TPTP Problem File: ITP225^1.p
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%------------------------------------------------------------------------------
% File : ITP225^1 : TPTP v8.2.0. Released v8.1.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_Member 00300_011794
% 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_00300_011794 [Des22]
% Status : Theorem
% Rating : 0.60 v8.2.0, 0.54 v8.1.0
% Syntax : Number of formulae : 11266 (6290 unt;1073 typ; 0 def)
% Number of atoms : 25699 (11743 equ; 0 cnn)
% Maximal formula atoms : 71 ( 2 avg)
% Number of connectives : 98527 (2371 ~; 490 |;1493 &;86321 @)
% ( 0 <=>;7852 =>; 0 <=; 0 <~>)
% Maximal formula depth : 39 ( 5 avg)
% Number of types : 72 ( 71 usr)
% Number of type conns : 3785 (3785 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1005 (1002 usr; 69 con; 0-8 aty)
% Number of variables : 22092 (1891 ^;19613 !; 588 ?;22092 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 18:47:27.061
%------------------------------------------------------------------------------
% Could-be-implicit typings (71)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Pr8693737435421807431at_nat: $tType ).
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thf(ty_n_t__Product____Type__Oprod_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(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
produc7822875418678951345atural: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
produc8923325533196201883nteger: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
produc7272778201969148633d_enat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
set_Pr958786334691620121nt_int: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Sum_sum_nat_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
list_P6285523579766656935_o_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
produc2504756804600209347T_VEBT: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
product_prod_int_int: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
list_P4002435161011370285od_o_o: $tType ).
thf(ty_n_t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
sum_sum_nat_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
list_VEBT_VEBT: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
product_prod_o_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
product_prod_o_int: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Code____Numeral__Ointeger_J,type,
list_Code_integer: $tType ).
thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
set_VEBT_VEBT: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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set_Product_unit: $tType ).
thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
product_prod_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
filter_real: $tType ).
thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
option_num: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
filter_nat: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Int__Oint_J,type,
filter_int: $tType ).
thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
set_char: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__VEBT____Definitions__OVEBT,type,
vEBT_VEBT: $tType ).
thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Code____Numeral__Onatural,type,
code_natural: $tType ).
thf(ty_n_t__Code____Numeral__Ointeger,type,
code_integer: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__Extended____Nat__Oenat,type,
extended_enat: $tType ).
thf(ty_n_t__List__Olist_I_Eo_J,type,
list_o: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__String__Oliteral,type,
literal: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Rat__Orat,type,
rat: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (1002)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
archim2889992004027027881ng_rat: rat > int ).
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
archim3151403230148437115or_rat: rat > int ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
archim6058952711729229775r_real: real > int ).
thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
archimedean_frac_rat: rat > rat ).
thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
archim2898591450579166408c_real: real > real ).
thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
archim7778729529865785530nd_rat: rat > int ).
thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
archim8280529875227126926d_real: real > int ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
bNF_re1962705104956426057at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
bNF_re895249473297799549at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo_001_Eo,type,
bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
bNF_re4695409256820837752l_real: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( real > real > real ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_Eo_J_001_062_It__Real__Oreal_M_Eo_J,type,
bNF_re4521903465945308077real_o: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > $o ) > ( real > $o ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( real > real > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
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,
bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001_062_It__Int__Oint_M_Eo_J_001_062_It__Code____Numeral__Ointeger_M_Eo_J,type,
bNF_re6321650412969554871eger_o: ( int > code_integer > $o ) > ( ( int > $o ) > ( code_integer > $o ) > $o ) > ( int > int > $o ) > ( code_integer > code_integer > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
bNF_re398004352372739002nteger: ( int > code_integer > $o ) > ( ( int > int ) > ( code_integer > code_integer ) > $o ) > ( int > int > int ) > ( code_integer > code_integer > code_integer ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001_Eo_001_Eo,type,
bNF_re6574881592172037608er_o_o: ( int > code_integer > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( code_integer > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
bNF_re3379532845092657523nteger: ( int > code_integer > $o ) > ( int > code_integer > $o ) > ( int > int ) > ( code_integer > code_integer ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
bNF_re1831474436612530402atural: ( int > code_integer > $o ) > ( nat > code_natural > $o ) > ( int > nat ) > ( code_integer > code_natural ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_re2807294637932363402at_nat: ( int > code_integer > $o ) > ( nat > nat > $o ) > ( int > nat ) > ( code_integer > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Num__Onum_001t__Num__Onum,type,
bNF_re6718328864250387230um_num: ( int > code_integer > $o ) > ( num > num > $o ) > ( int > num ) > ( code_integer > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
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 ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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 ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Rat__Orat_J,type,
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__Nat__Onat_001t__Nat__Onat,type,
bNF_re3715656647883201625at_nat: ( int > int > $o ) > ( nat > nat > $o ) > ( int > nat ) > ( int > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Num__Onum_001t__Num__Onum,type,
bNF_re7626690874201225453um_num: ( int > int > $o ) > ( num > num > $o ) > ( int > num ) > ( int > num ) > $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 ).
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__Rat__Orat,type,
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__Code____Numeral__Onatural_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Code____Numeral__Onatural_M_Eo_J,type,
bNF_re1639080489988575423ural_o: ( nat > code_natural > $o ) > ( ( nat > $o ) > ( code_natural > $o ) > $o ) > ( nat > nat > $o ) > ( code_natural > code_natural > $o ) > $o ).
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bNF_re88643428490162567atural: ( nat > code_natural > $o ) > ( ( nat > nat ) > ( code_natural > code_natural ) > $o ) > ( nat > nat > nat ) > ( code_natural > code_natural > code_natural ) > $o ).
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bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
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bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
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bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).
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bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).
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bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).
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bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
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bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
ord_Least_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Real__Oreal,type,
ord_Least_real: ( real > $o ) > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Onatural,type,
ord_le5570908160329646204atural: code_natural > code_natural > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
ord_less_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_set_num: set_num > set_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
ord_less_set_rat: set_rat > set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
ord_le3102999989581377725nteger: code_integer > code_integer > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Onatural,type,
ord_le1926595141338095240atural: code_natural > code_natural > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
ord_less_eq_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
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__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,
ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).
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__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_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_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $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__List__Olist_It__Nat__Onat_J_J,type,
top_top_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
top_to1996260823553986621t_unit: set_Product_unit ).
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,
top_top_set_char: set_char ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to6661820994512907621at_nat: set_Sum_sum_nat_nat ).
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__Code____Numeral__Onatural,type,
power_7079662738309270450atural: code_natural > nat > code_natural ).
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,
produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Onatural_001t__Code____Numeral__Onatural,type,
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thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
produc6639722614265839536atural: code_natural > produc7822875418678951345atural > produc5835291356934675326atural ).
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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).
thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).
thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat,type,
produc581526299967858633d_enat: vEBT_VEBT > extended_enat > produc7272778201969148633d_enat ).
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,
produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
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 ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
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produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
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 ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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,
produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
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,
produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_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__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__List__Olist_It__Nat__Onat_J,type,
produc2761476792215241774st_nat: ( nat > nat > list_nat ) > product_prod_nat_nat > list_nat ).
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_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_Product__Type_Oscomp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
produc5538323210962509403atural: ( produc7822875418678951345atural > produc5835291356934675326atural ) > ( code_natural > produc7822875418678951345atural > produc5835291356934675326atural ) > produc7822875418678951345atural > produc5835291356934675326atural ).
thf(sy_c_Quotient_OQuotient3_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
quotie8700032322157300518t_real: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > real ) > ( real > nat > rat ) > $o ).
thf(sy_c_Quotient_OQuotient3_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
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thf(sy_c_Random_Oinc__shift,type,
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thf(sy_c_Random_Oiterate_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
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thf(sy_c_Random_Olog,type,
log: code_natural > code_natural > code_natural ).
thf(sy_c_Random_Olog__rel,type,
log_rel: produc7822875418678951345atural > produc7822875418678951345atural > $o ).
thf(sy_c_Random_Ominus__shift,type,
minus_shift: code_natural > code_natural > code_natural > code_natural ).
thf(sy_c_Random_Onext,type,
next: produc7822875418678951345atural > produc5835291356934675326atural ).
thf(sy_c_Random_Orange,type,
range: code_natural > produc7822875418678951345atural > produc5835291356934675326atural ).
thf(sy_c_Rat_OAbs__Rat,type,
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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,
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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_ORatreal,type,
ratreal: rat > real ).
thf(sy_c_Real_OReal,type,
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thf(sy_c_Real_Ocauchy,type,
cauchy: ( nat > rat ) > $o ).
thf(sy_c_Real_Ocr__real,type,
cr_real: ( nat > rat ) > real > $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_Orealrel,type,
realrel: ( nat > rat ) > ( nat > rat ) > $o ).
thf(sy_c_Real_Orep__real,type,
rep_real: real > nat > rat ).
thf(sy_c_Real_Ovanishes,type,
vanishes: ( nat > rat ) > $o ).
thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
real_V2521375963428798218omplex: set_complex ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
real_V3694042436643373181omplex: complex > complex > real ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
real_V975177566351809787t_real: real > real > real ).
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,
real_V4546457046886955230omplex: real > complex ).
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,
real_V2046097035970521341omplex: real > complex > complex ).
thf(sy_c_Relation_OField_001t__Nat__Onat,type,
field_nat: set_Pr1261947904930325089at_nat > set_nat ).
thf(sy_c_Relation_Otransp_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
transp_nat_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > $o ).
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 ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Onatural,type,
divide5121882707175180666atural: code_natural > code_natural > code_natural ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
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,
divide_divide_real: real > real > real ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
dvd_dvd_Code_integer: code_integer > code_integer > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Onatural,type,
dvd_dvd_Code_natural: code_natural > code_natural > $o ).
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,
dvd_dvd_real: real > real > $o ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Onatural,type,
modulo8411746178871703098atural: code_natural > code_natural > code_natural ).
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_Ounit__factor__class_Ounit__factor_001t__Nat__Onat,type,
unit_f2748546683901255202or_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__Code____Numeral__Onatural,type,
zero_n8403883297036319079atural: $o > code_natural ).
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,
suminf_complex: ( nat > complex ) > complex ).
thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
suminf_int: ( nat > int ) > int ).
thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
suminf_nat: ( nat > nat ) > nat ).
thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
suminf_real: ( nat > real ) > real ).
thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
summable_complex: ( nat > complex ) > $o ).
thf(sy_c_Series_Osummable_001t__Int__Oint,type,
summable_int: ( nat > int ) > $o ).
thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
summable_nat: ( nat > nat ) > $o ).
thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
summable_real: ( nat > real ) > $o ).
thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
sums_complex: ( nat > complex ) > complex > $o ).
thf(sy_c_Series_Osums_001t__Int__Oint,type,
sums_int: ( nat > int ) > int > $o ).
thf(sy_c_Series_Osums_001t__Nat__Onat,type,
sums_nat: ( nat > nat ) > nat > $o ).
thf(sy_c_Series_Osums_001t__Real__Oreal,type,
sums_real: ( nat > real ) > real > $o ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
image_list_nat_nat: ( list_nat > nat ) > set_list_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
image_nat_list_nat: ( nat > list_nat ) > set_nat > set_list_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_5846123807819985514at_nat: ( nat > product_prod_nat_nat ) > set_nat > set_Pr1261947904930325089at_nat ).
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__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_678696785212003926at_nat: ( nat > sum_sum_nat_nat ) > set_nat > set_Sum_sum_nat_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_2486076414777270412at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > set_nat ).
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_Oimage_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_1320371278474632150at_nat: ( sum_sum_nat_nat > nat ) > set_Sum_sum_nat_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
insert_complex: complex > set_complex > set_complex ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Code____Numeral__Ointeger,type,
set_fo1084959871951514735nteger: ( nat > code_integer > code_integer ) > nat > nat > code_integer > code_integer ).
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__Nat__Onat,type,
set_ord_atMost_nat: nat > 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__Num__Onum,type,
set_or2392100141987894638an_num: num > num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Rat__Orat,type,
set_or5199638295745620268an_rat: rat > rat > set_rat ).
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_OgreaterThanLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or8625682525731655386et_nat: set_nat > set_nat > set_set_nat ).
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__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
set_or5984915006950818249n_real: real > set_real ).
thf(sy_c_String_OCode_Oabort_001t__Real__Oreal,type,
abort_real: literal > ( product_unit > real ) > real ).
thf(sy_c_String_OLiteral,type,
literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
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_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
unique3096191561947761185of_nat: nat > char ).
thf(sy_c_Sum__Type_OInl_001t__Nat__Onat_001t__Nat__Onat,type,
sum_Inl_nat_nat: nat > sum_sum_nat_nat ).
thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001t__Nat__Onat,type,
sum_Inr_nat_nat: nat > sum_sum_nat_nat ).
thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
sum_ca7763040182479039464nt_nat: ( nat > int ) > ( nat > int ) > sum_sum_nat_nat > int ).
thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
sum_ca6763686470577984908at_nat: ( nat > nat ) > ( nat > nat ) > sum_sum_nat_nat > nat ).
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__Real__Oreal,type,
topolo6980174941875973593q_real: ( nat > real ) > $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_Oconvergent_001t__Real__Oreal,type,
topolo7531315842566124627t_real: ( nat > real ) > $o ).
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__Complex__Ocomplex,type,
topolo6517432010174082258omplex: ( nat > complex ) > $o ).
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__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,
log2: 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_Opowr__real,type,
powr_real2: 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_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_Oelim__dead,type,
vEBT_VEBT_elim_dead: vEBT_VEBT > extended_enat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
vEBT_V312737461966249ad_rel: produc7272778201969148633d_enat > produc7272778201969148633d_enat > $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_OminNull,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(sy_c_VEBT__Member_Ovebt__member,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_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__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
accp_P8126237942716283194atural: ( produc7822875418678951345atural > produc7822875418678951345atural > $o ) > produc7822875418678951345atural > $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__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
accp_P6183159247885693666d_enat: ( produc7272778201969148633d_enat > produc7272778201969148633d_enat > $o ) > produc7272778201969148633d_enat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
thf(sy_c_Wellfounded_Oless__than,type,
less_than: set_Pr1261947904930325089at_nat ).
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__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(sy_c_member_001t__Rat__Orat,type,
member_rat: rat > set_rat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
thf(sy_v_deg____,type,
deg: nat ).
thf(sy_v_m____,type,
m: nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_summary____,type,
summary: vEBT_VEBT ).
thf(sy_v_treeList____,type,
treeList: list_VEBT_VEBT ).
thf(sy_v_xa____,type,
xa: nat ).
% Relevant facts (10152)
thf(fact_0_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_1_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_2_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_3_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_4__C3_Ohyps_C_I3_J,axiom,
( m
= ( suc @ na ) ) ).
% "3.hyps"(3)
thf(fact_5_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_6_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_7_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_8_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_9_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_10_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_11_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_12_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_13_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_14_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_15_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_16_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_17_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_18_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_19_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_20_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_21_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_22_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_23_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_24_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_25_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_26_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_27_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_28_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_29_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_30_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_31_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_32_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_33_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_34_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,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_35_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_36_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_37_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_38_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M3: nat] :
( ( M
= ( suc @ M3 ) )
& ( ord_less_nat @ N @ M3 ) ) ) ) ).
% Suc_less_eq2
thf(fact_39_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_40_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_41_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_42_mem__Collect__eq,axiom,
! [A: complex,P: complex > $o] :
( ( member_complex @ A @ ( collect_complex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_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_45_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_46_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A2: set_complex] :
( ( collect_complex
@ ^ [X4: complex] : ( member_complex @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A2: set_list_nat] :
( ( collect_list_nat
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X4: int] : ( member_int @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_54_Collect__cong,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X3: real] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_real @ P )
= ( collect_real @ Q ) ) ) ).
% Collect_cong
thf(fact_55_Collect__cong,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_nat @ P )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_56_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X3: set_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_57_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_58_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_59_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_60_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_61_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_62_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_63_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_64_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_65_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_66_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_67_lift__Suc__mono__less__iff,axiom,
! [F: nat > rat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_68_lift__Suc__mono__less__iff,axiom,
! [F: nat > num,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_69_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_70_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_71_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_72_lift__Suc__mono__less,axiom,
! [F: nat > rat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_rat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_73_lift__Suc__mono__less,axiom,
! [F: nat > num,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_74_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_75_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_76_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_77_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_78_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_79_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_80_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_81_zero__reorient,axiom,
! [X: literal] :
( ( zero_zero_literal = X )
= ( X = zero_zero_literal ) ) ).
% zero_reorient
thf(fact_82_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_83_zero__reorient,axiom,
! [X: rat] :
( ( zero_zero_rat = X )
= ( X = zero_zero_rat ) ) ).
% zero_reorient
thf(fact_84_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_85_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_86_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_87_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_88_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_89_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_90_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_91_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_92_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_93_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_94_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_95_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_96_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_97_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_98_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_99__C3_Ohyps_C_I4_J,axiom,
( deg
= ( plus_plus_nat @ na @ m ) ) ).
% "3.hyps"(4)
thf(fact_100_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_101_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_102_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_103_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_104_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_105_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_106_less__numeral__extra_I3_J,axiom,
~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% less_numeral_extra(3)
thf(fact_107_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_108_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_109__C3_Ohyps_C_I1_J,axiom,
vEBT_invar_vebt @ summary @ m ).
% "3.hyps"(1)
thf(fact_110_deg__not__0,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% deg_not_0
thf(fact_111_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_112_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_113_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_114_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_115_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_116_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_117_even__odd__cases,axiom,
! [X: nat] :
( ! [N2: nat] :
( X
!= ( plus_plus_nat @ N2 @ N2 ) )
=> ~ ! [N2: nat] :
( X
!= ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) ) ) ).
% even_odd_cases
thf(fact_118_valid__0__not,axiom,
! [T: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% valid_0_not
thf(fact_119_valid__tree__deg__neq__0,axiom,
! [T: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% valid_tree_deg_neq_0
thf(fact_120_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_121_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_122_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_123_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_124_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_125_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_126_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_127_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_128_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_129_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_130_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_131_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri4939895301339042750nteger @ M )
= ( semiri4939895301339042750nteger @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_132_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri681578069525770553at_rat @ M )
= ( semiri681578069525770553at_rat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_133_add_Oright__neutral,axiom,
! [A: literal] :
( ( plus_plus_literal @ A @ zero_zero_literal )
= A ) ).
% add.right_neutral
thf(fact_134_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_135_add_Oright__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.right_neutral
thf(fact_136_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_137_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_138_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_139_double__zero__sym,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( plus_plus_rat @ A @ A ) )
= ( A = zero_zero_rat ) ) ).
% double_zero_sym
thf(fact_140_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_141_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_142_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_143_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_144_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_145_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_146_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_147_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_148_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_149_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_150_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_151_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_152_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_153_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_154_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_155_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_156_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_157_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_158_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_159_add__0,axiom,
! [A: literal] :
( ( plus_plus_literal @ zero_zero_literal @ A )
= A ) ).
% add_0
thf(fact_160_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_161_add__0,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add_0
thf(fact_162_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_163_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_164_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_165_diff__self,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ A )
= zero_zero_rat ) ).
% diff_self
thf(fact_166_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_167_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_168_diff__0__right,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ zero_zero_rat )
= A ) ).
% diff_0_right
thf(fact_169_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_170_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_171_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_172_diff__zero,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ zero_zero_rat )
= A ) ).
% diff_zero
thf(fact_173_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_174_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_175_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_176_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_177_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_178_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_179_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_180_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_181_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_182_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_183_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_184_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_185_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_186_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_187_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_188_add__diff__cancel,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_189_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_190_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_191_diff__add__cancel,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_192_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_193_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_194_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_195_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_196_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_197_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_198_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_199_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_200_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_201_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_202_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_203_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_204_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_205_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_206_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_207_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_208_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_209_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_210_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_211_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_212_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_213_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_214_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_215_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_216_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_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_225_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_226_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_227_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_228_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_229_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_230_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_231_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_232_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_233_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_234_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_235_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_236_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_237_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_238_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_239_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_240_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_241_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_242_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_243_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_244_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_245_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_246_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_247_of__nat__0,axiom,
( ( semiri4939895301339042750nteger @ zero_zero_nat )
= zero_z3403309356797280102nteger ) ).
% of_nat_0
thf(fact_248_of__nat__0,axiom,
( ( semiri681578069525770553at_rat @ zero_zero_nat )
= zero_zero_rat ) ).
% of_nat_0
thf(fact_249_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_250_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_251_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_252_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_z3403309356797280102nteger
= ( semiri4939895301339042750nteger @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_253_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_rat
= ( semiri681578069525770553at_rat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_254_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_255_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_256_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_257_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri4939895301339042750nteger @ M )
= zero_z3403309356797280102nteger )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_258_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_259_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_260_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_261_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_262_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_263_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_264_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_265_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_add
thf(fact_266_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_267_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri4939895301339042750nteger @ ( plus_plus_nat @ M @ N ) )
= ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% of_nat_add
thf(fact_268_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% of_nat_add
thf(fact_269_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_270_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_271__C3_Ohyps_C_I5_J,axiom,
~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ summary @ X_1 ) ).
% "3.hyps"(5)
thf(fact_272_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_273_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_274_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_275_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_276_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_277_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_278_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_279_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_280_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_281_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_282_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_283_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_284_group__cancel_Oadd2,axiom,
! [B2: real,K: real,B: real,A: real] :
( ( B2
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_285_group__cancel_Oadd2,axiom,
! [B2: rat,K: rat,B: rat,A: rat] :
( ( B2
= ( plus_plus_rat @ K @ B ) )
=> ( ( plus_plus_rat @ A @ B2 )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_286_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_287_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_288_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_289_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_290_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_291_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_292_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_293_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_294_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_295_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_296_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_297_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_298_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_299_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_300_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_301_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_302_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_303_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_304_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_305_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_306_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_307_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_308_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_309_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_310_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_311_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_312_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_313_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_314_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_315_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_316_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_317_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_318_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_319_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B3: real] : ( plus_plus_real @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_320_add_Ocommute,axiom,
( plus_plus_rat
= ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_321_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_322_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_323_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_324_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_325_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_326_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_327_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_328_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_329_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_330_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_331_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_332_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_333_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_334_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_335_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_336_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_337_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_338_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_339_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_340_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_341_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_342_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_343_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_344_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_345_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_346_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_347_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_348_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_349_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_350_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_351_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_352_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_353_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_354_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_355_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_356_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_357_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_358_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_359_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_360_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_361_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_362_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_363_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_364_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_365_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_366_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_367_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_368_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_369_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_370_reals__Archimedean2,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% reals_Archimedean2
thf(fact_371_reals__Archimedean2,axiom,
! [X: rat] :
? [N2: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% reals_Archimedean2
thf(fact_372_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_373_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_374_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z: real] : Y4 = Z )
= ( ^ [A3: real,B3: real] :
( ( minus_minus_real @ A3 @ B3 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_375_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: rat,Z: rat] : Y4 = Z )
= ( ^ [A3: rat,B3: rat] :
( ( minus_minus_rat @ A3 @ B3 )
= zero_zero_rat ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_376_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z: int] : Y4 = Z )
= ( ^ [A3: int,B3: int] :
( ( minus_minus_int @ A3 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_377_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_378_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_379_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_380_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_381_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_382_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_383_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_384_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_385_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_386_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_387_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_388_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_389_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_390_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_391_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_392_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_393_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_394_add_Ocomm__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.comm_neutral
thf(fact_395_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_396_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_397_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_398_add_Ogroup__left__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add.group_left_neutral
thf(fact_399_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_400_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_401_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_402_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_403_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_404_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_405_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_406_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_407_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_408_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_409_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_410_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_411_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_412_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_413_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_414_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_415_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_416_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_417_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_418_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_419_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_420_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_421_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_422_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_423_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_424_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_425_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_426_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_427_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_428_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_429_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_430_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_431_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_432_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_433_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_434_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_435_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_436_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_437_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_438_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_439_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_440_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_441_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_442_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_443_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_444_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_445_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_446_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_447_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_448_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_449_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_450_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_451_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_452_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_453_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger ) ).
% of_nat_less_0_iff
thf(fact_454_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% of_nat_less_0_iff
thf(fact_455_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_456_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_457_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_458_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri4939895301339042750nteger @ ( suc @ N ) )
!= zero_z3403309356797280102nteger ) ).
% of_nat_neq_0
thf(fact_459_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri681578069525770553at_rat @ ( suc @ N ) )
!= zero_zero_rat ) ).
% of_nat_neq_0
thf(fact_460_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_461_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_462_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_463_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_464_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_465_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_466_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_467_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_468_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_469_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_470_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_471_less__iff__diff__less__0,axiom,
( ord_less_rat
= ( ^ [A3: rat,B3: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% less_iff_diff_less_0
thf(fact_472_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_473_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_474_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_475_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_476_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_477_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_478_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_479_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_480_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_481_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_482_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_483_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_484_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_485_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_486_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_487_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_488_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_489_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_490_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_491_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_492_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M4 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_493_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_494_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_495_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_496_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_497_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_498_buildup__gives__valid,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% buildup_gives_valid
thf(fact_499_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_500_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_501_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_502_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_503_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_504_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_505_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_506_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_507_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_508_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_509_valid__eq2,axiom,
! [T: vEBT_VEBT,D: nat] :
( ( vEBT_VEBT_valid @ T @ D )
=> ( vEBT_invar_vebt @ T @ D ) ) ).
% valid_eq2
thf(fact_510_valid__eq1,axiom,
! [T: vEBT_VEBT,D: nat] :
( ( vEBT_invar_vebt @ T @ D )
=> ( vEBT_VEBT_valid @ T @ D ) ) ).
% valid_eq1
thf(fact_511_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_512_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_513_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_514_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_515_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_516_triangle__0,axiom,
( ( nat_triangle @ zero_zero_nat )
= zero_zero_nat ) ).
% triangle_0
thf(fact_517_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_518_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_519_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_520_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M2: nat,N2: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_521_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_522_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_523_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W: int,Z3: int] :
? [N4: nat] :
( Z3
= ( plus_plus_int @ W @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_524_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_525_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_526_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_527_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_528_not__min__Null__member,axiom,
! [T: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ T )
=> ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 ) ) ).
% not_min_Null_member
thf(fact_529__C3_OIH_C_I2_J,axiom,
! [X: nat] :
( ( vEBT_V8194947554948674370ptions @ summary @ X )
=> ( vEBT_vebt_member @ summary @ X ) ) ).
% "3.IH"(2)
thf(fact_530_buildup__nothing__in__leaf,axiom,
! [N: nat,X: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_leaf
thf(fact_531_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_532_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_533_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_534_buildup__nothing__in__min__max,axiom,
! [N: nat,X: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_min_max
thf(fact_535_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_536_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_537_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_538_verit__sum__simplify,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% verit_sum_simplify
thf(fact_539_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_540_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_541_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_542_add__0__iff,axiom,
! [B: rat,A: rat] :
( ( B
= ( plus_plus_rat @ B @ A ) )
= ( A = zero_zero_rat ) ) ).
% add_0_iff
thf(fact_543_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_544_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_545_min__Null__member,axiom,
! [T: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_minNull @ T )
=> ~ ( vEBT_vebt_member @ T @ X ) ) ).
% min_Null_member
thf(fact_546_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T2: vEBT_VEBT,X4: nat] :
( ( vEBT_V5719532721284313246member @ T2 @ X4 )
| ( vEBT_VEBT_membermima @ T2 @ X4 ) ) ) ) ).
% both_member_options_def
thf(fact_547_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ Tree @ N )
=> ( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_V5719532721284313246member @ Tree @ X )
| ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).
% member_valid_both_member_options
thf(fact_548_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_549_verit__comp__simplify1_I1_J,axiom,
! [A: rat] :
~ ( ord_less_rat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_550_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_551_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_552_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_553_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [A3: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_554_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_555__C3_OIH_C_I1_J,axiom,
! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
=> ( ( vEBT_invar_vebt @ X5 @ na )
& ! [Xa: nat] :
( ( vEBT_V8194947554948674370ptions @ X5 @ Xa )
=> ( vEBT_vebt_member @ X5 @ Xa ) ) ) ) ).
% "3.IH"(1)
thf(fact_556_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_557_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
=> ? [Info: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info @ ( suc @ ( suc @ N ) ) @ TreeList @ S2 ) ) ) ).
% deg_SUcn_Node
thf(fact_558_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_559_Leaf__0__not,axiom,
! [A: $o,B: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).
% Leaf_0_not
thf(fact_560_ex__inverse__of__nat__less,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_561_ex__inverse__of__nat__less,axiom,
! [X: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ X ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_562_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_563_of__int__0__less__iff,axiom,
! [Z2: int] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_less_iff
thf(fact_564_of__int__0__less__iff,axiom,
! [Z2: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_less_iff
thf(fact_565_of__int__0__less__iff,axiom,
! [Z2: int] :
( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_less_iff
thf(fact_566_of__int__0__less__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_less_iff
thf(fact_567_of__int__less__0__iff,axiom,
! [Z2: int] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z2 ) @ zero_z3403309356797280102nteger )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_568_of__int__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_569_of__int__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_570_of__int__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_571_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_572_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_573_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M4: nat,N4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_574_deg__deg__n,axiom,
! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
=> ( Deg = N ) ) ).
% deg_deg_n
thf(fact_575__C3_Ohyps_C_I6_J,axiom,
! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ).
% "3.hyps"(6)
thf(fact_576_deg1Leaf,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T @ one_one_nat )
= ( ? [A3: $o,B3: $o] :
( T
= ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).
% deg1Leaf
thf(fact_577_deg__1__Leaf,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T @ one_one_nat )
=> ? [A4: $o,B4: $o] :
( T
= ( vEBT_Leaf @ A4 @ B4 ) ) ) ).
% deg_1_Leaf
thf(fact_578_deg__1__Leafy,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( N = one_one_nat )
=> ? [A4: $o,B4: $o] :
( T
= ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).
% deg_1_Leafy
thf(fact_579_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_580_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_581_verit__minus__simplify_I4_J,axiom,
! [B: complex] :
( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_582_verit__minus__simplify_I4_J,axiom,
! [B: code_integer] :
( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_583_verit__minus__simplify_I4_J,axiom,
! [B: rat] :
( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_584_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_585_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_586_neg__equal__iff__equal,axiom,
! [A: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= ( uminus1482373934393186551omplex @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_587_neg__equal__iff__equal,axiom,
! [A: code_integer,B: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= ( uminus1351360451143612070nteger @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_588_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_589_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_590_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_591_add_Oinverse__inverse,axiom,
! [A: complex] :
( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_592_add_Oinverse__inverse,axiom,
! [A: code_integer] :
( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_593_add_Oinverse__inverse,axiom,
! [A: rat] :
( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_594_of__int__eq__iff,axiom,
! [W2: int,Z2: int] :
( ( ( ring_1_of_int_real @ W2 )
= ( ring_1_of_int_real @ Z2 ) )
= ( W2 = Z2 ) ) ).
% of_int_eq_iff
thf(fact_595_of__int__eq__iff,axiom,
! [W2: int,Z2: int] :
( ( ( ring_18347121197199848620nteger @ W2 )
= ( ring_18347121197199848620nteger @ Z2 ) )
= ( W2 = Z2 ) ) ).
% of_int_eq_iff
thf(fact_596_of__int__eq__iff,axiom,
! [W2: int,Z2: int] :
( ( ( ring_1_of_int_rat @ W2 )
= ( ring_1_of_int_rat @ Z2 ) )
= ( W2 = Z2 ) ) ).
% of_int_eq_iff
thf(fact_597_of__int__eq__iff,axiom,
! [W2: int,Z2: int] :
( ( ( ring_17405671764205052669omplex @ W2 )
= ( ring_17405671764205052669omplex @ Z2 ) )
= ( W2 = Z2 ) ) ).
% of_int_eq_iff
thf(fact_598_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_599_VEBT_Oinject_I2_J,axiom,
! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
( ( ( vEBT_Leaf @ X21 @ X22 )
= ( vEBT_Leaf @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% VEBT.inject(2)
thf(fact_600_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_601_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_602_neg__equal__zero,axiom,
! [A: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= A )
= ( A = zero_z3403309356797280102nteger ) ) ).
% neg_equal_zero
thf(fact_603_neg__equal__zero,axiom,
! [A: rat] :
( ( ( uminus_uminus_rat @ A )
= A )
= ( A = zero_zero_rat ) ) ).
% neg_equal_zero
thf(fact_604_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_605_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_606_equal__neg__zero,axiom,
! [A: code_integer] :
( ( A
= ( uminus1351360451143612070nteger @ A ) )
= ( A = zero_z3403309356797280102nteger ) ) ).
% equal_neg_zero
thf(fact_607_equal__neg__zero,axiom,
! [A: rat] :
( ( A
= ( uminus_uminus_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% equal_neg_zero
thf(fact_608_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_609_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_610_neg__equal__0__iff__equal,axiom,
! [A: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% neg_equal_0_iff_equal
thf(fact_611_neg__equal__0__iff__equal,axiom,
! [A: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% neg_equal_0_iff_equal
thf(fact_612_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_613_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_614_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_615_neg__0__equal__iff__equal,axiom,
! [A: complex] :
( ( zero_zero_complex
= ( uminus1482373934393186551omplex @ A ) )
= ( zero_zero_complex = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_616_neg__0__equal__iff__equal,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger
= ( uminus1351360451143612070nteger @ A ) )
= ( zero_z3403309356797280102nteger = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_617_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_618_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_619_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_620_add_Oinverse__neutral,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% add.inverse_neutral
thf(fact_621_add_Oinverse__neutral,axiom,
( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% add.inverse_neutral
thf(fact_622_add_Oinverse__neutral,axiom,
( ( uminus_uminus_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% add.inverse_neutral
thf(fact_623_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_624_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_625_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_626_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_627_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_628_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_629_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_630_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_631_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_632_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_633_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_634_minus__add__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_635_minus__add__cancel,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_636_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_637_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_638_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_639_add__minus__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_640_add__minus__cancel,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_641_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_642_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_643_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_644_minus__diff__eq,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
= ( minus_minus_complex @ B @ A ) ) ).
% minus_diff_eq
thf(fact_645_minus__diff__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
= ( minus_8373710615458151222nteger @ B @ A ) ) ).
% minus_diff_eq
thf(fact_646_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_647_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_int @ Z2 )
= one_one_int )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_648_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_real @ Z2 )
= one_one_real )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_649_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_18347121197199848620nteger @ Z2 )
= one_one_Code_integer )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_650_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_rat @ Z2 )
= one_one_rat )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_651_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_17405671764205052669omplex @ Z2 )
= one_one_complex )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_652_of__int__1,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_1
thf(fact_653_of__int__1,axiom,
( ( ring_1_of_int_real @ one_one_int )
= one_one_real ) ).
% of_int_1
thf(fact_654_of__int__1,axiom,
( ( ring_18347121197199848620nteger @ one_one_int )
= one_one_Code_integer ) ).
% of_int_1
thf(fact_655_of__int__1,axiom,
( ( ring_1_of_int_rat @ one_one_int )
= one_one_rat ) ).
% of_int_1
thf(fact_656_of__int__1,axiom,
( ( ring_17405671764205052669omplex @ one_one_int )
= one_one_complex ) ).
% of_int_1
thf(fact_657_ceiling__one,axiom,
( ( archim2889992004027027881ng_rat @ one_one_rat )
= one_one_int ) ).
% ceiling_one
thf(fact_658_ceiling__one,axiom,
( ( archim7802044766580827645g_real @ one_one_real )
= one_one_int ) ).
% ceiling_one
thf(fact_659_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_660_ceiling__of__int,axiom,
! [Z2: int] :
( ( archim2889992004027027881ng_rat @ ( ring_1_of_int_rat @ Z2 ) )
= Z2 ) ).
% ceiling_of_int
thf(fact_661_ceiling__of__int,axiom,
! [Z2: int] :
( ( archim7802044766580827645g_real @ ( ring_1_of_int_real @ Z2 ) )
= Z2 ) ).
% ceiling_of_int
thf(fact_662_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_663_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_664_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_665_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_666_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_667_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_668_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_669_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_670_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_671_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_672_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_673_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_674_neg__less__pos,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_less_pos
thf(fact_675_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_676_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_677_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_678_less__neg__neg,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% less_neg_neg
thf(fact_679_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_680_diff__numeral__special_I9_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% diff_numeral_special(9)
thf(fact_681_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_682_diff__numeral__special_I9_J,axiom,
( ( minus_minus_rat @ one_one_rat @ one_one_rat )
= zero_zero_rat ) ).
% diff_numeral_special(9)
thf(fact_683_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_684_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_685_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_686_add_Oright__inverse,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
= zero_zero_complex ) ).
% add.right_inverse
thf(fact_687_add_Oright__inverse,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= zero_z3403309356797280102nteger ) ).
% add.right_inverse
thf(fact_688_add_Oright__inverse,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
= zero_zero_rat ) ).
% add.right_inverse
thf(fact_689_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_690_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_691_ab__left__minus,axiom,
! [A: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
= zero_zero_complex ) ).
% ab_left_minus
thf(fact_692_ab__left__minus,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= zero_z3403309356797280102nteger ) ).
% ab_left_minus
thf(fact_693_ab__left__minus,axiom,
! [A: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
= zero_zero_rat ) ).
% ab_left_minus
thf(fact_694_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_695_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_696_diff__0,axiom,
! [A: complex] :
( ( minus_minus_complex @ zero_zero_complex @ A )
= ( uminus1482373934393186551omplex @ A ) ) ).
% diff_0
thf(fact_697_diff__0,axiom,
! [A: code_integer] :
( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
= ( uminus1351360451143612070nteger @ A ) ) ).
% diff_0
thf(fact_698_diff__0,axiom,
! [A: rat] :
( ( minus_minus_rat @ zero_zero_rat @ A )
= ( uminus_uminus_rat @ A ) ) ).
% diff_0
thf(fact_699_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_700_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_701_verit__minus__simplify_I3_J,axiom,
! [B: complex] :
( ( minus_minus_complex @ zero_zero_complex @ B )
= ( uminus1482373934393186551omplex @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_702_verit__minus__simplify_I3_J,axiom,
! [B: code_integer] :
( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
= ( uminus1351360451143612070nteger @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_703_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_704_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_705_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_706_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_707_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_708_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_709_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_710_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_711_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_712_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_713_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_714_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_int @ Z2 )
= zero_zero_int )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_715_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_real @ Z2 )
= zero_zero_real )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_716_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_18347121197199848620nteger @ Z2 )
= zero_z3403309356797280102nteger )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_717_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_rat @ Z2 )
= zero_zero_rat )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_718_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_17405671764205052669omplex @ Z2 )
= zero_zero_complex )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_719_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_720_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_721_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_z3403309356797280102nteger
= ( ring_18347121197199848620nteger @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_722_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_zero_rat
= ( ring_1_of_int_rat @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_723_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_zero_complex
= ( ring_17405671764205052669omplex @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_724_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_725_of__int__0,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_0
thf(fact_726_of__int__0,axiom,
( ( ring_18347121197199848620nteger @ zero_zero_int )
= zero_z3403309356797280102nteger ) ).
% of_int_0
thf(fact_727_of__int__0,axiom,
( ( ring_1_of_int_rat @ zero_zero_int )
= zero_zero_rat ) ).
% of_int_0
thf(fact_728_of__int__0,axiom,
( ( ring_17405671764205052669omplex @ zero_zero_int )
= zero_zero_complex ) ).
% of_int_0
thf(fact_729_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri8010041392384452111omplex @ N )
= one_one_complex )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_730_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_731_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_732_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_733_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri4939895301339042750nteger @ N )
= one_one_Code_integer )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_734_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri681578069525770553at_rat @ N )
= one_one_rat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_735_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_complex
= ( semiri8010041392384452111omplex @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_736_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_737_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_738_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_739_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_Code_integer
= ( semiri4939895301339042750nteger @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_740_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_rat
= ( semiri681578069525770553at_rat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_741_of__nat__1,axiom,
( ( semiri8010041392384452111omplex @ one_one_nat )
= one_one_complex ) ).
% of_nat_1
thf(fact_742_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_743_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_744_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_745_of__nat__1,axiom,
( ( semiri4939895301339042750nteger @ one_one_nat )
= one_one_Code_integer ) ).
% of_nat_1
thf(fact_746_of__nat__1,axiom,
( ( semiri681578069525770553at_rat @ one_one_nat )
= one_one_rat ) ).
% of_nat_1
thf(fact_747_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_748_of__int__less__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ W2 ) @ ( ring_18347121197199848620nteger @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% of_int_less_iff
thf(fact_749_of__int__less__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% of_int_less_iff
thf(fact_750_of__int__less__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% of_int_less_iff
thf(fact_751_of__int__less__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% of_int_less_iff
thf(fact_752_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_753_of__int__add,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_int @ ( plus_plus_int @ W2 @ Z2 ) )
= ( plus_plus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_add
thf(fact_754_of__int__add,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_real @ ( plus_plus_int @ W2 @ Z2 ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_add
thf(fact_755_of__int__add,axiom,
! [W2: int,Z2: int] :
( ( ring_18347121197199848620nteger @ ( plus_plus_int @ W2 @ Z2 ) )
= ( plus_p5714425477246183910nteger @ ( ring_18347121197199848620nteger @ W2 ) @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).
% of_int_add
thf(fact_756_of__int__add,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_rat @ ( plus_plus_int @ W2 @ Z2 ) )
= ( plus_plus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% of_int_add
thf(fact_757_of__int__add,axiom,
! [W2: int,Z2: int] :
( ( ring_17405671764205052669omplex @ ( plus_plus_int @ W2 @ Z2 ) )
= ( plus_plus_complex @ ( ring_17405671764205052669omplex @ W2 ) @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% of_int_add
thf(fact_758_of__int__minus,axiom,
! [Z2: int] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z2 ) )
= ( uminus_uminus_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_minus
thf(fact_759_of__int__minus,axiom,
! [Z2: int] :
( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z2 ) )
= ( uminus_uminus_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_minus
thf(fact_760_of__int__minus,axiom,
! [Z2: int] :
( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ Z2 ) )
= ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% of_int_minus
thf(fact_761_of__int__minus,axiom,
! [Z2: int] :
( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ Z2 ) )
= ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).
% of_int_minus
thf(fact_762_of__int__minus,axiom,
! [Z2: int] :
( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z2 ) )
= ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% of_int_minus
thf(fact_763_ceiling__zero,axiom,
( ( archim2889992004027027881ng_rat @ zero_zero_rat )
= zero_zero_int ) ).
% ceiling_zero
thf(fact_764_ceiling__zero,axiom,
( ( archim7802044766580827645g_real @ zero_zero_real )
= zero_zero_int ) ).
% ceiling_zero
thf(fact_765_of__int__diff,axiom,
! [W2: int,Z2: int] :
( ( ring_18347121197199848620nteger @ ( minus_minus_int @ W2 @ Z2 ) )
= ( minus_8373710615458151222nteger @ ( ring_18347121197199848620nteger @ W2 ) @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).
