(fact_9912_DERIV__const__average,axiom,
! [A: real,B: real,V: real > real,K: real] :
( ( A != B )
=> ( ! [X2: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X2 @ 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_9913_DERIV__local__min,axiom,
! [F: real > real,L: real,X: real,D: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y2: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y2 ) ) @ D )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_min
thf(fact_9914_DERIV__local__max,axiom,
! [F: real > real,L: real,X: real,D: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y2: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y2 ) ) @ D )
=> ( ord_less_eq_real @ ( F @ Y2 ) @ ( F @ X ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_max
thf(fact_9915_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_9916_DERIV__pow,axiom,
! [N2: nat,X: real,S: set_real] :
( has_fi5821293074295781190e_real
@ ^ [X3: real] : ( power_power_real @ X3 @ N2 )
@ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
@ ( topolo2177554685111907308n_real @ X @ S ) ) ).
% DERIV_pow
thf(fact_9917_DERIV__fun__pow,axiom,
! [G: real > real,M: real,X: real,N2: nat] :
( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X3: real] : ( power_power_real @ ( G @ X3 ) @ N2 )
@ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) @ M )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_fun_pow
thf(fact_9918_has__real__derivative__powr,axiom,
! [Z: real,R4: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( has_fi5821293074295781190e_real
@ ^ [Z5: real] : ( powr_real @ Z5 @ R4 )
@ ( times_times_real @ R4 @ ( powr_real @ Z @ ( minus_minus_real @ R4 @ one_one_real ) ) )
@ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% has_real_derivative_powr
thf(fact_9919_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_9920_DERIV__fun__powr,axiom,
! [G: real > real,M: real,X: real,R4: real] :
( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ R4 )
@ ( times_times_real @ ( times_times_real @ R4 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R4 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_fun_powr
thf(fact_9921_DERIV__powr,axiom,
! [G: real > real,M: real,X: real,F: real > real,R4: 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 @ R4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ ( F @ X3 ) )
@ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R4 @ ( 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_9922_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_9923_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_9924_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_9925_DERIV__real__sqrt__generic,axiom,
! [X: real,D5: real] :
( ( X != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ X )
=> ( D5
= ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( ( ord_less_real @ X @ zero_zero_real )
=> ( D5
= ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( has_fi5821293074295781190e_real @ sqrt @ D5 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_9926_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_9927_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_9928_DERIV__power__series_H,axiom,
! [R: real,F: nat > real,X0: real] :
( ! [X2: real] :
( ( member_real @ X2 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
=> ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( F @ N4 ) @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ ( power_power_real @ X2 @ N4 ) ) ) )
=> ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
=> ( ( ord_less_real @ zero_zero_real @ R )
=> ( has_fi5821293074295781190e_real
@ ^ [X3: real] :
( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ X3 @ ( suc @ N4 ) ) ) )
@ ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( F @ N4 ) @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ ( power_power_real @ X0 @ N4 ) ) )
@ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_9929_DERIV__real__root,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_real_root
thf(fact_9930_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_9931_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_9932_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F: real > real,X: real,N2: nat] :
( ( ( ( Diff @ zero_zero_nat )
= F )
& ! [M5: nat,X2: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) )
=> ? [T: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T ) @ ( 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_9933_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F: real > real,X: real,N2: nat] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M5: nat,X2: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ? [T: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T ) @ ( 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_9934_DERIV__odd__real__root,axiom,
! [N2: nat,X: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( X != zero_zero_real )
=> ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_9935_Maclaurin__minus,axiom,
! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
( ( ord_less_real @ H2 @ zero_zero_real )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M5: nat,T: real] :
( ( ( ord_less_nat @ M5 @ N2 )
& ( ord_less_eq_real @ H2 @ T )
& ( ord_less_eq_real @ T @ zero_zero_real ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T ) @ ( topolo2177554685111907308n_real @ T @ top_top_set_real ) ) )
=> ? [T: real] :
( ( ord_less_real @ H2 @ T )
& ( ord_less_real @ T @ 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_9936_Maclaurin2,axiom,
! [H2: real,Diff: nat > real > real,F: real > real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ H2 )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M5: nat,T: real] :
( ( ( ord_less_nat @ M5 @ N2 )
& ( ord_less_eq_real @ zero_zero_real @ T )
& ( ord_less_eq_real @ T @ H2 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T ) @ ( topolo2177554685111907308n_real @ T @ top_top_set_real ) ) )
=> ? [T: real] :
( ( ord_less_real @ zero_zero_real @ T )
& ( ord_less_eq_real @ T @ 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_9937_Maclaurin,axiom,
! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
( ( ord_less_real @ zero_zero_real @ H2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M5: nat,T: real] :
( ( ( ord_less_nat @ M5 @ N2 )
& ( ord_less_eq_real @ zero_zero_real @ T )
& ( ord_less_eq_real @ T @ H2 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T ) @ ( topolo2177554685111907308n_real @ T @ top_top_set_real ) ) )
=> ? [T: real] :
( ( ord_less_real @ zero_zero_real @ T )
& ( ord_less_real @ T @ 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_9938_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F: real > real,N2: nat,X: real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( X != zero_zero_real )
=> ( ! [M5: nat,X2: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ? [T: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T ) )
& ( ord_less_real @ ( abs_abs_real @ T ) @ ( 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_9939_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F: real > real,N2: nat,X: real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M5: nat,T: real] :
( ( ( ord_less_nat @ M5 @ N2 )
& ( ord_less_eq_real @ ( abs_abs_real @ T ) @ ( abs_abs_real @ X ) ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T ) @ ( topolo2177554685111907308n_real @ T @ top_top_set_real ) ) )
=> ? [T: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T ) @ ( 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_9940_Taylor__down,axiom,
! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M5: nat,T: real] :
( ( ( ord_less_nat @ M5 @ N2 )
& ( ord_less_eq_real @ A @ T )
& ( ord_less_eq_real @ T @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T ) @ ( topolo2177554685111907308n_real @ T @ top_top_set_real ) ) )
=> ( ( ord_less_real @ A @ C )
=> ( ( ord_less_eq_real @ C @ B )
=> ? [T: real] :
( ( ord_less_real @ A @ T )
& ( ord_less_real @ T @ 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_9941_Taylor__up,axiom,
! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M5: nat,T: real] :
( ( ( ord_less_nat @ M5 @ N2 )
& ( ord_less_eq_real @ A @ T )
& ( ord_less_eq_real @ T @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T ) @ ( topolo2177554685111907308n_real @ T @ top_top_set_real ) ) )
=> ( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_real @ C @ B )
=> ? [T: real] :
( ( ord_less_real @ C @ T )
& ( ord_less_real @ T @ 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_9942_Taylor,axiom,
! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M5: nat,T: real] :
( ( ( ord_less_nat @ M5 @ N2 )
& ( ord_less_eq_real @ A @ T )
& ( ord_less_eq_real @ T @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T ) @ ( topolo2177554685111907308n_real @ T @ 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 )
=> ? [T: real] :
( ( ( ord_less_real @ X @ C )
=> ( ( ord_less_real @ X @ T )
& ( ord_less_real @ T @ C ) ) )
& ( ~ ( ord_less_real @ X @ C )
=> ( ( ord_less_real @ C @ T )
& ( ord_less_real @ T @ 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 @ N2 ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_9943_Maclaurin__lemma2,axiom,
! [N2: nat,H2: real,Diff: nat > real > real,K: nat,B3: real] :
( ! [M5: nat,T: real] :
( ( ( ord_less_nat @ M5 @ N2 )
& ( ord_less_eq_real @ zero_zero_real @ T )
& ( ord_less_eq_real @ T @ H2 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T ) @ ( topolo2177554685111907308n_real @ T @ top_top_set_real ) ) )
=> ( ( N2
= ( suc @ K ) )
=> ! [M3: nat,T4: real] :
( ( ( ord_less_nat @ M3 @ N2 )
& ( ord_less_eq_real @ zero_zero_real @ T4 )
& ( ord_less_eq_real @ T4 @ H2 ) )
=> ( has_fi5821293074295781190e_real
@ ^ [U2: real] :
( minus_minus_real @ ( Diff @ M3 @ U2 )
@ ( 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 @ U2 @ P4 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ M3 ) ) )
@ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N2 @ M3 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ M3 ) ) ) ) ) )
@ ( minus_minus_real @ ( Diff @ ( suc @ M3 ) @ T4 )
@ ( 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 @ T4 @ P4 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ M3 ) ) ) )
@ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ T4 @ ( minus_minus_nat @ N2 @ ( suc @ M3 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ ( suc @ M3 ) ) ) ) ) ) )
@ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_9944_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( has_fi5821293074295781190e_real
@ ^ [X9: real] :
( suminf_real
@ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
@ ( suminf_real
@ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( power_power_real @ X @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_arctan_series
thf(fact_9945_LIM__fun__less__zero,axiom,
! [F: real > real,L: real,C: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ! [X4: real] :
( ( ( X4 != C )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R2 ) )
=> ( ord_less_real @ ( F @ X4 ) @ zero_zero_real ) ) ) ) ) ).