% of_int_diff
thf(fact_766_of__int__diff,axiom,
! [W2: int,Z2: int] :
( ( ring_17405671764205052669omplex @ ( minus_minus_int @ W2 @ Z2 ) )
= ( minus_minus_complex @ ( ring_17405671764205052669omplex @ W2 ) @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% of_int_diff
thf(fact_767_of__int__diff,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_real @ ( minus_minus_int @ W2 @ Z2 ) )
= ( minus_minus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_diff
thf(fact_768_of__int__diff,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_rat @ ( minus_minus_int @ W2 @ Z2 ) )
= ( minus_minus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% of_int_diff
thf(fact_769_of__int__diff,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_int @ ( minus_minus_int @ W2 @ Z2 ) )
= ( minus_minus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_diff
thf(fact_770_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_771_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_17405671764205052669omplex @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri8010041392384452111omplex @ N ) ) ).
% of_int_of_nat_eq
thf(fact_772_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% of_int_of_nat_eq
thf(fact_773_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% of_int_of_nat_eq
thf(fact_774_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_18347121197199848620nteger @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri4939895301339042750nteger @ N ) ) ).
% of_int_of_nat_eq
thf(fact_775_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_rat @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri681578069525770553at_rat @ N ) ) ).
% of_int_of_nat_eq
thf(fact_776_ceiling__of__nat,axiom,
! [N: nat] :
( ( archim7802044766580827645g_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% ceiling_of_nat
thf(fact_777_ceiling__of__nat,axiom,
! [N: nat] :
( ( archim2889992004027027881ng_rat @ ( semiri681578069525770553at_rat @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% ceiling_of_nat
thf(fact_778_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_779_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_780_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_781_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_782_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_783_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_784_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_785_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_786_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_787_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_788_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_789_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_790_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_791_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_792_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_793_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ M ) )
= ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% of_nat_Suc
thf(fact_794_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_795_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ M ) )
= ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% of_nat_Suc
thf(fact_796_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_797_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri4939895301339042750nteger @ ( suc @ M ) )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M ) ) ) ).
% of_nat_Suc
thf(fact_798_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri681578069525770553at_rat @ ( suc @ M ) )
= ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% of_nat_Suc
thf(fact_799_of__int__less__1__iff,axiom,
! [Z2: int] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z2 ) @ one_one_Code_integer )
= ( ord_less_int @ Z2 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_800_of__int__less__1__iff,axiom,
! [Z2: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
= ( ord_less_int @ Z2 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_801_of__int__less__1__iff,axiom,
! [Z2: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
= ( ord_less_int @ Z2 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_802_of__int__less__1__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
= ( ord_less_int @ Z2 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_803_of__int__1__less__iff,axiom,
! [Z2: int] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% of_int_1_less_iff
thf(fact_804_of__int__1__less__iff,axiom,
! [Z2: int] :
( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% of_int_1_less_iff
thf(fact_805_of__int__1__less__iff,axiom,
! [Z2: int] :
( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% of_int_1_less_iff
thf(fact_806_of__int__1__less__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% of_int_1_less_iff
thf(fact_807_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_808_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_809_zless__nat__conj,axiom,
! [W2: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W2 @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_810_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_811_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_812_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_813_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_814_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_815_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_816_ceiling__add__of__int,axiom,
! [X: rat,Z2: int] :
( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) )
= ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 ) ) ).
% ceiling_add_of_int
thf(fact_817_ceiling__add__of__int,axiom,
! [X: real,Z2: int] :
( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z2 ) ) )
= ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ Z2 ) ) ).
% ceiling_add_of_int
thf(fact_818_ceiling__diff__of__int,axiom,
! [X: rat,Z2: int] :
( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) )
= ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 ) ) ).
% ceiling_diff_of_int
thf(fact_819_ceiling__diff__of__int,axiom,
! [X: real,Z2: int] :
( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z2 ) ) )
= ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ Z2 ) ) ).
% ceiling_diff_of_int
thf(fact_820_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_821_verit__negate__coefficient_I3_J,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_822_verit__negate__coefficient_I3_J,axiom,
! [A: code_integer,B: code_integer] :
( ( A = B )
=> ( ( uminus1351360451143612070nteger @ A )
= ( uminus1351360451143612070nteger @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_823_verit__negate__coefficient_I3_J,axiom,
! [A: rat,B: rat] :
( ( A = B )
=> ( ( uminus_uminus_rat @ A )
= ( uminus_uminus_rat @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_824_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_825_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_826_one__neq__neg__one,axiom,
( one_one_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% one_neq_neg_one
thf(fact_827_one__neq__neg__one,axiom,
( one_one_Code_integer
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% one_neq_neg_one
thf(fact_828_one__neq__neg__one,axiom,
( one_one_rat
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% one_neq_neg_one
thf(fact_829_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,X222: $o] :
( Y
!= ( vEBT_Leaf @ X212 @ X222 ) ) ) ).
% VEBT.exhaust
thf(fact_830_VEBT_Odistinct_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X22: $o] :
( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
!= ( vEBT_Leaf @ X21 @ X22 ) ) ).
% VEBT.distinct(1)
thf(fact_831_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_832_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_833_one__reorient,axiom,
! [X: rat] :
( ( one_one_rat = X )
= ( X = one_one_rat ) ) ).
% one_reorient
thf(fact_834_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_835_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_836_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_837_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_838_minus__equation__iff,axiom,
! [A: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= B )
= ( ( uminus1482373934393186551omplex @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_839_minus__equation__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= B )
= ( ( uminus1351360451143612070nteger @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_840_minus__equation__iff,axiom,
! [A: rat,B: rat] :
( ( ( uminus_uminus_rat @ A )
= B )
= ( ( uminus_uminus_rat @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_841_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_842_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_843_equation__minus__iff,axiom,
! [A: complex,B: complex] :
( ( A
= ( uminus1482373934393186551omplex @ B ) )
= ( B
= ( uminus1482373934393186551omplex @ A ) ) ) ).
% equation_minus_iff
thf(fact_844_equation__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( A
= ( uminus1351360451143612070nteger @ B ) )
= ( B
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% equation_minus_iff
thf(fact_845_equation__minus__iff,axiom,
! [A: rat,B: rat] :
( ( A
= ( uminus_uminus_rat @ B ) )
= ( B
= ( uminus_uminus_rat @ A ) ) ) ).
% equation_minus_iff
thf(fact_846_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_847_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_848_zero__neq__neg__one,axiom,
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% zero_neq_neg_one
thf(fact_849_zero__neq__neg__one,axiom,
( zero_z3403309356797280102nteger
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% zero_neq_neg_one
thf(fact_850_zero__neq__neg__one,axiom,
( zero_zero_rat
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% zero_neq_neg_one
thf(fact_851_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_852_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_853_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_854_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_855_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_856_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_857_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_858_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_859_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_860_of__int__of__nat,axiom,
( ring_17405671764205052669omplex
= ( ^ [K3: int] : ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri8010041392384452111omplex @ ( nat2 @ K3 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_861_of__int__of__nat,axiom,
( ring_1_of_int_int
= ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_862_of__int__of__nat,axiom,
( ring_1_of_int_real
= ( ^ [K3: int] : ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K3 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_863_of__int__of__nat,axiom,
( ring_18347121197199848620nteger
= ( ^ [K3: int] : ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri4939895301339042750nteger @ ( nat2 @ K3 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_864_of__int__of__nat,axiom,
( ring_1_of_int_rat
= ( ^ [K3: int] : ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri681578069525770553at_rat @ ( nat2 @ K3 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_865_less__ceiling__iff,axiom,
! [Z2: int,X: rat] :
( ( ord_less_int @ Z2 @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ) ).
% less_ceiling_iff
thf(fact_866_less__ceiling__iff,axiom,
! [Z2: int,X: real] :
( ( ord_less_int @ Z2 @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ) ).
% less_ceiling_iff
thf(fact_867_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_868_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_869_less__minus__one__simps_I1_J,axiom,
ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% less_minus_one_simps(1)
thf(fact_870_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_871_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_872_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_873_less__minus__one__simps_I3_J,axiom,
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% less_minus_one_simps(3)
thf(fact_874_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_875_ex__of__int__less,axiom,
! [X: real] :
? [Z4: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z4 ) @ X ) ).
% ex_of_int_less
thf(fact_876_ex__of__int__less,axiom,
! [X: rat] :
? [Z4: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z4 ) @ X ) ).
% ex_of_int_less
thf(fact_877_ex__less__of__int,axiom,
! [X: real] :
? [Z4: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z4 ) ) ).
% ex_less_of_int
thf(fact_878_ex__less__of__int,axiom,
! [X: rat] :
? [Z4: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z4 ) ) ).
% ex_less_of_int
thf(fact_879_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_880_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_881_forall__pos__mono__1,axiom,
! [P: real > $o,E2: real] :
( ! [D4: real,E: real] :
( ( ord_less_real @ D4 @ E )
=> ( ( P @ D4 )
=> ( P @ E ) ) )
=> ( ! [N2: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_882_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_883_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D4: real,E: real] :
( ( ord_less_real @ D4 @ E )
=> ( ( P @ D4 )
=> ( P @ E ) ) )
=> ( ! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_884_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_885_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_886_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_887_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_888_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_889_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_890_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_891_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_892_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_893_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_894_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_895_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_896_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_897_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_898_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_899_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_900_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_901_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_902_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_903_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_904_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_905_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_906_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_907_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_908_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_909_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_910_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_911_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_912_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_913_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_914_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_915_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_916_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_917_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_918_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_919_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_920_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_921_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_922_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_923_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_924_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
% zero_neq_one
thf(fact_925_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_926_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_927_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_928_less__numeral__extra_I4_J,axiom,
~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% less_numeral_extra(4)
thf(fact_929_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_930_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_931_int__cases2,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_932_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_933_VEBT__internal_OminNull_Osimps_I3_J,axiom,
! [Uu: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% VEBT_internal.minNull.simps(3)
thf(fact_934_VEBT__internal_OminNull_Osimps_I2_J,axiom,
! [Uv: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% VEBT_internal.minNull.simps(2)
thf(fact_935_VEBT__internal_OminNull_Osimps_I1_J,axiom,
vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% VEBT_internal.minNull.simps(1)
thf(fact_936_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_937_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_938_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_939_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_940_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_941_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_942_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_943_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_944_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_945_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_946_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_947_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_948_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_949_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_950_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_951_add_Oinverse__unique,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_952_add_Oinverse__unique,axiom,
! [A: code_integer,B: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger )
=> ( ( uminus1351360451143612070nteger @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_953_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_954_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_955_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_956_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_957_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_958_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_959_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_960_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_961_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_962_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_963_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_964_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_965_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A3: real,B3: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_966_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_complex
= ( ^ [A3: complex,B3: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_967_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_8373710615458151222nteger
= ( ^ [A3: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_968_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_969_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_970_diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A3: real,B3: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_971_diff__conv__add__uminus,axiom,
( minus_minus_complex
= ( ^ [A3: complex,B3: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_972_diff__conv__add__uminus,axiom,
( minus_8373710615458151222nteger
= ( ^ [A3: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_973_diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_974_group__cancel_Osub2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_975_group__cancel_Osub2,axiom,
! [B2: real,K: real,B: real,A: real] :
( ( B2
= ( plus_plus_real @ K @ B ) )
=> ( ( minus_minus_real @ A @ B2 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_976_group__cancel_Osub2,axiom,
! [B2: complex,K: complex,B: complex,A: complex] :
( ( B2
= ( plus_plus_complex @ K @ B ) )
=> ( ( minus_minus_complex @ A @ B2 )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_977_group__cancel_Osub2,axiom,
! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
( ( B2
= ( plus_p5714425477246183910nteger @ K @ B ) )
=> ( ( minus_8373710615458151222nteger @ A @ B2 )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_978_group__cancel_Osub2,axiom,
! [B2: rat,K: rat,B: rat,A: rat] :
( ( B2
= ( plus_plus_rat @ K @ B ) )
=> ( ( minus_minus_rat @ A @ B2 )
= ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_979_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ zero_zero_nat )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_980_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_981_not__one__less__zero,axiom,
~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_less_zero
thf(fact_982_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_983_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_984_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_985_zero__less__one,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one
thf(fact_986_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_987_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_988_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_989_less__numeral__extra_I1_J,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% less_numeral_extra(1)
thf(fact_990_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_991_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_992_int__of__nat__induct,axiom,
! [P: int > $o,Z2: int] :
( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_993_int__cases,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_994_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_995_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_996_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_997_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_998_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_999_less__add__one,axiom,
! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% less_add_one
thf(fact_1000_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1001_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_1002_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_1003_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1004_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_1005_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1006_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1007_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1008_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1009_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1010_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_1011_nat__mono__iff,axiom,
! [Z2: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_1012_zless__nat__eq__int__zless,axiom,
! [M: nat,Z2: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1013_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_1014_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_1015_int__minus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% int_minus
thf(fact_1016_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_1017_zero__less__two,axiom,
ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% zero_less_two
thf(fact_1018_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1019_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1020_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] :
( M
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_1021_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1022_reals__Archimedean,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N2: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ X ) ) ).
% reals_Archimedean
thf(fact_1023_reals__Archimedean,axiom,
! [X: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ? [N2: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ X ) ) ).
% reals_Archimedean
thf(fact_1024_of__int__pos,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).
% of_int_pos
thf(fact_1025_of__int__pos,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_pos
thf(fact_1026_of__int__pos,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% of_int_pos
thf(fact_1027_of__int__pos,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_pos
thf(fact_1028_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_1029_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_1030_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N ) @ ( ring_18347121197199848620nteger @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_1031_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_1032_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_1033_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_1034_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1035_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N2: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_1036_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1037_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1038__C3_Oprems_C,axiom,
vEBT_V8194947554948674370ptions @ ( vEBT_Node @ none_P5556105721700978146at_nat @ deg @ treeList @ summary ) @ xa ).
% "3.prems"
thf(fact_1039_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_1040_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_1041_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_1042_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_1043_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_1044_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_1045_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_1046_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_1047_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_1048_inverse__minus__eq,axiom,
! [A: complex] :
( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
= ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ).
% inverse_minus_eq
thf(fact_1049_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_1050_inverse__eq__1__iff,axiom,
! [X: real] :
( ( ( inverse_inverse_real @ X )
= one_one_real )
= ( X = one_one_real ) ) ).
% inverse_eq_1_iff
thf(fact_1051_inverse__eq__1__iff,axiom,
! [X: complex] :
( ( ( invers8013647133539491842omplex @ X )
= one_one_complex )
= ( X = one_one_complex ) ) ).
% inverse_eq_1_iff
thf(fact_1052_inverse__eq__1__iff,axiom,
! [X: rat] :
( ( ( inverse_inverse_rat @ X )
= one_one_rat )
= ( X = one_one_rat ) ) ).
% inverse_eq_1_iff
thf(fact_1053_inverse__1,axiom,
( ( inverse_inverse_real @ one_one_real )
= one_one_real ) ).
% inverse_1
thf(fact_1054_inverse__1,axiom,
( ( invers8013647133539491842omplex @ one_one_complex )
= one_one_complex ) ).
% inverse_1
thf(fact_1055_inverse__1,axiom,
( ( inverse_inverse_rat @ one_one_rat )
= one_one_rat ) ).
% inverse_1
thf(fact_1056_inverse__nonzero__iff__nonzero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_1057_inverse__nonzero__iff__nonzero,axiom,
! [A: complex] :
( ( ( invers8013647133539491842omplex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_1058_inverse__nonzero__iff__nonzero,axiom,
! [A: rat] :
( ( ( inverse_inverse_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_1059_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_1060_inverse__zero,axiom,
( ( invers8013647133539491842omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% inverse_zero
thf(fact_1061_inverse__zero,axiom,
( ( inverse_inverse_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% inverse_zero
thf(fact_1062_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_1063_inverse__inverse__eq,axiom,
! [A: real] :
( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_1064_inverse__inverse__eq,axiom,
! [A: complex] :
( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_1065_inverse__inverse__eq,axiom,
! [A: rat] :
( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_1066_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_1067_inverse__eq__iff__eq,axiom,
! [A: complex,B: complex] :
( ( ( invers8013647133539491842omplex @ A )
= ( invers8013647133539491842omplex @ B ) )
= ( A = B ) ) ).
% inverse_eq_iff_eq
thf(fact_1068_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_1069_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1070_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_1071_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_1072_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_1073_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1074_zless__add1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z2 )
| ( W2 = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1075_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_1076_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1077_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1078_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_1079_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_1080_VEBT__internal_OminNull_Osimps_I4_J,axiom,
! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% VEBT_internal.minNull.simps(4)
thf(fact_1081_linordered__field__no__lb,axiom,
! [X5: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X5 ) ).
% linordered_field_no_lb
thf(fact_1082_linordered__field__no__lb,axiom,
! [X5: rat] :
? [Y3: rat] : ( ord_less_rat @ Y3 @ X5 ) ).
% linordered_field_no_lb
thf(fact_1083_linordered__field__no__ub,axiom,
! [X5: real] :
? [X_12: real] : ( ord_less_real @ X5 @ X_12 ) ).
% linordered_field_no_ub
thf(fact_1084_linordered__field__no__ub,axiom,
! [X5: rat] :
? [X_12: rat] : ( ord_less_rat @ X5 @ X_12 ) ).
% linordered_field_no_ub
thf(fact_1085_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_1086_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1087_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_1088_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1089_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_1090_inverse__eq__imp__eq,axiom,
! [A: complex,B: complex] :
( ( ( invers8013647133539491842omplex @ A )
= ( invers8013647133539491842omplex @ B ) )
=> ( A = B ) ) ).
% inverse_eq_imp_eq
thf(fact_1091_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_1092_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% VEBT_internal.minNull.elims(2)
thf(fact_1093_Suc__as__int,axiom,
( suc
= ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_1094_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_1095_field__class_Ofield__inverse__zero,axiom,
( ( invers8013647133539491842omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% field_class.field_inverse_zero
thf(fact_1096_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% field_class.field_inverse_zero
thf(fact_1097_inverse__zero__imp__zero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_1098_inverse__zero__imp__zero,axiom,
! [A: complex] :
( ( ( invers8013647133539491842omplex @ A )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ).
% inverse_zero_imp_zero
thf(fact_1099_inverse__zero__imp__zero,axiom,
! [A: rat] :
( ( ( inverse_inverse_rat @ A )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ).
% inverse_zero_imp_zero
thf(fact_1100_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_1101_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_1102_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_1103_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_1104_nonzero__inverse__inverse__eq,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_1105_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_1106_nonzero__imp__inverse__nonzero,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_1107_nonzero__imp__inverse__nonzero,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ A )
!= zero_zero_complex ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_1108_nonzero__imp__inverse__nonzero,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( inverse_inverse_rat @ A )
!= zero_zero_rat ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_1109_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_1110_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_1111_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_1112_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_1113_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_1114_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_1115_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_1116_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_1117_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_1118_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_1119_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_1120_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_1121_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_1122_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_1123_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_1124_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_1125_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_1126_nonzero__inverse__minus__eq,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
= ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_1127_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_1128_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_1129_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_1130_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_1131_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_1132_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_1133_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_1134_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% dbl_dec_simps(2)
thf(fact_1135_dbl__dec__simps_I2_J,axiom,
( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% dbl_dec_simps(2)
thf(fact_1136_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% dbl_dec_simps(2)
thf(fact_1137_ceiling__eq,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
=> ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim7802044766580827645g_real @ X )
= ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% ceiling_eq
thf(fact_1138_ceiling__eq,axiom,
! [N: int,X: rat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X )
=> ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
=> ( ( archim2889992004027027881ng_rat @ X )
= ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% ceiling_eq
thf(fact_1139_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_1140_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_1141_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_1142_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_1143_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_1144_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_1145_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% dbl_inc_simps(4)
thf(fact_1146_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% dbl_inc_simps(4)
thf(fact_1147_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_1148_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1149_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1150_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1151_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1152_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1153_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1154_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_1155_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_1156_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_1157_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_1158_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_1159_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_1160_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_1161_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_1162_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_1163_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_1164_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_1165_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_1166_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_1167_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_1168_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_1169_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_1170_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_1171_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_1172_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1173_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1174_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_1175_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_1176_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_1177_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_1178_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6511756317524482435omplex @ one_one_complex )
= one_one_complex ) ).
% dbl_dec_simps(3)
thf(fact_1179_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6075765906172075777c_real @ one_one_real )
= one_one_real ) ).
% dbl_dec_simps(3)
thf(fact_1180_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
= one_one_rat ) ).
% dbl_dec_simps(3)
thf(fact_1181_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_1182_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_1183_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_1184_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_1185_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_1186_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_1187_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_1188_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_1189_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_1190_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_1191_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_1192_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_1193_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_1194_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_1195_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_1196_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_1197_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_1198_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_1199_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_1200_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_1201_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_1202_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_1203_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_1204_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_1205_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_1206_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_1207_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_1208_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_1209_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_1210_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_1211_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_1212_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_1213_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_1214_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_1215_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_1216_less__eq__neg__nonpos,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% less_eq_neg_nonpos
thf(fact_1217_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_1218_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_1219_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_1220_neg__less__eq__nonneg,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_1221_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_1222_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_1223_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_1224_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_1225_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_1226_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_1227_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_1228_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_1229_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_1230_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_1231_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_1232_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_1233_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_1234_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_1235_of__int__le__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% of_int_le_iff
thf(fact_1236_of__int__le__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ W2 ) @ ( ring_18347121197199848620nteger @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% of_int_le_iff
thf(fact_1237_of__int__le__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% of_int_le_iff
thf(fact_1238_of__int__le__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% of_int_le_iff
thf(fact_1239_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_1240_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_1241_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_1242_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
= one_one_complex ) ).
% dbl_inc_simps(2)
thf(fact_1243_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_1244_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
= one_one_rat ) ).
% dbl_inc_simps(2)
thf(fact_1245_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_1246_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_1247_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1248_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_1249_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_1250_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_1251_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_1252_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_1253_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_1254_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_1255_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1256_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ Z2 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1257_zle__add1__eq__le,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1258_int__nat__eq,axiom,
! [Z2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_1259_zle__diff1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1260_of__int__le__0__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
= ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_1261_of__int__le__0__iff,axiom,
! [Z2: int] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z2 ) @ zero_z3403309356797280102nteger )
= ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_1262_of__int__le__0__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
= ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_1263_of__int__le__0__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
= ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_1264_of__int__0__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_le_iff
thf(fact_1265_of__int__0__le__iff,axiom,
! [Z2: int] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z2 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_le_iff
thf(fact_1266_of__int__0__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_le_iff
thf(fact_1267_of__int__0__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_le_iff
thf(fact_1268_of__int__le__1__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
= ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_1269_of__int__le__1__iff,axiom,
! [Z2: int] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z2 ) @ one_one_Code_integer )
= ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_1270_of__int__le__1__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
= ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_1271_of__int__le__1__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
= ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_1272_of__int__1__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% of_int_1_le_iff
thf(fact_1273_of__int__1__le__iff,axiom,
! [Z2: int] :
( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z2 ) )
= ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% of_int_1_le_iff
thf(fact_1274_of__int__1__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% of_int_1_le_iff
thf(fact_1275_of__int__1__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% of_int_1_le_iff
thf(fact_1276_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_1277_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_1278_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_1279_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_1280_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_1281_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_1282_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_1283_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_1284_of__nat__nat,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri8010041392384452111omplex @ ( nat2 @ Z2 ) )
= ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% of_nat_nat
thf(fact_1285_of__nat__nat,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_nat_nat
thf(fact_1286_of__nat__nat,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri5074537144036343181t_real @ ( nat2 @ Z2 ) )
= ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_nat_nat
thf(fact_1287_of__nat__nat,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri4939895301339042750nteger @ ( nat2 @ Z2 ) )
= ( ring_18347121197199848620nteger @ Z2 ) ) ) ).
% of_nat_nat
thf(fact_1288_of__nat__nat,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri681578069525770553at_rat @ ( nat2 @ Z2 ) )
= ( ring_1_of_int_rat @ Z2 ) ) ) ).
% of_nat_nat
thf(fact_1289_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X4: real,Y5: real] : ( plus_plus_real @ X4 @ ( uminus_uminus_real @ Y5 ) ) ) ) ).
% minus_real_def
thf(fact_1290_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_1291_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_1292_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_1293_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_1294_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_1295_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1296_verit__comp__simplify1_I2_J,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1297_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1298_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1299_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1300_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1301_lift__Suc__mono__le,axiom,
! [F: nat > rat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1302_lift__Suc__mono__le,axiom,
! [F: nat > num,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1303_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1304_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1305_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1306_lift__Suc__antimono__le,axiom,
! [F: nat > rat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1307_lift__Suc__antimono__le,axiom,
! [F: nat > num,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1308_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1309_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1310_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_1311_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ I ) @ ( semiri4939895301339042750nteger @ J ) ) ) ).
% of_nat_mono
thf(fact_1312_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_1313_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_1314_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_1315_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1316_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1317_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_1318_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_1319_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_1320_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_1321_ceiling__le__iff,axiom,
! [X: real,Z2: int] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z2 )
= ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% ceiling_le_iff
thf(fact_1322_ceiling__le__iff,axiom,
! [X: rat,Z2: int] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 )
= ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% ceiling_le_iff
thf(fact_1323_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_1324_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_1325_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_1326_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_1327_le__numeral__extra_I3_J,axiom,
ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% le_numeral_extra(3)
thf(fact_1328_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1329_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_1330_verit__comp__simplify1_I3_J,axiom,
! [B5: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B5 @ A5 ) )
= ( ord_less_real @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1331_verit__comp__simplify1_I3_J,axiom,
! [B5: rat,A5: rat] :
( ( ~ ( ord_less_eq_rat @ B5 @ A5 ) )
= ( ord_less_rat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1332_verit__comp__simplify1_I3_J,axiom,
! [B5: num,A5: num] :
( ( ~ ( ord_less_eq_num @ B5 @ A5 ) )
= ( ord_less_num @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1333_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1334_verit__comp__simplify1_I3_J,axiom,
! [B5: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
= ( ord_less_int @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1335_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_1336_le__numeral__extra_I4_J,axiom,
ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% le_numeral_extra(4)
thf(fact_1337_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1338_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1339_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_1340_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_1341_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_1342_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_1343_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_1344_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_1345_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_1346_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_1347_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_1348_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_1349_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_1350_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_1351_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_1352_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_1353_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_1354_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_1355_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_1356_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_1357_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_1358_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_1359_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_1360_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_1361_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_1362_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_1363_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_1364_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_1365_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_1366_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_1367_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_1368_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_1369_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_1370_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_1371_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_1372_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_1373_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_1374_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_1375_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_1376_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_1377_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_1378_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_1379_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_1380_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_1381_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_1382_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_1383_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_1384_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_1385_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_1386_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_1387_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_1388_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_1389_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_1390_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_1391_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_1392_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_1393_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_1394_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_1395_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_1396_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_1397_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_1398_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_1399_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_1400_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1401_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1402_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1403_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M2: nat] :
( M6
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_1404_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1405_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1406_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1407_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1408_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1409_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z4: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z4 )
=> ( R @ X3 @ Z4 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1410_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1411_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_1412_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_1413_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1414_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N4: int,M4: int] : ( ord_less_real @ ( ring_1_of_int_real @ N4 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M4 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_1415_real__arch__simple,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% real_arch_simple
thf(fact_1416_real__arch__simple,axiom,
! [X: rat] :
? [N2: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% real_arch_simple
thf(fact_1417_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_1418_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_1419_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_1420_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1421_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1422_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1423_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1424_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1425_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_real @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_1426_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1427_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1428_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_1429_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_1430_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_1431_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_1432_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1433_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1434_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1435_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1436_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1437_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1438_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1439_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_1440_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1441_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1442_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1443_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1444_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1445_ex__le__of__int,axiom,
! [X: real] :
? [Z4: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z4 ) ) ).
% ex_le_of_int
thf(fact_1446_ex__le__of__int,axiom,
! [X: rat] :
? [Z4: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z4 ) ) ).
% ex_le_of_int
thf(fact_1447_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N4: int,M4: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N4 ) @ one_one_real ) @ ( ring_1_of_int_real @ M4 ) ) ) ) ).
% int_less_real_le
thf(fact_1448_of__int__nonneg,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_nonneg
thf(fact_1449_of__int__nonneg,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).
% of_int_nonneg
thf(fact_1450_of__int__nonneg,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% of_int_nonneg
thf(fact_1451_of__int__nonneg,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_nonneg
thf(fact_1452_nat__le__eq__zle,axiom,
! [W2: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_1453_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1454_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_1455_not__one__le__zero,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_le_zero
thf(fact_1456_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1457_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1458_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_1459_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_1460_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_1461_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_1462_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_1463_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_1464_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_1465_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_1466_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_1467_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_1468_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_1469_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_1470_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_1471_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_1472_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_1473_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_1474_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_1475_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_1476_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_1477_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_1478_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_1479_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_1480_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_1481_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_1482_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_1483_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_1484_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_1485_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_1486_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_1487_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_1488_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_1489_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_1490_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_1491_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_1492_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_1493_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_1494_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_1495_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_1496_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_1497_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_1498_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_1499_le__iff__diff__le__0,axiom,
( ord_less_eq_rat
= ( ^ [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% le_iff_diff_le_0
thf(fact_1500_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1501_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_1502_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_1503_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_1504_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_1505_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_1506_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_1507_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_1508_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_1509_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_1510_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_1511_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_1512_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_1513_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_1514_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_1515_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_1516_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_1517_add__le__imp__le__diff,axiom,
! [I: real,K: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1518_add__le__imp__le__diff,axiom,
! [I: rat,K: rat,N: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
=> ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1519_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1520_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1521_add__le__add__imp__diff__le,axiom,
! [I: real,K: real,N: real,J: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1522_add__le__add__imp__diff__le,axiom,
! [I: rat,K: rat,N: rat,J: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
=> ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
=> ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
=> ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1523_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1524_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1525_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_1526_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_1527_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_1528_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_1529_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_1530_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_1531_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_1532_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_1533_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_1534_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_1535_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_1536_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_1537_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_1538_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_1539_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_1540_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_1541_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_1542_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_1543_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_1544_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_1545_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_1546_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_1547_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_1548_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_1549_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_1550_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N ) ) ).
% of_nat_0_le_iff
thf(fact_1551_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_1552_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_1553_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_1554_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1555_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1556_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1557_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1558_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1559_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1560_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1561_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1562_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1563_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1564_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( F @ M2 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1565_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1566_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1567_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_1568_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_1569_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_1570_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_1571_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_1572_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_1573_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_1574_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1575_eq__nat__nat__iff,axiom,
! [Z2: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ( nat2 @ Z2 )
= ( nat2 @ Z5 ) )
= ( Z2 = Z5 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1576_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
! [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% all_nat
thf(fact_1577_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
& ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% ex_nat
thf(fact_1578_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N4: nat,M4: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M4 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1579_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N4: nat,M4: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N4 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M4 ) ) ) ) ).
% nat_less_real_le
thf(fact_1580_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_1581_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W: int,Z3: int] :
? [N4: nat] :
( Z3
= ( plus_plus_int @ W @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1582_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_1583_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_1584_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_1585_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_1586_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_1587_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_1588_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_1589_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_1590_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_1591_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_1592_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_1593_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_1594_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_1595_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_1596_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_1597_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_1598_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_1599_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_1600_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_1601_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_1602_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_1603_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_1604_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_1605_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_1606_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_1607_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_1608_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_1609_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_1610_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_1611_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_1612_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_1613_le__minus__one__simps_I1_J,axiom,
ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% le_minus_one_simps(1)
thf(fact_1614_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_1615_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_1616_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_1617_le__minus__one__simps_I3_J,axiom,
~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% le_minus_one_simps(3)
thf(fact_1618_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_1619_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_1620_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_1621_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_1622_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_1623_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_1624_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_1625_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_1626_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_1627_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_1628_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_1629_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_1630_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1631_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1632_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1633_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1634_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri4939895301339042750nteger @ ( minus_minus_nat @ M @ N ) )
= ( minus_8373710615458151222nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1635_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1636_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_1637_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1638_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1639_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1640_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1641_add1__zle__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1642_zless__imp__add1__zle,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1643_nat__0__le,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) ) ).
% nat_0_le
thf(fact_1644_int__eq__iff,axiom,
! [M: nat,Z2: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z2 )
= ( ( M
= ( nat2 @ Z2 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).
% int_eq_iff
thf(fact_1645_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_1646_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_1647_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_1648_of__nat__ceiling,axiom,
! [R2: real] : ( ord_less_eq_real @ R2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R2 ) ) ) ) ).
% of_nat_ceiling
thf(fact_1649_of__nat__ceiling,axiom,
! [R2: rat] : ( ord_less_eq_rat @ R2 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R2 ) ) ) ) ).
% of_nat_ceiling
thf(fact_1650_dbl__inc__def,axiom,
( neg_nu8557863876264182079omplex
= ( ^ [X4: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X4 @ X4 ) @ one_one_complex ) ) ) ).
% dbl_inc_def
thf(fact_1651_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X4: real] : ( plus_plus_real @ ( plus_plus_real @ X4 @ X4 ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_1652_dbl__inc__def,axiom,
( neg_nu5219082963157363817nc_rat
= ( ^ [X4: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X4 @ X4 ) @ one_one_rat ) ) ) ).
% dbl_inc_def
thf(fact_1653_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X4: int] : ( plus_plus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_1654_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_1655_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_1656_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_1657_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_1658_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_1659_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_1660_floor__exists1,axiom,
! [X: real] :
? [X3: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X3 @ one_one_int ) ) )
& ! [Y6: int] :
( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y6 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y6 @ one_one_int ) ) ) )
=> ( Y6 = X3 ) ) ) ).
% floor_exists1
thf(fact_1661_floor__exists1,axiom,
! [X: rat] :
? [X3: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
& ! [Y6: int] :
( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y6 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y6 @ one_one_int ) ) ) )
=> ( Y6 = X3 ) ) ) ).
% floor_exists1
thf(fact_1662_floor__exists,axiom,
! [X: real] :
? [Z4: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z4 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).
% floor_exists
thf(fact_1663_floor__exists,axiom,
! [X: rat] :
? [Z4: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z4 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).
% floor_exists
thf(fact_1664_of__int__ceiling__le__add__one,axiom,
! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% of_int_ceiling_le_add_one
thf(fact_1665_of__int__ceiling__le__add__one,axiom,
! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).
% of_int_ceiling_le_add_one
thf(fact_1666_of__int__ceiling__diff__one__le,axiom,
! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).
% of_int_ceiling_diff_one_le
thf(fact_1667_of__int__ceiling__diff__one__le,axiom,
! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).
% of_int_ceiling_diff_one_le
thf(fact_1668_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1669_nat__less__eq__zless,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_1670_nat__eq__iff,axiom,
! [W2: int,M: nat] :
( ( ( nat2 @ W2 )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1671_nat__eq__iff2,axiom,
! [M: nat,W2: int] :
( ( M
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1672_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1673_nat__add__distrib,axiom,
! [Z2: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( nat2 @ ( plus_plus_int @ Z2 @ Z5 ) )
= ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1674_nat__diff__distrib,axiom,
! [Z5: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ord_less_eq_int @ Z5 @ Z2 )
=> ( ( nat2 @ ( minus_minus_int @ Z2 @ Z5 ) )
= ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1675_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_1676_ceiling__split,axiom,
! [P: int > $o,T: real] :
( ( P @ ( archim7802044766580827645g_real @ T ) )
= ( ! [I3: int] :
( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) @ T )
& ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I3 ) ) )
=> ( P @ I3 ) ) ) ) ).
% ceiling_split
thf(fact_1677_ceiling__split,axiom,
! [P: int > $o,T: rat] :
( ( P @ ( archim2889992004027027881ng_rat @ T ) )
= ( ! [I3: int] :
( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) @ T )
& ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I3 ) ) )
=> ( P @ I3 ) ) ) ) ).
% ceiling_split
thf(fact_1678_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_1679_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_1680_ceiling__unique,axiom,
! [Z2: int,X: real] :
( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) )
=> ( ( archim7802044766580827645g_real @ X )
= Z2 ) ) ) ).
% ceiling_unique
thf(fact_1681_ceiling__unique,axiom,
! [Z2: int,X: rat] :
( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X )
=> ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) )
=> ( ( archim2889992004027027881ng_rat @ X )
= Z2 ) ) ) ).
% ceiling_unique
thf(fact_1682_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_1683_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_1684_le__ceiling__iff,axiom,
! [Z2: int,X: rat] :
( ( ord_less_eq_int @ Z2 @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X ) ) ).
% le_ceiling_iff
thf(fact_1685_le__ceiling__iff,axiom,
! [Z2: int,X: real] :
( ( ord_less_eq_int @ Z2 @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X ) ) ).
% le_ceiling_iff
thf(fact_1686_ceiling__less__iff,axiom,
! [X: real,Z2: int] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z2 )
= ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).
% ceiling_less_iff
thf(fact_1687_ceiling__less__iff,axiom,
! [X: rat,Z2: int] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 )
= ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).
% ceiling_less_iff
thf(fact_1688_Suc__nat__eq__nat__zadd1,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( suc @ ( nat2 @ Z2 ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1689_nat__less__iff,axiom,
! [W2: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_1690_dbl__dec__def,axiom,
( neg_nu6511756317524482435omplex
= ( ^ [X4: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X4 @ X4 ) @ one_one_complex ) ) ) ).
% dbl_dec_def
thf(fact_1691_dbl__dec__def,axiom,
( neg_nu6075765906172075777c_real
= ( ^ [X4: real] : ( minus_minus_real @ ( plus_plus_real @ X4 @ X4 ) @ one_one_real ) ) ) ).
% dbl_dec_def
thf(fact_1692_dbl__dec__def,axiom,
( neg_nu3179335615603231917ec_rat
= ( ^ [X4: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X4 @ X4 ) @ one_one_rat ) ) ) ).
% dbl_dec_def
thf(fact_1693_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X4: int] : ( minus_minus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_1694_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_1695_discrete,axiom,
( ord_less_nat
= ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).
% discrete
thf(fact_1696_discrete,axiom,
( ord_less_int
= ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).
% discrete
thf(fact_1697_conj__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1698_imp__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1699_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( 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 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1700_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1701_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1702_order__le__imp__less__or__eq,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1703_order__le__imp__less__or__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1704_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1705_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1706_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1707_linorder__le__less__linear,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
| ( ord_less_rat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1708_linorder__le__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1709_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1710_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1711_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1712_order__less__le__subst2,axiom,
! [A: rat,B: rat,F: rat > real,C: real] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1713_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > real,C: real] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1714_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1715_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1716_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > rat,C: rat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1717_order__less__le__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1718_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > rat,C: rat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1719_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1720_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > rat,C: rat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1721_abs__abs,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_abs
thf(fact_1722_abs__abs,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_abs
thf(fact_1723_abs__abs,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
= ( abs_abs_Code_integer @ A ) ) ).
% abs_abs
thf(fact_1724_abs__abs,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
= ( abs_abs_rat @ A ) ) ).
% abs_abs
thf(fact_1725_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_1726_abs__idempotent,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_idempotent
thf(fact_1727_abs__idempotent,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
= ( abs_abs_Code_integer @ A ) ) ).
% abs_idempotent
thf(fact_1728_abs__idempotent,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
= ( abs_abs_rat @ A ) ) ).
% abs_idempotent
thf(fact_1729_abs__0__eq,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger
= ( abs_abs_Code_integer @ A ) )
= ( A = zero_z3403309356797280102nteger ) ) ).
% abs_0_eq
thf(fact_1730_abs__0__eq,axiom,
! [A: real] :
( ( zero_zero_real
= ( abs_abs_real @ A ) )
= ( A = zero_zero_real ) ) ).
% abs_0_eq
thf(fact_1731_abs__0__eq,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( abs_abs_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% abs_0_eq
thf(fact_1732_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_1733_abs__eq__0,axiom,
! [A: code_integer] :
( ( ( abs_abs_Code_integer @ A )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% abs_eq_0
thf(fact_1734_abs__eq__0,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0
thf(fact_1735_abs__eq__0,axiom,
! [A: rat] :
( ( ( abs_abs_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% abs_eq_0
thf(fact_1736_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_1737_abs__zero,axiom,
( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% abs_zero
thf(fact_1738_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_zero
thf(fact_1739_abs__zero,axiom,
( ( abs_abs_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% abs_zero
thf(fact_1740_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_1741_abs__0,axiom,
( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% abs_0
thf(fact_1742_abs__0,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_0
thf(fact_1743_abs__0,axiom,
( ( abs_abs_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% abs_0
thf(fact_1744_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_1745_abs__1,axiom,
( ( abs_abs_Code_integer @ one_one_Code_integer )
= one_one_Code_integer ) ).
% abs_1
thf(fact_1746_abs__1,axiom,
( ( abs_abs_complex @ one_one_complex )
= one_one_complex ) ).
% abs_1
thf(fact_1747_abs__1,axiom,
( ( abs_abs_real @ one_one_real )
= one_one_real ) ).
% abs_1
thf(fact_1748_abs__1,axiom,
( ( abs_abs_rat @ one_one_rat )
= one_one_rat ) ).
% abs_1
thf(fact_1749_abs__1,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_1
thf(fact_1750_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_1751_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_1752_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_1753_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_1754_abs__minus,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus
thf(fact_1755_abs__minus,axiom,
! [A: real] :
( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_minus
thf(fact_1756_abs__minus,axiom,
! [A: complex] :
( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
= ( abs_abs_complex @ A ) ) ).
% abs_minus
thf(fact_1757_abs__minus,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
= ( abs_abs_Code_integer @ A ) ) ).
% abs_minus
thf(fact_1758_abs__minus,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
= ( abs_abs_rat @ A ) ) ).
% abs_minus
thf(fact_1759_abs__minus__cancel,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus_cancel
thf(fact_1760_abs__minus__cancel,axiom,
! [A: real] :
( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_minus_cancel
thf(fact_1761_abs__minus__cancel,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
= ( abs_abs_Code_integer @ A ) ) ).
% abs_minus_cancel
thf(fact_1762_abs__minus__cancel,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
= ( abs_abs_rat @ A ) ) ).
% abs_minus_cancel
thf(fact_1763_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_1764_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% abs_of_nat
thf(fact_1765_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
= ( semiri4939895301339042750nteger @ N ) ) ).
% abs_of_nat
thf(fact_1766_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
= ( semiri681578069525770553at_rat @ N ) ) ).
% abs_of_nat
thf(fact_1767_abs__inverse,axiom,
! [A: real] :
( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
= ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ).
% abs_inverse
thf(fact_1768_abs__inverse,axiom,
! [A: complex] :
( ( abs_abs_complex @ ( invers8013647133539491842omplex @ A ) )
= ( invers8013647133539491842omplex @ ( abs_abs_complex @ A ) ) ) ).
% abs_inverse
thf(fact_1769_abs__inverse,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
= ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ).
% abs_inverse
thf(fact_1770_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_1771_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_1772_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_1773_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_1774_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_1775_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_1776_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_1777_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_1778_abs__of__nonneg,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( abs_abs_Code_integer @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_1779_abs__of__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_1780_abs__of__nonneg,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( abs_abs_rat @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_1781_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_1782_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_1783_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_1784_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_1785_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_1786_abs__neg__one,axiom,
( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
= one_one_int ) ).
% abs_neg_one
thf(fact_1787_abs__neg__one,axiom,
( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
= one_one_real ) ).