% LIM_fun_less_zero
thf(fact_9946_LIM__fun__not__zero,axiom,
! [F: real > real,L: real,C: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
=> ( ( L != zero_zero_real )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ! [X4: real] :
( ( ( X4 != C )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R2 ) )
=> ( ( F @ X4 )
!= zero_zero_real ) ) ) ) ) ).
% LIM_fun_not_zero
thf(fact_9947_LIM__fun__gt__zero,axiom,
! [F: real > real,L: real,C: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ! [X4: real] :
( ( ( X4 != C )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R2 ) )
=> ( ord_less_real @ zero_zero_real @ ( F @ X4 ) ) ) ) ) ) ).
% LIM_fun_gt_zero
thf(fact_9948_LIM__cos__div__sin,axiom,
( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( cos_real @ X3 ) @ ( sin_real @ X3 ) )
@ ( 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_9949_binomial__def,axiom,
( binomial
= ( ^ [N4: nat,K2: nat] :
( finite_card_set_nat
@ ( collect_set_nat
@ ^ [K7: set_nat] :
( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) )
& ( ( finite_card_nat @ K7 )
= K2 ) ) ) ) ) ) ).
% binomial_def
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 ) )
=> ! [N8: nat] :
( member_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
@ ( set_or1222579329274155063t_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) @ 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 )
=> ! [N8: nat] :
( member_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
@ ( set_or1222579329274155063t_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) @ one_one_nat ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_9952_mult__nat__right__at__top,axiom,
! [C: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( filterlim_nat_nat
@ ^ [X3: nat] : ( times_times_nat @ X3 @ C )
@ at_top_nat
@ at_top_nat ) ) ).
% mult_nat_right_at_top
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_LIMSEQ__root,axiom,
( filterlim_nat_real
@ ^ [N4: nat] : ( root @ N4 @ ( semiri5074537144036343181t_real @ N4 ) )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat ) ).
% LIMSEQ_root
thf(fact_9955_nested__sequence__unique,axiom,
! [F: nat > real,G: nat > real] :
( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ! [N: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N ) ) @ ( G @ N ) )
=> ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( G @ N ) )
=> ( ( filterlim_nat_real
@ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( G @ N4 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat )
=> ? [L3: real] :
( ! [N8: nat] : ( ord_less_eq_real @ ( F @ N8 ) @ L3 )
& ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat )
& ! [N8: nat] : ( ord_less_eq_real @ L3 @ ( G @ N8 ) )
& ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_9956_LIMSEQ__inverse__zero,axiom,
! [X8: nat > real] :
( ! [R2: real] :
? [N6: nat] :
! [N: nat] :
( ( ord_less_eq_nat @ N6 @ N )
=> ( ord_less_real @ R2 @ ( X8 @ N ) ) )
=> ( filterlim_nat_real
@ ^ [N4: nat] : ( inverse_inverse_real @ ( X8 @ N4 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_inverse_zero
thf(fact_9957_lim__inverse__n_H,axiom,
( filterlim_nat_real
@ ^ [N4: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N4 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ).
% lim_inverse_n'
thf(fact_9958_LIMSEQ__root__const,axiom,
! [C: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( filterlim_nat_real
@ ^ [N4: nat] : ( root @ N4 @ C )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat ) ) ).
% LIMSEQ_root_const
thf(fact_9959_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim_nat_real
@ ^ [N4: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_9960_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R4: real] :
( filterlim_nat_real
@ ^ [N4: nat] : ( plus_plus_real @ R4 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
@ ( topolo2815343760600316023s_real @ R4 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_9961_increasing__LIMSEQ,axiom,
! [F: nat > real,L: real] :
( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ L )
=> ( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [N8: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N8 ) @ E ) ) )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_9962_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_9963_LIMSEQ__divide__realpow__zero,axiom,
! [X: real,A: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( filterlim_nat_real
@ ^ [N4: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N4 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_9964_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_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__inverse__realpow__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( filterlim_nat_real
@ ^ [N4: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N4 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_9967_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R4: real] :
( filterlim_nat_real
@ ^ [N4: nat] : ( plus_plus_real @ R4 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) ) )
@ ( topolo2815343760600316023s_real @ R4 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_9968_tendsto__exp__limit__sequentially,axiom,
! [X: real] :
( filterlim_nat_real
@ ^ [N4: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ) @ N4 )
@ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
@ at_top_nat ) ).
% tendsto_exp_limit_sequentially
thf(fact_9969_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R4: real] :
( filterlim_nat_real
@ ^ [N4: nat] : ( times_times_real @ R4 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) ) ) )
@ ( topolo2815343760600316023s_real @ R4 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_9970_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
@ ^ [N4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( A @ N4 ) ) ) ) ) ).
% summable_Leibniz(1)
thf(fact_9971_summable,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
=> ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
=> ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( A @ N4 ) ) ) ) ) ) ).
% summable
thf(fact_9972_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 )
=> ~ ! [K3: nat > int] :
~ ( filterlim_nat_real
@ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
@ ( topolo2815343760600316023s_real @ Theta2 )
@ at_top_nat ) ) ).
% cos_diff_limit_1
thf(fact_9973_cos__limit__1,axiom,
! [Theta: nat > real] :
( ( filterlim_nat_real
@ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat )
=> ? [K3: nat > int] :
( filterlim_nat_real
@ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% cos_limit_1
thf(fact_9974_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
@ ^ [N4: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
@ at_top_nat ) ) ) ).
% summable_Leibniz(4)
thf(fact_9975_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( filterlim_nat_real
@ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% zeroseq_arctan_series
thf(fact_9976_summable__Leibniz_H_I2_J,axiom,
! [A: nat > real,N2: nat] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
=> ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
=> ( ord_less_eq_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_9977_summable__Leibniz_H_I3_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
=> ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
=> ( filterlim_nat_real
@ ^ [N4: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
@ at_top_nat ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_9978_sums__alternating__upper__lower,axiom,
! [A: nat > real] :
( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
=> ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
=> ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ? [L3: real] :
( ! [N8: nat] :
( ord_less_eq_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) ) )
@ L3 )
& ( filterlim_nat_real
@ ^ [N4: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
@ ( topolo2815343760600316023s_real @ L3 )
@ at_top_nat )
& ! [N8: nat] :
( ord_less_eq_real @ L3
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) @ one_one_nat ) ) ) )
& ( filterlim_nat_real
@ ^ [N4: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real @ L3 )
@ at_top_nat ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_9979_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
@ ^ [N4: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
@ at_top_nat ) ) ) ).
% summable_Leibniz(5)
thf(fact_9980_summable__Leibniz_H_I5_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
=> ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
=> ( filterlim_nat_real
@ ^ [N4: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
@ at_top_nat ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_9981_summable__Leibniz_H_I4_J,axiom,
! [A: nat > real,N2: nat] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
=> ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
=> ( ord_less_eq_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_9982_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually_nat
@ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
@ at_top_nat )
= ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_sequentially_Suc
thf(fact_9983_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually_nat
@ ^ [N4: nat] : ( P @ ( plus_plus_nat @ N4 @ K ) )
@ at_top_nat )
= ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_sequentially_seg
thf(fact_9984_eventually__sequentiallyI,axiom,
! [C: nat,P: nat > $o] :
( ! [X2: nat] :
( ( ord_less_eq_nat @ C @ X2 )
=> ( P @ X2 ) )
=> ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_sequentiallyI
thf(fact_9985_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually_nat @ P @ at_top_nat )
= ( ? [N9: nat] :
! [N4: nat] :
( ( ord_less_eq_nat @ N9 @ N4 )
=> ( P @ N4 ) ) ) ) ).
% eventually_sequentially
thf(fact_9986_le__sequentially,axiom,
! [F4: filter_nat] :
( ( ord_le2510731241096832064er_nat @ F4 @ at_top_nat )
= ( ! [N9: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N9 ) @ F4 ) ) ) ).
% le_sequentially
thf(fact_9987_sequentially__offset,axiom,
! [P: nat > $o,K: nat] :
( ( eventually_nat @ P @ at_top_nat )
=> ( eventually_nat
@ ^ [I3: nat] : ( P @ ( plus_plus_nat @ I3 @ K ) )
@ at_top_nat ) ) ).