% abs_neg_one
thf(fact_1788_abs__neg__one,axiom,
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= one_one_Code_integer ) ).
% abs_neg_one
thf(fact_1789_abs__neg__one,axiom,
( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= one_one_rat ) ).
% abs_neg_one
thf(fact_1790_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_1791_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_1792_of__int__abs,axiom,
! [X: int] :
( ( ring_18347121197199848620nteger @ ( abs_abs_int @ X ) )
= ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ X ) ) ) ).
% of_int_abs
thf(fact_1793_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_1794_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_1795_abs__of__nonpos,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ A )
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% abs_of_nonpos
thf(fact_1796_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_1797_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_1798_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1799_complete__real,axiom,
! [S3: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S3 )
=> ( ? [Z6: real] :
! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ord_less_eq_real @ X3 @ Z6 ) )
=> ? [Y3: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S3 )
=> ( ord_less_eq_real @ X5 @ Y3 ) )
& ! [Z6: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ord_less_eq_real @ X3 @ Z6 ) )
=> ( ord_less_eq_real @ Y3 @ Z6 ) ) ) ) ) ).
% complete_real
thf(fact_1800_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1801_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1802_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1803_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1804_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_1805_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1806_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_1807_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_1808_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_1809_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_1810_abs__ge__self,axiom,
! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% abs_ge_self
thf(fact_1811_abs__ge__self,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_self
thf(fact_1812_abs__ge__self,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% abs_ge_self
thf(fact_1813_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_1814_abs__eq__0__iff,axiom,
! [A: code_integer] :
( ( ( abs_abs_Code_integer @ A )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% abs_eq_0_iff
thf(fact_1815_abs__eq__0__iff,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0_iff
thf(fact_1816_abs__eq__0__iff,axiom,
! [A: rat] :
( ( ( abs_abs_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% abs_eq_0_iff
thf(fact_1817_abs__eq__0__iff,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_1818_abs__one,axiom,
( ( abs_abs_Code_integer @ one_one_Code_integer )
= one_one_Code_integer ) ).
% abs_one
thf(fact_1819_abs__one,axiom,
( ( abs_abs_real @ one_one_real )
= one_one_real ) ).
% abs_one
thf(fact_1820_abs__one,axiom,
( ( abs_abs_rat @ one_one_rat )
= one_one_rat ) ).
% abs_one
thf(fact_1821_abs__one,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_one
thf(fact_1822_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_1823_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_1824_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_1825_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_1826_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_1827_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_1828_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_1829_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_1830_abs__ge__zero,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_zero
thf(fact_1831_abs__ge__zero,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% abs_ge_zero
thf(fact_1832_abs__ge__zero,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% abs_ge_zero
thf(fact_1833_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_1834_abs__of__pos,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( abs_abs_Code_integer @ A )
= A ) ) ).
% abs_of_pos
thf(fact_1835_abs__of__pos,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_pos
thf(fact_1836_abs__of__pos,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( abs_abs_rat @ A )
= A ) ) ).
% abs_of_pos
thf(fact_1837_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_1838_abs__not__less__zero,axiom,
! [A: code_integer] :
~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% abs_not_less_zero
thf(fact_1839_abs__not__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% abs_not_less_zero
thf(fact_1840_abs__not__less__zero,axiom,
! [A: rat] :
~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% abs_not_less_zero
thf(fact_1841_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_1842_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_1843_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_1844_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_1845_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_1846_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_1847_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_1848_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_1849_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_1850_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_1851_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_1852_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_1853_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_1854_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_1855_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_1856_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_1857_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_1858_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_1859_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_1860_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_1861_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_1862_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_1863_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_1864_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_1865_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_1866_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_1867_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_1868_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_1869_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_1870_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_1871_abs__ge__minus__self,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_minus_self
thf(fact_1872_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_1873_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_1874_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_1875_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_1876_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_1877_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_1878_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_1879_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_1880_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_1881_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_1882_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_1883_abs__minus__le__zero,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% abs_minus_le_zero
thf(fact_1884_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_1885_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_1886_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_1887_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_1888_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_1889_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_1890_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_1891_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_1892_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_1893_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_1894_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_1895_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_1896_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_1897_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_1898_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_1899_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_1900_abs__of__neg,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ A )
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% abs_of_neg
thf(fact_1901_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_1902_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_1903_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_1904_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_1905_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_1906_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_1907_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_1908_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_1909_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_1910_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_1911_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_1912_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_1913_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_1914_abs__diff__le__iff,axiom,
! [X: code_integer,A: code_integer,R2: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
= ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
& ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_1915_abs__diff__le__iff,axiom,
! [X: real,A: real,R2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
= ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X )
& ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_1916_abs__diff__le__iff,axiom,
! [X: rat,A: rat,R2: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
= ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
& ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_1917_abs__diff__le__iff,axiom,
! [X: int,A: int,R2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
= ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X )
& ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_1918_abs__diff__less__iff,axiom,
! [X: code_integer,A: code_integer,R2: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
= ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
& ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_1919_abs__diff__less__iff,axiom,
! [X: real,A: real,R2: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
= ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X )
& ( ord_less_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_1920_abs__diff__less__iff,axiom,
! [X: rat,A: rat,R2: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
= ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
& ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_1921_abs__diff__less__iff,axiom,
! [X: int,A: int,R2: int] :
( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
= ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
& ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_1922_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_1923_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_1924_minf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ~ ( ord_less_real @ T @ X5 ) ) ).
% minf(7)
thf(fact_1925_minf_I7_J,axiom,
! [T: rat] :
? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z4 )
=> ~ ( ord_less_rat @ T @ X5 ) ) ).
% minf(7)
thf(fact_1926_minf_I7_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z4 )
=> ~ ( ord_less_num @ T @ X5 ) ) ).
% minf(7)
thf(fact_1927_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_1928_minf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ~ ( ord_less_int @ T @ X5 ) ) ).
% minf(7)
thf(fact_1929_minf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( ord_less_real @ X5 @ T ) ) ).
% minf(5)
thf(fact_1930_minf_I5_J,axiom,
! [T: rat] :
? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z4 )
=> ( ord_less_rat @ X5 @ T ) ) ).
% minf(5)
thf(fact_1931_minf_I5_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z4 )
=> ( ord_less_num @ X5 @ T ) ) ).
% minf(5)
thf(fact_1932_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_1933_minf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ord_less_int @ X5 @ T ) ) ).
% minf(5)
thf(fact_1934_minf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1935_minf_I4_J,axiom,
! [T: rat] :
? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1936_minf_I4_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1937_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1938_minf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1939_minf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1940_minf_I3_J,axiom,
! [T: rat] :
? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1941_minf_I3_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1942_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1943_minf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1944_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
( ? [Z6: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1945_minf_I2_J,axiom,
! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
( ? [Z6: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1946_minf_I2_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
( ? [Z6: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1947_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1948_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1949_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
( ? [Z6: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1950_minf_I1_J,axiom,
! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
( ? [Z6: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1951_minf_I1_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
( ? [Z6: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1952_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1953_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1954_pinf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( ord_less_real @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1955_pinf_I7_J,axiom,
! [T: rat] :
? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z4 @ X5 )
=> ( ord_less_rat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1956_pinf_I7_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ Z4 @ X5 )
=> ( ord_less_num @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1957_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1958_pinf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ord_less_int @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1959_pinf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ~ ( ord_less_real @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1960_pinf_I5_J,axiom,
! [T: rat] :
? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z4 @ X5 )
=> ~ ( ord_less_rat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1961_pinf_I5_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ Z4 @ X5 )
=> ~ ( ord_less_num @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1962_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1963_pinf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ~ ( ord_less_int @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1964_pinf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1965_pinf_I4_J,axiom,
! [T: rat] :
? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1966_pinf_I4_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1967_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1968_pinf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1969_pinf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1970_pinf_I3_J,axiom,
! [T: rat] :
? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1971_pinf_I3_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1972_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1973_pinf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1974_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
( ? [Z6: real] :
! [X3: real] :
( ( ord_less_real @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: real] :
! [X3: real] :
( ( ord_less_real @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1975_pinf_I2_J,axiom,
! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
( ? [Z6: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1976_pinf_I2_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
( ? [Z6: num] :
! [X3: num] :
( ( ord_less_num @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: num] :
! [X3: num] :
( ( ord_less_num @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1977_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1978_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1979_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
( ? [Z6: real] :
! [X3: real] :
( ( ord_less_real @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: real] :
! [X3: real] :
( ( ord_less_real @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1980_pinf_I1_J,axiom,
! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
( ? [Z6: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1981_pinf_I1_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
( ? [Z6: num] :
! [X3: num] :
( ( ord_less_num @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: num] :
! [X3: num] :
( ( ord_less_num @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1982_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1983_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1984_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_1985_lt__ex,axiom,
! [X: rat] :
? [Y3: rat] : ( ord_less_rat @ Y3 @ X ) ).
% lt_ex
thf(fact_1986_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_1987_gt__ex,axiom,
! [X: real] :
? [X_12: real] : ( ord_less_real @ X @ X_12 ) ).
% gt_ex
thf(fact_1988_gt__ex,axiom,
! [X: rat] :
? [X_12: rat] : ( ord_less_rat @ X @ X_12 ) ).
% gt_ex
thf(fact_1989_gt__ex,axiom,
! [X: nat] :
? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).
% gt_ex
thf(fact_1990_gt__ex,axiom,
! [X: int] :
? [X_12: int] : ( ord_less_int @ X @ X_12 ) ).
% gt_ex
thf(fact_1991_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z4: real] :
( ( ord_less_real @ X @ Z4 )
& ( ord_less_real @ Z4 @ Y ) ) ) ).
% dense
thf(fact_1992_dense,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ? [Z4: rat] :
( ( ord_less_rat @ X @ Z4 )
& ( ord_less_rat @ Z4 @ Y ) ) ) ).
% dense
thf(fact_1993_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_1994_less__imp__neq,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_1995_less__imp__neq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_1996_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_1997_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_1998_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_1999_order_Oasym,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ B @ A ) ) ).
% order.asym
thf(fact_2000_order_Oasym,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order.asym
thf(fact_2001_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_2002_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_2003_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_2004_ord__eq__less__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_2005_ord__eq__less__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_2006_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_2007_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_2008_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_2009_ord__less__eq__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( B = C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_2010_ord__less__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_2011_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_2012_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_2013_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_2014_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_2015_antisym__conv3,axiom,
! [Y: rat,X: rat] :
( ~ ( ord_less_rat @ Y @ X )
=> ( ( ~ ( ord_less_rat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_2016_antisym__conv3,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_num @ Y @ X )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_2017_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_2018_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_2019_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_2020_linorder__cases,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_rat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_2021_linorder__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_2022_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_2023_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_2024_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_2025_dual__order_Oasym,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ~ ( ord_less_rat @ A @ B ) ) ).
% dual_order.asym
thf(fact_2026_dual__order_Oasym,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ A @ B ) ) ).
% dual_order.asym
thf(fact_2027_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_2028_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_2029_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_2030_dual__order_Oirrefl,axiom,
! [A: rat] :
~ ( ord_less_rat @ A @ A ) ).
% dual_order.irrefl
thf(fact_2031_dual__order_Oirrefl,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% dual_order.irrefl
thf(fact_2032_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_2033_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_2034_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N4: nat] :
( ( P3 @ N4 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ~ ( P3 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_2035_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_2036_linorder__less__wlog,axiom,
! [P: rat > rat > $o,A: rat,B: rat] :
( ! [A4: rat,B4: rat] :
( ( ord_less_rat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: rat] : ( P @ A4 @ A4 )
=> ( ! [A4: rat,B4: rat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_2037_linorder__less__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A4: num,B4: num] :
( ( ord_less_num @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: num] : ( P @ A4 @ A4 )
=> ( ! [A4: num,B4: num] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_2038_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_2039_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_2040_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_2041_order_Ostrict__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_2042_order_Ostrict__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_2043_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_2044_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_2045_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_2046_not__less__iff__gr__or__eq,axiom,
! [X: rat,Y: rat] :
( ( ~ ( ord_less_rat @ X @ Y ) )
= ( ( ord_less_rat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_2047_not__less__iff__gr__or__eq,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ( ord_less_num @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_2048_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_2049_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_2050_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_2051_dual__order_Ostrict__trans,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C @ B )
=> ( ord_less_rat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_2052_dual__order_Ostrict__trans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_2053_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_2054_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_2055_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_2056_order_Ostrict__implies__not__eq,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_2057_order_Ostrict__implies__not__eq,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_2058_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_2059_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_2060_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_2061_dual__order_Ostrict__implies__not__eq,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_2062_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_2063_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_2064_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_2065_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_2066_linorder__neqE,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
=> ( ~ ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_2067_linorder__neqE,axiom,
! [X: num,Y: num] :
( ( X != Y )
=> ( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_2068_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_2069_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_2070_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_2071_order__less__asym,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_asym
thf(fact_2072_order__less__asym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_asym
thf(fact_2073_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_2074_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_2075_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_2076_linorder__neq__iff,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
= ( ( ord_less_rat @ X @ Y )
| ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_2077_linorder__neq__iff,axiom,
! [X: num,Y: num] :
( ( X != Y )
= ( ( ord_less_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_2078_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_2079_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_2080_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_2081_order__less__asym_H,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ B @ A ) ) ).
% order_less_asym'
thf(fact_2082_order__less__asym_H,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order_less_asym'
thf(fact_2083_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_2084_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_2085_order__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_2086_order__less__trans,axiom,
! [X: rat,Y: rat,Z2: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ Z2 )
=> ( ord_less_rat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_2087_order__less__trans,axiom,
! [X: num,Y: num,Z2: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z2 )
=> ( ord_less_num @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_2088_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_2089_order__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_2090_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2091_ord__eq__less__subst,axiom,
! [A: rat,F: real > rat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2092_ord__eq__less__subst,axiom,
! [A: num,F: real > num,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2093_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2094_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2095_ord__eq__less__subst,axiom,
! [A: real,F: rat > real,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2096_ord__eq__less__subst,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2097_ord__eq__less__subst,axiom,
! [A: num,F: rat > num,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2098_ord__eq__less__subst,axiom,
! [A: nat,F: rat > nat,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2099_ord__eq__less__subst,axiom,
! [A: int,F: rat > int,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2100_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2101_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > rat,C: rat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2102_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > num,C: num] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2103_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2104_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2105_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > real,C: real] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2106_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2107_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2108_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2109_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2110_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_2111_order__less__irrefl,axiom,
! [X: rat] :
~ ( ord_less_rat @ X @ X ) ).
% order_less_irrefl
thf(fact_2112_order__less__irrefl,axiom,
! [X: num] :
~ ( ord_less_num @ X @ X ) ).
% order_less_irrefl
thf(fact_2113_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_2114_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_2115_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2116_order__less__subst1,axiom,
! [A: real,F: rat > real,B: rat,C: rat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2117_order__less__subst1,axiom,
! [A: real,F: num > real,B: num,C: num] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2118_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2119_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2120_order__less__subst1,axiom,
! [A: rat,F: real > rat,B: real,C: real] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2121_order__less__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2122_order__less__subst1,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2123_order__less__subst1,axiom,
! [A: rat,F: nat > rat,B: nat,C: nat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2124_order__less__subst1,axiom,
! [A: rat,F: int > rat,B: int,C: int] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2125_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2126_order__less__subst2,axiom,
! [A: real,B: real,F: real > rat,C: rat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2127_order__less__subst2,axiom,
! [A: real,B: real,F: real > num,C: num] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2128_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2129_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2130_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > real,C: real] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2131_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2132_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2133_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2134_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2135_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_2136_order__less__not__sym,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_2137_order__less__not__sym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_2138_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_2139_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_2140_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_2141_order__less__imp__triv,axiom,
! [X: rat,Y: rat,P: $o] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_2142_order__less__imp__triv,axiom,
! [X: num,Y: num,P: $o] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_2143_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_2144_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_2145_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_2146_linorder__less__linear,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
| ( X = Y )
| ( ord_less_rat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_2147_linorder__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
| ( X = Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_2148_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_2149_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_2150_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_2151_order__less__imp__not__eq,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_2152_order__less__imp__not__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_2153_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_2154_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_2155_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_2156_order__less__imp__not__eq2,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_2157_order__less__imp__not__eq2,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_2158_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_2159_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_2160_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_2161_order__less__imp__not__less,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_2162_order__less__imp__not__less,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_2163_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_2164_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_2165_subset__code_I1_J,axiom,
! [Xs: list_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ B2 )
= ( ! [X4: complex] :
( ( member_complex @ X4 @ ( set_complex2 @ Xs ) )
=> ( member_complex @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_2166_subset__code_I1_J,axiom,
! [Xs: list_real,B2: set_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B2 )
= ( ! [X4: real] :
( ( member_real @ X4 @ ( set_real2 @ Xs ) )
=> ( member_real @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_2167_subset__code_I1_J,axiom,
! [Xs: list_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B2 )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
=> ( member_set_nat @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_2168_subset__code_I1_J,axiom,
! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B2 )
= ( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( member_VEBT_VEBT @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_2169_subset__code_I1_J,axiom,
! [Xs: list_int,B2: set_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B2 )
= ( ! [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
=> ( member_int @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_2170_subset__code_I1_J,axiom,
! [Xs: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_2171_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_2172_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_2173_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_2174_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_2175_of__int__leD,axiom,
! [N: int,X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% of_int_leD
thf(fact_2176_of__int__leD,axiom,
! [N: int,X: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).
% of_int_leD
thf(fact_2177_of__int__leD,axiom,
! [N: int,X: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% of_int_leD
thf(fact_2178_of__int__leD,axiom,
! [N: int,X: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% of_int_leD
thf(fact_2179_of__int__lessD,axiom,
! [N: int,X: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).
% of_int_lessD
thf(fact_2180_of__int__lessD,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_real @ one_one_real @ X ) ) ) ).
% of_int_lessD
thf(fact_2181_of__int__lessD,axiom,
! [N: int,X: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% of_int_lessD
thf(fact_2182_of__int__lessD,axiom,
! [N: int,X: int] :
( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_int @ one_one_int @ X ) ) ) ).
% of_int_lessD
thf(fact_2183_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_2184_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_nat @ I2 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_2185_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( 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 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_2186_minf_I8_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% minf(8)
thf(fact_2187_minf_I8_J,axiom,
! [T: rat] :
? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z4 )
=> ~ ( ord_less_eq_rat @ T @ X5 ) ) ).
% minf(8)
thf(fact_2188_minf_I8_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z4 )
=> ~ ( ord_less_eq_num @ T @ X5 ) ) ).
% minf(8)
thf(fact_2189_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_2190_minf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% minf(8)
thf(fact_2191_minf_I6_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( ord_less_eq_real @ X5 @ T ) ) ).
% minf(6)
thf(fact_2192_minf_I6_J,axiom,
! [T: rat] :
? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z4 )
=> ( ord_less_eq_rat @ X5 @ T ) ) ).
% minf(6)
thf(fact_2193_minf_I6_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z4 )
=> ( ord_less_eq_num @ X5 @ T ) ) ).
% minf(6)
thf(fact_2194_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_2195_minf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ord_less_eq_int @ X5 @ T ) ) ).
% minf(6)
thf(fact_2196_pinf_I8_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( ord_less_eq_real @ T @ X5 ) ) ).
% pinf(8)
thf(fact_2197_pinf_I8_J,axiom,
! [T: rat] :
? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z4 @ X5 )
=> ( ord_less_eq_rat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_2198_pinf_I8_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ Z4 @ X5 )
=> ( ord_less_eq_num @ T @ X5 ) ) ).
% pinf(8)
thf(fact_2199_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_2200_pinf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ord_less_eq_int @ T @ X5 ) ) ).
% pinf(8)
thf(fact_2201_pinf_I6_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% pinf(6)
thf(fact_2202_pinf_I6_J,axiom,
! [T: rat] :
? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z4 @ X5 )
=> ~ ( ord_less_eq_rat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_2203_pinf_I6_J,axiom,
! [T: num] :
? [Z4: num] :
! [X5: num] :
( ( ord_less_num @ Z4 @ X5 )
=> ~ ( ord_less_eq_num @ X5 @ T ) ) ).
% pinf(6)
thf(fact_2204_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_2205_pinf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% pinf(6)
thf(fact_2206_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_2207_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_2208_leD,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ~ ( ord_less_rat @ X @ Y ) ) ).
% leD
thf(fact_2209_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_2210_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_2211_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_2212_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_2213_leI,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_rat @ X @ Y )
=> ( ord_less_eq_rat @ Y @ X ) ) ).
% leI
thf(fact_2214_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_2215_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_2216_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_2217_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_2218_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_2219_nless__le,axiom,
! [A: rat,B: rat] :
( ( ~ ( ord_less_rat @ A @ B ) )
= ( ~ ( ord_less_eq_rat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_2220_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_2221_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_2222_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_2223_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_2224_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_2225_antisym__conv1,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_2226_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_2227_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_2228_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_2229_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_2230_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_2231_antisym__conv2,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ~ ( ord_less_rat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_2232_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_2233_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_2234_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_2235_dense__ge,axiom,
! [Z2: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_2236_dense__ge,axiom,
! [Z2: rat,Y: rat] :
( ! [X3: rat] :
( ( ord_less_rat @ Z2 @ X3 )
=> ( ord_less_eq_rat @ Y @ X3 ) )
=> ( ord_less_eq_rat @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_2237_dense__le,axiom,
! [Y: real,Z2: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_le
thf(fact_2238_dense__le,axiom,
! [Y: rat,Z2: rat] :
( ! [X3: rat] :
( ( ord_less_rat @ X3 @ Y )
=> ( ord_less_eq_rat @ X3 @ Z2 ) )
=> ( ord_less_eq_rat @ Y @ Z2 ) ) ).
% dense_le
thf(fact_2239_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_2240_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y5 )
& ~ ( ord_less_eq_set_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_2241_less__le__not__le,axiom,
( ord_less_rat
= ( ^ [X4: rat,Y5: rat] :
( ( ord_less_eq_rat @ X4 @ Y5 )
& ~ ( ord_less_eq_rat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_2242_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
& ~ ( ord_less_eq_num @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_2243_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_2244_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y5: int] :
( ( ord_less_eq_int @ X4 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_2245_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_2246_not__le__imp__less,axiom,
! [Y: rat,X: rat] :
( ~ ( ord_less_eq_rat @ Y @ X )
=> ( ord_less_rat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_2247_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_eq_num @ Y @ X )
=> ( ord_less_num @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_2248_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_2249_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_2250_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_2251_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_2252_order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_2253_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A3: num,B3: num] :
( ( ord_less_num @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_2254_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_2255_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_2256_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_2257_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_2258_order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_2259_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_2260_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_2261_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_2262_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_2263_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_2264_order_Ostrict__trans1,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_2265_order_Ostrict__trans1,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_2266_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_2267_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_2268_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_2269_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_2270_order_Ostrict__trans2,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_2271_order_Ostrict__trans2,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_2272_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_2273_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_2274_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ~ ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_2275_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_2276_order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
& ~ ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_2277_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
& ~ ( ord_less_eq_num @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_2278_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_2279_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_2280_dense__ge__bounded,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ! [W3: real] :
( ( ord_less_real @ Z2 @ W3 )
=> ( ( ord_less_real @ W3 @ X )
=> ( ord_less_eq_real @ Y @ W3 ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_2281_dense__ge__bounded,axiom,
! [Z2: rat,X: rat,Y: rat] :
( ( ord_less_rat @ Z2 @ X )
=> ( ! [W3: rat] :
( ( ord_less_rat @ Z2 @ W3 )
=> ( ( ord_less_rat @ W3 @ X )
=> ( ord_less_eq_rat @ Y @ W3 ) ) )
=> ( ord_less_eq_rat @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_2282_dense__le__bounded,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W3: real] :
( ( ord_less_real @ X @ W3 )
=> ( ( ord_less_real @ W3 @ Y )
=> ( ord_less_eq_real @ W3 @ Z2 ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_2283_dense__le__bounded,axiom,
! [X: rat,Y: rat,Z2: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ! [W3: rat] :
( ( ord_less_rat @ X @ W3 )
=> ( ( ord_less_rat @ W3 @ Y )
=> ( ord_less_eq_rat @ W3 @ Z2 ) ) )
=> ( ord_less_eq_rat @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_2284_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_2285_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( ord_less_set_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_2286_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [B3: rat,A3: rat] :
( ( ord_less_rat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_2287_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B3: num,A3: num] :
( ( ord_less_num @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_2288_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_2289_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_2290_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_2291_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_2292_dual__order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [B3: rat,A3: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_2293_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B3: num,A3: num] :
( ( ord_less_eq_num @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_2294_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_2295_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_2296_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_2297_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_2298_dual__order_Ostrict__trans1,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_rat @ C @ B )
=> ( ord_less_rat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_2299_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_2300_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_2301_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_2302_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_2303_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_2304_dual__order_Ostrict__trans2,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_rat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_2305_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_2306_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_2307_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_2308_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_2309_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ~ ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_2310_dual__order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [B3: rat,A3: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
& ~ ( ord_less_eq_rat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_2311_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B3: num,A3: num] :
( ( ord_less_eq_num @ B3 @ A3 )
& ~ ( ord_less_eq_num @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_2312_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_2313_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_2314_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_2315_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_2316_order_Ostrict__implies__order,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_2317_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_2318_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_2319_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_2320_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_2321_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_2322_dual__order_Ostrict__implies__order,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ord_less_eq_rat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_2323_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_2324_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_2325_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_2326_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_real @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_2327_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_2328_order__le__less,axiom,
( ord_less_eq_rat
= ( ^ [X4: rat,Y5: rat] :
( ( ord_less_rat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_2329_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_2330_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_2331_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y5: int] :
( ( ord_less_int @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_2332_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_2333_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_2334_order__less__le,axiom,
( ord_less_rat
= ( ^ [X4: rat,Y5: rat] :
( ( ord_less_eq_rat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_2335_order__less__le,axiom,
( ord_less_num
= ( ^ [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_2336_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_2337_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y5: int] :
( ( ord_less_eq_int @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_2338_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_2339_linorder__not__le,axiom,
! [X: rat,Y: rat] :
( ( ~ ( ord_less_eq_rat @ X @ Y ) )
= ( ord_less_rat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_2340_linorder__not__le,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X @ Y ) )
= ( ord_less_num @ Y @ X ) ) ).
% linorder_not_le
thf(fact_2341_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_2342_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_2343_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_2344_linorder__not__less,axiom,
! [X: rat,Y: rat] :
( ( ~ ( ord_less_rat @ X @ Y ) )
= ( ord_less_eq_rat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_2345_linorder__not__less,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_not_less
thf(fact_2346_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_2347_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_2348_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_2349_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_2350_order__less__imp__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_2351_order__less__imp__le,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_2352_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_2353_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_2354_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_2355_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_2356_order__le__neq__trans,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( A != B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_2357_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_2358_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_2359_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_2360_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_2361_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_2362_order__neq__le__trans,axiom,
! [A: rat,B: rat] :
( ( A != B )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_2363_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_2364_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_2365_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_2366_order__le__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_2367_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_2368_order__le__less__trans,axiom,
! [X: rat,Y: rat,Z2: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ Z2 )
=> ( ord_less_rat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_2369_order__le__less__trans,axiom,
! [X: num,Y: num,Z2: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z2 )
=> ( ord_less_num @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_2370_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_2371_order__le__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_2372_order__less__le__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_2373_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_2374_order__less__le__trans,axiom,
! [X: rat,Y: rat,Z2: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ Y @ Z2 )
=> ( ord_less_rat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_2375_order__less__le__trans,axiom,
! [X: num,Y: num,Z2: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z2 )
=> ( ord_less_num @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_2376_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_2377_order__less__le__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_2378_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2379_order__le__less__subst1,axiom,
! [A: real,F: rat > real,B: rat,C: rat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2380_order__le__less__subst1,axiom,
! [A: real,F: num > real,B: num,C: num] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2381_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2382_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2383_order__le__less__subst1,axiom,
! [A: rat,F: real > rat,B: real,C: real] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2384_order__le__less__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2385_order__le__less__subst1,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2386_order__le__less__subst1,axiom,
! [A: rat,F: nat > rat,B: nat,C: nat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2387_order__le__less__subst1,axiom,
! [A: rat,F: int > rat,B: int,C: int] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2388_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > real,C: real] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2389_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2390_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2391_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2392_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2393_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2394_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2395_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2396_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2397_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2398_order__less__le__subst1,axiom,
! [A: real,F: rat > real,B: rat,C: rat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2399_order__less__le__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2400_order__less__le__subst1,axiom,
! [A: num,F: rat > num,B: rat,C: rat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2401_order__less__le__subst1,axiom,
! [A: nat,F: rat > nat,B: rat,C: rat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2402_order__less__le__subst1,axiom,
! [A: int,F: rat > int,B: rat,C: rat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2403_order__less__le__subst1,axiom,
! [A: real,F: num > real,B: num,C: num] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2404_order__less__le__subst1,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2405_order__less__le__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2406_order__less__le__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2407_order__less__le__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2408_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_2409_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B )
& ! [X5: real] :
( ( ( ord_less_eq_real @ A @ X5 )
& ( ord_less_real @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [D5: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ D5 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D5 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_2410_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [D5: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D5 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D5 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_2411_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X5: int] :
( ( ( ord_less_eq_int @ A @ X5 )
& ( ord_less_int @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [D5: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D5 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D5 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_2412_incr__lemma,axiom,
! [D: int,Z2: int,X: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ Z2 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) ) ) ).
% incr_lemma
thf(fact_2413_decr__lemma,axiom,
! [D: int,X: int,Z2: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) @ Z2 ) ) ).
% decr_lemma
thf(fact_2414_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_2415_VEBT_Osize__gen_I2_J,axiom,
! [X21: $o,X22: $o] :
( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
= zero_zero_nat ) ).
% VEBT.size_gen(2)
thf(fact_2416_bset_I6_J,axiom,
! [D6: int,B2: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less_eq_int @ X5 @ T )
=> ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D6 ) @ T ) ) ) ) ).
% bset(6)
thf(fact_2417_bset_I8_J,axiom,
! [D6: int,T: int,B2: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less_eq_int @ T @ X5 )
=> ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D6 ) ) ) ) ) ) ).
% bset(8)
thf(fact_2418_aset_I6_J,axiom,
! [D6: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less_eq_int @ X5 @ T )
=> ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D6 ) @ T ) ) ) ) ) ).
% aset(6)
thf(fact_2419_aset_I8_J,axiom,
! [D6: int,A2: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less_eq_int @ T @ X5 )
=> ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D6 ) ) ) ) ) ).
% aset(8)
thf(fact_2420_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_2421_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_2422_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_2423_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_2424_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_2425_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_2426_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_2427_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_2428_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_2429_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_2430_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_2431_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_2432_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_2433_mult__zero__right,axiom,
! [A: rat] :
( ( times_times_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% mult_zero_right
thf(fact_2434_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_2435_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_2436_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_2437_mult__zero__left,axiom,
! [A: rat] :
( ( times_times_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% mult_zero_left
thf(fact_2438_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_2439_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_2440_div__by__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% div_by_0
thf(fact_2441_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_2442_div__by__0,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% div_by_0
thf(fact_2443_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_2444_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_2445_div__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% div_0
thf(fact_2446_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_2447_div__0,axiom,
! [A: rat] :
( ( divide_divide_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% div_0
thf(fact_2448_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_2449_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_2450_division__ring__divide__zero,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% division_ring_divide_zero
thf(fact_2451_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_2452_division__ring__divide__zero,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% division_ring_divide_zero
thf(fact_2453_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_2454_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_2455_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_2456_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_2457_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_2458_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_2459_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_2460_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_2461_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_2462_mult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% mult_1
thf(fact_2463_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_2464_mult__1,axiom,
! [A: rat] :
( ( times_times_rat @ one_one_rat @ A )
= A ) ).
% mult_1
thf(fact_2465_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_2466_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_2467_mult_Oright__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.right_neutral
thf(fact_2468_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_2469_mult_Oright__neutral,axiom,
! [A: rat] :
( ( times_times_rat @ A @ one_one_rat )
= A ) ).
% mult.right_neutral
thf(fact_2470_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_2471_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_2472_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_2473_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_2474_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_2475_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_2476_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_2477_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_2478_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_2479_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_2480_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_2481_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_2482_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_2483_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_2484_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_2485_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_2486_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_2487_div__by__1,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ one_one_complex )
= A ) ).
% div_by_1
thf(fact_2488_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_2489_div__by__1,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ one_one_rat )
= A ) ).
% div_by_1
thf(fact_2490_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_2491_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_2492_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_2493_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_2494_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_2495_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri4939895301339042750nteger @ ( times_times_nat @ M @ N ) )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% of_nat_mult
thf(fact_2496_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% of_nat_mult
thf(fact_2497_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_2498_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_2499_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_2500_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_2501_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_2502_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_2503_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_2504_inverse__divide,axiom,
! [A: real,B: real] :
( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ B @ A ) ) ).
% inverse_divide
thf(fact_2505_inverse__divide,axiom,
! [A: complex,B: complex] :
( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ B @ A ) ) ).
% inverse_divide
thf(fact_2506_inverse__divide,axiom,
! [A: rat,B: rat] :
( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ B @ A ) ) ).
% inverse_divide
thf(fact_2507_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_2508_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_2509_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_2510_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_2511_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_2512_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_2513_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_2514_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_2515_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_2516_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_2517_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_2518_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_2519_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_2520_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_2521_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_2522_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_2523_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_2524_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_2525_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_2526_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_2527_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_2528_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_2529_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_2530_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_2531_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_2532_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_2533_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_2534_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_2535_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_2536_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_2537_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_2538_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_2539_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_2540_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_2541_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_2542_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_2543_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_2544_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_2545_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_2546_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_2547_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_2548_div__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% div_self
thf(fact_2549_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_2550_div__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% div_self
thf(fact_2551_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_2552_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_2553_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_2554_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_2555_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_2556_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_2557_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_2558_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_2559_divide__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% divide_self
thf(fact_2560_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_2561_divide__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% divide_self
thf(fact_2562_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_2563_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_2564_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_2565_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_2566_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_2567_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_2568_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_2569_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_2570_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_2571_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_2572_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_2573_mult__minus1__right,axiom,
! [Z2: int] :
( ( times_times_int @ Z2 @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1_right
thf(fact_2574_mult__minus1__right,axiom,
! [Z2: real] :
( ( times_times_real @ Z2 @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z2 ) ) ).
% mult_minus1_right
thf(fact_2575_mult__minus1__right,axiom,
! [Z2: complex] :
( ( times_times_complex @ Z2 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ Z2 ) ) ).
% mult_minus1_right
thf(fact_2576_mult__minus1__right,axiom,
! [Z2: code_integer] :
( ( times_3573771949741848930nteger @ Z2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ Z2 ) ) ).
% mult_minus1_right
thf(fact_2577_mult__minus1__right,axiom,
! [Z2: rat] :
( ( times_times_rat @ Z2 @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ Z2 ) ) ).
% mult_minus1_right
thf(fact_2578_mult__minus1,axiom,
! [Z2: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z2 )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1
thf(fact_2579_mult__minus1,axiom,
! [Z2: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z2 )
= ( uminus_uminus_real @ Z2 ) ) ).
% mult_minus1
thf(fact_2580_mult__minus1,axiom,
! [Z2: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z2 )
= ( uminus1482373934393186551omplex @ Z2 ) ) ).
% mult_minus1
thf(fact_2581_mult__minus1,axiom,
! [Z2: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z2 )
= ( uminus1351360451143612070nteger @ Z2 ) ) ).
% mult_minus1
thf(fact_2582_mult__minus1,axiom,
! [Z2: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z2 )
= ( uminus_uminus_rat @ Z2 ) ) ).
% mult_minus1
thf(fact_2583_divide__minus1,axiom,
! [X: real] :
( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ X ) ) ).
% divide_minus1
thf(fact_2584_divide__minus1,axiom,
! [X: complex] :
( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ X ) ) ).
% divide_minus1
thf(fact_2585_divide__minus1,axiom,
! [X: rat] :
( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ X ) ) ).
% divide_minus1
thf(fact_2586_of__int__mult,axiom,
! [W2: int,Z2: int] :
( ( ring_18347121197199848620nteger @ ( times_times_int @ W2 @ Z2 ) )
= ( times_3573771949741848930nteger @ ( ring_18347121197199848620nteger @ W2 ) @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).
% of_int_mult
thf(fact_2587_of__int__mult,axiom,
! [W2: int,Z2: int] :
( ( ring_17405671764205052669omplex @ ( times_times_int @ W2 @ Z2 ) )
= ( times_times_complex @ ( ring_17405671764205052669omplex @ W2 ) @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% of_int_mult
thf(fact_2588_of__int__mult,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_real @ ( times_times_int @ W2 @ Z2 ) )
= ( times_times_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_mult
thf(fact_2589_of__int__mult,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_rat @ ( times_times_int @ W2 @ Z2 ) )
= ( times_times_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% of_int_mult
thf(fact_2590_of__int__mult,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_int @ ( times_times_int @ W2 @ Z2 ) )
= ( times_times_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_mult
thf(fact_2591_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_2592_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_2593_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_2594_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_2595_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_2596_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_2597_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_2598_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_2599_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_2600_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_2601_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_2602_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_2603_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_2604_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_2605_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_2606_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_2607_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_2608_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_2609_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_2610_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_2611_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_2612_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_2613_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_2614_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_2615_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_2616_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_2617_left__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
= one_one_real ) ) ).
% left_inverse
thf(fact_2618_left__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
= one_one_complex ) ) ).
% left_inverse
thf(fact_2619_left__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
= one_one_rat ) ) ).
% left_inverse
thf(fact_2620_right__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
= one_one_real ) ) ).
% right_inverse
thf(fact_2621_right__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
= one_one_complex ) ) ).
% right_inverse
thf(fact_2622_right__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
= one_one_rat ) ) ).
% right_inverse
thf(fact_2623_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_2624_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_2625_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_2626_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_2627_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_2628_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_2629_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_2630_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_2631_field__class_Ofield__divide__inverse,axiom,
( divide_divide_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B3 ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_2632_field__class_Ofield__divide__inverse,axiom,
( divide1717551699836669952omplex
= ( ^ [A3: complex,B3: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B3 ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_2633_field__class_Ofield__divide__inverse,axiom,
( divide_divide_rat
= ( ^ [A3: rat,B3: rat] : ( times_times_rat @ A3 @ ( inverse_inverse_rat @ B3 ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_2634_divide__inverse,axiom,
( divide_divide_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B3 ) ) ) ) ).
% divide_inverse
thf(fact_2635_divide__inverse,axiom,
( divide1717551699836669952omplex
= ( ^ [A3: complex,B3: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B3 ) ) ) ) ).
% divide_inverse
thf(fact_2636_divide__inverse,axiom,
( divide_divide_rat
= ( ^ [A3: rat,B3: rat] : ( times_times_rat @ A3 @ ( inverse_inverse_rat @ B3 ) ) ) ) ).
% divide_inverse
thf(fact_2637_divide__inverse__commute,axiom,
( divide_divide_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ ( inverse_inverse_real @ B3 ) @ A3 ) ) ) ).
% divide_inverse_commute
thf(fact_2638_divide__inverse__commute,axiom,
( divide1717551699836669952omplex
= ( ^ [A3: complex,B3: complex] : ( times_times_complex @ ( invers8013647133539491842omplex @ B3 ) @ A3 ) ) ) ).
% divide_inverse_commute
thf(fact_2639_divide__inverse__commute,axiom,
( divide_divide_rat
= ( ^ [A3: rat,B3: rat] : ( times_times_rat @ ( inverse_inverse_rat @ B3 ) @ A3 ) ) ) ).
% divide_inverse_commute
thf(fact_2640_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_2641_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_2642_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_2643_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_2644_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_2645_mult_Ocommute,axiom,
( times_times_rat
= ( ^ [A3: rat,B3: rat] : ( times_times_rat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_2646_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_2647_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_2648_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_2649_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_2650_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_2651_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_2652_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_2653_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_2654_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_2655_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_2656_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_2657_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_2658_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_2659_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_2660_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_2661_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_2662_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_2663_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_2664_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_2665_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_2666_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_2667_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_2668_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_2669_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_2670_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_2671_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_2672_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_2673_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_2674_frac__eq__eq,axiom,
! [Y: complex,Z2: complex,X: complex,W2: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z2 != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ X @ Y )
= ( divide1717551699836669952omplex @ W2 @ Z2 ) )
= ( ( times_times_complex @ X @ Z2 )
= ( times_times_complex @ W2 @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2675_frac__eq__eq,axiom,
! [Y: real,Z2: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W2 @ Z2 ) )
= ( ( times_times_real @ X @ Z2 )
= ( times_times_real @ W2 @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2676_frac__eq__eq,axiom,
! [Y: rat,Z2: rat,X: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z2 != zero_zero_rat )
=> ( ( ( divide_divide_rat @ X @ Y )
= ( divide_divide_rat @ W2 @ Z2 ) )
= ( ( times_times_rat @ X @ Z2 )
= ( times_times_rat @ W2 @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2677_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_2678_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_2679_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_2680_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_2681_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_2682_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_2683_mult__imp__less__div__pos,axiom,
! [Y: real,Z2: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( times_times_real @ Z2 @ Y ) @ X )
=> ( ord_less_real @ Z2 @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_2684_mult__imp__less__div__pos,axiom,
! [Y: rat,Z2: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_rat @ ( times_times_rat @ Z2 @ Y ) @ X )
=> ( ord_less_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_2685_mult__imp__div__pos__less,axiom,
! [Y: real,X: real,Z2: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ ( times_times_real @ Z2 @ Y ) )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z2 ) ) ) ).
% mult_imp_div_pos_less
thf(fact_2686_mult__imp__div__pos__less,axiom,
! [Y: rat,X: rat,Z2: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_rat @ X @ ( times_times_rat @ Z2 @ Y ) )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 ) ) ) ).
% mult_imp_div_pos_less
thf(fact_2687_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_2688_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_2689_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_2690_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_2691_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_2692_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_2693_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_2694_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_2695_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_2696_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_2697_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_2698_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_2699_divide__add__eq__iff,axiom,
! [Z2: complex,X: complex,Y: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z2 ) @ Y )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% divide_add_eq_iff
thf(fact_2700_divide__add__eq__iff,axiom,
! [Z2: real,X: real,Y: real] :
( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Z2 ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% divide_add_eq_iff
thf(fact_2701_divide__add__eq__iff,axiom,
! [Z2: rat,X: rat,Y: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% divide_add_eq_iff
thf(fact_2702_add__divide__eq__iff,axiom,
! [Z2: complex,X: complex,Y: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z2 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% add_divide_eq_iff
thf(fact_2703_add__divide__eq__iff,axiom,
! [Z2: real,X: real,Y: real] :
( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% add_divide_eq_iff
thf(fact_2704_add__divide__eq__iff,axiom,
! [Z2: rat,X: rat,Y: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z2 ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% add_divide_eq_iff
thf(fact_2705_add__num__frac,axiom,
! [Y: complex,Z2: complex,X: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ X @ Y ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_2706_add__num__frac,axiom,
! [Y: real,Z2: real,X: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ Z2 @ ( divide_divide_real @ X @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_2707_add__num__frac,axiom,
! [Y: rat,Z2: rat,X: rat] :
( ( Y != zero_zero_rat )
=> ( ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_2708_add__frac__num,axiom,
! [Y: complex,X: complex,Z2: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z2 )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_2709_add__frac__num,axiom,
! [Y: real,X: real,Z2: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z2 )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_2710_add__frac__num,axiom,
! [Y: rat,X: rat,Z2: rat] :
( ( Y != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_2711_add__frac__eq,axiom,
! [Y: complex,Z2: complex,X: complex,W2: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z2 ) ) ) ) ) ).
% add_frac_eq
thf(fact_2712_add__frac__eq,axiom,
! [Y: real,Z2: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).
% add_frac_eq
thf(fact_2713_add__frac__eq,axiom,
! [Y: rat,Z2: rat,X: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) ) ) ) ).
% add_frac_eq
thf(fact_2714_add__divide__eq__if__simps_I1_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_2715_add__divide__eq__if__simps_I1_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_2716_add__divide__eq__if__simps_I1_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_2717_add__divide__eq__if__simps_I2_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
= B ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_2718_add__divide__eq__if__simps_I2_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
= B ) )
& ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_2719_add__divide__eq__if__simps_I2_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
= B ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
= ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_2720_divide__diff__eq__iff,axiom,
! [Z2: complex,X: complex,Y: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z2 ) @ Y )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% divide_diff_eq_iff
thf(fact_2721_divide__diff__eq__iff,axiom,
! [Z2: real,X: real,Y: real] :
( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Z2 ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% divide_diff_eq_iff
thf(fact_2722_divide__diff__eq__iff,axiom,
! [Z2: rat,X: rat,Y: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y )
= ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% divide_diff_eq_iff
thf(fact_2723_diff__divide__eq__iff,axiom,
! [Z2: complex,X: complex,Y: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% diff_divide_eq_iff
thf(fact_2724_diff__divide__eq__iff,axiom,
! [Z2: real,X: real,Y: real] :
( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% diff_divide_eq_iff
thf(fact_2725_diff__divide__eq__iff,axiom,
! [Z2: rat,X: rat,Y: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z2 ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% diff_divide_eq_iff
thf(fact_2726_diff__frac__eq,axiom,
! [Y: complex,Z2: complex,X: complex,W2: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z2 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_2727_diff__frac__eq,axiom,
! [Y: real,Z2: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_2728_diff__frac__eq,axiom,
! [Y: rat,Z2: rat,X: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_2729_add__divide__eq__if__simps_I4_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_2730_add__divide__eq__if__simps_I4_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_2731_add__divide__eq__if__simps_I4_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_2732_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_2733_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_2734_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_2735_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_2736_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_2737_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_2738_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_2739_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_2740_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_2741_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_2742_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_2743_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_2744_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_2745_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_2746_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_2747_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_2748_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_2749_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_2750_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_2751_minus__divide__right,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% minus_divide_right
thf(fact_2752_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_2753_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_2754_minus__divide__divide,axiom,
! [A: complex,B: complex] :
( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ).
% minus_divide_divide
thf(fact_2755_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_2756_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_2757_minus__divide__left,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_2758_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_2759_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_2760_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_2761_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_2762_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_2763_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_2764_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_2765_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_2766_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_2767_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_2768_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_2769_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_2770_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_2771_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_2772_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_2773_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_2774_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_2775_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_2776_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_2777_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_2778_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_2779_mult_Ocomm__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.comm_neutral
thf(fact_2780_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_2781_mult_Ocomm__neutral,axiom,
! [A: rat] :
( ( times_times_rat @ A @ one_one_rat )
= A ) ).
% mult.comm_neutral
thf(fact_2782_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_2783_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_2784_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_2785_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_2786_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_2787_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_2788_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_2789_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_2790_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_2791_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_2792_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_2793_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_2794_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_2795_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_2796_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_2797_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_2798_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_2799_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_2800_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_2801_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_2802_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_2803_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_2804_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_2805_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_2806_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_2807_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_2808_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_2809_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_2810_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_2811_crossproduct__eq,axiom,
! [W2: real,Y: real,X: real,Z2: real] :
( ( ( plus_plus_real @ ( times_times_real @ W2 @ Y ) @ ( times_times_real @ X @ Z2 ) )
= ( plus_plus_real @ ( times_times_real @ W2 @ Z2 ) @ ( times_times_real @ X @ Y ) ) )
= ( ( W2 = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_2812_crossproduct__eq,axiom,
! [W2: rat,Y: rat,X: rat,Z2: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ W2 @ Y ) @ ( times_times_rat @ X @ Z2 ) )
= ( plus_plus_rat @ ( times_times_rat @ W2 @ Z2 ) @ ( times_times_rat @ X @ Y ) ) )
= ( ( W2 = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_2813_crossproduct__eq,axiom,
! [W2: nat,Y: nat,X: nat,Z2: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y ) @ ( times_times_nat @ X @ Z2 ) )
= ( plus_plus_nat @ ( times_times_nat @ W2 @ Z2 ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W2 = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_2814_crossproduct__eq,axiom,
! [W2: int,Y: int,X: int,Z2: int] :
( ( ( plus_plus_int @ ( times_times_int @ W2 @ Y ) @ ( times_times_int @ X @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W2 @ Z2 ) @ ( times_times_int @ X @ Y ) ) )
= ( ( W2 = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_2815_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_2816_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_2817_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_2818_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_2819_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_2820_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_2821_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_2822_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_2823_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_2824_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_2825_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_2826_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_2827_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_2828_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_2829_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_2830_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_2831_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_2832_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_2833_inf__period_I2_J,axiom,
! [P: real > $o,D6: real,Q: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D6 ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D6 ) ) ) )
=> ! [X5: real,K4: real] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D6 ) ) )
| ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D6 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_2834_inf__period_I2_J,axiom,
! [P: rat > $o,D6: rat,Q: rat > $o] :
( ! [X3: rat,K2: rat] :
( ( P @ X3 )
= ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D6 ) ) ) )
=> ( ! [X3: rat,K2: rat] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D6 ) ) ) )
=> ! [X5: rat,K4: rat] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D6 ) ) )
| ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D6 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_2835_inf__period_I2_J,axiom,
! [P: int > $o,D6: int,Q: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D6 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D6 ) ) ) )
=> ! [X5: int,K4: int] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D6 ) ) )
| ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D6 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_2836_inf__period_I1_J,axiom,
! [P: real > $o,D6: real,Q: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D6 ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D6 ) ) ) )
=> ! [X5: real,K4: real] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D6 ) ) )
& ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D6 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_2837_inf__period_I1_J,axiom,
! [P: rat > $o,D6: rat,Q: rat > $o] :
( ! [X3: rat,K2: rat] :
( ( P @ X3 )
= ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D6 ) ) ) )
=> ( ! [X3: rat,K2: rat] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D6 ) ) ) )
=> ! [X5: rat,K4: rat] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D6 ) ) )
& ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D6 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_2838_inf__period_I1_J,axiom,
! [P: int > $o,D6: int,Q: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D6 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D6 ) ) ) )
=> ! [X5: int,K4: int] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D6 ) ) )
& ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D6 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_2839_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_2840_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_2841_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_2842_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_2843_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_2844_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_2845_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_2846_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_2847_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_2848_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_2849_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_2850_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_2851_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_2852_mult__of__nat__commute,axiom,
! [X: nat,Y: code_integer] :
( ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ X ) @ Y )
= ( times_3573771949741848930nteger @ Y @ ( semiri4939895301339042750nteger @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_2853_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_2854_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_2855_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_2856_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_2857_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_2858_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_2859_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_2860_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_2861_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_2862_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_2863_mult__of__int__commute,axiom,
! [X: int,Y: code_integer] :
( ( times_3573771949741848930nteger @ ( ring_18347121197199848620nteger @ X ) @ Y )
= ( times_3573771949741848930nteger @ Y @ ( ring_18347121197199848620nteger @ X ) ) ) ).
% mult_of_int_commute
thf(fact_2864_mult__of__int__commute,axiom,
! [X: int,Y: complex] :
( ( times_times_complex @ ( ring_17405671764205052669omplex @ X ) @ Y )
= ( times_times_complex @ Y @ ( ring_17405671764205052669omplex @ X ) ) ) ).
% mult_of_int_commute
thf(fact_2865_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_2866_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_2867_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_2868_int__distrib_I2_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_2869_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W2 )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(1)
thf(fact_2870_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_2871_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_2872_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_2873_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_2874_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_2875_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_2876_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_2877_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_2878_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_2879_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_2880_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_2881_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_2882_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_2883_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_2884_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_2885_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_2886_mult__imp__div__pos__le,axiom,
! [Y: real,X: real,Z2: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ ( times_times_real @ Z2 @ Y ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z2 ) ) ) ).
% mult_imp_div_pos_le
thf(fact_2887_mult__imp__div__pos__le,axiom,
! [Y: rat,X: rat,Z2: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z2 @ Y ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 ) ) ) ).
% mult_imp_div_pos_le
thf(fact_2888_mult__imp__le__div__pos,axiom,
! [Y: real,Z2: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ Y ) @ X )
=> ( ord_less_eq_real @ Z2 @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_2889_mult__imp__le__div__pos,axiom,
! [Y: rat,Z2: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ Y ) @ X )
=> ( ord_less_eq_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_2890_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_2891_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_2892_frac__le__eq,axiom,
! [Y: real,Z2: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
= ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% frac_le_eq
thf(fact_2893_frac__le__eq,axiom,
! [Y: rat,Z2: rat,X: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z2 != zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
= ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% frac_le_eq
thf(fact_2894_frac__less__eq,axiom,
! [Y: real,Z2: real,X: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
= ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% frac_less_eq
thf(fact_2895_frac__less__eq,axiom,
! [Y: rat,Z2: rat,X: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z2 != zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
= ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% frac_less_eq
thf(fact_2896_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_2897_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_2898_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_2899_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_2900_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_2901_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_2902_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_2903_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_2904_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_2905_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_2906_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_2907_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_2908_add__divide__eq__if__simps_I3_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
= B ) )
& ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_2909_add__divide__eq__if__simps_I3_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
= B ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_2910_add__divide__eq__if__simps_I3_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
= B ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
= ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_2911_minus__divide__add__eq__iff,axiom,
! [Z2: real,X: real,Y: real] :
( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z2 ) ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_2912_minus__divide__add__eq__iff,axiom,
! [Z2: complex,X: complex,Y: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z2 ) ) @ Y )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_2913_minus__divide__add__eq__iff,axiom,
! [Z2: rat,X: rat,Y: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y )
= ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_2914_minus__divide__diff__eq__iff,axiom,
! [Z2: real,X: real,Y: real] :
( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z2 ) ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_2915_minus__divide__diff__eq__iff,axiom,
! [Z2: complex,X: complex,Y: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z2 ) ) @ Y )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_2916_minus__divide__diff__eq__iff,axiom,
! [Z2: rat,X: rat,Y: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y )
= ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_2917_add__divide__eq__if__simps_I5_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_2918_add__divide__eq__if__simps_I5_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
= ( uminus1482373934393186551omplex @ B ) ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_2919_add__divide__eq__if__simps_I5_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
= ( uminus_uminus_rat @ B ) ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
= ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_2920_add__divide__eq__if__simps_I6_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_2921_add__divide__eq__if__simps_I6_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
= ( uminus1482373934393186551omplex @ B ) ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_2922_add__divide__eq__if__simps_I6_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
= ( uminus_uminus_rat @ B ) ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
= ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_2923_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_2924_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_2925_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_2926_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_2927_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_2928_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_2929_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_2930_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_2931_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_2932_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_2933_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_2934_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_2935_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_2936_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_2937_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_2938_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_2939_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_2940_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_2941_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_2942_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_2943_scaling__mono,axiom,
! [U: real,V: real,R2: real,S: real] :
( ( ord_less_eq_real @ U @ V )
=> ( ( ord_less_eq_real @ zero_zero_real @ R2 )
=> ( ( ord_less_eq_real @ R2 @ S )
=> ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% scaling_mono
thf(fact_2944_scaling__mono,axiom,
! [U: rat,V: rat,R2: rat,S: rat] :
( ( ord_less_eq_rat @ U @ V )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
=> ( ( ord_less_eq_rat @ R2 @ S )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% scaling_mono
thf(fact_2945_periodic__finite__ex,axiom,
! [D: int,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X4: int] :
( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
& ( P @ X4 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_2946_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_2947_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_2948_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_2949_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_2950_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_2951_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_2952_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_2953_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_2954_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_2955_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_2956_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_2957_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_2958_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_2959_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_2960_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_2961_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_2962_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_2963_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_2964_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N
= ( 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 )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_zdiv
thf(fact_2965_int__div__neg__eq,axiom,
! [A: int,B: int,Q4: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( divide_divide_int @ A @ B )
= Q4 ) ) ) ) ).
% int_div_neg_eq
thf(fact_2966_int__div__pos__eq,axiom,
! [A: int,B: int,Q4: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( divide_divide_int @ A @ B )
= Q4 ) ) ) ) ).
% int_div_pos_eq
thf(fact_2967_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_2968_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_2969_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_2970_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_2971_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_2972_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_2973_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_2974_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_2975_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_2976_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_2977_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_2978_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_2979_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_2980_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_2981_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_2982_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_2983_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_2984_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_2985_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_2986_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_2987_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_2988_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_2989_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_2990_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_2991_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_2992_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_2993_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_2994_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_2995_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_2996_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_2997_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_2998_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_2999_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_3000_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_3001_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_3002_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_3003_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_3004_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_3005_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_3006_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_3007_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_3008_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_3009_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_3010_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_3011_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_3012_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_3013_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_3014_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_3015_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_3016_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_3017_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_3018_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_3019_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_3020_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_3021_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_3022_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_3023_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_3024_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_3025_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_3026_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_3027_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_3028_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_3029_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_3030_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_3031_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_3032_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_3033_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_3034_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_3035_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_3036_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_3037_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_3038_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_3039_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_3040_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_3041_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_3042_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_3043_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_3044_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_3045_zero__le__square,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% zero_le_square
thf(fact_3046_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_3047_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_3048_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_3049_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_3050_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_3051_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_3052_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_3053_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_3054_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_3055_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_3056_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_3057_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_3058_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_3059_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_3060_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_3061_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_3062_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_3063_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_3064_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_3065_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_3066_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_3067_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_3068_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_3069_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_3070_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_3071_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_3072_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_3073_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_3074_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_3075_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_3076_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_3077_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_3078_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_3079_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_3080_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_3081_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_3082_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_3083_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_3084_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_3085_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_3086_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_3087_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_3088_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_3089_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_3090_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_3091_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_3092_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_3093_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_3094_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_3095_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_3096_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_3097_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_3098_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_3099_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_3100_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_3101_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_3102_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_3103_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_3104_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_3105_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_3106_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_3107_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_3108_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_3109_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_3110_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_3111_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_3112_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_3113_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_3114_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_3115_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_3116_not__square__less__zero,axiom,
! [A: rat] :
~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% not_square_less_zero
thf(fact_3117_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_3118_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_3119_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_3120_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_3121_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_3122_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_3123_add__scale__eq__noteq,axiom,
! [R2: real,A: real,B: real,C: real,D: real] :
( ( R2 != zero_zero_real )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_3124_add__scale__eq__noteq,axiom,
! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
( ( R2 != zero_zero_rat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
!= ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_3125_add__scale__eq__noteq,axiom,
! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_3126_add__scale__eq__noteq,axiom,
! [R2: int,A: int,B: int,C: int,D: int] :
( ( R2 != zero_zero_int )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_3127_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_3128_less__1__mult,axiom,
! [M: rat,N: rat] :
( ( ord_less_rat @ one_one_rat @ M )
=> ( ( ord_less_rat @ one_one_rat @ N )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_3129_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_3130_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_3131_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_3132_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_3133_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_3134_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_3135_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_3136_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_3137_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_3138_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_3139_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_3140_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_3141_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_3142_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_3143_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_3144_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_3145_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_3146_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_3147_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_3148_inverse__eq__divide,axiom,
( inverse_inverse_real
= ( divide_divide_real @ one_one_real ) ) ).
% inverse_eq_divide
thf(fact_3149_inverse__eq__divide,axiom,
( invers8013647133539491842omplex
= ( divide1717551699836669952omplex @ one_one_complex ) ) ).
% inverse_eq_divide
thf(fact_3150_inverse__eq__divide,axiom,
( inverse_inverse_rat
= ( divide_divide_rat @ one_one_rat ) ) ).
% inverse_eq_divide
thf(fact_3151_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_3152_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_3153_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_3154_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_3155_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_3156_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_3157_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_3158_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_3159_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_3160_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_3161_inverse__unique,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= one_one_real )
=> ( ( inverse_inverse_real @ A )
= B ) ) ).
% inverse_unique
thf(fact_3162_inverse__unique,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= one_one_complex )
=> ( ( invers8013647133539491842omplex @ A )
= B ) ) ).
% inverse_unique
thf(fact_3163_inverse__unique,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= one_one_rat )
=> ( ( inverse_inverse_rat @ A )
= B ) ) ).
% inverse_unique
thf(fact_3164_mult__inverse__of__nat__commute,axiom,
! [Xa3: nat,X: real] :
( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa3 ) ) @ X )
= ( times_times_real @ X @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa3 ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_3165_mult__inverse__of__nat__commute,axiom,
! [Xa3: nat,X: complex] :
( ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa3 ) ) @ X )
= ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa3 ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_3166_mult__inverse__of__nat__commute,axiom,
! [Xa3: nat,X: rat] :
( ( times_times_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa3 ) ) @ X )
= ( times_times_rat @ X @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa3 ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_3167_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_3168_mult__inverse__of__int__commute,axiom,
! [Xa3: int,X: real] :
( ( times_times_real @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa3 ) ) @ X )
= ( times_times_real @ X @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa3 ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_3169_mult__inverse__of__int__commute,axiom,
! [Xa3: int,X: complex] :
( ( times_times_complex @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa3 ) ) @ X )
= ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa3 ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_3170_mult__inverse__of__int__commute,axiom,
! [Xa3: int,X: rat] :
( ( times_times_rat @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa3 ) ) @ X )
= ( times_times_rat @ X @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa3 ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_3171_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_3172_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_3173_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_3174_ceiling__divide__upper,axiom,
! [Q4: real,P5: real] :
( ( ord_less_real @ zero_zero_real @ Q4 )
=> ( ord_less_eq_real @ P5 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P5 @ Q4 ) ) ) @ Q4 ) ) ) ).
% ceiling_divide_upper
thf(fact_3175_ceiling__divide__upper,axiom,
! [Q4: rat,P5: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q4 )
=> ( ord_less_eq_rat @ P5 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P5 @ Q4 ) ) ) @ Q4 ) ) ) ).
% ceiling_divide_upper
thf(fact_3176_cpmi,axiom,
! [D6: int,P: int > $o,P4: int > $o,B2: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B2 )
=> ( X3
!= ( plus_plus_int @ Xb2 @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D6 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D6 ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X4: int] :
( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
& ( P4 @ X4 ) )
| ? [X4: int] :
( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
& ? [Y5: int] :
( ( member_int @ Y5 @ B2 )
& ( P @ ( plus_plus_int @ Y5 @ X4 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_3177_cppi,axiom,
! [D6: int,P: int > $o,P4: int > $o,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X3
!= ( minus_minus_int @ Xb2 @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D6 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D6 ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X4: int] :
( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
& ( P4 @ X4 ) )
| ? [X4: int] :
( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
& ? [Y5: int] :
( ( member_int @ Y5 @ A2 )
& ( P @ ( minus_minus_int @ Y5 @ X4 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_3178_ceiling__divide__lower,axiom,
! [Q4: real,P5: real] :
( ( ord_less_real @ zero_zero_real @ Q4 )
=> ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P5 @ Q4 ) ) ) @ one_one_real ) @ Q4 ) @ P5 ) ) ).
% ceiling_divide_lower
thf(fact_3179_ceiling__divide__lower,axiom,
! [Q4: rat,P5: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q4 )
=> ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P5 @ Q4 ) ) ) @ one_one_rat ) @ Q4 ) @ P5 ) ) ).
% ceiling_divide_lower
thf(fact_3180_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_3181_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_3182_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_3183_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_3184_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_3185_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_3186_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_3187_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_3188_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_3189_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_3190_frac__less2,axiom,
! [X: real,Y: real,W2: real,Z2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_real @ W2 @ Z2 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_less2
thf(fact_3191_frac__less2,axiom,
! [X: rat,Y: rat,W2: rat,Z2: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_rat @ W2 @ Z2 )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% frac_less2
thf(fact_3192_frac__less,axiom,
! [X: real,Y: real,W2: real,Z2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z2 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_less
thf(fact_3193_frac__less,axiom,
! [X: rat,Y: rat,W2: rat,Z2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_eq_rat @ W2 @ Z2 )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% frac_less
thf(fact_3194_frac__le,axiom,
! [Y: real,X: real,W2: real,Z2: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_le
thf(fact_3195_frac__le,axiom,
! [Y: rat,X: rat,W2: rat,Z2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_eq_rat @ W2 @ Z2 )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% frac_le
thf(fact_3196_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_3197_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_3198_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_3199_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_3200_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_3201_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_3202_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_3203_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_3204_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_3205_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_3206_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_3207_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_3208_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_3209_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_3210_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_3211_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_3212_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_3213_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_3214_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_3215_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_3216_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_3217_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_3218_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_3219_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_3220_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_3221_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_3222_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_3223_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_3224_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_3225_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_3226_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_3227_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_3228_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_3229_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_3230_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_3231_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_3232_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_3233_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_3234_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_3235_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_3236_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_3237_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_3238_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_3239_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_3240_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_3241_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_3242_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_3243_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_3244_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_3245_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_3246_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_3247_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_3248_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_3249_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_3250_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_3251_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_3252_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_3253_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_3254_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_3255_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_3256_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_3257_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_3258_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_3259_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_3260_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_3261_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_3262_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_3263_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_3264_aset_I2_J,axiom,
! [D6: int,A2: set_int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X3
!= ( minus_minus_int @ Xb2 @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D6 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X3
!= ( minus_minus_int @ Xb2 @ Xa ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus_int @ X3 @ D6 ) ) ) )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
=> ( ( P @ ( plus_plus_int @ X5 @ D6 ) )
| ( Q @ ( plus_plus_int @ X5 @ D6 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_3265_aset_I1_J,axiom,
! [D6: int,A2: set_int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X3
!= ( minus_minus_int @ Xb2 @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D6 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X3
!= ( minus_minus_int @ Xb2 @ Xa ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus_int @ X3 @ D6 ) ) ) )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
=> ( ( P @ ( plus_plus_int @ X5 @ D6 ) )
& ( Q @ ( plus_plus_int @ X5 @ D6 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_3266_bset_I2_J,axiom,
! [D6: int,B2: set_int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B2 )
=> ( X3
!= ( plus_plus_int @ Xb2 @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D6 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B2 )
=> ( X3
!= ( plus_plus_int @ Xb2 @ Xa ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus_int @ X3 @ D6 ) ) ) )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
=> ( ( P @ ( minus_minus_int @ X5 @ D6 ) )
| ( Q @ ( minus_minus_int @ X5 @ D6 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_3267_bset_I1_J,axiom,
! [D6: int,B2: set_int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B2 )
=> ( X3
!= ( plus_plus_int @ Xb2 @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D6 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B2 )
=> ( X3
!= ( plus_plus_int @ Xb2 @ Xa ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus_int @ X3 @ D6 ) ) ) )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
=> ( ( P @ ( minus_minus_int @ X5 @ D6 ) )
& ( Q @ ( minus_minus_int @ X5 @ D6 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_3268_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_3269_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_3270_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_3271_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_3272_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_3273_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_3274_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_3275_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_3276_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_3277_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_3278_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_3279_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_3280_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_3281_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_3282_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_3283_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_3284_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_3285_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_3286_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_3287_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_3288_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_3289_nonzero__inverse__eq__divide,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ A )
= ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_3290_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_3291_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_3292_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_3293_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_3294_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_3295_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_3296_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_3297_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_3298_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_3299_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_3300_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_3301_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_3302_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_3303_ex__less__of__nat__mult,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N2: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_3304_ex__less__of__nat__mult,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ? [N2: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_3305_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_3306_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_3307_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_3308_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_3309_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_3310_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_3311_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_3312_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_3313_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_3314_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_3315_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_3316_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_3317_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_3318_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_3319_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_3320_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_3321_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_3322_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_3323_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_3324_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_3325_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_3326_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_3327_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_3328_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_3329_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_3330_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_3331_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_3332_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_3333_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_3334_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_3335_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_3336_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_3337_zdiv__mono2__neg,axiom,
! [A: int,B5: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B5 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_3338_zdiv__mono1__neg,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_3339_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_3340_zdiv__mono2,axiom,
! [A: int,B5: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B5 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_3341_zdiv__mono1,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A5 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_3342_ex__gt__or__lt,axiom,
! [A: real] :
? [B4: real] :
( ( ord_less_real @ A @ B4 )
| ( ord_less_real @ B4 @ A ) ) ).
% ex_gt_or_lt
thf(fact_3343_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_3344_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_3345_plusinfinity,axiom,
! [D: int,P4: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [X_1: int] : ( P4 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_3346_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_3347_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_3348_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_3349_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_3350_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_3351_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_3352_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_3353_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_3354_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_3355_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_3356_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_3357_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_3358_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_3359_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_3360_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_3361_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_3362_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_3363_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_3364_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_3365_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_3366_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_3367_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_3368_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_3369_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_3370_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_3371_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_3372_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_3373_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_3374_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_3375_field__le__mult__one__interval,axiom,
! [X: real,Y: real] :
( ! [Z4: real] :
( ( ord_less_real @ zero_zero_real @ Z4 )
=> ( ( ord_less_real @ Z4 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z4 @ X ) @ Y ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_3376_field__le__mult__one__interval,axiom,
! [X: rat,Y: rat] :
( ! [Z4: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z4 )
=> ( ( ord_less_rat @ Z4 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Z4 @ X ) @ Y ) ) )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_3377_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_3378_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_3379_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_3380_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_3381_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_3382_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_3383_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_3384_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_3385_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_3386_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_3387_verit__less__mono__div__int2,axiom,
! [A2: int,B2: int,N: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_3388_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_3389_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_3390_q__pos__lemma,axiom,
! [B5: int,Q5: int,R3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R3 ) )
=> ( ( ord_less_int @ R3 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% q_pos_lemma
thf(fact_3391_zdiv__mono2__lemma,axiom,
! [B: int,Q4: int,R2: int,B5: int,Q5: int,R3: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
= ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R3 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R3 ) )
=> ( ( ord_less_int @ R3 @ B5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ Q4 @ Q5 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_3392_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_3393_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q4: int,R2: int,B5: int,Q5: int,R3: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
= ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R3 ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R3 ) @ zero_zero_int )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R3 )
=> ( ( ord_less_int @ zero_zero_int @ B5 )
=> ( ( ord_less_eq_int @ B5 @ B )
=> ( ord_less_eq_int @ Q5 @ Q4 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_3394_unique__quotient__lemma,axiom,
! [B: int,Q5: int,R3: int,Q4: int,R2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R3 )
=> ( ( ord_less_int @ R3 @ B )
=> ( ( ord_less_int @ R2 @ B )
=> ( ord_less_eq_int @ Q5 @ Q4 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_3395_unique__quotient__lemma__neg,axiom,
! [B: int,Q5: int,R3: int,Q4: int,R2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( ord_less_int @ B @ R3 )
=> ( ord_less_eq_int @ Q4 @ Q5 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_3396_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_3397_nat__approx__posE,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ~ ! [N2: nat] :
~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ E2 ) ) ).
% nat_approx_posE
thf(fact_3398_nat__approx__posE,axiom,
! [E2: rat] :
( ( ord_less_rat @ zero_zero_rat @ E2 )
=> ~ ! [N2: nat] :
~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ E2 ) ) ).
% nat_approx_posE
thf(fact_3399_aset_I7_J,axiom,
! [D6: int,A2: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less_int @ T @ X5 )
=> ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D6 ) ) ) ) ) ).
% aset(7)
thf(fact_3400_aset_I5_J,axiom,
! [D6: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ T @ A2 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less_int @ X5 @ T )
=> ( ord_less_int @ ( plus_plus_int @ X5 @ D6 ) @ T ) ) ) ) ) ).
% aset(5)
thf(fact_3401_aset_I4_J,axiom,
! [D6: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ T @ A2 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( X5 != T )
=> ( ( plus_plus_int @ X5 @ D6 )
!= T ) ) ) ) ) ).
% aset(4)
thf(fact_3402_aset_I3_J,axiom,
! [D6: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( X5 = T )
=> ( ( plus_plus_int @ X5 @ D6 )
= T ) ) ) ) ) ).
% aset(3)
thf(fact_3403_bset_I7_J,axiom,
! [D6: int,T: int,B2: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ T @ B2 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less_int @ T @ X5 )
=> ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D6 ) ) ) ) ) ) ).
% bset(7)
thf(fact_3404_bset_I5_J,axiom,
! [D6: int,B2: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( ord_less_int @ X5 @ T )
=> ( ord_less_int @ ( minus_minus_int @ X5 @ D6 ) @ T ) ) ) ) ).
% bset(5)
thf(fact_3405_bset_I4_J,axiom,
! [D6: int,T: int,B2: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ T @ B2 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( X5 != T )
=> ( ( minus_minus_int @ X5 @ D6 )
!= T ) ) ) ) ) ).
% bset(4)
thf(fact_3406_bset_I3_J,axiom,
! [D6: int,T: int,B2: set_int] :
( ( ord_less_int @ zero_zero_int @ D6 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( X5 = T )
=> ( ( minus_minus_int @ X5 @ D6 )
= T ) ) ) ) ) ).
% bset(3)
thf(fact_3407_inverse__of__nat__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( N != zero_zero_nat )
=> ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_3408_inverse__of__nat__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( N != zero_zero_nat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_3409_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_3410_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_3411_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_3412_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_3413_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_3414_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_3415_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_3416_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_3417_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_3418_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_3419_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_3420_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_3421_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_3422_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_3423_div__minus1__right,axiom,
! [A: code_integer] :
( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ A ) ) ).
% div_minus1_right
thf(fact_3424_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_3425_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_3426_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_3427_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_3428_div__minus__minus,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
= ( divide6298287555418463151nteger @ A @ B ) ) ).
% div_minus_minus
thf(fact_3429_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_3430_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_3431_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_3432_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_3433_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_3434_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_3435_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_3436_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_3437_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_3438_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_3439_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_3440_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_3441_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_3442_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_3443_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_3444_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_3445_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_3446_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_3447_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_3448_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_3449_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_3450_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_3451_divide__real__def,axiom,
( divide_divide_real
= ( ^ [X4: real,Y5: real] : ( times_times_real @ X4 @ ( inverse_inverse_real @ Y5 ) ) ) ) ).
% divide_real_def
thf(fact_3452_div__less__iff__less__mult,axiom,
! [Q4: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q4 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q4 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q4 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_3453_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_3454_div__nat__eqI,axiom,
! [N: nat,Q4: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q4 ) ) ) ).
% div_nat_eqI
thf(fact_3455_less__eq__div__iff__mult__less__eq,axiom,
! [Q4: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q4 )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q4 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q4 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_3456_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I3: nat,J3: nat] :
( ( ord_less_nat @ J3 @ N )
=> ( ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) )
=> ( P @ I3 ) ) ) ) ) ) ).
% split_div
thf(fact_3457_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_3458_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_3459_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_3460_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_3461_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_3462_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_3463_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_3464_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_3465_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_3466_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_3467_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_3468_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_3469_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_3470_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_3471_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_3472_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_3473_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_3474_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q6: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q6 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q6 ) ) )
& ( P @ Q6 ) ) ) ) ).
% split_div'
thf(fact_3475_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_3476_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M4: nat] :
( ( ord_less_eq_nat @ M4 @ N )
& ( P @ M4 ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less
thf(fact_3477_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M4: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( P @ M4 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less
thf(fact_3478_real__of__nat__div3,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).
% real_of_nat_div3
thf(fact_3479_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_3480_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_3481_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_3482_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_3483_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_3484_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_3485_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_3486_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_3487_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_3488_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_3489_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_3490_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_3491_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_3492_div__if,axiom,
( divide_divide_nat
= ( ^ [M4: nat,N4: nat] :
( if_nat
@ ( ( ord_less_nat @ M4 @ N4 )
| ( N4 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M4 @ N4 ) @ N4 ) ) ) ) ) ).
% div_if
thf(fact_3493_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_3494_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_3495_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_3496_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_3497_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_3498_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y6: real] :
? [N2: nat] : ( ord_less_real @ Y6 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_3499_real__of__int__div4,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).
% real_of_int_div4
thf(fact_3500_nat__abs__mult__distrib,axiom,
! [W2: int,Z2: int] :
( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W2 @ Z2 ) ) )
= ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W2 ) ) @ ( nat2 @ ( abs_abs_int @ Z2 ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_3501_le__div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ M @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_3502_div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% div_geq
thf(fact_3503_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_3504_nat__mult__distrib,axiom,
! [Z2: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( nat2 @ ( times_times_int @ Z2 @ Z5 ) )
= ( times_times_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ).
% nat_mult_distrib
thf(fact_3505_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_3506_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 )
=> ( ! [M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_3507_nat__mult__distrib__neg,axiom,
! [Z2: int,Z5: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z2 @ Z5 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z5 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_3508_real__of__int__div2,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) ) ).
% real_of_int_div2
thf(fact_3509_real__of__int__div3,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) @ one_one_real ) ).
% real_of_int_div3
thf(fact_3510_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_3511_div__minus__right,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% div_minus_right
thf(fact_3512_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_3513_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_3514_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri4939895301339042750nteger @ ( divide_divide_nat @ M @ N ) )
= ( divide6298287555418463151nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_3515_div__mult2__eq_H,axiom,
! [A: int,M: nat,N: nat] :
( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% div_mult2_eq'
thf(fact_3516_div__mult2__eq_H,axiom,
! [A: nat,M: nat,N: nat] :
( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% div_mult2_eq'
thf(fact_3517_div__mult2__eq_H,axiom,
! [A: code_integer,M: nat,N: nat] :
( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
= ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% div_mult2_eq'
thf(fact_3518_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_3519_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_3520_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_3521_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_3522_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_3523_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_3524_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_3525_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_3526_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_3527_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_3528_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: 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 ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_3529_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: 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 ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_3530_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_3531_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_3532_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_3533_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_3534_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_3535_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_3536_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_3537_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_3538_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_3539_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_3540_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_3541_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_3542_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_3543_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_3544_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_3545_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_3546_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_3547_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_3548_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_3549_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_3550_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_3551_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: 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 ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_3552_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: 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 ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_3553_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: 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 ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_3554_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: 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 ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_3555_mult__le__cancel__iff2,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ ( times_times_real @ Z2 @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3556_mult__le__cancel__iff2,axiom,
! [Z2: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z2 )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ ( times_times_rat @ Z2 @ Y ) )
= ( ord_less_eq_rat @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3557_mult__le__cancel__iff2,axiom,
! [Z2: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X ) @ ( times_times_int @ Z2 @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3558_mult__le__cancel__iff1,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3559_mult__le__cancel__iff1,axiom,
! [Z2: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z2 )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ Z2 ) )
= ( ord_less_eq_rat @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3560_mult__le__cancel__iff1,axiom,
! [Z2: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3561_mult__less__iff1,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_3562_mult__less__iff1,axiom,
! [Z2: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z2 )
=> ( ( ord_less_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ Z2 ) )
= ( ord_less_rat @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_3563_mult__less__iff1,axiom,
! [Z2: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_3564_Compl__anti__mono,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B2 ) @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_3565_Compl__subset__Compl__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B2 ) )
= ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_3566_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_3567_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_3568_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_3569_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_3570_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_3571_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_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_3572_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_3573_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_3574_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_3575_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_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_3576_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_3577_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_3578_ComplI,axiom,
! [C: complex,A2: set_complex] :
( ~ ( member_complex @ C @ A2 )
=> ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) ) ) ).
% ComplI
thf(fact_3579_ComplI,axiom,
! [C: real,A2: set_real] :
( ~ ( member_real @ C @ A2 )
=> ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) ) ) ).
% ComplI
thf(fact_3580_ComplI,axiom,
! [C: set_nat,A2: set_set_nat] :
( ~ ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).
% ComplI
thf(fact_3581_ComplI,axiom,
! [C: nat,A2: set_nat] :
( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% ComplI
thf(fact_3582_ComplI,axiom,
! [C: int,A2: set_int] :
( ~ ( member_int @ C @ A2 )
=> ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% ComplI
thf(fact_3583_Compl__iff,axiom,
! [C: complex,A2: set_complex] :
( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) )
= ( ~ ( member_complex @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_3584_Compl__iff,axiom,
! [C: real,A2: set_real] :
( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
= ( ~ ( member_real @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_3585_Compl__iff,axiom,
! [C: set_nat,A2: set_set_nat] :
( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
= ( ~ ( member_set_nat @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_3586_Compl__iff,axiom,
! [C: nat,A2: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( ~ ( member_nat @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_3587_Compl__iff,axiom,
! [C: int,A2: set_int] :
( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
= ( ~ ( member_int @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_3588_power__one,axiom,
! [N: nat] :
( ( power_power_rat @ one_one_rat @ N )
= one_one_rat ) ).
% power_one
thf(fact_3589_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_3590_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_3591_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_3592_power__one,axiom,
! [N: nat] :
( ( power_power_complex @ one_one_complex @ N )
= one_one_complex ) ).
% power_one
thf(fact_3593_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ( semiri8010041392384452111omplex @ X )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W2 ) )
= ( X
= ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3594_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
= ( X
= ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3595_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ( semiri5074537144036343181t_real @ X )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
= ( X
= ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3596_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
= ( X
= ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3597_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ( semiri4939895301339042750nteger @ X )
= ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) )
= ( X
= ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3598_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ( semiri681578069525770553at_rat @ X )
= ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
= ( X
= ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3599_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W2 )
= ( semiri8010041392384452111omplex @ X ) )
= ( ( power_power_nat @ B @ W2 )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3600_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B @ W2 )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3601_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 )
= ( semiri5074537144036343181t_real @ X ) )
= ( ( power_power_nat @ B @ W2 )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3602_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B @ W2 )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3603_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 )
= ( semiri4939895301339042750nteger @ X ) )
= ( ( power_power_nat @ B @ W2 )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3604_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 )
= ( semiri681578069525770553at_rat @ X ) )
= ( ( power_power_nat @ B @ W2 )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3605_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% of_nat_power
thf(fact_3606_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% of_nat_power
thf(fact_3607_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% of_nat_power
thf(fact_3608_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_3609_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri4939895301339042750nteger @ ( power_power_nat @ M @ N ) )
= ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ M ) @ N ) ) ).
% of_nat_power
thf(fact_3610_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
= ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_3611_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_3612_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_3613_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_3614_power__one__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_3615_power__inject__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M )
= ( power_power_real @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_3616_power__inject__exp,axiom,
! [A: rat,M: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ( power_power_rat @ A @ M )
= ( power_power_rat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_3617_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_3618_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_3619_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
= zero_zero_rat ) ).
% power_0_Suc
thf(fact_3620_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_3621_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_3622_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_3623_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
= zero_zero_complex ) ).
% power_0_Suc
thf(fact_3624_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_3625_power__Suc0__right,axiom,
! [A: real] :
( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_3626_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_3627_power__Suc0__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_3628_abs__power__minus,axiom,
! [A: int,N: nat] :
( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
= ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% abs_power_minus
thf(fact_3629_abs__power__minus,axiom,
! [A: real,N: nat] :
( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
= ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).
% abs_power_minus
thf(fact_3630_abs__power__minus,axiom,
! [A: code_integer,N: nat] :
( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
= ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% abs_power_minus
thf(fact_3631_abs__power__minus,axiom,
! [A: rat,N: nat] :
( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
= ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).
% abs_power_minus
thf(fact_3632_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
= zero_zero_real ) ).
% gbinomial_0(2)
thf(fact_3633_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
= zero_zero_rat ) ).
% gbinomial_0(2)
thf(fact_3634_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% gbinomial_0(2)
thf(fact_3635_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
= zero_zero_int ) ).
% gbinomial_0(2)
thf(fact_3636_gbinomial__0_I1_J,axiom,
! [A: complex] :
( ( gbinomial_complex @ A @ zero_zero_nat )
= one_one_complex ) ).
% gbinomial_0(1)
thf(fact_3637_gbinomial__0_I1_J,axiom,
! [A: real] :
( ( gbinomial_real @ A @ zero_zero_nat )
= one_one_real ) ).
% gbinomial_0(1)
thf(fact_3638_gbinomial__0_I1_J,axiom,
! [A: rat] :
( ( gbinomial_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% gbinomial_0(1)
thf(fact_3639_gbinomial__0_I1_J,axiom,
! [A: nat] :
( ( gbinomial_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% gbinomial_0(1)
thf(fact_3640_gbinomial__0_I1_J,axiom,
! [A: int] :
( ( gbinomial_int @ A @ zero_zero_nat )
= one_one_int ) ).
% gbinomial_0(1)
thf(fact_3641_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ( ring_18347121197199848620nteger @ X )
= ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W2 ) )
= ( X
= ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_3642_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ( ring_1_of_int_rat @ X )
= ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
= ( X
= ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_3643_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ( ring_1_of_int_int @ X )
= ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
= ( X
= ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_3644_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ( ring_1_of_int_real @ X )
= ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
= ( X
= ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_3645_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ( ring_17405671764205052669omplex @ X )
= ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W2 ) )
= ( X
= ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_3646_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W2 )
= ( ring_18347121197199848620nteger @ X ) )
= ( ( power_power_int @ B @ W2 )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_3647_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 )
= ( ring_1_of_int_rat @ X ) )
= ( ( power_power_int @ B @ W2 )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_3648_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 )
= ( ring_1_of_int_int @ X ) )
= ( ( power_power_int @ B @ W2 )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_3649_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 )
= ( ring_1_of_int_real @ X ) )
= ( ( power_power_int @ B @ W2 )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_3650_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W2 )
= ( ring_17405671764205052669omplex @ X ) )
= ( ( power_power_int @ B @ W2 )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_3651_of__int__power,axiom,
! [Z2: int,N: nat] :
( ( ring_18347121197199848620nteger @ ( power_power_int @ Z2 @ N ) )
= ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ Z2 ) @ N ) ) ).
% of_int_power
thf(fact_3652_of__int__power,axiom,
! [Z2: int,N: nat] :
( ( ring_1_of_int_rat @ ( power_power_int @ Z2 @ N ) )
= ( power_power_rat @ ( ring_1_of_int_rat @ Z2 ) @ N ) ) ).
% of_int_power
thf(fact_3653_of__int__power,axiom,
! [Z2: int,N: nat] :
( ( ring_1_of_int_int @ ( power_power_int @ Z2 @ N ) )
= ( power_power_int @ ( ring_1_of_int_int @ Z2 ) @ N ) ) ).
% of_int_power
thf(fact_3654_of__int__power,axiom,
! [Z2: int,N: nat] :
( ( ring_1_of_int_real @ ( power_power_int @ Z2 @ N ) )
= ( power_power_real @ ( ring_1_of_int_real @ Z2 ) @ N ) ) ).