% sequentially_offset
thf(fact_9988_filterlim__Suc,axiom,
filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% filterlim_Suc
thf(fact_9989_sqrt__at__top,axiom,
filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).
% sqrt_at_top
thf(fact_9990_lhopital__left__at__top__at__top,axiom,
! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
=> ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
=> ( ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
@ at_top_real
@ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
=> ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ at_top_real
@ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_top
thf(fact_9991_lhopital__at__top__at__top,axiom,
! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
=> ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
=> ( ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
@ at_top_real
@ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ at_top_real
@ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% lhopital_at_top_at_top
thf(fact_9992_eventually__at__left__real,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( eventually_real
@ ^ [X3: real] : ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B @ A ) )
@ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).
% eventually_at_left_real
thf(fact_9993_lhopital__left__at__top,axiom,
! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
=> ( ( eventually_real
@ ^ [X3: real] :
( ( G2 @ X3 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
=> ( ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
@ ( topolo2815343760600316023s_real @ Y )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
=> ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2815343760600316023s_real @ Y )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top
thf(fact_9994_lhospital__at__top__at__top,axiom,
! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
=> ( ( eventually_real
@ ^ [X3: real] :
( ( G2 @ X3 )
!= zero_zero_real )
@ at_top_real )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ at_top_real )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ at_top_real )
=> ( ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
@ ( topolo2815343760600316023s_real @ X )
@ at_top_real )
=> ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2815343760600316023s_real @ X )
@ at_top_real ) ) ) ) ) ) ).
% lhospital_at_top_at_top
thf(fact_9995_lhopital__at__top,axiom,
! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X3: real] :
( ( G2 @ X3 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
@ ( topolo2815343760600316023s_real @ Y )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2815343760600316023s_real @ Y )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% lhopital_at_top
thf(fact_9996_tanh__real__at__top,axiom,
filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).
% tanh_real_at_top
thf(fact_9997_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_9998_ln__x__over__x__tendsto__0,axiom,
( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ X3 ) @ X3 )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_real ) ).
% ln_x_over_x_tendsto_0
thf(fact_9999_tendsto__power__div__exp__0,axiom,
! [K: nat] :
( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( power_power_real @ X3 @ K ) @ ( exp_real @ X3 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_real ) ).
% tendsto_power_div_exp_0
thf(fact_10000_lhopital__left,axiom,
! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
=> ( ( eventually_real
@ ^ [X3: real] :
( ( G @ X3 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
=> ( ( eventually_real
@ ^ [X3: real] :
( ( G2 @ X3 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
=> ( ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
@ F4
@ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
=> ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ F4
@ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% lhopital_left
thf(fact_10001_lhopital,axiom,
! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X3: real] :
( ( G @ X3 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X3: real] :
( ( G2 @ X3 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
@ F4
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ F4
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).
% lhopital
thf(fact_10002_tendsto__exp__limit__at__top,axiom,
! [X: real] :
( filterlim_real_real
@ ^ [Y4: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y4 ) ) @ Y4 )
@ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
@ at_top_real ) ).
% tendsto_exp_limit_at_top
thf(fact_10003_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_10004_DERIV__neg__imp__decreasing__at__top,axiom,
! [B: real,F: real > real,Flim: real] :
( ! [X2: real] :
( ( ord_less_eq_real @ B @ X2 )
=> ? [Y3: real] :
( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
& ( ord_less_real @ Y3 @ zero_zero_real ) ) )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
=> ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% DERIV_neg_imp_decreasing_at_top
thf(fact_10005_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_10006_filterlim__pow__at__bot__even,axiom,
! [N2: nat,F: real > real,F4: filter_real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( filterlim_real_real
@ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N2 )
@ at_top_real
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_10007_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_10008_lhopital__at__top__at__bot,axiom,
! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
=> ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
=> ( ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
@ at_bot_real
@ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ at_bot_real
@ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% lhopital_at_top_at_bot
thf(fact_10009_DERIV__pos__imp__increasing__at__bot,axiom,
! [B: real,F: real > real,Flim: real] :
( ! [X2: real] :
( ( ord_less_eq_real @ X2 @ B )
=> ? [Y3: real] :
( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
& ( ord_less_real @ zero_zero_real @ Y3 ) ) )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
=> ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% DERIV_pos_imp_increasing_at_bot
thf(fact_10010_filterlim__pow__at__bot__odd,axiom,
! [N2: nat,F: real > real,F4: filter_real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( filterlim_real_real
@ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N2 )
@ at_bot_real
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_10011_lhopital__left__at__top__at__bot,axiom,
! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
=> ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
=> ( ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
@ at_bot_real
@ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
=> ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ at_bot_real
@ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_bot
thf(fact_10012_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_10013_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_10014_tendsto__exp__limit__at__right,axiom,
! [X: real] :
( filterlim_real_real
@ ^ [Y4: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y4 ) ) @ ( divide_divide_real @ one_one_real @ Y4 ) )
@ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% tendsto_exp_limit_at_right
thf(fact_10015_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_10016_eventually__at__right__to__0,axiom,
! [P: real > $o,A: real] :
( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
= ( eventually_real
@ ^ [X3: real] : ( P @ ( plus_plus_real @ X3 @ A ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% eventually_at_right_to_0
thf(fact_10017_eventually__at__right__real,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( eventually_real
@ ^ [X3: real] : ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
@ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).
% eventually_at_right_real
thf(fact_10018_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_10019_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_10020_lhopital__right__at__top__at__top,axiom,
! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
=> ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
=> ( ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
@ at_top_real
@ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
=> ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ at_top_real
@ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_top
thf(fact_10021_lhopital__right,axiom,
! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
=> ( ( eventually_real
@ ^ [X3: real] :
( ( G @ X3 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
=> ( ( eventually_real
@ ^ [X3: real] :
( ( G2 @ X3 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
=> ( ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
@ F4
@ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
=> ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ F4
@ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right
thf(fact_10022_lhopital__right__0,axiom,
! [F0: real > real,G0: real > real,G2: real > real,F3: real > real,F4: filter_real] :
( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X3: real] :
( ( G0 @ X3 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X3: real] :
( ( G2 @ X3 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
@ F4
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F0 @ X3 ) @ ( G0 @ X3 ) )
@ F4
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0
thf(fact_10023_lhopital__right__at__top__at__bot,axiom,
! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
=> ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
=> ( ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
@ at_bot_real
@ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
=> ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ at_bot_real
@ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_bot
thf(fact_10024_lhopital__right__at__top,axiom,
! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
=> ( ( eventually_real
@ ^ [X3: real] :
( ( G2 @ X3 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
=> ( ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
@ ( topolo2815343760600316023s_real @ Y )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
=> ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2815343760600316023s_real @ Y )
@ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top
thf(fact_10025_lhopital__right__0__at__top,axiom,
! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X3: real] :
( ( G2 @ X3 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
@ ( topolo2815343760600316023s_real @ X )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( filterlim_real_real
@ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2815343760600316023s_real @ X )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).
% lhopital_right_0_at_top
thf(fact_10026_atLeast__0,axiom,
( ( set_ord_atLeast_nat @ zero_zero_nat )
= top_top_set_nat ) ).
% atLeast_0
thf(fact_10027_atLeast__Suc__greaterThan,axiom,
! [K: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K ) )
= ( set_or1210151606488870762an_nat @ K ) ) ).
% atLeast_Suc_greaterThan
thf(fact_10028_greaterThan__0,axiom,
( ( set_or1210151606488870762an_nat @ zero_zero_nat )
= ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% greaterThan_0
thf(fact_10029_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_10030_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_10031_isCont__real__sqrt,axiom,
! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ sqrt ) ).
% isCont_real_sqrt
thf(fact_10032_isCont__real__root,axiom,
! [X: real,N2: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( root @ N2 ) ) ).
% isCont_real_root
thf(fact_10033_isCont__inverse__function2,axiom,
! [A: real,X: real,B: real,G: real > real,F: real > real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ( ! [Z2: real] :
( ( ord_less_eq_real @ A @ Z2 )
=> ( ( ord_less_eq_real @ Z2 @ B )
=> ( ( G @ ( F @ Z2 ) )
= Z2 ) ) )
=> ( ! [Z2: real] :
( ( ord_less_eq_real @ A @ Z2 )
=> ( ( ord_less_eq_real @ Z2 @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% isCont_inverse_function2
thf(fact_10034_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_10035_DERIV__inverse__function,axiom,
! [F: real > real,D5: real,G: real > real,X: real,A: real,B: real] :
( ( has_fi5821293074295781190e_real @ F @ D5 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
=> ( ( D5 != zero_zero_real )
=> ( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ( ! [Y2: real] :
( ( ord_less_real @ A @ Y2 )
=> ( ( ord_less_real @ Y2 @ B )
=> ( ( F @ ( G @ Y2 ) )
= Y2 ) ) )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
=> ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D5 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).