% of_int_power
thf(fact_3655_of__int__power,axiom,
! [Z2: int,N: nat] :
( ( ring_17405671764205052669omplex @ ( power_power_int @ Z2 @ N ) )
= ( power_power_complex @ ( ring_17405671764205052669omplex @ Z2 ) @ N ) ) ).
% of_int_power
thf(fact_3656_left__minus__one__mult__self,axiom,
! [N: nat,A: int] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_3657_left__minus__one__mult__self,axiom,
! [N: nat,A: real] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_3658_left__minus__one__mult__self,axiom,
! [N: nat,A: complex] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_3659_left__minus__one__mult__self,axiom,
! [N: nat,A: code_integer] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_3660_left__minus__one__mult__self,axiom,
! [N: nat,A: rat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_3661_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
= one_one_int ) ).
% minus_one_mult_self
thf(fact_3662_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
= one_one_real ) ).
% minus_one_mult_self
thf(fact_3663_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
= one_one_complex ) ).
% minus_one_mult_self
thf(fact_3664_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
= one_one_Code_integer ) ).
% minus_one_mult_self
thf(fact_3665_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
= one_one_rat ) ).
% minus_one_mult_self
thf(fact_3666_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_3667_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_3668_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_3669_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_3670_power__eq__0__iff,axiom,
! [A: rat,N: nat] :
( ( ( power_power_rat @ A @ N )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_3671_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_3672_power__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( power_power_real @ A @ N )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_3673_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_3674_power__eq__0__iff,axiom,
! [A: complex,N: nat] :
( ( ( power_power_complex @ A @ N )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_3675_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_3676_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_3677_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_3678_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) @ ( semiri4939895301339042750nteger @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_3679_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_3680_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_3681_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_3682_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_3683_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_3684_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_3685_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_3686_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) @ ( semiri4939895301339042750nteger @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_3687_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_3688_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_3689_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_3690_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_3691_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_3692_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_3693_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_3694_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_3695_power__strict__decreasing__iff,axiom,
! [B: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_3696_power__strict__decreasing__iff,axiom,
! [B: rat,M: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_rat @ B @ one_one_rat )
=> ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_3697_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_3698_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_3699_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_3700_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_3701_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_3702_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_3703_power__mono__iff,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_3704_power__mono__iff,axiom,
! [A: rat,B: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_3705_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_3706_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_3707_zero__less__power__abs__iff,axiom,
! [A: code_integer,N: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
= ( ( A != zero_z3403309356797280102nteger )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_3708_zero__less__power__abs__iff,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
= ( ( A != zero_zero_real )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_3709_zero__less__power__abs__iff,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
= ( ( A != zero_zero_rat )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_3710_zero__less__power__abs__iff,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
= ( ( A != zero_zero_int )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_3711_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ X ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W2 ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_3712_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_3713_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_3714_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_3715_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W2 ) @ ( ring_18347121197199848620nteger @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_3716_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_3717_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_3718_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_3719_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W2 ) @ ( ring_18347121197199848620nteger @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_3720_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_3721_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_3722_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_3723_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ X ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W2 ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_3724_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_3725_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_3726_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_3727_power__decreasing__iff,axiom,
! [B: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_3728_power__decreasing__iff,axiom,
! [B: rat,M: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_rat @ B @ one_one_rat )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_3729_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_3730_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_3731_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_3732_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_3733_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_3734_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_3735_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_3736_ComplD,axiom,
! [C: complex,A2: set_complex] :
( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) )
=> ~ ( member_complex @ C @ A2 ) ) ).
% ComplD
thf(fact_3737_ComplD,axiom,
! [C: real,A2: set_real] :
( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
=> ~ ( member_real @ C @ A2 ) ) ).
% ComplD
thf(fact_3738_ComplD,axiom,
! [C: set_nat,A2: set_set_nat] :
( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
=> ~ ( member_set_nat @ C @ A2 ) ) ).
% ComplD
thf(fact_3739_ComplD,axiom,
! [C: nat,A2: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
=> ~ ( member_nat @ C @ A2 ) ) ).
% ComplD
thf(fact_3740_ComplD,axiom,
! [C: int,A2: set_int] :
( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
=> ~ ( member_int @ C @ A2 ) ) ).
% ComplD
thf(fact_3741_Ints__power,axiom,
! [A: int,N: nat] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( member_int @ ( power_power_int @ A @ N ) @ ring_1_Ints_int ) ) ).
% Ints_power
thf(fact_3742_Ints__power,axiom,
! [A: real,N: nat] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( member_real @ ( power_power_real @ A @ N ) @ ring_1_Ints_real ) ) ).
% Ints_power
thf(fact_3743_Ints__power,axiom,
! [A: complex,N: nat] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( member_complex @ ( power_power_complex @ A @ N ) @ ring_1_Ints_complex ) ) ).
% Ints_power
thf(fact_3744_of__nat__gbinomial,axiom,
! [N: nat,K: nat] :
( ( semiri5074537144036343181t_real @ ( gbinomial_nat @ N @ K ) )
= ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K ) ) ).
% of_nat_gbinomial
thf(fact_3745_of__nat__gbinomial,axiom,
! [N: nat,K: nat] :
( ( semiri681578069525770553at_rat @ ( gbinomial_nat @ N @ K ) )
= ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K ) ) ).
% of_nat_gbinomial
thf(fact_3746_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_3747_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_3748_power__not__zero,axiom,
! [A: rat,N: nat] :
( ( A != zero_zero_rat )
=> ( ( power_power_rat @ A @ N )
!= zero_zero_rat ) ) ).
% power_not_zero
thf(fact_3749_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_3750_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_3751_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_3752_power__not__zero,axiom,
! [A: complex,N: nat] :
( ( A != zero_zero_complex )
=> ( ( power_power_complex @ A @ N )
!= zero_zero_complex ) ) ).
% power_not_zero
thf(fact_3753_Ints__0,axiom,
member_complex @ zero_zero_complex @ ring_1_Ints_complex ).
% Ints_0
thf(fact_3754_Ints__0,axiom,
member_real @ zero_zero_real @ ring_1_Ints_real ).
% Ints_0
thf(fact_3755_Ints__0,axiom,
member_rat @ zero_zero_rat @ ring_1_Ints_rat ).
% Ints_0
thf(fact_3756_Ints__0,axiom,
member_int @ zero_zero_int @ ring_1_Ints_int ).
% Ints_0
thf(fact_3757_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_3758_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_3759_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_3760_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_3761_Ints__1,axiom,
member_rat @ one_one_rat @ ring_1_Ints_rat ).
% Ints_1
thf(fact_3762_Ints__1,axiom,
member_int @ one_one_int @ ring_1_Ints_int ).
% Ints_1
thf(fact_3763_Ints__1,axiom,
member_real @ one_one_real @ ring_1_Ints_real ).
% Ints_1
thf(fact_3764_Ints__1,axiom,
member_complex @ one_one_complex @ ring_1_Ints_complex ).
% Ints_1
thf(fact_3765_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_3766_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_3767_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_3768_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_3769_power__inverse,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( inverse_inverse_real @ A ) @ N )
= ( inverse_inverse_real @ ( power_power_real @ A @ N ) ) ) ).
% power_inverse
thf(fact_3770_power__inverse,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ ( invers8013647133539491842omplex @ A ) @ N )
= ( invers8013647133539491842omplex @ ( power_power_complex @ A @ N ) ) ) ).
% power_inverse
thf(fact_3771_power__inverse,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ ( inverse_inverse_rat @ A ) @ N )
= ( inverse_inverse_rat @ ( power_power_rat @ A @ N ) ) ) ).
% power_inverse
thf(fact_3772_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_3773_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_3774_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_3775_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_3776_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_3777_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_3778_Ints__minus,axiom,
! [A: complex] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( member_complex @ ( uminus1482373934393186551omplex @ A ) @ ring_1_Ints_complex ) ) ).
% Ints_minus
thf(fact_3779_Ints__minus,axiom,
! [A: code_integer] :
( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
=> ( member_Code_integer @ ( uminus1351360451143612070nteger @ A ) @ ring_11222124179247155820nteger ) ) ).
% Ints_minus
thf(fact_3780_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_3781_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_3782_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_3783_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_3784_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_3785_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_3786_Ints__of__nat,axiom,
! [N: nat] : ( member_complex @ ( semiri8010041392384452111omplex @ N ) @ ring_1_Ints_complex ) ).
% Ints_of_nat
thf(fact_3787_Ints__of__nat,axiom,
! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ring_1_Ints_int ) ).
% Ints_of_nat
thf(fact_3788_Ints__of__nat,axiom,
! [N: nat] : ( member_real @ ( semiri5074537144036343181t_real @ N ) @ ring_1_Ints_real ) ).
% Ints_of_nat
thf(fact_3789_Ints__of__nat,axiom,
! [N: nat] : ( member_Code_integer @ ( semiri4939895301339042750nteger @ N ) @ ring_11222124179247155820nteger ) ).
% Ints_of_nat
thf(fact_3790_Ints__of__nat,axiom,
! [N: nat] : ( member_rat @ ( semiri681578069525770553at_rat @ N ) @ ring_1_Ints_rat ) ).
% Ints_of_nat
thf(fact_3791_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_3792_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_3793_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_3794_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_3795_Ints__cases,axiom,
! [Q4: int] :
( ( member_int @ Q4 @ ring_1_Ints_int )
=> ~ ! [Z4: int] :
( Q4
!= ( ring_1_of_int_int @ Z4 ) ) ) ).
% Ints_cases
thf(fact_3796_Ints__cases,axiom,
! [Q4: real] :
( ( member_real @ Q4 @ ring_1_Ints_real )
=> ~ ! [Z4: int] :
( Q4
!= ( ring_1_of_int_real @ Z4 ) ) ) ).
% Ints_cases
thf(fact_3797_Ints__cases,axiom,
! [Q4: code_integer] :
( ( member_Code_integer @ Q4 @ ring_11222124179247155820nteger )
=> ~ ! [Z4: int] :
( Q4
!= ( ring_18347121197199848620nteger @ Z4 ) ) ) ).
% Ints_cases
thf(fact_3798_Ints__cases,axiom,
! [Q4: rat] :
( ( member_rat @ Q4 @ ring_1_Ints_rat )
=> ~ ! [Z4: int] :
( Q4
!= ( ring_1_of_int_rat @ Z4 ) ) ) ).
% Ints_cases
thf(fact_3799_Ints__cases,axiom,
! [Q4: complex] :
( ( member_complex @ Q4 @ ring_1_Ints_complex )
=> ~ ! [Z4: int] :
( Q4
!= ( ring_17405671764205052669omplex @ Z4 ) ) ) ).
% Ints_cases
thf(fact_3800_Ints__induct,axiom,
! [Q4: int,P: int > $o] :
( ( member_int @ Q4 @ ring_1_Ints_int )
=> ( ! [Z4: int] : ( P @ ( ring_1_of_int_int @ Z4 ) )
=> ( P @ Q4 ) ) ) ).
% Ints_induct
thf(fact_3801_Ints__induct,axiom,
! [Q4: real,P: real > $o] :
( ( member_real @ Q4 @ ring_1_Ints_real )
=> ( ! [Z4: int] : ( P @ ( ring_1_of_int_real @ Z4 ) )
=> ( P @ Q4 ) ) ) ).
% Ints_induct
thf(fact_3802_Ints__induct,axiom,
! [Q4: code_integer,P: code_integer > $o] :
( ( member_Code_integer @ Q4 @ ring_11222124179247155820nteger )
=> ( ! [Z4: int] : ( P @ ( ring_18347121197199848620nteger @ Z4 ) )
=> ( P @ Q4 ) ) ) ).
% Ints_induct
thf(fact_3803_Ints__induct,axiom,
! [Q4: rat,P: rat > $o] :
( ( member_rat @ Q4 @ ring_1_Ints_rat )
=> ( ! [Z4: int] : ( P @ ( ring_1_of_int_rat @ Z4 ) )
=> ( P @ Q4 ) ) ) ).
% Ints_induct
thf(fact_3804_Ints__induct,axiom,
! [Q4: complex,P: complex > $o] :
( ( member_complex @ Q4 @ ring_1_Ints_complex )
=> ( ! [Z4: int] : ( P @ ( ring_17405671764205052669omplex @ Z4 ) )
=> ( P @ Q4 ) ) ) ).
% Ints_induct
thf(fact_3805_Ints__of__int,axiom,
! [Z2: int] : ( member_int @ ( ring_1_of_int_int @ Z2 ) @ ring_1_Ints_int ) ).
% Ints_of_int
thf(fact_3806_Ints__of__int,axiom,
! [Z2: int] : ( member_real @ ( ring_1_of_int_real @ Z2 ) @ ring_1_Ints_real ) ).
% Ints_of_int
thf(fact_3807_Ints__of__int,axiom,
! [Z2: int] : ( member_Code_integer @ ( ring_18347121197199848620nteger @ Z2 ) @ ring_11222124179247155820nteger ) ).
% Ints_of_int
thf(fact_3808_Ints__of__int,axiom,
! [Z2: int] : ( member_rat @ ( ring_1_of_int_rat @ Z2 ) @ ring_1_Ints_rat ) ).
% Ints_of_int
thf(fact_3809_Ints__of__int,axiom,
! [Z2: int] : ( member_complex @ ( ring_17405671764205052669omplex @ Z2 ) @ ring_1_Ints_complex ) ).
% Ints_of_int
thf(fact_3810_gbinomial__index__swap,axiom,
! [K: nat,N: nat] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K ) )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N ) ) ) ).
% gbinomial_index_swap
thf(fact_3811_gbinomial__index__swap,axiom,
! [K: nat,N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ K ) )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ N ) ) ) ).
% gbinomial_index_swap
thf(fact_3812_gbinomial__index__swap,axiom,
! [K: nat,N: nat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ K ) )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ N ) ) ) ).
% gbinomial_index_swap
thf(fact_3813_gbinomial__negated__upper,axiom,
( gbinomial_complex
= ( ^ [A3: complex,K3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A3 ) @ one_one_complex ) @ K3 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_3814_gbinomial__negated__upper,axiom,
( gbinomial_real
= ( ^ [A3: real,K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A3 ) @ one_one_real ) @ K3 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_3815_gbinomial__negated__upper,axiom,
( gbinomial_rat
= ( ^ [A3: rat,K3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A3 ) @ one_one_rat ) @ K3 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_3816_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_3817_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_3818_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_3819_power__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono
thf(fact_3820_power__mono,axiom,
! [A: rat,B: rat,N: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_3821_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_3822_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_3823_zero__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_power
thf(fact_3824_zero__le__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_3825_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_3826_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_3827_zero__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_less_power
thf(fact_3828_zero__less__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_3829_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_3830_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_3831_one__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% one_le_power
thf(fact_3832_one__le__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% one_le_power
thf(fact_3833_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_3834_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_3835_left__right__inverse__power,axiom,
! [X: complex,Y: complex,N: nat] :
( ( ( times_times_complex @ X @ Y )
= one_one_complex )
=> ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
= one_one_complex ) ) ).
% left_right_inverse_power
thf(fact_3836_left__right__inverse__power,axiom,
! [X: real,Y: real,N: nat] :
( ( ( times_times_real @ X @ Y )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_3837_left__right__inverse__power,axiom,
! [X: rat,Y: rat,N: nat] :
( ( ( times_times_rat @ X @ Y )
= one_one_rat )
=> ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
= one_one_rat ) ) ).
% left_right_inverse_power
thf(fact_3838_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_3839_left__right__inverse__power,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_3840_power__one__over,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
= ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% power_one_over
thf(fact_3841_power__one__over,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% power_one_over
thf(fact_3842_power__one__over,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
= ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% power_one_over
thf(fact_3843_power__Suc,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( suc @ N ) )
= ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% power_Suc
thf(fact_3844_power__Suc,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_Suc
thf(fact_3845_power__Suc,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% power_Suc
thf(fact_3846_power__Suc,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_Suc
thf(fact_3847_power__Suc,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_Suc
thf(fact_3848_power__Suc2,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( suc @ N ) )
= ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_3849_power__Suc2,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_3850_power__Suc2,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_3851_power__Suc2,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_3852_power__Suc2,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_3853_power__0,axiom,
! [A: rat] :
( ( power_power_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% power_0
thf(fact_3854_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_3855_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_3856_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_3857_power__0,axiom,
! [A: complex] :
( ( power_power_complex @ A @ zero_zero_nat )
= one_one_complex ) ).
% power_0
thf(fact_3858_gbinomial__of__nat__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
= ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_3859_gbinomial__of__nat__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K )
= ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_3860_power__add,axiom,
! [A: complex,M: nat,N: nat] :
( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% power_add
thf(fact_3861_power__add,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% power_add
thf(fact_3862_power__add,axiom,
! [A: rat,M: nat,N: nat] :
( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% power_add
thf(fact_3863_power__add,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_3864_power__add,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% power_add
thf(fact_3865_power__mult__power__inverse__commute,axiom,
! [X: real,M: nat,N: nat] :
( ( times_times_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) )
= ( times_times_real @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) @ ( power_power_real @ X @ M ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_3866_power__mult__power__inverse__commute,axiom,
! [X: complex,M: nat,N: nat] :
( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) )
= ( times_times_complex @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) @ ( power_power_complex @ X @ M ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_3867_power__mult__power__inverse__commute,axiom,
! [X: rat,M: nat,N: nat] :
( ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) )
= ( times_times_rat @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) @ ( power_power_rat @ X @ M ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_3868_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_3869_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_3870_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_3871_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_3872_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_3873_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_3874_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_3875_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_3876_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_3877_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_3878_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_3879_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_3880_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_3881_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_3882_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_3883_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_3884_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_3885_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_3886_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_3887_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_3888_power__less__imp__less__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_3889_power__less__imp__less__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_3890_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_3891_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_3892_power__le__one,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_3893_power__le__one,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% power_le_one
thf(fact_3894_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_3895_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_3896_power__le__imp__le__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_3897_power__le__imp__le__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_3898_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_3899_power__le__imp__le__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_3900_power__inject__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ ( suc @ N ) )
= ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_3901_power__inject__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ( power_power_rat @ A @ ( suc @ N ) )
= ( power_power_rat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_3902_power__inject__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_3903_power__inject__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ ( suc @ N ) )
= ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_3904_power__less__power__Suc,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_3905_power__less__power__Suc,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_3906_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_3907_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_3908_power__gt1__lemma,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_3909_power__gt1__lemma,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_3910_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_3911_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_3912_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ) ).
% power_0_left
thf(fact_3913_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_3914_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_3915_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_3916_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= one_one_complex ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ) ).
% power_0_left
thf(fact_3917_power__gt1,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_3918_power__gt1,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_3919_power__gt1,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_3920_power__gt1,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_3921_power__minus,axiom,
! [A: int,N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% power_minus
thf(fact_3922_power__minus,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% power_minus
thf(fact_3923_power__minus,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% power_minus
thf(fact_3924_power__minus,axiom,
! [A: code_integer,N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% power_minus
thf(fact_3925_power__minus,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% power_minus
thf(fact_3926_zero__le__power__abs,axiom,
! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_3927_zero__le__power__abs,axiom,
! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_3928_zero__le__power__abs,axiom,
! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_3929_zero__le__power__abs,axiom,
! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_3930_power__less__imp__less__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_3931_power__less__imp__less__exp,axiom,
! [A: rat,M: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_3932_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_3933_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_3934_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: real] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_3935_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: rat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_3936_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_3937_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_3938_power__increasing,axiom,
! [N: nat,N5: nat,A: real] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_3939_power__increasing,axiom,
! [N: nat,N5: nat,A: rat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_3940_power__increasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_3941_power__increasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_3942_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ).
% zero_power
thf(fact_3943_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_3944_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% zero_power
thf(fact_3945_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_3946_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ).
% zero_power
thf(fact_3947_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_3948_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_3949_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_3950_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_3951_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_3952_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_3953_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_3954_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_3955_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_3956_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_3957_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_3958_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_3959_power__Suc__less,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_3960_power__Suc__less,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_3961_power__Suc__less,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_3962_power__Suc__less,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_3963_power__Suc__le__self,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_3964_power__Suc__le__self,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_3965_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_3966_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_3967_power__Suc__less__one,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% power_Suc_less_one
thf(fact_3968_power__Suc__less__one,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% power_Suc_less_one
thf(fact_3969_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_3970_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_3971_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: real] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_3972_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: rat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_3973_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_3974_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_3975_power__decreasing,axiom,
! [N: nat,N5: nat,A: real] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_3976_power__decreasing,axiom,
! [N: nat,N5: nat,A: rat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_3977_power__decreasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_3978_power__decreasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_3979_power__le__imp__le__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_3980_power__le__imp__le__exp,axiom,
! [A: rat,M: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_3981_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_3982_power__le__imp__le__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_3983_power__eq__iff__eq__base,axiom,
! [N: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_3984_power__eq__iff__eq__base,axiom,
! [N: nat,A: rat,B: rat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ( power_power_rat @ A @ N )
= ( power_power_rat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_3985_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_3986_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_3987_power__eq__imp__eq__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_3988_power__eq__imp__eq__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ( power_power_rat @ A @ N )
= ( power_power_rat @ B @ N ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_3989_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_3990_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_3991_self__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_3992_self__le__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_3993_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_3994_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_3995_one__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_3996_one__less__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_3997_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_3998_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_3999_power__diff,axiom,
! [A: complex,N: nat,M: nat] :
( ( A != zero_zero_complex )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_4000_power__diff,axiom,
! [A: real,N: nat,M: nat] :
( ( A != zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_4001_power__diff,axiom,
! [A: rat,N: nat,M: nat] :
( ( A != zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_4002_power__diff,axiom,
! [A: nat,N: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_4003_power__diff,axiom,
! [A: int,N: nat,M: nat] :
( ( A != zero_zero_int )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_4004_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_4005_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_4006_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_4007_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_4008_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_4009_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_4010_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_4011_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_4012_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_4013_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_4014_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_4015_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_4016_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_4017_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_4018_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_4019_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_4020_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_4021_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_4022_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_4023_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_4024_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_4025_power__strict__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_4026_power__strict__mono,axiom,
! [A: rat,B: rat,N: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_4027_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_4028_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_4029_power__diff__power__eq,axiom,
! [A: nat,N: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
= ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
= ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_4030_power__diff__power__eq,axiom,
! [A: int,N: nat,M: nat] :
( ( A != zero_zero_int )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
= ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
= ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_4031_power__diff__conv__inverse,axiom,
! [X: real,M: nat,N: nat] :
( ( X != zero_zero_real )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( power_power_real @ X @ ( minus_minus_nat @ N @ M ) )
= ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ M ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_4032_power__diff__conv__inverse,axiom,
! [X: complex,M: nat,N: nat] :
( ( X != zero_zero_complex )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( power_power_complex @ X @ ( minus_minus_nat @ N @ M ) )
= ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ M ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_4033_power__diff__conv__inverse,axiom,
! [X: rat,M: nat,N: nat] :
( ( X != zero_zero_rat )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( power_power_rat @ X @ ( minus_minus_nat @ N @ M ) )
= ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ M ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_4034_power__eq__if,axiom,
( power_power_complex
= ( ^ [P6: complex,M4: nat] : ( if_complex @ ( M4 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P6 @ ( power_power_complex @ P6 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_4035_power__eq__if,axiom,
( power_power_real
= ( ^ [P6: real,M4: nat] : ( if_real @ ( M4 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P6 @ ( power_power_real @ P6 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_4036_power__eq__if,axiom,
( power_power_rat
= ( ^ [P6: rat,M4: nat] : ( if_rat @ ( M4 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P6 @ ( power_power_rat @ P6 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_4037_power__eq__if,axiom,
( power_power_nat
= ( ^ [P6: nat,M4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_4038_power__eq__if,axiom,
( power_power_int
= ( ^ [P6: int,M4: nat] : ( if_int @ ( M4 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_4039_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_4040_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_4041_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_4042_power__minus__mult,axiom,
! [N: nat,A: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_complex @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_4043_power__minus__mult,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_real @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_4044_power__minus__mult,axiom,
! [N: nat,A: rat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_rat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_4045_power__minus__mult,axiom,
! [N: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_4046_power__minus__mult,axiom,
! [N: nat,A: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_int @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_4047_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4048_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4049_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4050_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
= ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4051_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4052_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_4053_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_4054_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_4055_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_4056_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_4057_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_4058_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_4059_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_4060_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_4061_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_4062_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_4063_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_4064_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_4065_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_4066_Bernoulli__inequality,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_4067_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_4068_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_4069_frac__in__Ints__iff,axiom,
! [X: real] :
( ( member_real @ ( archim2898591450579166408c_real @ X ) @ ring_1_Ints_real )
= ( member_real @ X @ ring_1_Ints_real ) ) ).
% frac_in_Ints_iff
thf(fact_4070_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_4071_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_4072_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_4073_pochhammer__0,axiom,
! [A: complex] :
( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
= one_one_complex ) ).
% pochhammer_0
thf(fact_4074_pochhammer__0,axiom,
! [A: real] :
( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
= one_one_real ) ).
% pochhammer_0
thf(fact_4075_pochhammer__0,axiom,
! [A: rat] :
( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% pochhammer_0
thf(fact_4076_pochhammer__0,axiom,
! [A: nat] :
( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% pochhammer_0
thf(fact_4077_pochhammer__0,axiom,
! [A: int] :
( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
= one_one_int ) ).
% pochhammer_0
thf(fact_4078_frac__of__int,axiom,
! [Z2: int] :
( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z2 ) )
= zero_zero_real ) ).
% frac_of_int
thf(fact_4079_frac__of__int,axiom,
! [Z2: int] :
( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z2 ) )
= zero_zero_rat ) ).
% frac_of_int
thf(fact_4080_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_4081_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_4082_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_4083_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_4084_pochhammer__of__nat,axiom,
! [X: nat,N: nat] :
( ( comm_s4660882817536571857er_int @ ( semiri1314217659103216013at_int @ X ) @ N )
= ( semiri1314217659103216013at_int @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% pochhammer_of_nat
thf(fact_4085_pochhammer__of__nat,axiom,
! [X: nat,N: nat] :
( ( comm_s7457072308508201937r_real @ ( semiri5074537144036343181t_real @ X ) @ N )
= ( semiri5074537144036343181t_real @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% pochhammer_of_nat
thf(fact_4086_pochhammer__of__nat,axiom,
! [X: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ ( semiri1316708129612266289at_nat @ X ) @ N )
= ( semiri1316708129612266289at_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% pochhammer_of_nat
thf(fact_4087_pochhammer__of__nat,axiom,
! [X: nat,N: nat] :
( ( comm_s8582702949713902594nteger @ ( semiri4939895301339042750nteger @ X ) @ N )
= ( semiri4939895301339042750nteger @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% pochhammer_of_nat
thf(fact_4088_pochhammer__of__nat,axiom,
! [X: nat,N: nat] :
( ( comm_s4028243227959126397er_rat @ ( semiri681578069525770553at_rat @ X ) @ N )
= ( semiri681578069525770553at_rat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% pochhammer_of_nat
thf(fact_4089_pochhammer__of__int,axiom,
! [X: int,N: nat] :
( ( comm_s7457072308508201937r_real @ ( ring_1_of_int_real @ X ) @ N )
= ( ring_1_of_int_real @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% pochhammer_of_int
thf(fact_4090_pochhammer__of__int,axiom,
! [X: int,N: nat] :
( ( comm_s8582702949713902594nteger @ ( ring_18347121197199848620nteger @ X ) @ N )
= ( ring_18347121197199848620nteger @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% pochhammer_of_int
thf(fact_4091_pochhammer__of__int,axiom,
! [X: int,N: nat] :
( ( comm_s4028243227959126397er_rat @ ( ring_1_of_int_rat @ X ) @ N )
= ( ring_1_of_int_rat @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% pochhammer_of_int
thf(fact_4092_pochhammer__of__int,axiom,
! [X: int,N: nat] :
( ( comm_s2602460028002588243omplex @ ( ring_17405671764205052669omplex @ X ) @ N )
= ( ring_17405671764205052669omplex @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% pochhammer_of_int
thf(fact_4093_arctan__minus,axiom,
! [X: real] :
( ( arctan @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( arctan @ X ) ) ) ).
% arctan_minus
thf(fact_4094_pochhammer__pos,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% pochhammer_pos
thf(fact_4095_pochhammer__pos,axiom,
! [X: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% pochhammer_pos
thf(fact_4096_pochhammer__pos,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% pochhammer_pos
thf(fact_4097_pochhammer__pos,axiom,
! [X: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% pochhammer_pos
thf(fact_4098_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_4099_pochhammer__neq__0__mono,axiom,
! [A: real,M: nat,N: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ M )
!= zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s7457072308508201937r_real @ A @ N )
!= zero_zero_real ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_4100_pochhammer__neq__0__mono,axiom,
! [A: rat,M: nat,N: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ M )
!= zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s4028243227959126397er_rat @ A @ N )
!= zero_zero_rat ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_4101_pochhammer__eq__0__mono,axiom,
! [A: real,N: nat,M: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ N )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s7457072308508201937r_real @ A @ M )
= zero_zero_real ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_4102_pochhammer__eq__0__mono,axiom,
! [A: rat,N: nat,M: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ N )
= zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s4028243227959126397er_rat @ A @ M )
= zero_zero_rat ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_4103_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N2: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ).
% real_arch_pow
thf(fact_4104_frac__ge__0,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).
% frac_ge_0
thf(fact_4105_frac__ge__0,axiom,
! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).
% frac_ge_0
thf(fact_4106_frac__lt__1,axiom,
! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).
% frac_lt_1
thf(fact_4107_frac__lt__1,axiom,
! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).
% frac_lt_1
thf(fact_4108_frac__1__eq,axiom,
! [X: real] :
( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ one_one_real ) )
= ( archim2898591450579166408c_real @ X ) ) ).
% frac_1_eq
thf(fact_4109_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_4110_pochhammer__nonneg,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_4111_pochhammer__nonneg,axiom,
! [X: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_4112_pochhammer__nonneg,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_4113_pochhammer__nonneg,axiom,
! [X: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_4114_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_4115_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_4116_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
= one_one_complex ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ) ).
% pochhammer_0_left
thf(fact_4117_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% pochhammer_0_left
thf(fact_4118_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
= one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ) ).
% pochhammer_0_left
thf(fact_4119_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% pochhammer_0_left
thf(fact_4120_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% pochhammer_0_left
thf(fact_4121_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_4122_pochhammer__rec,axiom,
! [A: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
= ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_4123_pochhammer__rec,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_4124_pochhammer__rec,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_4125_pochhammer__rec,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_4126_pochhammer__rec,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_4127_pochhammer__rec_H,axiom,
! [Z2: int,N: nat] :
( ( comm_s4660882817536571857er_int @ Z2 @ ( suc @ N ) )
= ( times_times_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z2 @ N ) ) ) ).
% pochhammer_rec'
thf(fact_4128_pochhammer__rec_H,axiom,
! [Z2: real,N: nat] :
( ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N ) )
= ( times_times_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z2 @ N ) ) ) ).
% pochhammer_rec'
thf(fact_4129_pochhammer__rec_H,axiom,
! [Z2: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ Z2 @ ( suc @ N ) )
= ( times_times_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z2 @ N ) ) ) ).
% pochhammer_rec'
thf(fact_4130_pochhammer__rec_H,axiom,
! [Z2: code_integer,N: nat] :
( ( comm_s8582702949713902594nteger @ Z2 @ ( suc @ N ) )
= ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ Z2 @ ( semiri4939895301339042750nteger @ N ) ) @ ( comm_s8582702949713902594nteger @ Z2 @ N ) ) ) ).
% pochhammer_rec'
thf(fact_4131_pochhammer__rec_H,axiom,
! [Z2: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N ) )
= ( times_times_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) ) ) ).
% pochhammer_rec'
thf(fact_4132_pochhammer__Suc,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_4133_pochhammer__Suc,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_4134_pochhammer__Suc,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_4135_pochhammer__Suc,axiom,
! [A: code_integer,N: nat] :
( ( comm_s8582702949713902594nteger @ A @ ( suc @ N ) )
= ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ A @ N ) @ ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_4136_pochhammer__Suc,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_4137_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
= zero_zero_complex ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_4138_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
= zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_4139_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
= zero_zero_real ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_4140_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
= zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_4141_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
= zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_4142_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
= zero_zero_complex )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_4143_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
= zero_zero_int )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_4144_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
= zero_zero_real )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_4145_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
= zero_z3403309356797280102nteger )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_4146_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
= zero_zero_rat )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_4147_pochhammer__eq__0__iff,axiom,
! [A: complex,N: nat] :
( ( ( comm_s2602460028002588243omplex @ A @ N )
= zero_zero_complex )
= ( ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ( A
= ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_4148_pochhammer__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ N )
= zero_zero_real )
= ( ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ( A
= ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_4149_pochhammer__eq__0__iff,axiom,
! [A: rat,N: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ N )
= zero_zero_rat )
= ( ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ( A
= ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_4150_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
!= zero_zero_complex ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_4151_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
!= zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_4152_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
!= zero_zero_real ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_4153_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
!= zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_4154_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
!= zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_4155_pochhammer__product_H,axiom,
! [Z2: int,N: nat,M: nat] :
( ( comm_s4660882817536571857er_int @ Z2 @ ( plus_plus_nat @ N @ M ) )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_4156_pochhammer__product_H,axiom,
! [Z2: real,N: nat,M: nat] :
( ( comm_s7457072308508201937r_real @ Z2 @ ( plus_plus_nat @ N @ M ) )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_4157_pochhammer__product_H,axiom,
! [Z2: nat,N: nat,M: nat] :
( ( comm_s4663373288045622133er_nat @ Z2 @ ( plus_plus_nat @ N @ M ) )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_4158_pochhammer__product_H,axiom,
! [Z2: code_integer,N: nat,M: nat] :
( ( comm_s8582702949713902594nteger @ Z2 @ ( plus_plus_nat @ N @ M ) )
= ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z2 @ N ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z2 @ ( semiri4939895301339042750nteger @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_4159_pochhammer__product_H,axiom,
! [Z2: rat,N: nat,M: nat] :
( ( comm_s4028243227959126397er_rat @ Z2 @ ( plus_plus_nat @ N @ M ) )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_4160_nat__power__eq,axiom,
! [Z2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( nat2 @ ( power_power_int @ Z2 @ N ) )
= ( power_power_nat @ ( nat2 @ Z2 ) @ N ) ) ) ).
% nat_power_eq
thf(fact_4161_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_4162_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_4163_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_4164_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_4165_pochhammer__product,axiom,
! [M: nat,N: nat,Z2: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s4660882817536571857er_int @ Z2 @ N )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_4166_pochhammer__product,axiom,
! [M: nat,N: nat,Z2: real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s7457072308508201937r_real @ Z2 @ N )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_4167_pochhammer__product,axiom,
! [M: nat,N: nat,Z2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s4663373288045622133er_nat @ Z2 @ N )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_4168_pochhammer__product,axiom,
! [M: nat,N: nat,Z2: code_integer] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s8582702949713902594nteger @ Z2 @ N )
= ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z2 @ M ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z2 @ ( semiri4939895301339042750nteger @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_4169_pochhammer__product,axiom,
! [M: nat,N: nat,Z2: rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s4028243227959126397er_rat @ Z2 @ N )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_4170_pochhammer__absorb__comp,axiom,
! [R2: complex,K: nat] :
( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K ) )
= ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_4171_pochhammer__absorb__comp,axiom,
! [R2: int,K: nat] :
( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
= ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_4172_pochhammer__absorb__comp,axiom,
! [R2: real,K: nat] :
( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K ) )
= ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_4173_pochhammer__absorb__comp,axiom,
! [R2: code_integer,K: nat] :
( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K ) )
= ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_4174_pochhammer__absorb__comp,axiom,
! [R2: rat,K: nat] :
( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
= ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_4175_linear__plus__1__le__power,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_4176_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_4177_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A )
& ! [Y6: real] :
( ( ( ord_less_real @ zero_zero_real @ Y6 )
& ( ( power_power_real @ Y6 @ N )
= A ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_4178_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R4: real] :
( ( ord_less_real @ zero_zero_real @ R4 )
& ( ( power_power_real @ R4 @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_4179_realpow__pos__nth2,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R4: real] :
( ( ord_less_real @ zero_zero_real @ R4 )
& ( ( power_power_real @ R4 @ ( suc @ N ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_4180_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_4181_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_4182_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_4183_arsinh__minus__real,axiom,
! [X: real] :
( ( arsinh_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( arsinh_real @ X ) ) ) ).
% arsinh_minus_real
thf(fact_4184_powr__0,axiom,
! [Z2: real] :
( ( powr_real @ zero_zero_real @ Z2 )
= zero_zero_real ) ).
% powr_0
thf(fact_4185_powr__eq__0__iff,axiom,
! [W2: real,Z2: real] :
( ( ( powr_real @ W2 @ Z2 )
= zero_zero_real )
= ( W2 = zero_zero_real ) ) ).
% powr_eq_0_iff
thf(fact_4186_powr__one__eq__one,axiom,
! [A: real] :
( ( powr_real @ one_one_real @ A )
= one_one_real ) ).
% powr_one_eq_one
thf(fact_4187_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_4188_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_4189_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_4190_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_4191_powr__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ one_one_real )
= X ) ) ).
% powr_one
thf(fact_4192_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_4193_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_4194_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_4195_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_4196_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_4197_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_4198_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_4199_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_4200_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_4201_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_4202_inverse__powr,axiom,
! [Y: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( powr_real @ ( inverse_inverse_real @ Y ) @ A )
= ( inverse_inverse_real @ ( powr_real @ Y @ A ) ) ) ) ).
% inverse_powr
thf(fact_4203_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_4204_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_4205_powr__diff,axiom,
! [W2: real,Z1: real,Z22: real] :
( ( powr_real @ W2 @ ( minus_minus_real @ Z1 @ Z22 ) )
= ( divide_divide_real @ ( powr_real @ W2 @ Z1 ) @ ( powr_real @ W2 @ Z22 ) ) ) ).
% powr_diff
thf(fact_4206_powr__minus,axiom,
! [X: real,A: real] :
( ( powr_real @ X @ ( uminus_uminus_real @ A ) )
= ( inverse_inverse_real @ ( powr_real @ X @ A ) ) ) ).
% powr_minus
thf(fact_4207_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_4208_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_4209_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_4210_powr__real__of__int,axiom,
! [X: real,N: int] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
= ( power_power_real @ X @ ( nat2 @ N ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
= ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% powr_real_of_int
thf(fact_4211_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 @ ( log2 @ 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_4212_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_4213_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N: nat,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_4214_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_4215_root__powr__inverse,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( root @ N @ X )
= ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_4216_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_4217_gbinomial__pochhammer_H,axiom,
( gbinomial_complex
= ( ^ [A3: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_4218_gbinomial__pochhammer_H,axiom,
( gbinomial_rat
= ( ^ [A3: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_4219_gbinomial__pochhammer_H,axiom,
( gbinomial_real
= ( ^ [A3: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_4220_gbinomial__pochhammer,axiom,
( gbinomial_complex
= ( ^ [A3: complex,K3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A3 ) @ K3 ) ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_4221_gbinomial__pochhammer,axiom,
( gbinomial_rat
= ( ^ [A3: rat,K3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A3 ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_4222_gbinomial__pochhammer,axiom,
( gbinomial_real
= ( ^ [A3: real,K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A3 ) @ K3 ) ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_4223_of__nat__fact,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( semiri1408675320244567234ct_nat @ N ) )
= ( semiri1406184849735516958ct_int @ N ) ) ).
% of_nat_fact
thf(fact_4224_of__nat__fact,axiom,
! [N: nat] :
( ( semiri4939895301339042750nteger @ ( semiri1408675320244567234ct_nat @ N ) )
= ( semiri3624122377584611663nteger @ N ) ) ).
% of_nat_fact
thf(fact_4225_of__nat__fact,axiom,
! [N: nat] :
( ( semiri681578069525770553at_rat @ ( semiri1408675320244567234ct_nat @ N ) )
= ( semiri773545260158071498ct_rat @ N ) ) ).
% of_nat_fact
thf(fact_4226_of__nat__fact,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( semiri1408675320244567234ct_nat @ N ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ).
% of_nat_fact
thf(fact_4227_of__nat__fact,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( semiri1408675320244567234ct_nat @ N ) )
= ( semiri2265585572941072030t_real @ N ) ) ).
% of_nat_fact
thf(fact_4228_greaterThanLessThan__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or8625682525731655386et_nat @ L @ U ) )
= ( ( ord_less_set_nat @ L @ I )
& ( ord_less_set_nat @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_4229_greaterThanLessThan__iff,axiom,
! [I: rat,L: rat,U: rat] :
( ( member_rat @ I @ ( set_or5199638295745620268an_rat @ L @ U ) )
= ( ( ord_less_rat @ L @ I )
& ( ord_less_rat @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_4230_greaterThanLessThan__iff,axiom,
! [I: num,L: num,U: num] :
( ( member_num @ I @ ( set_or2392100141987894638an_num @ L @ U ) )
= ( ( ord_less_num @ L @ I )
& ( ord_less_num @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_4231_greaterThanLessThan__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or5832277885323065728an_int @ L @ U ) )
= ( ( ord_less_int @ L @ I )
& ( ord_less_int @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_4232_greaterThanLessThan__iff,axiom,
! [I: real,L: real,U: real] :
( ( member_real @ I @ ( set_or1633881224788618240n_real @ L @ U ) )
= ( ( ord_less_real @ L @ I )
& ( ord_less_real @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_4233_greaterThanLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_4234_of__int__fact,axiom,
! [N: nat] :
( ( ring_18347121197199848620nteger @ ( semiri1406184849735516958ct_int @ N ) )
= ( semiri3624122377584611663nteger @ N ) ) ).
% of_int_fact
thf(fact_4235_of__int__fact,axiom,
! [N: nat] :
( ( ring_1_of_int_rat @ ( semiri1406184849735516958ct_int @ N ) )
= ( semiri773545260158071498ct_rat @ N ) ) ).
% of_int_fact
thf(fact_4236_of__int__fact,axiom,
! [N: nat] :
( ( ring_17405671764205052669omplex @ ( semiri1406184849735516958ct_int @ N ) )
= ( semiri5044797733671781792omplex @ N ) ) ).
% of_int_fact
thf(fact_4237_of__int__fact,axiom,
! [N: nat] :
( ( ring_1_of_int_real @ ( semiri1406184849735516958ct_int @ N ) )
= ( semiri2265585572941072030t_real @ N ) ) ).
% of_int_fact
thf(fact_4238_exp__zero,axiom,
( ( exp_complex @ zero_zero_complex )
= one_one_complex ) ).
% exp_zero
thf(fact_4239_exp__zero,axiom,
( ( exp_real @ zero_zero_real )
= one_one_real ) ).
% exp_zero
thf(fact_4240_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_4241_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_4242_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_4243_fact__0,axiom,
( ( semiri5044797733671781792omplex @ zero_zero_nat )
= one_one_complex ) ).
% fact_0
thf(fact_4244_fact__0,axiom,
( ( semiri773545260158071498ct_rat @ zero_zero_nat )
= one_one_rat ) ).
% fact_0
thf(fact_4245_fact__0,axiom,
( ( semiri1406184849735516958ct_int @ zero_zero_nat )
= one_one_int ) ).
% fact_0
thf(fact_4246_fact__0,axiom,
( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
= one_one_nat ) ).
% fact_0
thf(fact_4247_fact__0,axiom,
( ( semiri2265585572941072030t_real @ zero_zero_nat )
= one_one_real ) ).
% fact_0
thf(fact_4248_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ zero_zero_nat ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_4249_fact__1,axiom,
( ( semiri5044797733671781792omplex @ one_one_nat )
= one_one_complex ) ).
% fact_1
thf(fact_4250_fact__1,axiom,
( ( semiri773545260158071498ct_rat @ one_one_nat )
= one_one_rat ) ).
% fact_1
thf(fact_4251_fact__1,axiom,
( ( semiri1406184849735516958ct_int @ one_one_nat )
= one_one_int ) ).
% fact_1
thf(fact_4252_fact__1,axiom,
( ( semiri1408675320244567234ct_nat @ one_one_nat )
= one_one_nat ) ).
% fact_1
thf(fact_4253_fact__1,axiom,
( ( semiri2265585572941072030t_real @ one_one_nat )
= one_one_real ) ).
% fact_1
thf(fact_4254_root__0,axiom,
! [X: real] :
( ( root @ zero_zero_nat @ X )
= zero_zero_real ) ).
% root_0
thf(fact_4255_real__root__eq__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= ( root @ N @ Y ) )
= ( X = Y ) ) ) ).
% real_root_eq_iff
thf(fact_4256_log__one,axiom,
! [A: real] :
( ( log2 @ A @ one_one_real )
= zero_zero_real ) ).
% log_one
thf(fact_4257_exp__eq__one__iff,axiom,
! [X: real] :
( ( ( exp_real @ X )
= one_one_real )
= ( X = zero_zero_real ) ) ).
% exp_eq_one_iff
thf(fact_4258_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_4259_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_4260_fact__Suc__0,axiom,
( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
= one_one_complex ) ).
% fact_Suc_0
thf(fact_4261_fact__Suc__0,axiom,
( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
= one_one_rat ) ).
% fact_Suc_0
thf(fact_4262_fact__Suc__0,axiom,
( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
% fact_Suc_0
thf(fact_4263_fact__Suc__0,axiom,
( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% fact_Suc_0
thf(fact_4264_fact__Suc__0,axiom,
( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
= one_one_real ) ).
% fact_Suc_0
thf(fact_4265_fact__Suc,axiom,
! [N: nat] :
( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% fact_Suc
thf(fact_4266_fact__Suc,axiom,
! [N: nat] :
( ( semiri3624122377584611663nteger @ ( suc @ N ) )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( suc @ N ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% fact_Suc
thf(fact_4267_fact__Suc,axiom,
! [N: nat] :
( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% fact_Suc
thf(fact_4268_fact__Suc,axiom,
! [N: nat] :
( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% fact_Suc
thf(fact_4269_fact__Suc,axiom,
! [N: nat] :
( ( semiri2265585572941072030t_real @ ( suc @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% fact_Suc
thf(fact_4270_real__root__eq__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ) ).
% real_root_eq_0_iff
thf(fact_4271_real__root__less__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% real_root_less_iff
thf(fact_4272_real__root__le__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% real_root_le_iff
thf(fact_4273_real__root__eq__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= one_one_real )
= ( X = one_one_real ) ) ) ).
% real_root_eq_1_iff
thf(fact_4274_real__root__one,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ one_one_real )
= one_one_real ) ) ).
% real_root_one
thf(fact_4275_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 @ ( log2 @ A @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_4276_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 @ ( log2 @ A @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_4277_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 @ ( log2 @ A @ X ) )
= ( ord_less_real @ A @ X ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_4278_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 @ ( log2 @ A @ X ) @ one_one_real )
= ( ord_less_real @ X @ A ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_4279_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 @ ( log2 @ A @ X ) @ ( log2 @ A @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_4280_log__eq__one,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log2 @ A @ A )
= one_one_real ) ) ) ).
% log_eq_one
thf(fact_4281_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_4282_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_4283_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_4284_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_4285_real__root__lt__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% real_root_lt_0_iff
thf(fact_4286_real__root__gt__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
= ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_4287_real__root__le__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% real_root_le_0_iff
thf(fact_4288_real__root__ge__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
= ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_4289_real__root__lt__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% real_root_lt_1_iff
thf(fact_4290_real__root__gt__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
= ( ord_less_real @ one_one_real @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_4291_real__root__le__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% real_root_le_1_iff
thf(fact_4292_real__root__ge__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
= ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_4293_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 @ ( log2 @ A @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_4294_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 @ ( log2 @ A @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_4295_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 @ ( log2 @ A @ X ) )
= ( ord_less_eq_real @ A @ X ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_4296_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 @ ( log2 @ A @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ A ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_4297_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 @ ( log2 @ A @ X ) @ ( log2 @ A @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_4298_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 @ ( log2 @ A @ X ) )
= X ) ) ) ) ).
% powr_log_cancel
thf(fact_4299_log__powr__cancel,axiom,
! [A: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log2 @ A @ ( powr_real @ A @ Y ) )
= Y ) ) ) ).
% log_powr_cancel
thf(fact_4300_real__root__pow__pos2,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_4301_log__pow__cancel,axiom,
! [A: real,B: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log2 @ A @ ( power_power_real @ A @ B ) )
= ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% log_pow_cancel
thf(fact_4302_log__ln,axiom,
( ln_ln_real
= ( log2 @ ( exp_real @ one_one_real ) ) ) ).
% log_ln
thf(fact_4303_real__root__minus,axiom,
! [N: nat,X: real] :
( ( root @ N @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( root @ N @ X ) ) ) ).
% real_root_minus
thf(fact_4304_exp__not__eq__zero,axiom,
! [X: complex] :
( ( exp_complex @ X )
!= zero_zero_complex ) ).
% exp_not_eq_zero
thf(fact_4305_exp__not__eq__zero,axiom,
! [X: real] :
( ( exp_real @ X )
!= zero_zero_real ) ).
% exp_not_eq_zero
thf(fact_4306_fact__nonzero,axiom,
! [N: nat] :
( ( semiri773545260158071498ct_rat @ N )
!= zero_zero_rat ) ).
% fact_nonzero
thf(fact_4307_fact__nonzero,axiom,
! [N: nat] :
( ( semiri1406184849735516958ct_int @ N )
!= zero_zero_int ) ).
% fact_nonzero
thf(fact_4308_fact__nonzero,axiom,
! [N: nat] :
( ( semiri1408675320244567234ct_nat @ N )
!= zero_zero_nat ) ).
% fact_nonzero
thf(fact_4309_fact__nonzero,axiom,
! [N: nat] :
( ( semiri2265585572941072030t_real @ N )
!= zero_zero_real ) ).
% fact_nonzero
thf(fact_4310_real__root__inverse,axiom,
! [N: nat,X: real] :
( ( root @ N @ ( inverse_inverse_real @ X ) )
= ( inverse_inverse_real @ ( root @ N @ X ) ) ) ).
% real_root_inverse
thf(fact_4311_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_4312_fact__less__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% fact_less_mono_nat
thf(fact_4313_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% fact_ge_zero
thf(fact_4314_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_ge_zero
thf(fact_4315_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_zero
thf(fact_4316_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% fact_ge_zero
thf(fact_4317_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% fact_not_neg
thf(fact_4318_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% fact_not_neg
thf(fact_4319_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% fact_not_neg
thf(fact_4320_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% fact_not_neg
thf(fact_4321_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% fact_gt_zero
thf(fact_4322_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_gt_zero
thf(fact_4323_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_gt_zero
thf(fact_4324_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% fact_gt_zero
thf(fact_4325_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_4326_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_4327_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_4328_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_4329_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% fact_ge_1
thf(fact_4330_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_ge_1
thf(fact_4331_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_1
thf(fact_4332_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% fact_ge_1
thf(fact_4333_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_4334_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_4335_powr__def,axiom,
( powr_real
= ( ^ [X4: real,A3: real] : ( if_real @ ( X4 = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A3 @ ( ln_ln_real @ X4 ) ) ) ) ) ) ).
% powr_def
thf(fact_4336_exp__minus,axiom,
! [X: real] :
( ( exp_real @ ( uminus_uminus_real @ X ) )
= ( inverse_inverse_real @ ( exp_real @ X ) ) ) ).
% exp_minus
thf(fact_4337_exp__minus,axiom,
! [X: complex] :
( ( exp_complex @ ( uminus1482373934393186551omplex @ X ) )
= ( invers8013647133539491842omplex @ ( exp_complex @ X ) ) ) ).
% exp_minus
thf(fact_4338_pochhammer__fact,axiom,
( semiri5044797733671781792omplex
= ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).
% pochhammer_fact
thf(fact_4339_pochhammer__fact,axiom,
( semiri773545260158071498ct_rat
= ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).
% pochhammer_fact
thf(fact_4340_pochhammer__fact,axiom,
( semiri1406184849735516958ct_int
= ( comm_s4660882817536571857er_int @ one_one_int ) ) ).
% pochhammer_fact
thf(fact_4341_pochhammer__fact,axiom,
( semiri1408675320244567234ct_nat
= ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).
% pochhammer_fact
thf(fact_4342_pochhammer__fact,axiom,
( semiri2265585572941072030t_real
= ( comm_s7457072308508201937r_real @ one_one_real ) ) ).
% pochhammer_fact
thf(fact_4343_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 )
=> ( ( log2 @ A @ X )
= ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log2 @ B @ X ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_4344_ln__root,axiom,
! [N: nat,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ln_ln_real @ ( root @ N @ B ) )
= ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% ln_root
thf(fact_4345_log__root,axiom,
! [N: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( log2 @ B @ ( root @ N @ A ) )
= ( divide_divide_real @ ( log2 @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_root
thf(fact_4346_log__base__root,axiom,
! [N: nat,B: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( log2 @ ( root @ N @ B ) @ X )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ B @ X ) ) ) ) ) ).
% log_base_root
thf(fact_4347_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_4348_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_4349_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_4350_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_4351_real__root__less__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_4352_real__root__le__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_4353_fact__ge__Suc__0__nat,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_Suc_0_nat
thf(fact_4354_real__root__power,axiom,
! [N: nat,X: real,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( power_power_real @ X @ K ) )
= ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).
% real_root_power
thf(fact_4355_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_4356_real__root__abs,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( abs_abs_real @ X ) )
= ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).
% real_root_abs
thf(fact_4357_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_4358_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_4359_exp__minus__inverse,axiom,
! [X: complex] :
( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
= one_one_complex ) ).
% exp_minus_inverse
thf(fact_4360_exp__of__nat2__mult,axiom,
! [X: complex,N: nat] :
( ( exp_complex @ ( times_times_complex @ X @ ( semiri8010041392384452111omplex @ N ) ) )
= ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% exp_of_nat2_mult
thf(fact_4361_exp__of__nat2__mult,axiom,
! [X: real,N: nat] :
( ( exp_real @ ( times_times_real @ X @ ( semiri5074537144036343181t_real @ N ) ) )
= ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% exp_of_nat2_mult
thf(fact_4362_exp__of__nat__mult,axiom,
! [N: nat,X: complex] :
( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X ) )
= ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% exp_of_nat_mult
thf(fact_4363_exp__of__nat__mult,axiom,
! [N: nat,X: real] :
( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) )
= ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% exp_of_nat_mult
thf(fact_4364_fact__less__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% fact_less_mono
thf(fact_4365_fact__less__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% fact_less_mono
thf(fact_4366_fact__less__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% fact_less_mono
thf(fact_4367_fact__less__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% fact_less_mono
thf(fact_4368_fact__le__power,axiom,
! [N: nat] : ( ord_le3102999989581377725nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri4939895301339042750nteger @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_4369_fact__le__power,axiom,
! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_4370_fact__le__power,axiom,
! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_4371_fact__le__power,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_4372_fact__le__power,axiom,
! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_4373_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or5199638295745620268an_rat @ A @ B ) @ ( set_or5199638295745620268an_rat @ C @ D ) )
= ( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ A )
& ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_4374_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1633881224788618240n_real @ A @ B ) @ ( set_or1633881224788618240n_real @ C @ D ) )
= ( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_4375_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_4376_real__root__gt__zero,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_4377_real__root__strict__decreasing,axiom,
! [N: nat,N5: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_4378_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_4379_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_4380_log__base__change,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log2 @ B @ X )
= ( divide_divide_real @ ( log2 @ A @ X ) @ ( log2 @ A @ B ) ) ) ) ) ).
% log_base_change
thf(fact_4381_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_4382_root__abs__power,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y @ N ) ) )
= ( abs_abs_real @ Y ) ) ) ).
% root_abs_power
thf(fact_4383_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_4384_lemma__exp__total,axiom,
! [Y: real] :
( ( ord_less_eq_real @ one_one_real @ Y )
=> ? [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
& ( ord_less_eq_real @ X3 @ ( minus_minus_real @ Y @ one_one_real ) )
& ( ( exp_real @ X3 )
= Y ) ) ) ).
% lemma_exp_total
thf(fact_4385_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 @ ( log2 @ B @ X ) ) ) ) ) ).
% powr_less_iff
thf(fact_4386_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 @ ( log2 @ B @ X ) @ Y ) ) ) ) ).
% less_powr_iff
thf(fact_4387_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 @ ( log2 @ B @ X ) @ Y )
= ( ord_less_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% log_less_iff
thf(fact_4388_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 @ ( log2 @ B @ X ) )
= ( ord_less_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% less_log_iff
thf(fact_4389_log__of__power__eq,axiom,
! [M: nat,B: real,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( semiri5074537144036343181t_real @ N )
= ( log2 @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% log_of_power_eq
thf(fact_4390_less__log__of__power,axiom,
! [B: real,N: nat,M: real] :
( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ B @ M ) ) ) ) ).
% less_log_of_power
thf(fact_4391_fact__div__fact__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq_nat @ R2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% fact_div_fact_le_pow
thf(fact_4392_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or5199638295745620268an_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
= ( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ A )
& ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_4393_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1633881224788618240n_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_4394_real__root__pos__pos,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_4395_real__root__strict__increasing,axiom,
! [N: nat,N5: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_4396_real__root__decreasing,axiom,
! [N: nat,N5: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_4397_real__root__pow__pos,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos
thf(fact_4398_real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( root @ N @ ( power_power_real @ X @ N ) )
= X ) ) ) ).
% real_root_power_cancel
thf(fact_4399_real__root__pos__unique,axiom,
! [N: nat,Y: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( power_power_real @ Y @ N )
= X )
=> ( ( root @ N @ X )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_4400_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_4401_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_4402_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 )
=> ( ( log2 @ A @ ( times_times_real @ X @ Y ) )
= ( plus_plus_real @ ( log2 @ A @ X ) @ ( log2 @ A @ Y ) ) ) ) ) ) ) ).
% log_mult
thf(fact_4403_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 @ ( log2 @ B @ X ) )
= ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% le_log_iff
thf(fact_4404_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 @ ( log2 @ B @ X ) @ Y )
= ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% log_le_iff
thf(fact_4405_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 @ ( log2 @ B @ X ) @ Y ) ) ) ) ).
% le_powr_iff
thf(fact_4406_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 @ ( log2 @ B @ X ) ) ) ) ) ).
% powr_le_iff
thf(fact_4407_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 )
=> ( ( log2 @ A @ ( divide_divide_real @ X @ Y ) )
= ( minus_minus_real @ ( log2 @ A @ X ) @ ( log2 @ A @ Y ) ) ) ) ) ) ) ).
% log_divide
thf(fact_4408_le__log__of__power,axiom,
! [B: real,N: nat,M: real] :
( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ B @ M ) ) ) ) ).
% le_log_of_power
thf(fact_4409_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 )
=> ( ( log2 @ A @ ( inverse_inverse_real @ X ) )
= ( uminus_uminus_real @ ( log2 @ A @ X ) ) ) ) ) ) ).
% log_inverse
thf(fact_4410_exp__divide__power__eq,axiom,
! [N: nat,X: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
= ( exp_complex @ X ) ) ) ).
% exp_divide_power_eq
thf(fact_4411_exp__divide__power__eq,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
= ( exp_real @ X ) ) ) ).
% exp_divide_power_eq
thf(fact_4412_real__root__increasing,axiom,
! [N: nat,N5: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_4413_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_4414_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_4415_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_4416_log__of__power__less,axiom,
! [M: nat,B: real,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_real @ ( log2 @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_4417_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 @ ( log2 @ B @ X ) )
= ( log2 @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_4418_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 @ ( log2 @ B @ X ) @ Y )
= ( log2 @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_4419_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 @ ( log2 @ B @ X ) )
= ( log2 @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_4420_fact__num__eq__if,axiom,
( semiri5044797733671781792omplex
= ( ^ [M4: nat] : ( if_complex @ ( M4 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M4 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_4421_fact__num__eq__if,axiom,
( semiri1406184849735516958ct_int
= ( ^ [M4: nat] : ( if_int @ ( M4 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_4422_fact__num__eq__if,axiom,
( semiri3624122377584611663nteger
= ( ^ [M4: nat] : ( if_Code_integer @ ( M4 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M4 ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_4423_fact__num__eq__if,axiom,
( semiri773545260158071498ct_rat
= ( ^ [M4: nat] : ( if_rat @ ( M4 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M4 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_4424_fact__num__eq__if,axiom,
( semiri1408675320244567234ct_nat
= ( ^ [M4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_4425_fact__num__eq__if,axiom,
( semiri2265585572941072030t_real
= ( ^ [M4: nat] : ( if_real @ ( M4 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_4426_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri1406184849735516958ct_int @ N )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_4427_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri3624122377584611663nteger @ N )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_4428_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri773545260158071498ct_rat @ N )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_4429_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri1408675320244567234ct_nat @ N )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_4430_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri2265585572941072030t_real @ N )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_4431_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ N )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% pochhammer_same
thf(fact_4432_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% pochhammer_same
thf(fact_4433_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% pochhammer_same
thf(fact_4434_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% pochhammer_same
thf(fact_4435_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ N )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% pochhammer_same
thf(fact_4436_log__of__power__le,axiom,
! [M: nat,B: real,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_eq_real @ ( log2 @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_4437_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 @ ( log2 @ B @ X ) @ Y )
= ( log2 @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_4438_fact__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
= ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_4439_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 @ ( log2 @ 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_4440_tanh__altdef,axiom,
( tanh_real
= ( ^ [X4: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X4 ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) @ ( plus_plus_real @ ( exp_real @ X4 ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_4441_tanh__altdef,axiom,
( tanh_complex
= ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_4442_split__root,axiom,
! [P: real > $o,N: nat,X: real] :
( ( P @ ( root @ N @ X ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_real ) )
& ( ( ord_less_nat @ zero_zero_nat @ N )
=> ! [Y5: real] :
( ( ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N ) )
= X )
=> ( P @ Y5 ) ) ) ) ) ).
% split_root
thf(fact_4443_floor__divide__upper,axiom,
! [Q4: real,P5: real] :
( ( ord_less_real @ zero_zero_real @ Q4 )
=> ( ord_less_real @ P5 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P5 @ Q4 ) ) ) @ one_one_real ) @ Q4 ) ) ) ).
% floor_divide_upper
thf(fact_4444_floor__divide__upper,axiom,
! [Q4: rat,P5: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q4 )
=> ( ord_less_rat @ P5 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P5 @ Q4 ) ) ) @ one_one_rat ) @ Q4 ) ) ) ).
% floor_divide_upper
thf(fact_4445_split__pos__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
= ( ! [I3: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_4446_split__neg__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
= ( ! [I3: int,J3: int] :
( ( ( ord_less_int @ K @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_4447_sgn__sgn,axiom,
! [A: real] :
( ( sgn_sgn_real @ ( sgn_sgn_real @ A ) )
= ( sgn_sgn_real @ A ) ) ).
% sgn_sgn
thf(fact_4448_sgn__sgn,axiom,
! [A: int] :
( ( sgn_sgn_int @ ( sgn_sgn_int @ A ) )
= ( sgn_sgn_int @ A ) ) ).
% sgn_sgn
thf(fact_4449_sgn__sgn,axiom,
! [A: complex] :
( ( sgn_sgn_complex @ ( sgn_sgn_complex @ A ) )
= ( sgn_sgn_complex @ A ) ) ).
% sgn_sgn
thf(fact_4450_sgn__sgn,axiom,
! [A: code_integer] :
( ( sgn_sgn_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
= ( sgn_sgn_Code_integer @ A ) ) ).
% sgn_sgn
thf(fact_4451_sgn__sgn,axiom,
! [A: rat] :
( ( sgn_sgn_rat @ ( sgn_sgn_rat @ A ) )
= ( sgn_sgn_rat @ A ) ) ).
% sgn_sgn
thf(fact_4452_mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mod_0
thf(fact_4453_mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mod_0
thf(fact_4454_mod__0,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
= zero_z3403309356797280102nteger ) ).
% mod_0
thf(fact_4455_mod__0,axiom,
! [A: code_natural] :
( ( modulo8411746178871703098atural @ zero_z2226904508553997617atural @ A )
= zero_z2226904508553997617atural ) ).
% mod_0
thf(fact_4456_mod__by__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ zero_zero_int )
= A ) ).
% mod_by_0
thf(fact_4457_mod__by__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ zero_zero_nat )
= A ) ).
% mod_by_0
thf(fact_4458_mod__by__0,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
= A ) ).
% mod_by_0
thf(fact_4459_mod__by__0,axiom,
! [A: code_natural] :
( ( modulo8411746178871703098atural @ A @ zero_z2226904508553997617atural )
= A ) ).
% mod_by_0
thf(fact_4460_mod__self,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ A )
= zero_zero_int ) ).
% mod_self
thf(fact_4461_mod__self,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ A )
= zero_zero_nat ) ).
% mod_self
thf(fact_4462_mod__self,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ A )
= zero_z3403309356797280102nteger ) ).
% mod_self
thf(fact_4463_mod__self,axiom,
! [A: code_natural] :
( ( modulo8411746178871703098atural @ A @ A )
= zero_z2226904508553997617atural ) ).
% mod_self
thf(fact_4464_bits__mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_mod_0
thf(fact_4465_bits__mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_4466_bits__mod__0,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
= zero_z3403309356797280102nteger ) ).
% bits_mod_0
thf(fact_4467_bits__mod__0,axiom,
! [A: code_natural] :
( ( modulo8411746178871703098atural @ zero_z2226904508553997617atural @ A )
= zero_z2226904508553997617atural ) ).
% bits_mod_0
thf(fact_4468_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_4469_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_4470_mod__add__self2,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_add_self2
thf(fact_4471_mod__add__self2,axiom,
! [A: code_natural,B: code_natural] :
( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ B )
= ( modulo8411746178871703098atural @ A @ B ) ) ).
% mod_add_self2
thf(fact_4472_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_4473_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_4474_mod__add__self1,axiom,
! [B: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_add_self1
thf(fact_4475_mod__add__self1,axiom,
! [B: code_natural,A: code_natural] :
( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ B @ A ) @ B )
= ( modulo8411746178871703098atural @ A @ B ) ) ).
% mod_add_self1
thf(fact_4476_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_4477_minus__mod__self2,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% minus_mod_self2
thf(fact_4478_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_4479_mod__minus__minus,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% mod_minus_minus
thf(fact_4480_sgn__0,axiom,
( ( sgn_sgn_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% sgn_0
thf(fact_4481_sgn__0,axiom,
( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% sgn_0
thf(fact_4482_sgn__0,axiom,
( ( sgn_sgn_real @ zero_zero_real )
= zero_zero_real ) ).
% sgn_0
thf(fact_4483_sgn__0,axiom,
( ( sgn_sgn_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% sgn_0
thf(fact_4484_sgn__0,axiom,
( ( sgn_sgn_int @ zero_zero_int )
= zero_zero_int ) ).
% sgn_0
thf(fact_4485_sgn__1,axiom,
( ( sgn_sgn_real @ one_one_real )
= one_one_real ) ).
% sgn_1
thf(fact_4486_sgn__1,axiom,
( ( sgn_sgn_int @ one_one_int )
= one_one_int ) ).
% sgn_1
thf(fact_4487_sgn__1,axiom,
( ( sgn_sgn_complex @ one_one_complex )
= one_one_complex ) ).
% sgn_1
thf(fact_4488_sgn__1,axiom,
( ( sgn_sgn_Code_integer @ one_one_Code_integer )
= one_one_Code_integer ) ).
% sgn_1
thf(fact_4489_sgn__1,axiom,
( ( sgn_sgn_rat @ one_one_rat )
= one_one_rat ) ).
% sgn_1
thf(fact_4490_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_4491_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_4492_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_4493_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_4494_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_4495_inverse__sgn,axiom,
! [A: real] :
( ( inverse_inverse_real @ ( sgn_sgn_real @ A ) )
= ( sgn_sgn_real @ A ) ) ).
% inverse_sgn
thf(fact_4496_inverse__sgn,axiom,
! [A: rat] :
( ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) )
= ( sgn_sgn_rat @ A ) ) ).
% inverse_sgn
thf(fact_4497_sgn__inverse,axiom,
! [A: real] :
( ( sgn_sgn_real @ ( inverse_inverse_real @ A ) )
= ( inverse_inverse_real @ ( sgn_sgn_real @ A ) ) ) ).
% sgn_inverse
thf(fact_4498_sgn__inverse,axiom,
! [A: complex] :
( ( sgn_sgn_complex @ ( invers8013647133539491842omplex @ A ) )
= ( invers8013647133539491842omplex @ ( sgn_sgn_complex @ A ) ) ) ).
% sgn_inverse
thf(fact_4499_sgn__inverse,axiom,
! [A: rat] :
( ( sgn_sgn_rat @ ( inverse_inverse_rat @ A ) )
= ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) ) ) ).
% sgn_inverse
thf(fact_4500_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_4501_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_4502_floor__of__int,axiom,
! [Z2: int] :
( ( archim6058952711729229775r_real @ ( ring_1_of_int_real @ Z2 ) )
= Z2 ) ).
% floor_of_int
thf(fact_4503_floor__of__int,axiom,
! [Z2: int] :
( ( archim3151403230148437115or_rat @ ( ring_1_of_int_rat @ Z2 ) )
= Z2 ) ).
% floor_of_int
thf(fact_4504_tanh__0,axiom,
( ( tanh_real @ zero_zero_real )
= zero_zero_real ) ).
% tanh_0
thf(fact_4505_tanh__minus,axiom,
! [X: real] :
( ( tanh_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( tanh_real @ X ) ) ) ).
% tanh_minus
thf(fact_4506_tanh__minus,axiom,
! [X: complex] :
( ( tanh_complex @ ( uminus1482373934393186551omplex @ X ) )
= ( uminus1482373934393186551omplex @ ( tanh_complex @ X ) ) ) ).
% tanh_minus
thf(fact_4507_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_4508_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_4509_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_4510_mod__mult__self1__is__0,axiom,
! [B: code_natural,A: code_natural] :
( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ B @ A ) @ B )
= zero_z2226904508553997617atural ) ).
% mod_mult_self1_is_0
thf(fact_4511_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_4512_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_4513_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_4514_mod__mult__self2__is__0,axiom,
! [A: code_natural,B: code_natural] :
( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ B )
= zero_z2226904508553997617atural ) ).
% mod_mult_self2_is_0
thf(fact_4515_mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_4516_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_4517_mod__by__1,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% mod_by_1
thf(fact_4518_mod__by__1,axiom,
! [A: code_natural] :
( ( modulo8411746178871703098atural @ A @ one_one_Code_natural )
= zero_z2226904508553997617atural ) ).
% mod_by_1
thf(fact_4519_bits__mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_4520_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_4521_bits__mod__by__1,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% bits_mod_by_1
thf(fact_4522_bits__mod__by__1,axiom,
! [A: code_natural] :
( ( modulo8411746178871703098atural @ A @ one_one_Code_natural )
= zero_z2226904508553997617atural ) ).
% bits_mod_by_1
thf(fact_4523_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_4524_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_4525_mod__div__trivial,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
= zero_z3403309356797280102nteger ) ).
% mod_div_trivial
thf(fact_4526_mod__div__trivial,axiom,
! [A: code_natural,B: code_natural] :
( ( divide5121882707175180666atural @ ( modulo8411746178871703098atural @ A @ B ) @ B )
= zero_z2226904508553997617atural ) ).
% mod_div_trivial
thf(fact_4527_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_4528_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_4529_bits__mod__div__trivial,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
= zero_z3403309356797280102nteger ) ).
% bits_mod_div_trivial
thf(fact_4530_bits__mod__div__trivial,axiom,
! [A: code_natural,B: code_natural] :
( ( divide5121882707175180666atural @ ( modulo8411746178871703098atural @ A @ B ) @ B )
= zero_z2226904508553997617atural ) ).
% bits_mod_div_trivial
thf(fact_4531_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_4532_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_4533_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_4534_mod__mult__self4,axiom,
! [B: code_natural,C: code_natural,A: code_natural] :
( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ C ) @ A ) @ B )
= ( modulo8411746178871703098atural @ A @ B ) ) ).
% mod_mult_self4
thf(fact_4535_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_4536_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_4537_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_4538_mod__mult__self3,axiom,
! [C: code_natural,B: code_natural,A: code_natural] :
( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ C @ B ) @ A ) @ B )
= ( modulo8411746178871703098atural @ A @ B ) ) ).
% mod_mult_self3
thf(fact_4539_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_4540_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_4541_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_4542_mod__mult__self2,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ B @ C ) ) @ B )
= ( modulo8411746178871703098atural @ A @ B ) ) ).
% mod_mult_self2
thf(fact_4543_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_4544_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_4545_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_4546_mod__mult__self1,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ C @ B ) ) @ B )
= ( modulo8411746178871703098atural @ A @ B ) ) ).
% mod_mult_self1
thf(fact_4547_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_4548_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_4549_sgn__less,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
= ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% sgn_less
thf(fact_4550_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_4551_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_4552_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_4553_sgn__greater,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% sgn_greater
thf(fact_4554_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_4555_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_4556_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_4557_floor__zero,axiom,
( ( archim6058952711729229775r_real @ zero_zero_real )
= zero_zero_int ) ).
% floor_zero
thf(fact_4558_floor__zero,axiom,
( ( archim3151403230148437115or_rat @ zero_zero_rat )
= zero_zero_int ) ).
% floor_zero
thf(fact_4559_floor__one,axiom,
( ( archim6058952711729229775r_real @ one_one_real )
= one_one_int ) ).
% floor_one
thf(fact_4560_floor__one,axiom,
( ( archim3151403230148437115or_rat @ one_one_rat )
= one_one_int ) ).
% floor_one
thf(fact_4561_floor__of__nat,axiom,
! [N: nat] :
( ( archim6058952711729229775r_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% floor_of_nat
thf(fact_4562_floor__of__nat,axiom,
! [N: nat] :
( ( archim3151403230148437115or_rat @ ( semiri681578069525770553at_rat @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% floor_of_nat
thf(fact_4563_mod__minus1__right,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% mod_minus1_right
thf(fact_4564_mod__minus1__right,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% mod_minus1_right
thf(fact_4565_sgn__pos,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( sgn_sgn_Code_integer @ A )
= one_one_Code_integer ) ) ).
% sgn_pos
thf(fact_4566_sgn__pos,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( sgn_sgn_real @ A )
= one_one_real ) ) ).
% sgn_pos
thf(fact_4567_sgn__pos,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( sgn_sgn_rat @ A )
= one_one_rat ) ) ).
% sgn_pos
thf(fact_4568_sgn__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( sgn_sgn_int @ A )
= one_one_int ) ) ).
% sgn_pos
thf(fact_4569_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_4570_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_4571_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_4572_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_4573_floor__uminus__of__int,axiom,
! [Z2: int] :
( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( ring_1_of_int_real @ Z2 ) ) )
= ( uminus_uminus_int @ Z2 ) ) ).
% floor_uminus_of_int
thf(fact_4574_floor__uminus__of__int,axiom,
! [Z2: int] :
( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z2 ) ) )
= ( uminus_uminus_int @ Z2 ) ) ).
% floor_uminus_of_int
thf(fact_4575_floor__diff__of__int,axiom,
! [X: real,Z2: int] :
( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z2 ) ) )
= ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ Z2 ) ) ).
% floor_diff_of_int
thf(fact_4576_floor__diff__of__int,axiom,
! [X: rat,Z2: int] :
( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) )
= ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 ) ) ).
% floor_diff_of_int
thf(fact_4577_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_4578_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_4579_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_4580_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_4581_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_4582_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_4583_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_4584_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_4585_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_4586_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_4587_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_4588_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_4589_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_4590_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_4591_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_4592_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_4593_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_4594_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_4595_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_4596_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_4597_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_4598_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_4599_of__nat__mod,axiom,
! [M: nat,N: nat] :
( ( semiri3763490453095760265atural @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo8411746178871703098atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) ) ).
% of_nat_mod
thf(fact_4600_of__nat__mod,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mod
thf(fact_4601_of__nat__mod,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mod
thf(fact_4602_of__nat__mod,axiom,
! [M: nat,N: nat] :
( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% of_nat_mod
thf(fact_4603_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_4604_sgn__0__0,axiom,
! [A: code_integer] :
( ( ( sgn_sgn_Code_integer @ A )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% sgn_0_0
thf(fact_4605_sgn__0__0,axiom,
! [A: real] :
( ( ( sgn_sgn_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% sgn_0_0
thf(fact_4606_sgn__0__0,axiom,
! [A: rat] :
( ( ( sgn_sgn_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% sgn_0_0
thf(fact_4607_sgn__0__0,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% sgn_0_0
thf(fact_4608_sgn__eq__0__iff,axiom,
! [A: complex] :
( ( ( sgn_sgn_complex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% sgn_eq_0_iff
thf(fact_4609_sgn__eq__0__iff,axiom,
! [A: code_integer] :
( ( ( sgn_sgn_Code_integer @ A )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% sgn_eq_0_iff
thf(fact_4610_sgn__eq__0__iff,axiom,
! [A: real] :
( ( ( sgn_sgn_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% sgn_eq_0_iff
thf(fact_4611_sgn__eq__0__iff,axiom,
! [A: rat] :
( ( ( sgn_sgn_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% sgn_eq_0_iff
thf(fact_4612_sgn__eq__0__iff,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% sgn_eq_0_iff
thf(fact_4613_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_4614_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_4615_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_4616_mod__add__eq,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C ) @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
= ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_4617_mod__add__cong,axiom,
! [A: int,C: int,A5: int,B: int,B5: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A5 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B5 @ C ) )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_4618_mod__add__cong,axiom,
! [A: nat,C: nat,A5: nat,B: nat,B5: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A5 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B5 @ C ) )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_4619_mod__add__cong,axiom,
! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ A5 @ C ) )
=> ( ( ( modulo364778990260209775nteger @ B @ C )
= ( modulo364778990260209775nteger @ B5 @ C ) )
=> ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_4620_mod__add__cong,axiom,
! [A: code_natural,C: code_natural,A5: code_natural,B: code_natural,B5: code_natural] :
( ( ( modulo8411746178871703098atural @ A @ C )
= ( modulo8411746178871703098atural @ A5 @ C ) )
=> ( ( ( modulo8411746178871703098atural @ B @ C )
= ( modulo8411746178871703098atural @ B5 @ C ) )
=> ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
= ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_4621_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_4622_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_4623_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_4624_mod__add__left__eq,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C ) @ B ) @ C )
= ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_4625_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_4626_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_4627_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_4628_mod__add__right__eq,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
= ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_4629_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_4630_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_4631_mod__diff__cong,axiom,
! [A: int,C: int,A5: int,B: int,B5: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A5 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B5 @ C ) )
=> ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_diff_cong
thf(fact_4632_mod__diff__cong,axiom,
! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ A5 @ C ) )
=> ( ( ( modulo364778990260209775nteger @ B @ C )
= ( modulo364778990260209775nteger @ B5 @ C ) )
=> ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
= ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_diff_cong
thf(fact_4633_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_4634_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_4635_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_4636_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_4637_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_4638_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_4639_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_4640_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_4641_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_4642_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_4643_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_4644_mod__minus__cong,axiom,
! [A: int,B: int,A5: int] :
( ( ( modulo_modulo_int @ A @ B )
= ( modulo_modulo_int @ A5 @ B ) )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A5 ) @ B ) ) ) ).
% mod_minus_cong
thf(fact_4645_mod__minus__cong,axiom,
! [A: code_integer,B: code_integer,A5: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ B )
= ( modulo364778990260209775nteger @ A5 @ B ) )
=> ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A5 ) @ B ) ) ) ).
% mod_minus_cong
thf(fact_4646_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_4647_mod__minus__right,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% mod_minus_right
thf(fact_4648_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_4649_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_4650_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_4651_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_4652_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_4653_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_4654_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_4655_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_4656_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_4657_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_4658_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_4659_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_4660_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_4661_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_4662_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_4663_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_4664_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_4665_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_4666_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_4667_mod__eq__self__iff__div__eq__0,axiom,
! [A: code_natural,B: code_natural] :
( ( ( modulo8411746178871703098atural @ A @ B )
= A )
= ( ( divide5121882707175180666atural @ A @ B )
= zero_z2226904508553997617atural ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_4668_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_4669_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_4670_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_4671_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_4672_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_4673_div__add1__eq,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
= ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ C ) @ ( divide5121882707175180666atural @ B @ C ) ) @ ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C ) @ ( modulo8411746178871703098atural @ B @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_4674_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_4675_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_4676_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_4677_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_4678_sgn__minus__1,axiom,
( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% sgn_minus_1
thf(fact_4679_sgn__minus__1,axiom,
( ( sgn_sgn_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% sgn_minus_1
thf(fact_4680_sgn__minus__1,axiom,
( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% sgn_minus_1
thf(fact_4681_sgn__minus__1,axiom,
( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% sgn_minus_1
thf(fact_4682_sgn__minus__1,axiom,
( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% sgn_minus_1
thf(fact_4683_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_4684_mult__sgn__abs,axiom,
! [X: real] :
( ( times_times_real @ ( sgn_sgn_real @ X ) @ ( abs_abs_real @ X ) )
= X ) ).
% mult_sgn_abs
thf(fact_4685_mult__sgn__abs,axiom,
! [X: rat] :
( ( times_times_rat @ ( sgn_sgn_rat @ X ) @ ( abs_abs_rat @ X ) )
= X ) ).
% mult_sgn_abs
thf(fact_4686_mult__sgn__abs,axiom,
! [X: int] :
( ( times_times_int @ ( sgn_sgn_int @ X ) @ ( abs_abs_int @ X ) )
= X ) ).
% mult_sgn_abs
thf(fact_4687_sgn__mult__abs,axiom,
! [A: complex] :
( ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( abs_abs_complex @ A ) )
= A ) ).
% sgn_mult_abs
thf(fact_4688_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_4689_sgn__mult__abs,axiom,
! [A: real] :
( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( abs_abs_real @ A ) )
= A ) ).
% sgn_mult_abs
thf(fact_4690_sgn__mult__abs,axiom,
! [A: rat] :
( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( abs_abs_rat @ A ) )
= A ) ).
% sgn_mult_abs
thf(fact_4691_sgn__mult__abs,axiom,
! [A: int] :
( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
= A ) ).
% sgn_mult_abs
thf(fact_4692_abs__mult__sgn,axiom,
! [A: complex] :
( ( times_times_complex @ ( abs_abs_complex @ A ) @ ( sgn_sgn_complex @ A ) )
= A ) ).
% abs_mult_sgn
thf(fact_4693_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_4694_abs__mult__sgn,axiom,
! [A: real] :
( ( times_times_real @ ( abs_abs_real @ A ) @ ( sgn_sgn_real @ A ) )
= A ) ).
% abs_mult_sgn
thf(fact_4695_abs__mult__sgn,axiom,
! [A: rat] :
( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( sgn_sgn_rat @ A ) )
= A ) ).
% abs_mult_sgn
thf(fact_4696_abs__mult__sgn,axiom,
! [A: int] :
( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
= A ) ).
% abs_mult_sgn
thf(fact_4697_linordered__idom__class_Oabs__sgn,axiom,
( abs_abs_Code_integer
= ( ^ [K3: code_integer] : ( times_3573771949741848930nteger @ K3 @ ( sgn_sgn_Code_integer @ K3 ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_4698_linordered__idom__class_Oabs__sgn,axiom,
( abs_abs_real
= ( ^ [K3: real] : ( times_times_real @ K3 @ ( sgn_sgn_real @ K3 ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_4699_linordered__idom__class_Oabs__sgn,axiom,
( abs_abs_rat
= ( ^ [K3: rat] : ( times_times_rat @ K3 @ ( sgn_sgn_rat @ K3 ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_4700_linordered__idom__class_Oabs__sgn,axiom,
( abs_abs_int
= ( ^ [K3: int] : ( times_times_int @ K3 @ ( sgn_sgn_int @ K3 ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_4701_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_4702_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_4703_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_4704_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_4705_floor__le__ceiling,axiom,
! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% floor_le_ceiling
thf(fact_4706_floor__le__ceiling,axiom,
! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim2889992004027027881ng_rat @ X ) ) ).
% floor_le_ceiling
thf(fact_4707_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_4708_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_4709_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_4710_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_4711_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_4712_zmod__eq__0__iff,axiom,
! [M: int,D: int] :
( ( ( modulo_modulo_int @ M @ D )
= zero_zero_int )
= ( ? [Q6: int] :
( M
= ( times_times_int @ D @ Q6 ) ) ) ) ).
% zmod_eq_0_iff
thf(fact_4713_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_4714_tanh__real__lt__1,axiom,
! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).
% tanh_real_lt_1
thf(fact_4715_le__floor__iff,axiom,
! [Z2: int,X: real] :
( ( ord_less_eq_int @ Z2 @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ) ).
% le_floor_iff
thf(fact_4716_le__floor__iff,axiom,
! [Z2: int,X: rat] :
( ( ord_less_eq_int @ Z2 @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ) ).
% le_floor_iff
thf(fact_4717_floor__less__iff,axiom,
! [X: real,Z2: int] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ Z2 )
= ( ord_less_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% floor_less_iff
thf(fact_4718_floor__less__iff,axiom,
! [X: rat,Z2: int] :
( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 )
= ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% floor_less_iff
thf(fact_4719_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_4720_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_4721_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_4722_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_4723_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_4724_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_4725_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_4726_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_4727_int__add__floor,axiom,
! [Z2: int,X: real] :
( ( plus_plus_int @ Z2 @ ( archim6058952711729229775r_real @ X ) )
= ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ) ) ).
% int_add_floor
thf(fact_4728_int__add__floor,axiom,
! [Z2: int,X: rat] :
( ( plus_plus_int @ Z2 @ ( archim3151403230148437115or_rat @ X ) )
= ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ) ) ).
% int_add_floor
thf(fact_4729_floor__add__int,axiom,
! [X: real,Z2: int] :
( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ Z2 )
= ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ) ) ).
% floor_add_int
thf(fact_4730_floor__add__int,axiom,
! [X: rat,Z2: int] :
( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 )
= ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ) ) ).