% DERIV_inverse_function
thf(fact_10036_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_10037_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_10038_LIM__less__bound,axiom,
! [B: real,X: real,F: real > real] :
( ( ord_less_real @ B @ X )
=> ( ! [X2: real] :
( ( member_real @ X2 @ ( set_or1633881224788618240n_real @ B @ X ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).
% LIM_less_bound
thf(fact_10039_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_10040_isCont__inverse__function,axiom,
! [D: real,X: real,G: real > real,F: real > real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Z2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X ) ) @ D )
=> ( ( G @ ( F @ Z2 ) )
= Z2 ) )
=> ( ! [Z2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X ) ) @ D )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).
% isCont_inverse_function
thf(fact_10041_GMVT_H,axiom,
! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F3: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [Z2: real] :
( ( ord_less_eq_real @ A @ Z2 )
=> ( ( ord_less_eq_real @ Z2 @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
=> ( ! [Z2: real] :
( ( ord_less_eq_real @ A @ Z2 )
=> ( ( ord_less_eq_real @ Z2 @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ G ) ) )
=> ( ! [Z2: real] :
( ( ord_less_real @ A @ Z2 )
=> ( ( ord_less_real @ Z2 @ B )
=> ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
=> ( ! [Z2: real] :
( ( ord_less_real @ A @ Z2 )
=> ( ( ord_less_real @ Z2 @ B )
=> ( has_fi5821293074295781190e_real @ F @ ( F3 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ 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 ) ) @ ( F3 @ C3 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_10042_GMVT,axiom,
! [A: real,B: real,F: real > real,G: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X2: real] :
( ( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_eq_real @ X2 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ F ) )
=> ( ! [X2: real] :
( ( ( ord_less_real @ A @ X2 )
& ( ord_less_real @ X2 @ B ) )
=> ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) )
=> ( ! [X2: real] :
( ( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_eq_real @ X2 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ G ) )
=> ( ! [X2: real] :
( ( ( ord_less_real @ A @ X2 )
& ( ord_less_real @ X2 @ B ) )
=> ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X2 @ 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_10043_MVT,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X2: real] :
( ( ord_less_real @ A @ X2 )
=> ( ( ord_less_real @ X2 @ B )
=> ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) )
=> ? [L3: real,Z2: real] :
( ( ord_less_real @ A @ Z2 )
& ( ord_less_real @ Z2 @ B )
& ( has_fi5821293074295781190e_real @ F @ L3 @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) )
& ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
= ( times_times_real @ ( minus_minus_real @ B @ A ) @ L3 ) ) ) ) ) ) ).
% MVT
thf(fact_10044_mono__times__nat,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( order_mono_nat_nat @ ( times_times_nat @ N2 ) ) ) ).
% mono_times_nat
thf(fact_10045_mono__Suc,axiom,
order_mono_nat_nat @ suc ).
% mono_Suc
thf(fact_10046_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_10047_continuous__on__arcosh_H,axiom,
! [A2: set_real,F: real > real] :
( ( topolo5044208981011980120l_real @ A2 @ F )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X2 ) ) )
=> ( topolo5044208981011980120l_real @ A2
@ ^ [X3: real] : ( arcosh_real @ ( F @ X3 ) ) ) ) ) ).
% continuous_on_arcosh'
thf(fact_10048_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_10049_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_10050_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_10051_continuous__on__artanh_H,axiom,
! [A2: set_real,F: real > real] :
( ( topolo5044208981011980120l_real @ A2 @ F )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( member_real @ ( F @ X2 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
=> ( topolo5044208981011980120l_real @ A2
@ ^ [X3: real] : ( artanh_real @ ( F @ X3 ) ) ) ) ) ).
% continuous_on_artanh'
thf(fact_10052_Rolle__deriv,axiom,
! [A: real,B: real,F: real > real,F3: real > real > real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ A )
= ( F @ B ) )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X2: real] :
( ( ord_less_real @ A @ X2 )
=> ( ( ord_less_real @ X2 @ B )
=> ( has_de1759254742604945161l_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) )
=> ? [Z2: real] :
( ( ord_less_real @ A @ Z2 )
& ( ord_less_real @ Z2 @ B )
& ( ( F3 @ Z2 )
= ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).
% Rolle_deriv
thf(fact_10053_mvt,axiom,
! [A: real,B: real,F: real > real,F3: real > real > real] :
( ( ord_less_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X2: real] :
( ( ord_less_real @ A @ X2 )
=> ( ( ord_less_real @ X2 @ B )
=> ( has_de1759254742604945161l_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) )
=> ~ ! [Xi: real] :
( ( ord_less_real @ A @ Xi )
=> ( ( ord_less_real @ Xi @ B )
=> ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
!= ( F3 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% mvt
thf(fact_10054_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_10055_DERIV__pos__imp__increasing__open,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X2: real] :
( ( ord_less_real @ A @ X2 )
=> ( ( ord_less_real @ X2 @ B )
=> ? [Y3: real] :
( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
& ( ord_less_real @ zero_zero_real @ Y3 ) ) ) )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).
% DERIV_pos_imp_increasing_open
thf(fact_10056_DERIV__neg__imp__decreasing__open,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X2: real] :
( ( ord_less_real @ A @ X2 )
=> ( ( ord_less_real @ X2 @ B )
=> ? [Y3: real] :
( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
& ( ord_less_real @ Y3 @ zero_zero_real ) ) ) )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).
% DERIV_neg_imp_decreasing_open
thf(fact_10057_DERIV__isconst__end,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X2: real] :
( ( ord_less_real @ A @ X2 )
=> ( ( ord_less_real @ X2 @ B )
=> ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) )
=> ( ( F @ B )
= ( F @ A ) ) ) ) ) ).
% DERIV_isconst_end
thf(fact_10058_DERIV__isconst2,axiom,
! [A: real,B: real,F: real > real,X: real] :
( ( ord_less_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X2: real] :
( ( ord_less_real @ A @ X2 )
=> ( ( ord_less_real @ X2 @ B )
=> ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) )
=> ( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ X @ B )
=> ( ( F @ X )
= ( F @ A ) ) ) ) ) ) ) ).
% DERIV_isconst2
thf(fact_10059_Rolle,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ A )
= ( F @ B ) )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X2: real] :
( ( ord_less_real @ A @ X2 )
=> ( ( ord_less_real @ X2 @ B )
=> ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) )
=> ? [Z2: real] :
( ( ord_less_real @ A @ Z2 )
& ( ord_less_real @ Z2 @ B )
& ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ) ) ) ) ).
% Rolle
thf(fact_10060_inj__sgn__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( inj_on_real_real
@ ^ [Y4: real] : ( times_times_real @ ( sgn_sgn_real @ Y4 ) @ ( power_power_real @ ( abs_abs_real @ Y4 ) @ N2 ) )
@ top_top_set_real ) ) ).
% inj_sgn_power
thf(fact_10061_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_10062_inj__on__diff__nat,axiom,
! [N3: set_nat,K: nat] :
( ! [N: nat] :
( ( member_nat @ N @ N3 )
=> ( ord_less_eq_nat @ K @ N ) )
=> ( inj_on_nat_nat
@ ^ [N4: nat] : ( minus_minus_nat @ N4 @ K )
@ N3 ) ) ).
% inj_on_diff_nat
thf(fact_10063_inj__Suc,axiom,
! [N3: set_nat] : ( inj_on_nat_nat @ suc @ N3 ) ).
% inj_Suc
thf(fact_10064_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_10065_sup__int__def,axiom,
sup_sup_int = ord_max_int ).
% sup_int_def
thf(fact_10066_sup__enat__def,axiom,
sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).
% sup_enat_def
thf(fact_10067_sup__nat__def,axiom,
sup_sup_nat = ord_max_nat ).
% sup_nat_def
thf(fact_10068_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_10069_powr__real__of__int_H,axiom,
! [X: real,N2: int] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( X != zero_zero_real )
| ( ord_less_int @ zero_zero_int @ N2 ) )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ N2 ) )
= ( power_int_real @ X @ N2 ) ) ) ) ).
% powr_real_of_int'
thf(fact_10070_pred__nat__def,axiom,
( pred_nat
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [M2: nat,N4: nat] :
( N4
= ( suc @ M2 ) ) ) ) ) ).