% floor_add_int
thf(fact_4731_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_4732_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_4733_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_4734_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_4735_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_4736_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_4737_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_4738_div__mult1__eq,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ C )
= ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ C ) ) @ ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_4739_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_4740_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_4741_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_4742_mult__div__mod__eq,axiom,
! [B: code_natural,A: code_natural] :
( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) @ ( modulo8411746178871703098atural @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_4743_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_4744_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_4745_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_4746_mod__mult__div__eq,axiom,
! [A: code_natural,B: code_natural] :
( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ B ) @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_4747_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_4748_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_4749_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_4750_mod__div__mult__eq,axiom,
! [A: code_natural,B: code_natural] :
( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ B ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_4751_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_4752_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_4753_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_4754_div__mult__mod__eq,axiom,
! [A: code_natural,B: code_natural] :
( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_4755_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_4756_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_4757_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_4758_mod__div__decomp,axiom,
! [A: code_natural,B: code_natural] :
( A
= ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_4759_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_4760_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_4761_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_4762_cancel__div__mod__rules_I1_J,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C )
= ( plus_p4538020629002901425atural @ A @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_4763_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_4764_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_4765_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_4766_cancel__div__mod__rules_I2_J,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C )
= ( plus_p4538020629002901425atural @ A @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_4767_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_4768_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_4769_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_4770_minus__mult__div__eq__mod,axiom,
! [A: code_natural,B: code_natural] :
( ( minus_7197305767214868737atural @ A @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) )
= ( modulo8411746178871703098atural @ A @ B ) ) ).
% minus_mult_div_eq_mod
thf(fact_4771_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_4772_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_4773_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_4774_minus__mod__eq__mult__div,axiom,
! [A: code_natural,B: code_natural] :
( ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) )
= ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_4775_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_4776_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_4777_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_4778_minus__mod__eq__div__mult,axiom,
! [A: code_natural,B: code_natural] :
( ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) )
= ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) ) ).
% minus_mod_eq_div_mult
thf(fact_4779_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_4780_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_4781_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_4782_minus__div__mult__eq__mod,axiom,
! [A: code_natural,B: code_natural] :
( ( minus_7197305767214868737atural @ A @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) )
= ( modulo8411746178871703098atural @ A @ B ) ) ).
% minus_div_mult_eq_mod
thf(fact_4783_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_4784_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_4785_ceiling__def,axiom,
( archim7802044766580827645g_real
= ( ^ [X4: real] : ( uminus_uminus_int @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X4 ) ) ) ) ) ).
% ceiling_def
thf(fact_4786_ceiling__def,axiom,
( archim2889992004027027881ng_rat
= ( ^ [X4: rat] : ( uminus_uminus_int @ ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ X4 ) ) ) ) ) ).
% ceiling_def
thf(fact_4787_floor__minus,axiom,
! [X: real] :
( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_int @ ( archim7802044766580827645g_real @ X ) ) ) ).
% floor_minus
thf(fact_4788_floor__minus,axiom,
! [X: rat] :
( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ X ) )
= ( uminus_uminus_int @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% floor_minus
thf(fact_4789_ceiling__minus,axiom,
! [X: real] :
( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_int @ ( archim6058952711729229775r_real @ X ) ) ) ).
% ceiling_minus
thf(fact_4790_ceiling__minus,axiom,
! [X: rat] :
( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ X ) )
= ( uminus_uminus_int @ ( archim3151403230148437115or_rat @ X ) ) ) ).
% ceiling_minus
thf(fact_4791_sgn__root,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( sgn_sgn_real @ ( root @ N @ X ) )
= ( sgn_sgn_real @ X ) ) ) ).
% sgn_root
thf(fact_4792_floor__power,axiom,
! [X: real,N: nat] :
( ( X
= ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
=> ( ( archim6058952711729229775r_real @ ( power_power_real @ X @ N ) )
= ( power_power_int @ ( archim6058952711729229775r_real @ X ) @ N ) ) ) ).
% floor_power
thf(fact_4793_floor__power,axiom,
! [X: rat,N: nat] :
( ( X
= ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
=> ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X @ N ) )
= ( power_power_int @ ( archim3151403230148437115or_rat @ X ) @ N ) ) ) ).
% floor_power
thf(fact_4794_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_4795_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_4796_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_4797_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_4798_fact__mod,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
= zero_zero_int ) ) ).
% fact_mod
thf(fact_4799_fact__mod,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
= zero_z3403309356797280102nteger ) ) ).
% fact_mod
thf(fact_4800_fact__mod,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo8411746178871703098atural @ ( semiri2447717529341329178atural @ N ) @ ( semiri2447717529341329178atural @ M ) )
= zero_z2226904508553997617atural ) ) ).
% fact_mod
thf(fact_4801_fact__mod,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
= zero_zero_nat ) ) ).
% fact_mod
thf(fact_4802_frac__def,axiom,
( archim2898591450579166408c_real
= ( ^ [X4: real] : ( minus_minus_real @ X4 @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) ) ) ) ).
% frac_def
thf(fact_4803_frac__def,axiom,
( archimedean_frac_rat
= ( ^ [X4: rat] : ( minus_minus_rat @ X4 @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) ) ) ) ).
% frac_def
thf(fact_4804_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_4805_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_4806_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_4807_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_4808_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_4809_zdiv__mono__strict,axiom,
! [A2: int,B2: int,N: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( modulo_modulo_int @ A2 @ N )
= zero_zero_int )
=> ( ( ( modulo_modulo_int @ B2 @ N )
= zero_zero_int )
=> ( ord_less_int @ ( divide_divide_int @ A2 @ N ) @ ( divide_divide_int @ B2 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_4810_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_4811_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_4812_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_4813_div__mod__decomp__int,axiom,
! [A2: int,N: int] :
( A2
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N ) @ N ) @ ( modulo_modulo_int @ A2 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_4814_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_4815_of__nat__floor,axiom,
! [R2: real] :
( ( ord_less_eq_real @ zero_zero_real @ R2 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R2 ) ) ) @ R2 ) ) ).
% of_nat_floor
thf(fact_4816_of__nat__floor,axiom,
! [R2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
=> ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R2 ) ) ) @ R2 ) ) ).
% of_nat_floor
thf(fact_4817_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_4818_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_4819_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_4820_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_4821_floor__divide__of__nat__eq,axiom,
! [M: nat,N: nat] :
( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% floor_divide_of_nat_eq
thf(fact_4822_floor__divide__of__nat__eq,axiom,
! [M: nat,N: nat] :
( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% floor_divide_of_nat_eq
thf(fact_4823_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_4824_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_4825_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_4826_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_4827_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_4828_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_4829_sgn__if,axiom,
( sgn_sgn_int
= ( ^ [X4: int] : ( if_int @ ( X4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% sgn_if
thf(fact_4830_sgn__if,axiom,
( sgn_sgn_real
= ( ^ [X4: real] : ( if_real @ ( X4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% sgn_if
thf(fact_4831_sgn__if,axiom,
( sgn_sgn_Code_integer
= ( ^ [X4: code_integer] : ( if_Code_integer @ ( X4 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X4 ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).
% sgn_if
thf(fact_4832_sgn__if,axiom,
( sgn_sgn_rat
= ( ^ [X4: rat] : ( if_rat @ ( X4 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X4 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% sgn_if
thf(fact_4833_mod__mult2__eq_H,axiom,
! [A: code_natural,M: nat,N: nat] :
( ( modulo8411746178871703098atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) )
= ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M ) @ ( modulo8411746178871703098atural @ ( divide5121882707175180666atural @ A @ ( semiri3763490453095760265atural @ M ) ) @ ( semiri3763490453095760265atural @ N ) ) ) @ ( modulo8411746178871703098atural @ A @ ( semiri3763490453095760265atural @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_4834_mod__mult2__eq_H,axiom,
! [A: int,M: nat,N: nat] :
( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_4835_mod__mult2__eq_H,axiom,
! [A: nat,M: nat,N: nat] :
( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_4836_mod__mult2__eq_H,axiom,
! [A: code_integer,M: nat,N: nat] :
( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_4837_floor__eq3,axiom,
! [N: nat,X: real] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= N ) ) ) ).
% floor_eq3
thf(fact_4838_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_4839_ceiling__altdef,axiom,
( archim7802044766580827645g_real
= ( ^ [X4: real] :
( if_int
@ ( X4
= ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) )
@ ( archim6058952711729229775r_real @ X4 )
@ ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int ) ) ) ) ).
% ceiling_altdef
thf(fact_4840_ceiling__altdef,axiom,
( archim2889992004027027881ng_rat
= ( ^ [X4: rat] :
( if_int
@ ( X4
= ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) )
@ ( archim3151403230148437115or_rat @ X4 )
@ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int ) ) ) ) ).
% ceiling_altdef
thf(fact_4841_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_4842_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_4843_floor__eq,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= N ) ) ) ).
% floor_eq
thf(fact_4844_real__of__int__floor__add__one__gt,axiom,
! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_gt
thf(fact_4845_real__of__int__floor__add__one__ge,axiom,
! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_ge
thf(fact_4846_real__of__int__floor__gt__diff__one,axiom,
! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% real_of_int_floor_gt_diff_one
thf(fact_4847_real__of__int__floor__ge__diff__one,axiom,
! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% real_of_int_floor_ge_diff_one
thf(fact_4848_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_4849_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_4850_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_4851_floor__unique,axiom,
! [Z2: int,X: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= Z2 ) ) ) ).
% floor_unique
thf(fact_4852_floor__unique,axiom,
! [Z2: int,X: rat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
=> ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) )
=> ( ( archim3151403230148437115or_rat @ X )
= Z2 ) ) ) ).
% floor_unique
thf(fact_4853_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_4854_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_4855_floor__split,axiom,
! [P: int > $o,T: real] :
( ( P @ ( archim6058952711729229775r_real @ T ) )
= ( ! [I3: int] :
( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I3 ) @ T )
& ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) ) )
=> ( P @ I3 ) ) ) ) ).
% floor_split
thf(fact_4856_floor__split,axiom,
! [P: int > $o,T: rat] :
( ( P @ ( archim3151403230148437115or_rat @ T ) )
= ( ! [I3: int] :
( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I3 ) @ T )
& ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) ) )
=> ( P @ I3 ) ) ) ) ).
% floor_split
thf(fact_4857_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_4858_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_4859_less__floor__iff,axiom,
! [Z2: int,X: real] :
( ( ord_less_int @ Z2 @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X ) ) ).
% less_floor_iff
thf(fact_4860_less__floor__iff,axiom,
! [Z2: int,X: rat] :
( ( ord_less_int @ Z2 @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X ) ) ).
% less_floor_iff
thf(fact_4861_floor__le__iff,axiom,
! [X: real,Z2: int] :
( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z2 )
= ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).
% floor_le_iff
thf(fact_4862_floor__le__iff,axiom,
! [X: rat,Z2: int] :
( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z2 )
= ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).
% floor_le_iff
thf(fact_4863_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_4864_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_4865_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_4866_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_4867_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_4868_floor__eq4,axiom,
! [N: nat,X: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= N ) ) ) ).
% floor_eq4
thf(fact_4869_sgn__power__injE,axiom,
! [A: real,N: nat,X: real,B: real] :
( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
= X )
=> ( ( X
= ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ).
% sgn_power_injE
thf(fact_4870_floor__eq2,axiom,
! [N: int,X: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= N ) ) ) ).
% floor_eq2
thf(fact_4871_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_4872_int__mod__pos__eq,axiom,
! [A: int,B: int,Q4: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( modulo_modulo_int @ A @ B )
= R2 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_4873_int__mod__neg__eq,axiom,
! [A: int,B: int,Q4: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( modulo_modulo_int @ A @ B )
= R2 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_4874_split__zmod,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( modulo_modulo_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ N ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N
= ( 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 )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
=> ( P @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_4875_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_4876_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_4877_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_4878_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_4879_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_4880_floor__divide__lower,axiom,
! [Q4: real,P5: real] :
( ( ord_less_real @ zero_zero_real @ Q4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P5 @ Q4 ) ) ) @ Q4 ) @ P5 ) ) ).
% floor_divide_lower
thf(fact_4881_floor__divide__lower,axiom,
! [Q4: rat,P5: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q4 )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P5 @ Q4 ) ) ) @ Q4 ) @ P5 ) ) ).
% floor_divide_lower
thf(fact_4882_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_4883_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_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_18347121197199848620nteger @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_4884_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_4885_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_4886_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_4887_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_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_18347121197199848620nteger @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_4888_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_4889_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_4890_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_4891_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_4892_root__sgn__power,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) ) )
= Y ) ) ).
% root_sgn_power
thf(fact_4893_sgn__power__root,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
= X ) ) ).
% sgn_power_root
thf(fact_4894_verit__le__mono__div__int,axiom,
! [A2: int,B2: int,N: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int
@ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
@ ( if_int
@ ( ( modulo_modulo_int @ B2 @ N )
= zero_zero_int )
@ one_one_int
@ zero_zero_int ) )
@ ( divide_divide_int @ B2 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_4895_sgn__one,axiom,
( ( sgn_sgn_real @ one_one_real )
= one_one_real ) ).
% sgn_one
thf(fact_4896_sgn__one,axiom,
( ( sgn_sgn_complex @ one_one_complex )
= one_one_complex ) ).
% sgn_one
thf(fact_4897_sgn__zero,axiom,
( ( sgn_sgn_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% sgn_zero
thf(fact_4898_sgn__zero,axiom,
( ( sgn_sgn_real @ zero_zero_real )
= zero_zero_real ) ).
% sgn_zero
thf(fact_4899_Real__Vector__Spaces_Osgn__minus,axiom,
! [X: real] :
( ( sgn_sgn_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( sgn_sgn_real @ X ) ) ) ).
% Real_Vector_Spaces.sgn_minus
thf(fact_4900_Real__Vector__Spaces_Osgn__minus,axiom,
! [X: complex] :
( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ X ) )
= ( uminus1482373934393186551omplex @ ( sgn_sgn_complex @ X ) ) ) ).
% Real_Vector_Spaces.sgn_minus
thf(fact_4901_sgn__zero__iff,axiom,
! [X: complex] :
( ( ( sgn_sgn_complex @ X )
= zero_zero_complex )
= ( X = zero_zero_complex ) ) ).
% sgn_zero_iff
thf(fact_4902_sgn__zero__iff,axiom,
! [X: real] :
( ( ( sgn_sgn_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% sgn_zero_iff
thf(fact_4903_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_4904_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_4905_verit__le__mono__div,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B2 @ N )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B2 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_4906_Cauchy__iff2,axiom,
( topolo4055970368930404560y_real
= ( ^ [X8: nat > real] :
! [J3: nat] :
? [M7: nat] :
! [M4: nat] :
( ( ord_less_eq_nat @ M7 @ M4 )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M7 @ N4 )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X8 @ M4 ) @ ( X8 @ N4 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_4907_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( modulo_modulo_nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_4908_cosh__minus,axiom,
! [X: real] :
( ( cosh_real @ ( uminus_uminus_real @ X ) )
= ( cosh_real @ X ) ) ).
% cosh_minus
thf(fact_4909_cosh__minus,axiom,
! [X: complex] :
( ( cosh_complex @ ( uminus1482373934393186551omplex @ X ) )
= ( cosh_complex @ X ) ) ).
% cosh_minus
thf(fact_4910_cosh__0,axiom,
( ( cosh_complex @ zero_zero_complex )
= one_one_complex ) ).
% cosh_0
thf(fact_4911_cosh__0,axiom,
( ( cosh_real @ zero_zero_real )
= one_one_real ) ).
% cosh_0
thf(fact_4912_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_4913_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_4914_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_4915_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_4916_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_4917_mod__Suc__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_4918_mod__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% mod_Suc_eq
thf(fact_4919_int__sgnE,axiom,
! [K: int] :
~ ! [N2: nat,L2: int] :
( K
!= ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_sgnE
thf(fact_4920_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
!= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_4921_mod__induct,axiom,
! [P: nat > $o,N: nat,P5: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ N @ P5 )
=> ( ( ord_less_nat @ M @ P5 )
=> ( ! [N2: nat] :
( ( ord_less_nat @ N2 @ P5 )
=> ( ( P @ N2 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P5 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_4922_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M2: nat] : ( P @ M2 @ zero_zero_nat )
=> ( ! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 @ ( modulo_modulo_nat @ M2 @ N2 ) )
=> ( P @ M2 @ N2 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_4923_mod__less__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_4924_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_4925_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_4926_mod__geq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( modulo_modulo_nat @ M @ N )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% mod_geq
thf(fact_4927_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M4: nat,N4: nat] : ( if_nat @ ( ord_less_nat @ M4 @ N4 ) @ M4 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M4 @ N4 ) @ N4 ) ) ) ) ).
% mod_if
thf(fact_4928_le__mod__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( modulo_modulo_nat @ M @ N )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% le_mod_geq
thf(fact_4929_cosh__real__ge__1,axiom,
! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).
% cosh_real_ge_1
thf(fact_4930_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X @ N )
= ( modulo_modulo_nat @ Y @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
= ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_4931_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_4932_mod__le__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_4933_div__less__mono,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( modulo_modulo_nat @ A2 @ N )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B2 @ N )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B2 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_4934_mod__eq__nat1E,axiom,
! [M: nat,Q4: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q4 )
= ( modulo_modulo_nat @ N @ Q4 ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ~ ! [S2: nat] :
( M
!= ( plus_plus_nat @ N @ ( times_times_nat @ Q4 @ S2 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_4935_mod__eq__nat2E,axiom,
! [M: nat,Q4: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q4 )
= ( modulo_modulo_nat @ N @ Q4 ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ~ ! [S2: nat] :
( N
!= ( plus_plus_nat @ M @ ( times_times_nat @ Q4 @ S2 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_4936_nat__mod__eq__lemma,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X @ N )
= ( modulo_modulo_nat @ Y @ N ) )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ? [Q2: nat] :
( X
= ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q2 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_4937_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q4 ) )
= ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q4 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% mod_mult2_eq
thf(fact_4938_div__mod__decomp,axiom,
! [A2: nat,N: nat] :
( A2
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N ) @ N ) @ ( modulo_modulo_nat @ A2 @ N ) ) ) ).
% div_mod_decomp
thf(fact_4939_modulo__nat__def,axiom,
( modulo_modulo_nat
= ( ^ [M4: nat,N4: nat] : ( minus_minus_nat @ M4 @ ( times_times_nat @ ( divide_divide_nat @ M4 @ N4 ) @ N4 ) ) ) ) ).
% modulo_nat_def
thf(fact_4940_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_4941_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_4942_field__char__0__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_4943_field__char__0__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_4944_field__char__0__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_4945_split__mod,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( modulo_modulo_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ M ) )
& ( ( N != zero_zero_nat )
=> ! [I3: nat,J3: nat] :
( ( ord_less_nat @ J3 @ N )
=> ( ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_4946_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_4947_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_4948_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_4949_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_4950_le__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: real,C: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_4951_le__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: rat,C: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_4952_divide__le__eq__numeral_I2_J,axiom,
! [B: real,C: real,W2: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_4953_divide__le__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W2: num] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_4954_norm__power__diff,axiom,
! [Z2: real,W2: real,M: nat] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W2 ) @ one_one_real )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z2 @ M ) @ ( power_power_real @ W2 @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z2 @ W2 ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_4955_norm__power__diff,axiom,
! [Z2: complex,W2: complex,M: nat] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W2 ) @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z2 @ M ) @ ( power_power_complex @ W2 @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z2 @ W2 ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_4956_count__notin,axiom,
! [X: complex,Xs: list_complex] :
( ~ ( member_complex @ X @ ( set_complex2 @ Xs ) )
=> ( ( count_list_complex @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_4957_count__notin,axiom,
! [X: real,Xs: list_real] :
( ~ ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( ( count_list_real @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_4958_count__notin,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
=> ( ( count_list_set_nat @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_4959_count__notin,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ~ ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( ( count_list_VEBT_VEBT @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_4960_count__notin,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( count_list_nat @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_4961_count__notin,axiom,
! [X: int,Xs: list_int] :
( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( ( count_list_int @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_4962_metric__Cauchy__iff2,axiom,
( topolo4055970368930404560y_real
= ( ^ [X8: nat > real] :
! [J3: nat] :
? [M7: nat] :
! [M4: nat] :
( ( ord_less_eq_nat @ M7 @ M4 )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M7 @ N4 )
=> ( ord_less_real @ ( real_V975177566351809787t_real @ ( X8 @ M4 ) @ ( X8 @ N4 ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% metric_Cauchy_iff2
thf(fact_4963_powr__real__of__int_H,axiom,
! [X: real,N: int] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( X != zero_zero_real )
| ( ord_less_int @ zero_zero_int @ N ) )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
= ( power_int_real @ X @ N ) ) ) ) ).
% powr_real_of_int'
thf(fact_4964_round__diff__minimal,axiom,
! [Z2: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z2 ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ M ) ) ) ) ).
% round_diff_minimal
thf(fact_4965_round__diff__minimal,axiom,
! [Z2: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z2 ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ M ) ) ) ) ).
% round_diff_minimal
thf(fact_4966_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numera6690914467698888265omplex @ M )
= ( numera6690914467698888265omplex @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_4967_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_4968_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_rat @ M )
= ( numeral_numeral_rat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_4969_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_4970_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_4971_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_4972_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% nat_numeral
thf(fact_4973_power__int__mult__numeral,axiom,
! [X: real,M: num,N: num] :
( ( power_int_real @ ( power_int_real @ X @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( power_int_real @ X @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% power_int_mult_numeral
thf(fact_4974_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_4975_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_4976_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_4977_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_4978_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_4979_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_4980_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_4981_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_4982_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
= ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_4983_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_4984_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_4985_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_4986_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_4987_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z2: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ Z2 ) )
= ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_4988_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z2: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ Z2 ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_4989_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z2: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ Z2 ) )
= ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_4990_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W2 ) @ Z2 ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_4991_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W2 ) @ Z2 ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_4992_add__numeral__left,axiom,
! [V: num,W2: num,Z2: complex] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W2 ) @ Z2 ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_4993_add__numeral__left,axiom,
! [V: num,W2: num,Z2: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W2 ) @ Z2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_4994_add__numeral__left,axiom,
! [V: num,W2: num,Z2: rat] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W2 ) @ Z2 ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_4995_add__numeral__left,axiom,
! [V: num,W2: num,Z2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_4996_add__numeral__left,axiom,
! [V: num,W2: num,Z2: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W2 ) @ Z2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_4997_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_4998_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_4999_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_5000_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_5001_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_5002_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
= zero_zero_rat ) ).
% power_zero_numeral
thf(fact_5003_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_5004_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
= zero_zero_real ) ).
% power_zero_numeral
thf(fact_5005_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_5006_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
= zero_zero_complex ) ).
% power_zero_numeral
thf(fact_5007_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_5008_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_5009_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_5010_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_5011_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_5012_of__nat__numeral,axiom,
! [N: num] :
( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
= ( numera6690914467698888265omplex @ N ) ) ).
% of_nat_numeral
thf(fact_5013_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% of_nat_numeral
thf(fact_5014_of__nat__numeral,axiom,
! [N: num] :
( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_real @ N ) ) ).
% of_nat_numeral
thf(fact_5015_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ N ) ) ).
% of_nat_numeral
thf(fact_5016_of__nat__numeral,axiom,
! [N: num] :
( ( semiri4939895301339042750nteger @ ( numeral_numeral_nat @ N ) )
= ( numera6620942414471956472nteger @ N ) ) ).
% of_nat_numeral
thf(fact_5017_of__nat__numeral,axiom,
! [N: num] :
( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_rat @ N ) ) ).
% of_nat_numeral
thf(fact_5018_abs__numeral,axiom,
! [N: num] :
( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
= ( numera6620942414471956472nteger @ N ) ) ).
% abs_numeral
thf(fact_5019_abs__numeral,axiom,
! [N: num] :
( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ N ) ) ).
% abs_numeral
thf(fact_5020_abs__numeral,axiom,
! [N: num] :
( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ N ) ) ).
% abs_numeral
thf(fact_5021_abs__numeral,axiom,
! [N: num] :
( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% abs_numeral
thf(fact_5022_of__int__eq__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ( ring_18347121197199848620nteger @ Z2 )
= ( numera6620942414471956472nteger @ N ) )
= ( Z2
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_5023_of__int__eq__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ( ring_17405671764205052669omplex @ Z2 )
= ( numera6690914467698888265omplex @ N ) )
= ( Z2
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_5024_of__int__eq__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ( ring_1_of_int_real @ Z2 )
= ( numeral_numeral_real @ N ) )
= ( Z2
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_5025_of__int__eq__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ( ring_1_of_int_rat @ Z2 )
= ( numeral_numeral_rat @ N ) )
= ( Z2
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_5026_of__int__eq__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ( ring_1_of_int_int @ Z2 )
= ( numeral_numeral_int @ N ) )
= ( Z2
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_5027_of__int__numeral,axiom,
! [K: num] :
( ( ring_18347121197199848620nteger @ ( numeral_numeral_int @ K ) )
= ( numera6620942414471956472nteger @ K ) ) ).
% of_int_numeral
thf(fact_5028_of__int__numeral,axiom,
! [K: num] :
( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
= ( numera6690914467698888265omplex @ K ) ) ).
% of_int_numeral
thf(fact_5029_of__int__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_real @ K ) ) ).
% of_int_numeral
thf(fact_5030_of__int__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_rat @ K ) ) ).
% of_int_numeral
thf(fact_5031_of__int__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ K ) ) ).
% of_int_numeral
thf(fact_5032_norm__minus__cancel,axiom,
! [X: real] :
( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ X ) )
= ( real_V7735802525324610683m_real @ X ) ) ).
% norm_minus_cancel
thf(fact_5033_norm__minus__cancel,axiom,
! [X: complex] :
( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ X ) )
= ( real_V1022390504157884413omplex @ X ) ) ).
% norm_minus_cancel
thf(fact_5034_floor__numeral,axiom,
! [V: num] :
( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_int @ V ) ) ).
% floor_numeral
thf(fact_5035_floor__numeral,axiom,
! [V: num] :
( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
= ( numeral_numeral_int @ V ) ) ).
% floor_numeral
thf(fact_5036_power__int__add__numeral,axiom,
! [X: real,M: num,N: num] :
( ( times_times_real @ ( power_int_real @ X @ ( numeral_numeral_int @ M ) ) @ ( power_int_real @ X @ ( numeral_numeral_int @ N ) ) )
= ( power_int_real @ X @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_int_add_numeral
thf(fact_5037_power__int__add__numeral,axiom,
! [X: rat,M: num,N: num] :
( ( times_times_rat @ ( power_int_rat @ X @ ( numeral_numeral_int @ M ) ) @ ( power_int_rat @ X @ ( numeral_numeral_int @ N ) ) )
= ( power_int_rat @ X @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_int_add_numeral
thf(fact_5038_power__int__add__numeral2,axiom,
! [X: real,M: num,N: num,B: real] :
( ( times_times_real @ ( power_int_real @ X @ ( numeral_numeral_int @ M ) ) @ ( times_times_real @ ( power_int_real @ X @ ( numeral_numeral_int @ N ) ) @ B ) )
= ( times_times_real @ ( power_int_real @ X @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_int_add_numeral2
thf(fact_5039_power__int__add__numeral2,axiom,
! [X: rat,M: num,N: num,B: rat] :
( ( times_times_rat @ ( power_int_rat @ X @ ( numeral_numeral_int @ M ) ) @ ( times_times_rat @ ( power_int_rat @ X @ ( numeral_numeral_int @ N ) ) @ B ) )
= ( times_times_rat @ ( power_int_rat @ X @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_int_add_numeral2
thf(fact_5040_power__int__1__left,axiom,
! [N: int] :
( ( power_int_complex @ one_one_complex @ N )
= one_one_complex ) ).
% power_int_1_left
thf(fact_5041_power__int__1__left,axiom,
! [N: int] :
( ( power_int_real @ one_one_real @ N )
= one_one_real ) ).
% power_int_1_left
thf(fact_5042_power__int__1__left,axiom,
! [N: int] :
( ( power_int_rat @ one_one_rat @ N )
= one_one_rat ) ).
% power_int_1_left
thf(fact_5043_dist__add__cancel,axiom,
! [A: real,B: real,C: real] :
( ( real_V975177566351809787t_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ A @ C ) )
= ( real_V975177566351809787t_real @ B @ C ) ) ).
% dist_add_cancel
thf(fact_5044_dist__add__cancel2,axiom,
! [B: real,A: real,C: real] :
( ( real_V975177566351809787t_real @ ( plus_plus_real @ B @ A ) @ ( plus_plus_real @ C @ A ) )
= ( real_V975177566351809787t_real @ B @ C ) ) ).
% dist_add_cancel2
thf(fact_5045_power__int__numeral,axiom,
! [X: real,N: num] :
( ( power_int_real @ X @ ( numeral_numeral_int @ N ) )
= ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ).
% power_int_numeral
thf(fact_5046_power__int__numeral,axiom,
! [X: complex,N: num] :
( ( power_int_complex @ X @ ( numeral_numeral_int @ N ) )
= ( power_power_complex @ X @ ( numeral_numeral_nat @ N ) ) ) ).
% power_int_numeral
thf(fact_5047_ceiling__numeral,axiom,
! [V: num] :
( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_int @ V ) ) ).
% ceiling_numeral
thf(fact_5048_ceiling__numeral,axiom,
! [V: num] :
( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
= ( numeral_numeral_int @ V ) ) ).
% ceiling_numeral
thf(fact_5049_power__int__1__right,axiom,
! [Y: real] :
( ( power_int_real @ Y @ one_one_int )
= Y ) ).
% power_int_1_right
thf(fact_5050_power__int__sgn,axiom,
! [A: real,N: int] :
( ( sgn_sgn_real @ ( power_int_real @ A @ N ) )
= ( power_int_real @ ( sgn_sgn_real @ A ) @ N ) ) ).
% power_int_sgn
thf(fact_5051_power__int__sgn,axiom,
! [A: rat,N: int] :
( ( sgn_sgn_rat @ ( power_int_rat @ A @ N ) )
= ( power_int_rat @ ( sgn_sgn_rat @ A ) @ N ) ) ).
% power_int_sgn
thf(fact_5052_round__numeral,axiom,
! [N: num] :
( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% round_numeral
thf(fact_5053_round__numeral,axiom,
! [N: num] :
( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% round_numeral
thf(fact_5054_round__of__int,axiom,
! [N: int] :
( ( archim8280529875227126926d_real @ ( ring_1_of_int_real @ N ) )
= N ) ).
% round_of_int
thf(fact_5055_round__of__int,axiom,
! [N: int] :
( ( archim7778729529865785530nd_rat @ ( ring_1_of_int_rat @ N ) )
= N ) ).
% round_of_int
thf(fact_5056_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_5057_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_5058_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_5059_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_5060_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_5061_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_5062_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_5063_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_5064_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_5065_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_5066_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_5067_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_5068_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_5069_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_5070_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_5071_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_5072_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_5073_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_5074_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_5075_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_5076_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_5077_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_5078_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_5079_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_5080_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_5081_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_5082_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_5083_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_5084_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_5085_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_5086_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_5087_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_5088_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_5089_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_5090_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_5091_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_5092_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_5093_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_5094_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_5095_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_5096_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_5097_semiring__norm_I172_J,axiom,
! [V: num,W2: num,Y: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) @ Y ) ) ).
% semiring_norm(172)
thf(fact_5098_semiring__norm_I172_J,axiom,
! [V: num,W2: num,Y: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ Y ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W2 ) ) @ Y ) ) ).
% semiring_norm(172)
thf(fact_5099_semiring__norm_I172_J,axiom,
! [V: num,W2: num,Y: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ Y ) )
= ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W2 ) ) @ Y ) ) ).
% semiring_norm(172)
thf(fact_5100_semiring__norm_I172_J,axiom,
! [V: num,W2: num,Y: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y ) )
= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W2 ) ) @ Y ) ) ).
% semiring_norm(172)
thf(fact_5101_semiring__norm_I172_J,axiom,
! [V: num,W2: num,Y: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y ) )
= ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) @ Y ) ) ).
% semiring_norm(172)
thf(fact_5102_semiring__norm_I171_J,axiom,
! [V: num,W2: num,Y: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_5103_semiring__norm_I171_J,axiom,
! [V: num,W2: num,Y: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ Y ) )
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_5104_semiring__norm_I171_J,axiom,
! [V: num,W2: num,Y: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ Y ) )
= ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_5105_semiring__norm_I171_J,axiom,
! [V: num,W2: num,Y: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y ) )
= ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_5106_semiring__norm_I171_J,axiom,
! [V: num,W2: num,Y: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y ) )
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_5107_semiring__norm_I170_J,axiom,
! [V: num,W2: num,Y: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W2 ) @ Y ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_5108_semiring__norm_I170_J,axiom,
! [V: num,W2: num,Y: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ Y ) )
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_5109_semiring__norm_I170_J,axiom,
! [V: num,W2: num,Y: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ Y ) )
= ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_5110_semiring__norm_I170_J,axiom,
! [V: num,W2: num,Y: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W2 ) @ Y ) )
= ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_5111_semiring__norm_I170_J,axiom,
! [V: num,W2: num,Y: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ Y ) )
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_5112_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_5113_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_5114_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_5115_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_5116_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_5117_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_5118_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ Y ) )
= ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_5119_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ Y ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_5120_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_5121_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_5122_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_5123_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_5124_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_5125_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_5126_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_5127_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_5128_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_5129_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_5130_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_5131_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_5132_abs__neg__numeral,axiom,
! [N: num] :
( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ N ) ) ).
% abs_neg_numeral
thf(fact_5133_abs__neg__numeral,axiom,
! [N: num] :
( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( numeral_numeral_real @ N ) ) ).
% abs_neg_numeral
thf(fact_5134_abs__neg__numeral,axiom,
! [N: num] :
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ N ) ) ).
% abs_neg_numeral
thf(fact_5135_abs__neg__numeral,axiom,
! [N: num] :
( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ N ) ) ).
% abs_neg_numeral
thf(fact_5136_norm__zero,axiom,
( ( real_V7735802525324610683m_real @ zero_zero_real )
= zero_zero_real ) ).
% norm_zero
thf(fact_5137_norm__zero,axiom,
( ( real_V1022390504157884413omplex @ zero_zero_complex )
= zero_zero_real ) ).
% norm_zero
thf(fact_5138_norm__eq__zero,axiom,
! [X: real] :
( ( ( real_V7735802525324610683m_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_eq_zero
thf(fact_5139_norm__eq__zero,axiom,
! [X: complex] :
( ( ( real_V1022390504157884413omplex @ X )
= zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_eq_zero
thf(fact_5140_norm__neg__numeral,axiom,
! [W2: num] :
( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( numeral_numeral_real @ W2 ) ) ).
% norm_neg_numeral
thf(fact_5141_norm__neg__numeral,axiom,
! [W2: num] :
( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= ( numeral_numeral_real @ W2 ) ) ).
% norm_neg_numeral
thf(fact_5142_norm__one,axiom,
( ( real_V7735802525324610683m_real @ one_one_real )
= one_one_real ) ).
% norm_one
thf(fact_5143_norm__one,axiom,
( ( real_V1022390504157884413omplex @ one_one_complex )
= one_one_real ) ).
% norm_one
thf(fact_5144_power__int__mult__distrib__numeral2,axiom,
! [X: complex,W2: num,M: int] :
( ( power_int_complex @ ( times_times_complex @ X @ ( numera6690914467698888265omplex @ W2 ) ) @ M )
= ( times_times_complex @ ( power_int_complex @ X @ M ) @ ( power_int_complex @ ( numera6690914467698888265omplex @ W2 ) @ M ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_5145_power__int__mult__distrib__numeral2,axiom,
! [X: real,W2: num,M: int] :
( ( power_int_real @ ( times_times_real @ X @ ( numeral_numeral_real @ W2 ) ) @ M )
= ( times_times_real @ ( power_int_real @ X @ M ) @ ( power_int_real @ ( numeral_numeral_real @ W2 ) @ M ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_5146_power__int__mult__distrib__numeral2,axiom,
! [X: rat,W2: num,M: int] :
( ( power_int_rat @ ( times_times_rat @ X @ ( numeral_numeral_rat @ W2 ) ) @ M )
= ( times_times_rat @ ( power_int_rat @ X @ M ) @ ( power_int_rat @ ( numeral_numeral_rat @ W2 ) @ M ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_5147_power__int__mult__distrib__numeral1,axiom,
! [W2: num,Y: complex,M: int] :
( ( power_int_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ Y ) @ M )
= ( times_times_complex @ ( power_int_complex @ ( numera6690914467698888265omplex @ W2 ) @ M ) @ ( power_int_complex @ Y @ M ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_5148_power__int__mult__distrib__numeral1,axiom,
! [W2: num,Y: real,M: int] :
( ( power_int_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ Y ) @ M )
= ( times_times_real @ ( power_int_real @ ( numeral_numeral_real @ W2 ) @ M ) @ ( power_int_real @ Y @ M ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_5149_power__int__mult__distrib__numeral1,axiom,
! [W2: num,Y: rat,M: int] :
( ( power_int_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ Y ) @ M )
= ( times_times_rat @ ( power_int_rat @ ( numeral_numeral_rat @ W2 ) @ M ) @ ( power_int_rat @ Y @ M ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_5150_power__int__0__left,axiom,
! [M: int] :
( ( M != zero_zero_int )
=> ( ( power_int_real @ zero_zero_real @ M )
= zero_zero_real ) ) ).
% power_int_0_left
thf(fact_5151_power__int__0__left,axiom,
! [M: int] :
( ( M != zero_zero_int )
=> ( ( power_int_rat @ zero_zero_rat @ M )
= zero_zero_rat ) ) ).
% power_int_0_left
thf(fact_5152_power__int__eq__0__iff,axiom,
! [X: real,N: int] :
( ( ( power_int_real @ X @ N )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( N != zero_zero_int ) ) ) ).
% power_int_eq_0_iff
thf(fact_5153_power__int__eq__0__iff,axiom,
! [X: rat,N: int] :
( ( ( power_int_rat @ X @ N )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( N != zero_zero_int ) ) ) ).
% power_int_eq_0_iff
thf(fact_5154_numeral__less__real__of__nat__iff,axiom,
! [W2: num,N: nat] :
( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ W2 ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_5155_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W2: num] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W2 ) )
= ( ord_less_nat @ N @ ( numeral_numeral_nat @ W2 ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_5156_power__int__0__right,axiom,
! [X: complex] :
( ( power_int_complex @ X @ zero_zero_int )
= one_one_complex ) ).
% power_int_0_right
thf(fact_5157_power__int__0__right,axiom,
! [X: real] :
( ( power_int_real @ X @ zero_zero_int )
= one_one_real ) ).
% power_int_0_right
thf(fact_5158_power__int__0__right,axiom,
! [X: rat] :
( ( power_int_rat @ X @ zero_zero_int )
= one_one_rat ) ).
% power_int_0_right
thf(fact_5159_numeral__le__real__of__nat__iff,axiom,
! [N: num,M: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_5160_dist__0__norm,axiom,
! [X: real] :
( ( real_V975177566351809787t_real @ zero_zero_real @ X )
= ( real_V7735802525324610683m_real @ X ) ) ).
% dist_0_norm
thf(fact_5161_dist__0__norm,axiom,
! [X: complex] :
( ( real_V3694042436643373181omplex @ zero_zero_complex @ X )
= ( real_V1022390504157884413omplex @ X ) ) ).
% dist_0_norm
thf(fact_5162_abs__power__int__minus,axiom,
! [A: real,N: int] :
( ( abs_abs_real @ ( power_int_real @ ( uminus_uminus_real @ A ) @ N ) )
= ( abs_abs_real @ ( power_int_real @ A @ N ) ) ) ).
% abs_power_int_minus
thf(fact_5163_abs__power__int__minus,axiom,
! [A: rat,N: int] :
( ( abs_abs_rat @ ( power_int_rat @ ( uminus_uminus_rat @ A ) @ N ) )
= ( abs_abs_rat @ ( power_int_rat @ A @ N ) ) ) ).
% abs_power_int_minus
thf(fact_5164_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= zero_zero_nat ) ).
% nat_neg_numeral
thf(fact_5165_norm__of__nat,axiom,
! [N: nat] :
( ( real_V7735802525324610683m_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% norm_of_nat
thf(fact_5166_norm__of__nat,axiom,
! [N: nat] :
( ( real_V1022390504157884413omplex @ ( semiri8010041392384452111omplex @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% norm_of_nat
thf(fact_5167_dist__diff_I2_J,axiom,
! [A: real,B: real] :
( ( real_V975177566351809787t_real @ ( minus_minus_real @ A @ B ) @ A )
= ( real_V7735802525324610683m_real @ B ) ) ).
% dist_diff(2)
thf(fact_5168_dist__diff_I2_J,axiom,
! [A: complex,B: complex] :
( ( real_V3694042436643373181omplex @ ( minus_minus_complex @ A @ B ) @ A )
= ( real_V1022390504157884413omplex @ B ) ) ).
% dist_diff(2)
thf(fact_5169_dist__diff_I1_J,axiom,
! [A: real,B: real] :
( ( real_V975177566351809787t_real @ A @ ( minus_minus_real @ A @ B ) )
= ( real_V7735802525324610683m_real @ B ) ) ).
% dist_diff(1)
thf(fact_5170_dist__diff_I1_J,axiom,
! [A: complex,B: complex] :
( ( real_V3694042436643373181omplex @ A @ ( minus_minus_complex @ A @ B ) )
= ( real_V1022390504157884413omplex @ B ) ) ).
% dist_diff(1)
thf(fact_5171_power__int__of__nat,axiom,
! [X: real,N: nat] :
( ( power_int_real @ X @ ( semiri1314217659103216013at_int @ N ) )
= ( power_power_real @ X @ N ) ) ).
% power_int_of_nat
thf(fact_5172_power__int__of__nat,axiom,
! [X: complex,N: nat] :
( ( power_int_complex @ X @ ( semiri1314217659103216013at_int @ N ) )
= ( power_power_complex @ X @ N ) ) ).
% power_int_of_nat
thf(fact_5173_diff__nat__numeral,axiom,
! [V: num,V2: num] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V2 ) )
= ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V2 ) ) ) ) ).
% diff_nat_numeral
thf(fact_5174_numeral__power__eq__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= ( nat2 @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_5175_nat__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( nat2 @ Y )
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_5176_round__0,axiom,
( ( archim8280529875227126926d_real @ zero_zero_real )
= zero_zero_int ) ).
% round_0
thf(fact_5177_round__0,axiom,
( ( archim7778729529865785530nd_rat @ zero_zero_rat )
= zero_zero_int ) ).
% round_0
thf(fact_5178_round__1,axiom,
( ( archim8280529875227126926d_real @ one_one_real )
= one_one_int ) ).
% round_1
thf(fact_5179_round__1,axiom,
( ( archim7778729529865785530nd_rat @ one_one_rat )
= one_one_int ) ).