% pred_nat_def
thf(fact_10071_less__eq,axiom,
! [M: nat,N2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N2 ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% less_eq
thf(fact_10072_pred__nat__trancl__eq__le,axiom,
! [M: nat,N2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N2 ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% pred_nat_trancl_eq_le
thf(fact_10073_pos__deriv__imp__strict__mono,axiom,
! [F: real > real,F3: real > real] :
( ! [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( F3 @ X2 ) )
=> ( order_7092887310737990675l_real @ F ) ) ) ).
% pos_deriv_imp_strict_mono
thf(fact_10074_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N2: nat] :
( ( order_5726023648592871131at_nat @ F )
=> ( ord_less_eq_nat @ N2 @ ( F @ N2 ) ) ) ).
% strict_mono_imp_increasing
thf(fact_10075_rat__floor__code,axiom,
( archim3151403230148437115or_rat
= ( ^ [P4: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P4 ) ) ) ) ).
% rat_floor_code
thf(fact_10076_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_10077_list__encode_Opelims,axiom,
! [X: list_nat,Y: nat] :
( ( ( nat_list_encode @ X )
= Y )
=> ( ( accp_list_nat @ nat_list_encode_rel @ X )
=> ( ( ( X = nil_nat )
=> ( ( Y = zero_zero_nat )
=> ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
=> ~ ! [X2: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs2 ) )
=> ( ( Y
= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X2 @ ( nat_list_encode @ Xs2 ) ) ) ) )
=> ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X2 @ Xs2 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_10078_remdups__upt,axiom,
! [M: nat,N2: nat] :
( ( remdups_nat @ ( upt @ M @ N2 ) )
= ( upt @ M @ N2 ) ) ).
% remdups_upt
thf(fact_10079_tl__upt,axiom,
! [M: nat,N2: nat] :
( ( tl_nat @ ( upt @ M @ N2 ) )
= ( upt @ ( suc @ M ) @ N2 ) ) ).
% tl_upt
thf(fact_10080_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( hd_nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_10081_drop__upt,axiom,
! [M: nat,I: nat,J: nat] :
( ( drop_nat @ M @ ( upt @ I @ J ) )
= ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
% drop_upt
thf(fact_10082_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_10083_take__upt,axiom,
! [I: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N2 )
=> ( ( take_nat @ M @ ( upt @ I @ N2 ) )
= ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% take_upt
thf(fact_10084_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_10085_sorted__list__of__set__range,axiom,
! [M: nat,N2: nat] :
( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
= ( upt @ M @ N2 ) ) ).
% sorted_list_of_set_range
thf(fact_10086_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_10087_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
=> ( ( nth_nat @ ( upt @ I @ J ) @ K )
= ( plus_plus_nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_10088_upt__rec__numeral,axiom,
! [M: num,N2: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
= ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_10089_map__Suc__upt,axiom,
! [M: nat,N2: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N2 ) )
= ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% map_Suc_upt
thf(fact_10090_distinct__upt,axiom,
! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).
% distinct_upt
thf(fact_10091_atLeastAtMost__upt,axiom,
( set_or1269000886237332187st_nat
= ( ^ [N4: nat,M2: nat] : ( set_nat2 @ ( upt @ N4 @ ( suc @ M2 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_10092_greaterThanLessThan__upt,axiom,
( set_or5834768355832116004an_nat
= ( ^ [N4: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N4 ) @ M2 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_10093_atLeastLessThan__upt,axiom,
( set_or4665077453230672383an_nat
= ( ^ [I3: nat,J3: nat] : ( set_nat2 @ ( upt @ I3 @ J3 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_10094_greaterThanAtMost__upt,axiom,
( set_or6659071591806873216st_nat
= ( ^ [N4: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N4 ) @ ( suc @ M2 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_10095_upt__conv__Cons__Cons,axiom,
! [M: nat,N2: nat,Ns: list_nat,Q3: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N2 @ Ns ) )
= ( upt @ M @ Q3 ) )
= ( ( cons_nat @ N2 @ Ns )
= ( upt @ ( suc @ M ) @ Q3 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_10096_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_10097_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N4 ) ) ) ) ).
% atLeast_upt
thf(fact_10098_atMost__upto,axiom,
( set_ord_atMost_nat
= ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N4 ) ) ) ) ) ).
% atMost_upto
thf(fact_10099_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_10100_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
= ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_10101_map__add__upt,axiom,
! [N2: nat,M: nat] :
( ( map_nat_nat
@ ^ [I3: nat] : ( plus_plus_nat @ I3 @ N2 )
@ ( upt @ zero_zero_nat @ M ) )
= ( upt @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% map_add_upt
thf(fact_10102_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list_nat] :
( ( ( upt @ I @ J )
= ( cons_nat @ X @ Xs ) )
= ( ( ord_less_nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_10103_upt__rec,axiom,
( upt
= ( ^ [I3: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I3 @ J3 ) @ ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_10104_map__decr__upt,axiom,
! [M: nat,N2: nat] :
( ( map_nat_nat
@ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( upt @ M @ N2 ) ) ).
% map_decr_upt
thf(fact_10105_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_10106_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_10107_sum__list__upt,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( groups4561878855575611511st_nat @ ( upt @ M @ N2 ) )
= ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ).
% sum_list_upt
thf(fact_10108_card__length__sum__list__rec,axiom,
! [M: nat,N3: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= M )
& ( ( groups4561878855575611511st_nat @ L2 )
= N3 ) ) ) )
= ( plus_plus_nat
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= ( minus_minus_nat @ M @ one_one_nat ) )
& ( ( groups4561878855575611511st_nat @ L2 )
= N3 ) ) ) )
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= M )
& ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
= N3 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_10109_card__length__sum__list,axiom,
! [M: nat,N3: nat] :
( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= M )
& ( ( groups4561878855575611511st_nat @ L2 )
= N3 ) ) ) )
= ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M ) @ one_one_nat ) @ N3 ) ) ).
% card_length_sum_list
thf(fact_10110_sorted__wrt__upt,axiom,
! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N2 ) ) ).
% sorted_wrt_upt
thf(fact_10111_sorted__upt,axiom,
! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N2 ) ) ).
% sorted_upt
thf(fact_10112_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_10113_sorted__upto,axiom,
! [M: int,N2: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N2 ) ) ).
% sorted_upto
thf(fact_10114_sorted__wrt__upto,axiom,
! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).
% sorted_wrt_upto
thf(fact_10115_pairs__le__eq__Sigma,axiom,
! [M: nat] :
( ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ 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_10116_quotient__of__def,axiom,
( quotient_of
= ( ^ [X3: rat] :
( the_Pr4378521158711661632nt_int
@ ^ [Pair: product_prod_int_int] :
( ( X3
= ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
& ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
& ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% quotient_of_def
thf(fact_10117_coprime__abs__left__iff,axiom,
! [K: int,L: int] :
( ( algebr932160517623751201me_int @ ( abs_abs_int @ K ) @ L )
= ( algebr932160517623751201me_int @ K @ L ) ) ).
% coprime_abs_left_iff
thf(fact_10118_coprime__abs__right__iff,axiom,
! [K: int,L: int] :
( ( algebr932160517623751201me_int @ K @ ( abs_abs_int @ L ) )
= ( algebr932160517623751201me_int @ K @ L ) ) ).
% coprime_abs_right_iff
thf(fact_10119_normalize__stable,axiom,
! [Q3: int,P3: int] :
( ( ord_less_int @ zero_zero_int @ Q3 )
=> ( ( algebr932160517623751201me_int @ P3 @ Q3 )
=> ( ( normalize @ ( product_Pair_int_int @ P3 @ Q3 ) )
= ( product_Pair_int_int @ P3 @ Q3 ) ) ) ) ).
% normalize_stable
thf(fact_10120_coprime__common__divisor__int,axiom,
! [A: int,B: int,X: int] :
( ( algebr932160517623751201me_int @ A @ B )
=> ( ( dvd_dvd_int @ X @ A )
=> ( ( dvd_dvd_int @ X @ B )
=> ( ( abs_abs_int @ X )
= one_one_int ) ) ) ) ).
% coprime_common_divisor_int
thf(fact_10121_Rat__cases,axiom,
! [Q3: rat] :
~ ! [A5: int,B5: int] :
( ( Q3
= ( fract @ A5 @ B5 ) )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ~ ( algebr932160517623751201me_int @ A5 @ B5 ) ) ) ).
% Rat_cases
thf(fact_10122_Rat__induct,axiom,
! [P: rat > $o,Q3: rat] :
( ! [A5: int,B5: int] :
( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( algebr932160517623751201me_int @ A5 @ B5 )
=> ( P @ ( fract @ A5 @ B5 ) ) ) )
=> ( P @ Q3 ) ) ).
% Rat_induct
thf(fact_10123_Rat__cases__nonzero,axiom,
! [Q3: rat] :
( ! [A5: int,B5: int] :
( ( Q3
= ( fract @ A5 @ B5 ) )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( A5 != zero_zero_int )
=> ~ ( algebr932160517623751201me_int @ A5 @ B5 ) ) ) )
=> ( Q3 = zero_zero_rat ) ) ).
% Rat_cases_nonzero
thf(fact_10124_quotient__of__unique,axiom,
! [R4: rat] :
? [X2: product_prod_int_int] :
( ( R4
= ( fract @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) )
& ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X2 ) )
& ( algebr932160517623751201me_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) )
& ! [Y3: product_prod_int_int] :
( ( ( R4
= ( fract @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ Y3 ) ) )
& ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y3 ) )
& ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ Y3 ) ) )
=> ( Y3 = X2 ) ) ) ).
% quotient_of_unique
thf(fact_10125_coprime__int__iff,axiom,
! [M: nat,N2: nat] :
( ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( algebr934650988132801477me_nat @ M @ N2 ) ) ).
% coprime_int_iff
thf(fact_10126_coprime__nat__abs__right__iff,axiom,
! [N2: nat,K: int] :
( ( algebr934650988132801477me_nat @ N2 @ ( nat2 @ ( abs_abs_int @ K ) ) )
= ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ).
% coprime_nat_abs_right_iff
thf(fact_10127_coprime__nat__abs__left__iff,axiom,
! [K: int,N2: nat] :
( ( algebr934650988132801477me_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N2 )
= ( algebr932160517623751201me_int @ K @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% coprime_nat_abs_left_iff
thf(fact_10128_coprime__Suc__0__left,axiom,
! [N2: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N2 ) ).
% coprime_Suc_0_left
thf(fact_10129_coprime__Suc__0__right,axiom,
! [N2: nat] : ( algebr934650988132801477me_nat @ N2 @ ( suc @ zero_zero_nat ) ) ).
% coprime_Suc_0_right
thf(fact_10130_coprime__Suc__right__nat,axiom,
! [N2: nat] : ( algebr934650988132801477me_nat @ N2 @ ( suc @ N2 ) ) ).
% coprime_Suc_right_nat
thf(fact_10131_coprime__Suc__left__nat,axiom,
! [N2: nat] : ( algebr934650988132801477me_nat @ ( suc @ N2 ) @ N2 ) ).
% coprime_Suc_left_nat
thf(fact_10132_coprime__common__divisor__nat,axiom,
! [A: nat,B: nat,X: nat] :
( ( algebr934650988132801477me_nat @ A @ B )
=> ( ( dvd_dvd_nat @ X @ A )
=> ( ( dvd_dvd_nat @ X @ B )
=> ( X = one_one_nat ) ) ) ) ).
% coprime_common_divisor_nat
thf(fact_10133_coprime__diff__one__right__nat,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( algebr934650988132801477me_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).
% coprime_diff_one_right_nat
thf(fact_10134_coprime__diff__one__left__nat,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ N2 ) ) ).
% coprime_diff_one_left_nat
thf(fact_10135_Rats__abs__nat__div__natE,axiom,
! [X: real] :
( ( member_real @ X @ field_5140801741446780682s_real )
=> ~ ! [M5: nat,N: nat] :
( ( N != zero_zero_nat )
=> ( ( ( abs_abs_real @ X )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ M5 ) @ ( semiri5074537144036343181t_real @ N ) ) )
=> ~ ( algebr934650988132801477me_nat @ M5 @ N ) ) ) ) ).
% Rats_abs_nat_div_natE
thf(fact_10136_Gcd__nat__set__eq__fold,axiom,
! [Xs: list_nat] :
( ( gcd_Gcd_nat @ ( set_nat2 @ Xs ) )
= ( fold_nat_nat @ gcd_gcd_nat @ Xs @ zero_zero_nat ) ) ).
% Gcd_nat_set_eq_fold
thf(fact_10137_Gcd__int__set__eq__fold,axiom,
! [Xs: list_int] :
( ( gcd_Gcd_int @ ( set_int2 @ Xs ) )
= ( fold_int_int @ gcd_gcd_int @ Xs @ zero_zero_int ) ) ).
% Gcd_int_set_eq_fold
thf(fact_10138_sort__upt,axiom,
! [M: nat,N2: nat] :
( ( linord738340561235409698at_nat
@ ^ [X3: nat] : X3
@ ( upt @ M @ N2 ) )
= ( upt @ M @ N2 ) ) ).
% sort_upt
thf(fact_10139_sort__upto,axiom,
! [I: int,J: int] :
( ( linord1735203802627413978nt_int
@ ^ [X3: int] : X3
@ ( upto @ I @ J ) )
= ( upto @ I @ J ) ) ).
% sort_upto
thf(fact_10140_Rat_Opositive_Orsp,axiom,
( bNF_re8699439704749558557nt_o_o @ ratrel
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
@ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ).
% Rat.positive.rsp
thf(fact_10141_vanishes__mult__bounded,axiom,
! [X8: nat > rat,Y6: nat > rat] :
( ? [A7: rat] :
( ( ord_less_rat @ zero_zero_rat @ A7 )
& ! [N: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N ) ) @ A7 ) )
=> ( ( vanishes @ Y6 )
=> ( vanishes
@ ^ [N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y6 @ N4 ) ) ) ) ) ).
% vanishes_mult_bounded
thf(fact_10142_vanishes__const,axiom,
! [C: rat] :
( ( vanishes
@ ^ [N4: nat] : C )
= ( C = zero_zero_rat ) ) ).
% vanishes_const
thf(fact_10143_vanishes__minus,axiom,
! [X8: nat > rat] :
( ( vanishes @ X8 )
=> ( vanishes
@ ^ [N4: nat] : ( uminus_uminus_rat @ ( X8 @ N4 ) ) ) ) ).
% vanishes_minus
thf(fact_10144_vanishes__add,axiom,
! [X8: nat > rat,Y6: nat > rat] :
( ( vanishes @ X8 )
=> ( ( vanishes @ Y6 )
=> ( vanishes
@ ^ [N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y6 @ N4 ) ) ) ) ) ).
% vanishes_add
thf(fact_10145_vanishes__diff,axiom,
! [X8: nat > rat,Y6: nat > rat] :
( ( vanishes @ X8 )
=> ( ( vanishes @ Y6 )
=> ( vanishes
@ ^ [N4: nat] : ( minus_minus_rat @ ( X8 @ N4 ) @ ( Y6 @ N4 ) ) ) ) ) ).
% vanishes_diff
thf(fact_10146_vanishesD,axiom,
! [X8: nat > rat,R4: rat] :
( ( vanishes @ X8 )
=> ( ( ord_less_rat @ zero_zero_rat @ R4 )
=> ? [K3: nat] :
! [N8: nat] :
( ( ord_less_eq_nat @ K3 @ N8 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N8 ) ) @ R4 ) ) ) ) ).
% vanishesD
thf(fact_10147_vanishesI,axiom,
! [X8: nat > rat] :
( ! [R2: rat] :
( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ? [K4: nat] :
! [N: nat] :
( ( ord_less_eq_nat @ K4 @ N )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N ) ) @ R2 ) ) )
=> ( vanishes @ X8 ) ) ).
% vanishesI
thf(fact_10148_vanishes__def,axiom,
( vanishes
= ( ^ [X6: nat > rat] :
! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K2: nat] :
! [N4: nat] :
( ( ord_less_eq_nat @ K2 @ N4 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N4 ) ) @ R5 ) ) ) ) ) ).
% vanishes_def
thf(fact_10149_and__not__num_Opelims,axiom,
! [X: num,Xa: num,Y: option_num] :
( ( ( bit_and_not_num @ X @ Xa )
= Y )
=> ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X @ Xa ) )
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( ( Y = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X = one )
=> ! [N: num] :
( ( Xa
= ( bit0 @ N ) )
=> ( ( Y
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N ) ) ) ) ) )
=> ( ( ( X = one )
=> ! [N: num] :
( ( Xa
= ( bit1 @ N ) )
=> ( ( Y = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one )
=> ( ( Y
= ( some_num @ ( bit0 @ M5 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit0 @ N ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit1 @ N ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one )
=> ( ( Y
= ( some_num @ ( bit0 @ M5 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit0 @ N ) )
=> ( ( Y
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M5 @ N ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N ) ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit1 @ N ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
thf(fact_10150_plus__rat_Orsp,axiom,
( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
@ ^ [X3: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y4 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y4 ) ) )
@ ^ [X3: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y4 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y4 ) ) ) ) ).
% plus_rat.rsp
thf(fact_10151_Fract_Orsp,axiom,
( bNF_re157797125943740599nt_int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_re6250860962936578807nt_int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ratrel )
@ ^ [A3: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B2 ) )
@ ^ [A3: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B2 ) ) ) ).
% Fract.rsp
thf(fact_10152_less__eq__integer_Orsp,axiom,
( bNF_re3403563459893282935_int_o
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_re5089333283451836215nt_o_o
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ord_less_eq_int
@ ord_less_eq_int ) ).
% less_eq_integer.rsp
thf(fact_10153_less__eq__natural_Orsp,axiom,
( bNF_re578469030762574527_nat_o
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_re4705727531993890431at_o_o
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ord_less_eq_nat
@ ord_less_eq_nat ) ).
% less_eq_natural.rsp
thf(fact_10154_less__natural_Orsp,axiom,
( bNF_re578469030762574527_nat_o
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_re4705727531993890431at_o_o
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ord_less_nat
@ ord_less_nat ) ).
% less_natural.rsp
thf(fact_10155_less__integer_Orsp,axiom,
( bNF_re3403563459893282935_int_o
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_re5089333283451836215nt_o_o
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ord_less_int
@ ord_less_int ) ).
% less_integer.rsp
thf(fact_10156_sub_Orsp,axiom,
( bNF_re8402795839162346335um_int
@ ^ [Y5: num,Z4: num] : Y5 = Z4
@ ( bNF_re1822329894187522285nt_int
@ ^ [Y5: num,Z4: num] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ^ [M2: num,N4: num] : ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N4 ) )
@ ^ [M2: num,N4: num] : ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N4 ) ) ) ).
% sub.rsp
thf(fact_10157_dup_Orsp,axiom,
( bNF_re4712519889275205905nt_int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [K2: int] : ( plus_plus_int @ K2 @ K2 )
@ ^ [K2: int] : ( plus_plus_int @ K2 @ K2 ) ) ).
% dup.rsp
thf(fact_10158_plus__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ plus_plus_nat
@ plus_plus_nat ) ).
% plus_natural.rsp
thf(fact_10159_plus__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ plus_plus_int
@ plus_plus_int ) ).
% plus_integer.rsp
thf(fact_10160_Suc_Orsp,axiom,
( bNF_re5653821019739307937at_nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ suc
@ suc ) ).
% Suc.rsp
thf(fact_10161_divide__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ divide_divide_nat
@ divide_divide_nat ) ).
% divide_natural.rsp
thf(fact_10162_divide__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ divide_divide_int
@ divide_divide_int ) ).
% divide_integer.rsp
thf(fact_10163_inverse__rat_Orsp,axiom,
( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
@ ^ [X3: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X3 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
@ ^ [X3: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X3 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ).
% inverse_rat.rsp
thf(fact_10164_and__num_Opelims,axiom,
! [X: num,Xa: num,Y: option_num] :
( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
= Y )
=> ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X @ Xa ) )
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( ( Y
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X = one )
=> ! [N: num] :
( ( Xa
= ( bit0 @ N ) )
=> ( ( Y = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N ) ) ) ) ) )
=> ( ( ( X = one )
=> ! [N: num] :
( ( Xa
= ( bit1 @ N ) )
=> ( ( Y
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one )
=> ( ( Y = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit0 @ N ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit1 @ N ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one )
=> ( ( Y
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit0 @ N ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N ) ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit1 @ N ) )
=> ( ( Y
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M5 @ N ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
thf(fact_10165_xor__num_Opelims,axiom,
! [X: num,Xa: num,Y: option_num] :
( ( ( bit_un2480387367778600638or_num @ X @ Xa )
= Y )
=> ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X @ Xa ) )
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( ( Y = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X = one )
=> ! [N: num] :
( ( Xa
= ( bit0 @ N ) )
=> ( ( Y
= ( some_num @ ( bit1 @ N ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N ) ) ) ) ) )
=> ( ( ( X = one )
=> ! [N: num] :
( ( Xa
= ( bit1 @ N ) )
=> ( ( Y
= ( some_num @ ( bit0 @ N ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one )
=> ( ( Y
= ( some_num @ ( bit1 @ M5 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit0 @ N ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit1 @ N ) )
=> ( ( Y
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N ) ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one )
=> ( ( Y
= ( some_num @ ( bit0 @ M5 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit0 @ N ) )
=> ( ( Y
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N ) ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N ) ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N: num] :
( ( Xa
= ( bit1 @ N ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.pelims
thf(fact_10166_or__not__num__neg_Opelims,axiom,
! [X: num,Xa: num,Y: num] :
( ( ( bit_or_not_num_neg @ X @ Xa )
= Y )
=> ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X @ Xa ) )
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( ( Y = one )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X = one )
=> ! [M5: num] :
( ( Xa
= ( bit0 @ M5 ) )
=> ( ( Y
= ( bit1 @ M5 ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M5 ) ) ) ) ) )
=> ( ( ( X = one )
=> ! [M5: num] :
( ( Xa
= ( bit1 @ M5 ) )
=> ( ( Y
= ( bit1 @ M5 ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M5 ) ) ) ) ) )
=> ( ! [N: num] :
( ( X
= ( bit0 @ N ) )
=> ( ( Xa = one )
=> ( ( Y
= ( bit0 @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N ) @ one ) ) ) ) )
=> ( ! [N: num] :
( ( X
= ( bit0 @ N ) )
=> ! [M5: num] :
( ( Xa
= ( bit0 @ M5 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N @ M5 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N ) @ ( bit0 @ M5 ) ) ) ) ) )
=> ( ! [N: num] :
( ( X
= ( bit0 @ N ) )
=> ! [M5: num] :
( ( Xa
= ( bit1 @ M5 ) )
=> ( ( Y
= ( bit0 @ ( bit_or_not_num_neg @ N @ M5 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N ) @ ( bit1 @ M5 ) ) ) ) ) )
=> ( ! [N: num] :
( ( X
= ( bit1 @ N ) )
=> ( ( Xa = one )
=> ( ( Y = one )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N ) @ one ) ) ) ) )
=> ( ! [N: num] :
( ( X
= ( bit1 @ N ) )
=> ! [M5: num] :
( ( Xa
= ( bit0 @ M5 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N @ M5 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N ) @ ( bit0 @ M5 ) ) ) ) ) )
=> ~ ! [N: num] :
( ( X
= ( bit1 @ N ) )
=> ! [M5: num] :
( ( Xa
= ( bit1 @ M5 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N @ M5 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N ) @ ( bit1 @ M5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.pelims
thf(fact_10167_and__num__rel__dict,axiom,
bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).
% and_num_rel_dict
thf(fact_10168_xor__num__rel__dict,axiom,
bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).
% xor_num_rel_dict
thf(fact_10169_plus__rat_Otransfer,axiom,
( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
@ ^ [X3: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y4 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y4 ) ) )
@ plus_plus_rat ) ).
% plus_rat.transfer
thf(fact_10170_one__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).
% one_rat.transfer
thf(fact_10171_zero__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).
% zero_rat.transfer
thf(fact_10172_Fract_Otransfer,axiom,
( bNF_re3461391660133120880nt_rat
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_re2214769303045360666nt_rat
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ pcr_rat )
@ ^ [A3: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B2 ) )
@ fract ) ).
% Fract.transfer
thf(fact_10173_Rat_Opositive_Otransfer,axiom,
( bNF_re1494630372529172596at_o_o @ pcr_rat
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
@ positive ) ).
% Rat.positive.transfer
thf(fact_10174_inverse__rat_Otransfer,axiom,
( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
@ ^ [X3: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X3 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
@ inverse_inverse_rat ) ).
% inverse_rat.transfer
thf(fact_10175_times__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y4 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y4 @ U2 ) ) ) ) )
@ times_times_int ) ).
% times_int.transfer
thf(fact_10176_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).
% zero_int.transfer
thf(fact_10177_int__transfer,axiom,
( bNF_re6830278522597306478at_int
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ pcr_int
@ ^ [N4: nat] : ( product_Pair_nat_nat @ N4 @ zero_zero_nat )
@ semiri1314217659103216013at_int ) ).
% int_transfer
thf(fact_10178_uminus__int_Otransfer,axiom,
( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
@ ( produc2626176000494625587at_nat
@ ^ [X3: nat,Y4: nat] : ( product_Pair_nat_nat @ Y4 @ X3 ) )
@ uminus_uminus_int ) ).
% uminus_int.transfer
thf(fact_10179_nat_Otransfer,axiom,
( bNF_re4555766996558763186at_nat @ pcr_int
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( produc6842872674320459806at_nat @ minus_minus_nat )
@ nat2 ) ).
% nat.transfer
thf(fact_10180_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% one_int.transfer
thf(fact_10181_less__int_Otransfer,axiom,
( bNF_re717283939379294677_int_o @ pcr_int
@ ( bNF_re6644619430987730960nt_o_o @ pcr_int
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y4: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) )
@ ord_less_int ) ).
% less_int.transfer
thf(fact_10182_less__eq__int_Otransfer,axiom,
( bNF_re717283939379294677_int_o @ pcr_int
@ ( bNF_re6644619430987730960nt_o_o @ pcr_int
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y4: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) )
@ ord_less_eq_int ) ).
% less_eq_int.transfer
thf(fact_10183_plus__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y4 @ V4 ) ) ) )
@ plus_plus_int ) ).
% plus_int.transfer
thf(fact_10184_minus__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y4 @ U2 ) ) ) )
@ minus_minus_int ) ).
% minus_int.transfer
thf(fact_10185_inverse__real_Otransfer,axiom,
( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
@ ^ [X6: nat > rat] :
( if_nat_rat @ ( vanishes @ X6 )
@ ^ [N4: nat] : zero_zero_rat
@ ^ [N4: nat] : ( inverse_inverse_rat @ ( X6 @ N4 ) ) )
@ inverse_inverse_real ) ).
% inverse_real.transfer
thf(fact_10186_times__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y4 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y4 @ U2 ) ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y4 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y4 @ U2 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_10187_intrel__iff,axiom,
! [X: nat,Y: nat,U: nat,V: nat] :
( ( intrel @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ U @ V ) )
= ( ( plus_plus_nat @ X @ V )
= ( plus_plus_nat @ U @ Y ) ) ) ).
% intrel_iff
thf(fact_10188_int_Orel__eq__transfer,axiom,
( bNF_re717283939379294677_int_o @ pcr_int
@ ( bNF_re6644619430987730960nt_o_o @ pcr_int
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ intrel
@ ^ [Y5: int,Z4: int] : Y5 = Z4 ) ).
% int.rel_eq_transfer
thf(fact_10189_nat_Orsp,axiom,
( bNF_re8246922863344978751at_nat @ intrel
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( produc6842872674320459806at_nat @ minus_minus_nat )
@ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).
% nat.rsp
thf(fact_10190_uminus__int_Orsp,axiom,
( bNF_re2241393799969408733at_nat @ intrel @ intrel
@ ( produc2626176000494625587at_nat
@ ^ [X3: nat,Y4: nat] : ( product_Pair_nat_nat @ Y4 @ X3 ) )
@ ( produc2626176000494625587at_nat
@ ^ [X3: nat,Y4: nat] : ( product_Pair_nat_nat @ Y4 @ X3 ) ) ) ).
% uminus_int.rsp
thf(fact_10191_zero__real_Otransfer,axiom,
( pcr_real
@ ^ [N4: nat] : zero_zero_rat
@ zero_zero_real ) ).
% zero_real.transfer
thf(fact_10192_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_10193_int_Oabs__eq__iff,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( ( abs_Integ @ X )
= ( abs_Integ @ Y ) )
= ( intrel @ X @ Y ) ) ).
% int.abs_eq_iff
thf(fact_10194_one__real_Otransfer,axiom,
( pcr_real
@ ^ [N4: nat] : one_one_rat
@ one_one_real ) ).
% one_real.transfer
thf(fact_10195_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_10196_uminus__real_Otransfer,axiom,
( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
@ ^ [X6: nat > rat,N4: nat] : ( uminus_uminus_rat @ ( X6 @ N4 ) )
@ uminus_uminus_real ) ).
% uminus_real.transfer
thf(fact_10197_plus__real_Otransfer,axiom,
( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
@ ^ [X6: nat > rat,Y7: nat > rat,N4: nat] : ( plus_plus_rat @ ( X6 @ N4 ) @ ( Y7 @ N4 ) )
@ plus_plus_real ) ).
% plus_real.transfer
thf(fact_10198_times__real_Otransfer,axiom,
( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
@ ^ [X6: nat > rat,Y7: nat > rat,N4: nat] : ( times_times_rat @ ( X6 @ N4 ) @ ( Y7 @ N4 ) )
@ times_times_real ) ).
% times_real.transfer
thf(fact_10199_intrel__def,axiom,
( intrel
= ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y4: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] :
( ( plus_plus_nat @ X3 @ V4 )
= ( plus_plus_nat @ U2 @ Y4 ) ) ) ) ) ).
% intrel_def
thf(fact_10200_less__int_Orsp,axiom,
( bNF_re4202695980764964119_nat_o @ intrel
@ ( bNF_re3666534408544137501at_o_o @ intrel
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y4: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y4: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) ) ) ).
% less_int.rsp
thf(fact_10201_less__eq__int_Orsp,axiom,
( bNF_re4202695980764964119_nat_o @ intrel
@ ( bNF_re3666534408544137501at_o_o @ intrel
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y4: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y4: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_10202_plus__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y4 @ V4 ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y4 @ V4 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_10203_minus__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y4 @ U2 ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y4 @ U2 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_10204_Real_Opositive_Otransfer,axiom,
( bNF_re4297313714947099218al_o_o @ pcr_real
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ^ [X6: nat > rat] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K2: nat] :
! [N4: nat] :
( ( ord_less_eq_nat @ K2 @ N4 )
=> ( ord_less_rat @ R5 @ ( X6 @ N4 ) ) ) )
@ positive2 ) ).
% Real.positive.transfer
thf(fact_10205_rcis__inverse,axiom,
! [R4: real,A: real] :
( ( invers8013647133539491842omplex @ ( rcis @ R4 @ A ) )
= ( rcis @ ( divide_divide_real @ one_one_real @ R4 ) @ ( uminus_uminus_real @ A ) ) ) ).
% rcis_inverse
thf(fact_10206_less__real__def,axiom,
( ord_less_real
= ( ^ [X3: real,Y4: real] : ( positive2 @ ( minus_minus_real @ Y4 @ X3 ) ) ) ) ).
% less_real_def
thf(fact_10207_Real_Opositive__minus,axiom,
! [X: real] :
( ~ ( positive2 @ X )
=> ( ( X != zero_zero_real )
=> ( positive2 @ ( uminus_uminus_real @ X ) ) ) ) ).
% Real.positive_minus
% Helper facts (40)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ( if_num @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ( if_num @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
! [X: rat,Y: rat] :
( ( if_rat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
! [X: rat,Y: rat] :
( ( if_rat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
! [P: real > $o] :
( ( P @ ( fChoice_real @ P ) )
= ( ? [X6: real] : ( P @ X6 ) ) ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y: code_integer] :
( ( if_Code_integer @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y: code_integer] :
( ( if_Code_integer @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X: set_int,Y: set_int] :
( ( if_set_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X: set_int,Y: set_int] :
( ( if_set_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: int > int,Y: int > int] :
( ( if_int_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: int > int,Y: int > int] :
( ( if_int_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
! [X: nat > rat,Y: nat > rat] :
( ( if_nat_rat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
! [X: nat > rat,Y: nat > rat] :
( ( if_nat_rat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
! [X: option_num,Y: option_num] :
( ( if_option_num @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
! [X: option_num,Y: option_num] :
( ( if_option_num @ $true @ X @ Y )
= 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,Y: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
= 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,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
! [X: nat > int > int,Y: nat > int > int] :
( ( if_nat_int_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
! [X: nat > int > int,Y: nat > int > int] :
( ( if_nat_int_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [X: nat > nat > nat,Y: nat > nat > nat] :
( ( if_nat_nat_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [X: nat > nat > nat,Y: nat > nat > nat] :
( ( if_nat_nat_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
= 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,Y: produc8923325533196201883nteger] :
( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
= X ) ).
% Conjectures (10)
thf(conj_0,hypothesis,
info = none_P5556105721700978146at_nat ).
thf(conj_1,hypothesis,
zero_zero_nat = deg ).
thf(conj_2,hypothesis,
! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ treeList ) )
=> ( vEBT_invar_vebt @ X4 @ na ) ) ).
thf(conj_3,hypothesis,
vEBT_invar_vebt @ summary @ m ).
thf(conj_4,hypothesis,
( ( size_s6755466524823107622T_VEBT @ treeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
thf(conj_5,hypothesis,
m = na ).
thf(conj_6,hypothesis,
( deg
= ( plus_plus_nat @ na @ m ) ) ).
thf(conj_7,hypothesis,
~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ summary @ X_12 ) ).
thf(conj_8,hypothesis,
! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ treeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ).
thf(conj_9,conjecture,
ord_less_eq_nat @ n @ zero_zero_nat ).
%------------------------------------------------------------------------------