% round_1
thf(fact_5180_round__of__nat,axiom,
! [N: nat] :
( ( archim8280529875227126926d_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% round_of_nat
thf(fact_5181_round__of__nat,axiom,
! [N: nat] :
( ( archim7778729529865785530nd_rat @ ( semiri681578069525770553at_rat @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% round_of_nat
thf(fact_5182_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_5183_divide__le__eq__numeral1_I1_J,axiom,
! [B: real,W2: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_5184_divide__le__eq__numeral1_I1_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_5185_le__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W2: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_5186_le__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_5187_eq__divide__eq__numeral1_I1_J,axiom,
! [A: complex,B: complex,W2: num] :
( ( A
= ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) ) )
= ( ( ( ( numera6690914467698888265omplex @ W2 )
!= zero_zero_complex )
=> ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) )
= B ) )
& ( ( ( numera6690914467698888265omplex @ W2 )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_5188_eq__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W2: num] :
( ( A
= ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( ( numeral_numeral_real @ W2 )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) )
= B ) )
& ( ( ( numeral_numeral_real @ W2 )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_5189_eq__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( A
= ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( ( numeral_numeral_rat @ W2 )
!= zero_zero_rat )
=> ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) )
= B ) )
& ( ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_5190_divide__eq__eq__numeral1_I1_J,axiom,
! [B: complex,W2: num,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) )
= A )
= ( ( ( ( numera6690914467698888265omplex @ W2 )
!= zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) ) ) )
& ( ( ( numera6690914467698888265omplex @ W2 )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_5191_divide__eq__eq__numeral1_I1_J,axiom,
! [B: real,W2: num,A: real] :
( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) )
= A )
= ( ( ( ( numeral_numeral_real @ W2 )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) )
& ( ( ( numeral_numeral_real @ W2 )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_5192_divide__eq__eq__numeral1_I1_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) )
= A )
= ( ( ( ( numeral_numeral_rat @ W2 )
!= zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) )
& ( ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_5193_less__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W2: num] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
= ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_5194_less__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
= ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_5195_divide__less__eq__numeral1_I1_J,axiom,
! [B: real,W2: num,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_5196_divide__less__eq__numeral1_I1_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_5197_inverse__eq__divide__numeral,axiom,
! [W2: num] :
( ( inverse_inverse_real @ ( numeral_numeral_real @ W2 ) )
= ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ W2 ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_5198_inverse__eq__divide__numeral,axiom,
! [W2: num] :
( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ W2 ) )
= ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ W2 ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_5199_inverse__eq__divide__numeral,axiom,
! [W2: num] :
( ( inverse_inverse_rat @ ( numeral_numeral_rat @ W2 ) )
= ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ W2 ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_5200_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_5201_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_5202_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_5203_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_5204_of__int__numeral__le__iff,axiom,
! [N: num,Z2: int] :
( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z2 ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_le_iff
thf(fact_5205_of__int__numeral__le__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_le_iff
thf(fact_5206_of__int__numeral__le__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_le_iff
thf(fact_5207_of__int__numeral__le__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_le_iff
thf(fact_5208_of__int__le__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z2 ) @ ( numera6620942414471956472nteger @ N ) )
= ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_5209_of__int__le__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_5210_of__int__le__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_5211_of__int__le__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_5212_of__int__less__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z2 ) @ ( numera6620942414471956472nteger @ N ) )
= ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_5213_of__int__less__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_5214_of__int__less__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_5215_of__int__less__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_5216_of__int__numeral__less__iff,axiom,
! [N: num,Z2: int] :
( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z2 ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_less_iff
thf(fact_5217_of__int__numeral__less__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_less_iff
thf(fact_5218_of__int__numeral__less__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_less_iff
thf(fact_5219_of__int__numeral__less__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_less_iff
thf(fact_5220_power__int__minus__one__mult__self_H,axiom,
! [M: int,B: real] :
( ( times_times_real @ ( power_int_real @ ( uminus_uminus_real @ one_one_real ) @ M ) @ ( times_times_real @ ( power_int_real @ ( uminus_uminus_real @ one_one_real ) @ M ) @ B ) )
= B ) ).
% power_int_minus_one_mult_self'
thf(fact_5221_power__int__minus__one__mult__self_H,axiom,
! [M: int,B: complex] :
( ( times_times_complex @ ( power_int_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ M ) @ ( times_times_complex @ ( power_int_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ M ) @ B ) )
= B ) ).
% power_int_minus_one_mult_self'
thf(fact_5222_power__int__minus__one__mult__self_H,axiom,
! [M: int,B: rat] :
( ( times_times_rat @ ( power_int_rat @ ( uminus_uminus_rat @ one_one_rat ) @ M ) @ ( times_times_rat @ ( power_int_rat @ ( uminus_uminus_rat @ one_one_rat ) @ M ) @ B ) )
= B ) ).
% power_int_minus_one_mult_self'
thf(fact_5223_power__int__minus__one__mult__self,axiom,
! [M: int] :
( ( times_times_real @ ( power_int_real @ ( uminus_uminus_real @ one_one_real ) @ M ) @ ( power_int_real @ ( uminus_uminus_real @ one_one_real ) @ M ) )
= one_one_real ) ).
% power_int_minus_one_mult_self
thf(fact_5224_power__int__minus__one__mult__self,axiom,
! [M: int] :
( ( times_times_complex @ ( power_int_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ M ) @ ( power_int_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ M ) )
= one_one_complex ) ).
% power_int_minus_one_mult_self
thf(fact_5225_power__int__minus__one__mult__self,axiom,
! [M: int] :
( ( times_times_rat @ ( power_int_rat @ ( uminus_uminus_rat @ one_one_rat ) @ M ) @ ( power_int_rat @ ( uminus_uminus_rat @ one_one_rat ) @ M ) )
= one_one_rat ) ).
% power_int_minus_one_mult_self
thf(fact_5226_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( ( semiri8010041392384452111omplex @ Y )
= ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_5227_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( ( semiri1314217659103216013at_int @ Y )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_5228_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( ( semiri5074537144036343181t_real @ Y )
= ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_5229_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( ( semiri1316708129612266289at_nat @ Y )
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_5230_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( ( semiri4939895301339042750nteger @ Y )
= ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_5231_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( ( semiri681578069525770553at_rat @ Y )
= ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_5232_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
= ( semiri8010041392384452111omplex @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_5233_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= ( semiri1314217659103216013at_int @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_5234_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
= ( semiri5074537144036343181t_real @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_5235_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= ( semiri1316708129612266289at_nat @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_5236_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N )
= ( semiri4939895301339042750nteger @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_5237_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
= ( semiri681578069525770553at_rat @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_5238_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_5239_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_5240_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_5241_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_5242_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_5243_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_5244_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_5245_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_5246_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_5247_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_5248_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_5249_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_5250_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_5251_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_5252_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_5253_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_5254_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N )
= ( ring_18347121197199848620nteger @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_5255_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
= ( ring_17405671764205052669omplex @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_5256_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
= ( ring_1_of_int_real @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_5257_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
= ( ring_1_of_int_rat @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_5258_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= ( ring_1_of_int_int @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_5259_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_18347121197199848620nteger @ Y )
= ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_5260_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_17405671764205052669omplex @ Y )
= ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_5261_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int_real @ Y )
= ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_5262_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int_rat @ Y )
= ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_5263_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int_int @ Y )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_5264_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_5265_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_5266_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral_nat @ K )
!= one_one_nat )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
= one_one_nat ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_5267_floor__numeral__power,axiom,
! [X: num,N: nat] :
( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% floor_numeral_power
thf(fact_5268_floor__numeral__power,axiom,
! [X: num,N: nat] :
( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% floor_numeral_power
thf(fact_5269_ceiling__numeral__power,axiom,
! [X: num,N: nat] :
( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% ceiling_numeral_power
thf(fact_5270_ceiling__numeral__power,axiom,
! [X: num,N: nat] :
( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% ceiling_numeral_power
thf(fact_5271_power__int__minus1__right,axiom,
! [Y: real] :
( ( power_int_real @ Y @ ( uminus_uminus_int @ one_one_int ) )
= ( inverse_inverse_real @ Y ) ) ).
% power_int_minus1_right
thf(fact_5272_power__int__minus1__right,axiom,
! [Y: complex] :
( ( power_int_complex @ Y @ ( uminus_uminus_int @ one_one_int ) )
= ( invers8013647133539491842omplex @ Y ) ) ).
% power_int_minus1_right
thf(fact_5273_power__int__minus1__right,axiom,
! [Y: rat] :
( ( power_int_rat @ Y @ ( uminus_uminus_int @ one_one_int ) )
= ( inverse_inverse_rat @ Y ) ) ).
% power_int_minus1_right
thf(fact_5274_round__neg__numeral,axiom,
! [N: num] :
( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% round_neg_numeral
thf(fact_5275_round__neg__numeral,axiom,
! [N: num] :
( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% round_neg_numeral
thf(fact_5276_norm__of__int,axiom,
! [Z2: int] :
( ( real_V7735802525324610683m_real @ ( ring_1_of_int_real @ Z2 ) )
= ( abs_abs_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% norm_of_int
thf(fact_5277_norm__of__int,axiom,
! [Z2: int] :
( ( real_V1022390504157884413omplex @ ( ring_17405671764205052669omplex @ Z2 ) )
= ( abs_abs_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% norm_of_int
thf(fact_5278_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_5279_divide__le__eq__numeral1_I2_J,axiom,
! [B: real,W2: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_5280_divide__le__eq__numeral1_I2_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_5281_le__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W2: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_5282_le__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_5283_divide__eq__eq__numeral1_I2_J,axiom,
! [B: real,W2: num,A: real] :
( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= A )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_5284_divide__eq__eq__numeral1_I2_J,axiom,
! [B: complex,W2: num,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= A )
= ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
!= zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) )
& ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_5285_divide__eq__eq__numeral1_I2_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= A )
= ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
!= zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) )
& ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_5286_eq__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W2: num] :
( ( A
= ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= B ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_5287_eq__divide__eq__numeral1_I2_J,axiom,
! [A: complex,B: complex,W2: num] :
( ( A
= ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) )
= ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
!= zero_zero_complex )
=> ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= B ) )
& ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_5288_eq__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( A
= ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
= ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
!= zero_zero_rat )
=> ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= B ) )
& ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_5289_divide__less__eq__numeral1_I2_J,axiom,
! [B: real,W2: num,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_5290_divide__less__eq__numeral1_I2_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
= ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_5291_less__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W2: num] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_5292_less__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_5293_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_5294_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_5295_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
= ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_5296_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_5297_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_5298_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_5299_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_5300_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
= ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_5301_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_5302_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_5303_inverse__eq__divide__neg__numeral,axiom,
! [W2: num] :
( ( inverse_inverse_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( divide_divide_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_5304_inverse__eq__divide__neg__numeral,axiom,
! [W2: num] :
( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= ( divide1717551699836669952omplex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_5305_inverse__eq__divide__neg__numeral,axiom,
! [W2: num] :
( ( inverse_inverse_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= ( divide_divide_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_5306_power__int__mono__iff,axiom,
! [A: real,B: real,N: int] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ord_less_eq_real @ ( power_int_real @ A @ N ) @ ( power_int_real @ B @ N ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_int_mono_iff
thf(fact_5307_power__int__mono__iff,axiom,
! [A: rat,B: rat,N: int] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ord_less_eq_rat @ ( power_int_rat @ A @ N ) @ ( power_int_rat @ B @ N ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% power_int_mono_iff
thf(fact_5308_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_5309_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_5310_nat__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_5311_nat__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_5312_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_5313_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_5314_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_5315_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_5316_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_5317_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_5318_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_5319_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_5320_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_5321_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_5322_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_5323_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_5324_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) @ ( semiri4939895301339042750nteger @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_5325_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_5326_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_5327_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) @ ( semiri4939895301339042750nteger @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_5328_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_5329_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_5330_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_5331_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_5332_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_5333_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_5334_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_5335_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_5336_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_5337_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_5338_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_5339_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_5340_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_5341_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_5342_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_5343_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_5344_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_5345_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_5346_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_5347_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_5348_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_5349_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_5350_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_5351_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_5352_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_5353_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_5354_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_5355_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_5356_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_5357_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_5358_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_5359_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_5360_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_5361_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_5362_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_5363_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_5364_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_5365_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_5366_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_5367_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_5368_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= ( ring_1_of_int_int @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_5369_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N )
= ( ring_1_of_int_real @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_5370_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N )
= ( ring_17405671764205052669omplex @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_5371_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N )
= ( ring_18347121197199848620nteger @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_5372_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
= ( ring_1_of_int_rat @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_5373_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int_int @ Y )
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_5374_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int_real @ Y )
= ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_5375_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_17405671764205052669omplex @ Y )
= ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_5376_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_18347121197199848620nteger @ Y )
= ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_5377_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int_rat @ Y )
= ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_5378_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_5379_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_5380_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_5381_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_5382_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_5383_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_5384_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_5385_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_5386_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_5387_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5388_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5389_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5390_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5391_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5392_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5393_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5394_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5395_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5396_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5397_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5398_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5399_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5400_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5401_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5402_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5403_dist__norm,axiom,
( real_V975177566351809787t_real
= ( ^ [X4: real,Y5: real] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y5 ) ) ) ) ).
% dist_norm
thf(fact_5404_dist__norm,axiom,
( real_V3694042436643373181omplex
= ( ^ [X4: complex,Y5: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y5 ) ) ) ) ).
% dist_norm
thf(fact_5405_norm__conv__dist,axiom,
( real_V7735802525324610683m_real
= ( ^ [X4: real] : ( real_V975177566351809787t_real @ X4 @ zero_zero_real ) ) ) ).
% norm_conv_dist
thf(fact_5406_norm__conv__dist,axiom,
( real_V1022390504157884413omplex
= ( ^ [X4: complex] : ( real_V3694042436643373181omplex @ X4 @ zero_zero_complex ) ) ) ).
% norm_conv_dist
thf(fact_5407_int__ops_I3_J,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% int_ops(3)
thf(fact_5408_nat__numeral__as__int,axiom,
( numeral_numeral_nat
= ( ^ [I3: num] : ( nat2 @ ( numeral_numeral_int @ I3 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_5409_power__int__commutes,axiom,
! [X: real,N: int] :
( ( times_times_real @ ( power_int_real @ X @ N ) @ X )
= ( times_times_real @ X @ ( power_int_real @ X @ N ) ) ) ).
% power_int_commutes
thf(fact_5410_power__int__commutes,axiom,
! [X: rat,N: int] :
( ( times_times_rat @ ( power_int_rat @ X @ N ) @ X )
= ( times_times_rat @ X @ ( power_int_rat @ X @ N ) ) ) ).
% power_int_commutes
thf(fact_5411_power__int__mult__distrib,axiom,
! [X: real,Y: real,M: int] :
( ( power_int_real @ ( times_times_real @ X @ Y ) @ M )
= ( times_times_real @ ( power_int_real @ X @ M ) @ ( power_int_real @ Y @ M ) ) ) ).
% power_int_mult_distrib
thf(fact_5412_power__int__mult__distrib,axiom,
! [X: rat,Y: rat,M: int] :
( ( power_int_rat @ ( times_times_rat @ X @ Y ) @ M )
= ( times_times_rat @ ( power_int_rat @ X @ M ) @ ( power_int_rat @ Y @ M ) ) ) ).
% power_int_mult_distrib
thf(fact_5413_power__int__divide__distrib,axiom,
! [X: complex,Y: complex,M: int] :
( ( power_int_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ M )
= ( divide1717551699836669952omplex @ ( power_int_complex @ X @ M ) @ ( power_int_complex @ Y @ M ) ) ) ).
% power_int_divide_distrib
thf(fact_5414_power__int__divide__distrib,axiom,
! [X: real,Y: real,M: int] :
( ( power_int_real @ ( divide_divide_real @ X @ Y ) @ M )
= ( divide_divide_real @ ( power_int_real @ X @ M ) @ ( power_int_real @ Y @ M ) ) ) ).
% power_int_divide_distrib
thf(fact_5415_power__int__divide__distrib,axiom,
! [X: rat,Y: rat,M: int] :
( ( power_int_rat @ ( divide_divide_rat @ X @ Y ) @ M )
= ( divide_divide_rat @ ( power_int_rat @ X @ M ) @ ( power_int_rat @ Y @ M ) ) ) ).
% power_int_divide_distrib
thf(fact_5416_power__int__abs,axiom,
! [A: real,N: int] :
( ( abs_abs_real @ ( power_int_real @ A @ N ) )
= ( power_int_real @ ( abs_abs_real @ A ) @ N ) ) ).
% power_int_abs
thf(fact_5417_power__int__abs,axiom,
! [A: rat,N: int] :
( ( abs_abs_rat @ ( power_int_rat @ A @ N ) )
= ( power_int_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% power_int_abs
thf(fact_5418_power__int__inverse,axiom,
! [X: real,N: int] :
( ( power_int_real @ ( inverse_inverse_real @ X ) @ N )
= ( inverse_inverse_real @ ( power_int_real @ X @ N ) ) ) ).
% power_int_inverse
thf(fact_5419_power__int__inverse,axiom,
! [X: complex,N: int] :
( ( power_int_complex @ ( invers8013647133539491842omplex @ X ) @ N )
= ( invers8013647133539491842omplex @ ( power_int_complex @ X @ N ) ) ) ).
% power_int_inverse
thf(fact_5420_power__int__inverse,axiom,
! [X: rat,N: int] :
( ( power_int_rat @ ( inverse_inverse_rat @ X ) @ N )
= ( inverse_inverse_rat @ ( power_int_rat @ X @ N ) ) ) ).
% power_int_inverse
thf(fact_5421_power__int__mult,axiom,
! [X: real,M: int,N: int] :
( ( power_int_real @ X @ ( times_times_int @ M @ N ) )
= ( power_int_real @ ( power_int_real @ X @ M ) @ N ) ) ).
% power_int_mult
thf(fact_5422_norm__minus__commute,axiom,
! [A: real,B: real] :
( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) )
= ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ A ) ) ) ).
% norm_minus_commute
thf(fact_5423_norm__minus__commute,axiom,
! [A: complex,B: complex] :
( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) )
= ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ A ) ) ) ).
% norm_minus_commute
thf(fact_5424_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_complex
!= ( numera6690914467698888265omplex @ N ) ) ).
% zero_neq_numeral
thf(fact_5425_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_5426_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_rat
!= ( numeral_numeral_rat @ N ) ) ).
% zero_neq_numeral
thf(fact_5427_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_5428_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_5429_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
!= ( numeral_numeral_int @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_5430_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
!= ( numeral_numeral_real @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_5431_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
!= ( numera6690914467698888265omplex @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_5432_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
!= ( numera6620942414471956472nteger @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_5433_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
!= ( numeral_numeral_rat @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_5434_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_int @ M )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_5435_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_real @ M )
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_5436_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numera6690914467698888265omplex @ M )
!= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_5437_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numera6620942414471956472nteger @ M )
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_5438_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_rat @ M )
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_5439_Ints__numeral,axiom,
! [N: num] : ( member_complex @ ( numera6690914467698888265omplex @ N ) @ ring_1_Ints_complex ) ).
% Ints_numeral
thf(fact_5440_Ints__numeral,axiom,
! [N: num] : ( member_real @ ( numeral_numeral_real @ N ) @ ring_1_Ints_real ) ).
% Ints_numeral
thf(fact_5441_Ints__numeral,axiom,
! [N: num] : ( member_rat @ ( numeral_numeral_rat @ N ) @ ring_1_Ints_rat ) ).
% Ints_numeral
thf(fact_5442_Ints__numeral,axiom,
! [N: num] : ( member_int @ ( numeral_numeral_int @ N ) @ ring_1_Ints_int ) ).
% Ints_numeral
thf(fact_5443_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_5444_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_5445_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_5446_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_5447_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_5448_zero__le__power__int,axiom,
! [X: real,N: int] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_int_real @ X @ N ) ) ) ).
% zero_le_power_int
thf(fact_5449_zero__le__power__int,axiom,
! [X: rat,N: int] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( power_int_rat @ X @ N ) ) ) ).
% zero_le_power_int
thf(fact_5450_zero__less__power__int,axiom,
! [X: real,N: int] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( power_int_real @ X @ N ) ) ) ).
% zero_less_power_int
thf(fact_5451_zero__less__power__int,axiom,
! [X: rat,N: int] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ord_less_rat @ zero_zero_rat @ ( power_int_rat @ X @ N ) ) ) ).
% zero_less_power_int
thf(fact_5452_power__int__one__over,axiom,
! [X: complex,N: int] :
( ( power_int_complex @ ( divide1717551699836669952omplex @ one_one_complex @ X ) @ N )
= ( divide1717551699836669952omplex @ one_one_complex @ ( power_int_complex @ X @ N ) ) ) ).
% power_int_one_over
thf(fact_5453_power__int__one__over,axiom,
! [X: real,N: int] :
( ( power_int_real @ ( divide_divide_real @ one_one_real @ X ) @ N )
= ( divide_divide_real @ one_one_real @ ( power_int_real @ X @ N ) ) ) ).
% power_int_one_over
thf(fact_5454_power__int__one__over,axiom,
! [X: rat,N: int] :
( ( power_int_rat @ ( divide_divide_rat @ one_one_rat @ X ) @ N )
= ( divide_divide_rat @ one_one_rat @ ( power_int_rat @ X @ N ) ) ) ).
% power_int_one_over
thf(fact_5455_power__int__not__zero,axiom,
! [X: real,N: int] :
( ( ( X != zero_zero_real )
| ( N = zero_zero_int ) )
=> ( ( power_int_real @ X @ N )
!= zero_zero_real ) ) ).
% power_int_not_zero
thf(fact_5456_power__int__not__zero,axiom,
! [X: rat,N: int] :
( ( ( X != zero_zero_rat )
| ( N = zero_zero_int ) )
=> ( ( power_int_rat @ X @ N )
!= zero_zero_rat ) ) ).
% power_int_not_zero
thf(fact_5457_power__int__minus,axiom,
! [X: real,N: int] :
( ( power_int_real @ X @ ( uminus_uminus_int @ N ) )
= ( inverse_inverse_real @ ( power_int_real @ X @ N ) ) ) ).
% power_int_minus
thf(fact_5458_power__int__minus,axiom,
! [X: complex,N: int] :
( ( power_int_complex @ X @ ( uminus_uminus_int @ N ) )
= ( invers8013647133539491842omplex @ ( power_int_complex @ X @ N ) ) ) ).
% power_int_minus
thf(fact_5459_power__int__minus,axiom,
! [X: rat,N: int] :
( ( power_int_rat @ X @ ( uminus_uminus_int @ N ) )
= ( inverse_inverse_rat @ ( power_int_rat @ X @ N ) ) ) ).
% power_int_minus
thf(fact_5460_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_le_numeral
thf(fact_5461_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% zero_le_numeral
thf(fact_5462_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_5463_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_5464_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_5465_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% not_numeral_le_zero
thf(fact_5466_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_5467_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_5468_zero__less__numeral,axiom,
! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_less_numeral
thf(fact_5469_zero__less__numeral,axiom,
! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% zero_less_numeral
thf(fact_5470_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_5471_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_5472_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_less_zero
thf(fact_5473_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% not_numeral_less_zero
thf(fact_5474_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_5475_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_5476_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% one_le_numeral
thf(fact_5477_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% one_le_numeral
thf(fact_5478_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_5479_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_5480_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% not_numeral_less_one
thf(fact_5481_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% not_numeral_less_one
thf(fact_5482_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_5483_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_5484_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_5485_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_5486_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_5487_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_5488_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_5489_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_5490_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_5491_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_5492_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5493_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5494_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5495_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_z3403309356797280102nteger
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5496_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_rat
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5497_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_5498_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_5499_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_5500_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_5501_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_5502_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_5503_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_5504_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_5505_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_5506_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_5507_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_5508_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_5509_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_5510_numeral__times__minus__swap,axiom,
! [W2: num,X: int] :
( ( times_times_int @ ( numeral_numeral_int @ W2 ) @ ( uminus_uminus_int @ X ) )
= ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_5511_numeral__times__minus__swap,axiom,
! [W2: num,X: real] :
( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ ( uminus_uminus_real @ X ) )
= ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_5512_numeral__times__minus__swap,axiom,
! [W2: num,X: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ ( uminus1482373934393186551omplex @ X ) )
= ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_5513_numeral__times__minus__swap,axiom,
! [W2: num,X: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W2 ) @ ( uminus1351360451143612070nteger @ X ) )
= ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_5514_numeral__times__minus__swap,axiom,
! [W2: num,X: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ ( uminus_uminus_rat @ X ) )
= ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_5515_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_5516_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_real @ N )
!= ( uminus_uminus_real @ one_one_real ) ) ).
% numeral_neq_neg_one
thf(fact_5517_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ N )
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% numeral_neq_neg_one
thf(fact_5518_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numera6620942414471956472nteger @ N )
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% numeral_neq_neg_one
thf(fact_5519_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_rat @ N )
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% numeral_neq_neg_one
thf(fact_5520_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_5521_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_5522_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_complex
!= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_5523_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_Code_integer
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_5524_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_rat
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_5525_norm__inverse,axiom,
! [A: real] :
( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
= ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% norm_inverse
thf(fact_5526_norm__inverse,axiom,
! [A: complex] :
( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
= ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% norm_inverse
thf(fact_5527_power__int__0__left__If,axiom,
! [M: int] :
( ( ( M = zero_zero_int )
=> ( ( power_int_complex @ zero_zero_complex @ M )
= one_one_complex ) )
& ( ( M != zero_zero_int )
=> ( ( power_int_complex @ zero_zero_complex @ M )
= zero_zero_complex ) ) ) ).
% power_int_0_left_If
thf(fact_5528_power__int__0__left__If,axiom,
! [M: int] :
( ( ( M = zero_zero_int )
=> ( ( power_int_real @ zero_zero_real @ M )
= one_one_real ) )
& ( ( M != zero_zero_int )
=> ( ( power_int_real @ zero_zero_real @ M )
= zero_zero_real ) ) ) ).
% power_int_0_left_If
thf(fact_5529_power__int__0__left__If,axiom,
! [M: int] :
( ( ( M = zero_zero_int )
=> ( ( power_int_rat @ zero_zero_rat @ M )
= one_one_rat ) )
& ( ( M != zero_zero_int )
=> ( ( power_int_rat @ zero_zero_rat @ M )
= zero_zero_rat ) ) ) ).
% power_int_0_left_If
thf(fact_5530_power__int__increasing,axiom,
! [N: int,N5: int,A: real] :
( ( ord_less_eq_int @ N @ N5 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_int_real @ A @ N ) @ ( power_int_real @ A @ N5 ) ) ) ) ).
% power_int_increasing
thf(fact_5531_power__int__increasing,axiom,
! [N: int,N5: int,A: rat] :
( ( ord_less_eq_int @ N @ N5 )
=> ( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ord_less_eq_rat @ ( power_int_rat @ A @ N ) @ ( power_int_rat @ A @ N5 ) ) ) ) ).
% power_int_increasing
thf(fact_5532_power__int__strict__increasing,axiom,
! [N: int,N5: int,A: real] :
( ( ord_less_int @ N @ N5 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_int_real @ A @ N ) @ ( power_int_real @ A @ N5 ) ) ) ) ).
% power_int_strict_increasing
thf(fact_5533_power__int__strict__increasing,axiom,
! [N: int,N5: int,A: rat] :
( ( ord_less_int @ N @ N5 )
=> ( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ ( power_int_rat @ A @ N ) @ ( power_int_rat @ A @ N5 ) ) ) ) ).
% power_int_strict_increasing
thf(fact_5534_power__int__minus__one__minus,axiom,
! [N: int] :
( ( power_int_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_int @ N ) )
= ( power_int_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% power_int_minus_one_minus
thf(fact_5535_power__int__minus__one__minus,axiom,
! [N: int] :
( ( power_int_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus_uminus_int @ N ) )
= ( power_int_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% power_int_minus_one_minus
thf(fact_5536_power__int__minus__one__minus,axiom,
! [N: int] :
( ( power_int_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_int @ N ) )
= ( power_int_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) ) ).
% power_int_minus_one_minus
thf(fact_5537_power__int__diff,axiom,
! [X: complex,M: int,N: int] :
( ( ( X != zero_zero_complex )
| ( M != N ) )
=> ( ( power_int_complex @ X @ ( minus_minus_int @ M @ N ) )
= ( divide1717551699836669952omplex @ ( power_int_complex @ X @ M ) @ ( power_int_complex @ X @ N ) ) ) ) ).
% power_int_diff
thf(fact_5538_power__int__diff,axiom,
! [X: real,M: int,N: int] :
( ( ( X != zero_zero_real )
| ( M != N ) )
=> ( ( power_int_real @ X @ ( minus_minus_int @ M @ N ) )
= ( divide_divide_real @ ( power_int_real @ X @ M ) @ ( power_int_real @ X @ N ) ) ) ) ).
% power_int_diff
thf(fact_5539_power__int__diff,axiom,
! [X: rat,M: int,N: int] :
( ( ( X != zero_zero_rat )
| ( M != N ) )
=> ( ( power_int_rat @ X @ ( minus_minus_int @ M @ N ) )
= ( divide_divide_rat @ ( power_int_rat @ X @ M ) @ ( power_int_rat @ X @ N ) ) ) ) ).
% power_int_diff
thf(fact_5540_power__int__minus__one__diff__commute,axiom,
! [A: int,B: int] :
( ( power_int_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_int @ A @ B ) )
= ( power_int_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_int @ B @ A ) ) ) ).
% power_int_minus_one_diff_commute
thf(fact_5541_power__int__minus__one__diff__commute,axiom,
! [A: int,B: int] :
( ( power_int_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_int @ A @ B ) )
= ( power_int_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_int @ B @ A ) ) ) ).
% power_int_minus_one_diff_commute
thf(fact_5542_power__int__minus__one__diff__commute,axiom,
! [A: int,B: int] :
( ( power_int_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_int @ A @ B ) )
= ( power_int_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_int @ B @ A ) ) ) ).
% power_int_minus_one_diff_commute
thf(fact_5543_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_5544_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_5545_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_5546_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_5547_power__eq__imp__eq__norm,axiom,
! [W2: real,N: nat,Z2: real] :
( ( ( power_power_real @ W2 @ N )
= ( power_power_real @ Z2 @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( real_V7735802525324610683m_real @ W2 )
= ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_5548_power__eq__imp__eq__norm,axiom,
! [W2: complex,N: nat,Z2: complex] :
( ( ( power_power_complex @ W2 @ N )
= ( power_power_complex @ Z2 @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( real_V1022390504157884413omplex @ W2 )
= ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_5549_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_5550_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_5551_norm__add__less,axiom,
! [X: real,R2: real,Y: real,S: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% norm_add_less
thf(fact_5552_norm__add__less,axiom,
! [X: complex,R2: real,Y: complex,S: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% norm_add_less
thf(fact_5553_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_5554_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_5555_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_5556_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_5557_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_5558_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_5559_norm__triangle__mono,axiom,
! [A: real,R2: real,B: real,S: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% norm_triangle_mono
thf(fact_5560_norm__triangle__mono,axiom,
! [A: complex,R2: real,B: complex,S: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% norm_triangle_mono
thf(fact_5561_norm__diff__triangle__less,axiom,
! [X: real,Y: real,E1: real,Z2: 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 @ Z2 ) ) @ E22 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_5562_norm__diff__triangle__less,axiom,
! [X: complex,Y: complex,E1: real,Z2: complex,E22: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z2 ) ) @ E22 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_5563_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_5564_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_5565_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_5566_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_5567_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_5568_norm__diff__triangle__le,axiom,
! [X: real,Y: real,E1: real,Z2: 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 @ Z2 ) ) @ E22 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_5569_norm__diff__triangle__le,axiom,
! [X: complex,Y: complex,E1: real,Z2: 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 @ Z2 ) ) @ E22 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_5570_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_5571_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_5572_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% neg_numeral_le_zero
thf(fact_5573_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% neg_numeral_le_zero
thf(fact_5574_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% neg_numeral_le_zero
thf(fact_5575_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_le_zero
thf(fact_5576_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_5577_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_5578_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_5579_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_5580_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_less_zero
thf(fact_5581_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% neg_numeral_less_zero
thf(fact_5582_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% neg_numeral_less_zero
thf(fact_5583_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% neg_numeral_less_zero
thf(fact_5584_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5585_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5586_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5587_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5588_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_5589_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_5590_eq__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: complex,C: complex] :
( ( ( numera6690914467698888265omplex @ W2 )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C )
= B ) )
& ( ( C = zero_zero_complex )
=> ( ( numera6690914467698888265omplex @ W2 )
= zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_5591_eq__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: real,C: real] :
( ( ( numeral_numeral_real @ W2 )
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W2 )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_5592_eq__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: rat,C: rat] :
( ( ( numeral_numeral_rat @ W2 )
= ( divide_divide_rat @ B @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C )
= B ) )
& ( ( C = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_5593_divide__eq__eq__numeral_I1_J,axiom,
! [B: complex,C: complex,W2: num] :
( ( ( divide1717551699836669952omplex @ B @ C )
= ( numera6690914467698888265omplex @ W2 ) )
= ( ( ( C != zero_zero_complex )
=> ( B
= ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( ( numera6690914467698888265omplex @ W2 )
= zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_5594_divide__eq__eq__numeral_I1_J,axiom,
! [B: real,C: real,W2: num] :
( ( ( divide_divide_real @ B @ C )
= ( numeral_numeral_real @ W2 ) )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W2 )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_5595_divide__eq__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W2: num] :
( ( ( divide_divide_rat @ B @ C )
= ( numeral_numeral_rat @ W2 ) )
= ( ( ( C != zero_zero_rat )
=> ( B
= ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_5596_norm__triangle__ineq2,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% norm_triangle_ineq2
thf(fact_5597_norm__triangle__ineq2,axiom,
! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% norm_triangle_ineq2
thf(fact_5598_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_5599_neg__numeral__le__one,axiom,
! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% neg_numeral_le_one
thf(fact_5600_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_5601_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_5602_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_5603_neg__one__le__numeral,axiom,
! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% neg_one_le_numeral
thf(fact_5604_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_5605_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_5606_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_5607_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_5608_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_5609_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_5610_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_5611_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% not_numeral_le_neg_one
thf(fact_5612_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_5613_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_5614_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_5615_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_5616_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_5617_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_5618_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_5619_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_5620_neg__numeral__less__one,axiom,
! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% neg_numeral_less_one
thf(fact_5621_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_5622_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_5623_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_5624_neg__one__less__numeral,axiom,
! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% neg_one_less_numeral
thf(fact_5625_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_5626_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_5627_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_5628_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% not_numeral_less_neg_one
thf(fact_5629_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_5630_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_5631_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_5632_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_5633_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_5634_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_5635_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_5636_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_5637_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_5638_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_5639_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_5640_floor__le__round,axiom,
! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim8280529875227126926d_real @ X ) ) ).
% floor_le_round
thf(fact_5641_floor__le__round,axiom,
! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim7778729529865785530nd_rat @ X ) ) ).
% floor_le_round
thf(fact_5642_ceiling__ge__round,axiom,
! [X: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% ceiling_ge_round
thf(fact_5643_powr__neg__numeral,axiom,
! [X: real,N: num] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_5644_power__int__strict__decreasing,axiom,
! [N: int,N5: int,A: real] :
( ( ord_less_int @ N @ N5 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_int_real @ A @ N5 ) @ ( power_int_real @ A @ N ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_5645_power__int__strict__decreasing,axiom,
! [N: int,N5: int,A: rat] :
( ( ord_less_int @ N @ N5 )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( power_int_rat @ A @ N5 ) @ ( power_int_rat @ A @ N ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_5646_power__int__mono,axiom,
! [X: real,Y: real,N: int] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( power_int_real @ X @ N ) @ ( power_int_real @ Y @ N ) ) ) ) ) ).
% power_int_mono
thf(fact_5647_power__int__mono,axiom,
! [X: rat,Y: rat,N: int] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ord_less_eq_rat @ ( power_int_rat @ X @ N ) @ ( power_int_rat @ Y @ N ) ) ) ) ) ).
% power_int_mono
thf(fact_5648_power__int__strict__antimono,axiom,
! [A: real,B: real,N: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_int @ N @ zero_zero_int )
=> ( ord_less_real @ ( power_int_real @ B @ N ) @ ( power_int_real @ A @ N ) ) ) ) ) ).
% power_int_strict_antimono
thf(fact_5649_power__int__strict__antimono,axiom,
! [A: rat,B: rat,N: int] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_int @ N @ zero_zero_int )
=> ( ord_less_rat @ ( power_int_rat @ B @ N ) @ ( power_int_rat @ A @ N ) ) ) ) ) ).
% power_int_strict_antimono
thf(fact_5650_one__le__power__int,axiom,
! [X: real,N: int] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ord_less_eq_real @ one_one_real @ ( power_int_real @ X @ N ) ) ) ) ).
% one_le_power_int
thf(fact_5651_one__le__power__int,axiom,
! [X: rat,N: int] :
( ( ord_less_eq_rat @ one_one_rat @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ord_less_eq_rat @ one_one_rat @ ( power_int_rat @ X @ N ) ) ) ) ).
% one_le_power_int
thf(fact_5652_one__less__power__int,axiom,
! [A: real,N: int] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_real @ one_one_real @ ( power_int_real @ A @ N ) ) ) ) ).
% one_less_power_int
thf(fact_5653_one__less__power__int,axiom,
! [A: rat,N: int] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_rat @ one_one_rat @ ( power_int_rat @ A @ N ) ) ) ) ).
% one_less_power_int
thf(fact_5654_power__int__add,axiom,
! [X: real,M: int,N: int] :
( ( ( X != zero_zero_real )
| ( ( plus_plus_int @ M @ N )
!= zero_zero_int ) )
=> ( ( power_int_real @ X @ ( plus_plus_int @ M @ N ) )
= ( times_times_real @ ( power_int_real @ X @ M ) @ ( power_int_real @ X @ N ) ) ) ) ).
% power_int_add
thf(fact_5655_power__int__add,axiom,
! [X: rat,M: int,N: int] :
( ( ( X != zero_zero_rat )
| ( ( plus_plus_int @ M @ N )
!= zero_zero_int ) )
=> ( ( power_int_rat @ X @ ( plus_plus_int @ M @ N ) )
= ( times_times_rat @ ( power_int_rat @ X @ M ) @ ( power_int_rat @ X @ N ) ) ) ) ).
% power_int_add
thf(fact_5656_power__eq__1__iff,axiom,
! [W2: real,N: nat] :
( ( ( power_power_real @ W2 @ N )
= one_one_real )
=> ( ( ( real_V7735802525324610683m_real @ W2 )
= one_one_real )
| ( N = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_5657_power__eq__1__iff,axiom,
! [W2: complex,N: nat] :
( ( ( power_power_complex @ W2 @ N )
= one_one_complex )
=> ( ( ( real_V1022390504157884413omplex @ W2 )
= one_one_real )
| ( N = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_5658_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_5659_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_5660_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_5661_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_5662_divide__less__eq__numeral_I1_J,axiom,
! [B: real,C: real,W2: num] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_5663_divide__less__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W2: num] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_5664_less__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: real,C: real] :
( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_5665_less__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: rat,C: rat] :
( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_5666_dist__of__int,axiom,
! [M: int,N: int] :
( ( real_V975177566351809787t_real @ ( ring_1_of_int_real @ M ) @ ( ring_1_of_int_real @ N ) )
= ( ring_1_of_int_real @ ( abs_abs_int @ ( minus_minus_int @ M @ N ) ) ) ) ).
% dist_of_int
thf(fact_5667_dist__of__int,axiom,
! [M: int,N: int] :
( ( real_V3694042436643373181omplex @ ( ring_17405671764205052669omplex @ M ) @ ( ring_17405671764205052669omplex @ N ) )
= ( ring_1_of_int_real @ ( abs_abs_int @ ( minus_minus_int @ M @ N ) ) ) ) ).
% dist_of_int
thf(fact_5668_divide__eq__eq__numeral_I2_J,axiom,
! [B: real,C: real,W2: num] :
( ( ( divide_divide_real @ B @ C )
= ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_5669_divide__eq__eq__numeral_I2_J,axiom,
! [B: complex,C: complex,W2: num] :
( ( ( divide1717551699836669952omplex @ B @ C )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= ( ( ( C != zero_zero_complex )
=> ( B
= ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_5670_divide__eq__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W2: num] :
( ( ( divide_divide_rat @ B @ C )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( C != zero_zero_rat )
=> ( B
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_5671_eq__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: real,C: real] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_5672_eq__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: complex,C: complex] :
( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C )
= B ) )
& ( ( C = zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_5673_eq__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: rat,C: rat] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= ( divide_divide_rat @ B @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C )
= B ) )
& ( ( C = zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_5674_norm__triangle__ineq3,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% norm_triangle_ineq3
thf(fact_5675_norm__triangle__ineq3,axiom,
! [A: complex,B: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% norm_triangle_ineq3
thf(fact_5676_power__int__antimono,axiom,
! [A: real,B: real,N: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_int @ N @ zero_zero_int )
=> ( ord_less_eq_real @ ( power_int_real @ B @ N ) @ ( power_int_real @ A @ N ) ) ) ) ) ).
% power_int_antimono
thf(fact_5677_power__int__antimono,axiom,
! [A: rat,B: rat,N: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_int @ N @ zero_zero_int )
=> ( ord_less_eq_rat @ ( power_int_rat @ B @ N ) @ ( power_int_rat @ A @ N ) ) ) ) ) ).
% power_int_antimono
thf(fact_5678_power__int__strict__mono,axiom,
! [A: real,B: real,N: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_real @ ( power_int_real @ A @ N ) @ ( power_int_real @ B @ N ) ) ) ) ) ).
% power_int_strict_mono
thf(fact_5679_power__int__strict__mono,axiom,
! [A: rat,B: rat,N: int] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_rat @ ( power_int_rat @ A @ N ) @ ( power_int_rat @ B @ N ) ) ) ) ) ).
% power_int_strict_mono
thf(fact_5680_power__int__decreasing,axiom,
! [N: int,N5: int,A: real] :
( ( ord_less_eq_int @ N @ N5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ( A != zero_zero_real )
| ( N5 != zero_zero_int )
| ( N = zero_zero_int ) )
=> ( ord_less_eq_real @ ( power_int_real @ A @ N5 ) @ ( power_int_real @ A @ N ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_5681_power__int__decreasing,axiom,
! [N: int,N5: int,A: rat] :
( ( ord_less_eq_int @ N @ N5 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ( ( A != zero_zero_rat )
| ( N5 != zero_zero_int )
| ( N = zero_zero_int ) )
=> ( ord_less_eq_rat @ ( power_int_rat @ A @ N5 ) @ ( power_int_rat @ A @ N ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_5682_power__int__le__one,axiom,
! [X: real,N: int] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( power_int_real @ X @ N ) @ one_one_real ) ) ) ) ).
% power_int_le_one
thf(fact_5683_power__int__le__one,axiom,
! [X: rat,N: int] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( ord_less_eq_rat @ X @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_int_rat @ X @ N ) @ one_one_rat ) ) ) ) ).
% power_int_le_one
thf(fact_5684_power__int__le__imp__le__exp,axiom,
! [X: real,M: int,N: int] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ ( power_int_real @ X @ M ) @ ( power_int_real @ X @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ M @ N ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_5685_power__int__le__imp__le__exp,axiom,
! [X: rat,M: int,N: int] :
( ( ord_less_rat @ one_one_rat @ X )
=> ( ( ord_less_eq_rat @ ( power_int_rat @ X @ M ) @ ( power_int_rat @ X @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ M @ N ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_5686_power__int__le__imp__less__exp,axiom,
! [X: real,M: int,N: int] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ ( power_int_real @ X @ M ) @ ( power_int_real @ X @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ord_less_int @ M @ N ) ) ) ) ).
% power_int_le_imp_less_exp
thf(fact_5687_power__int__le__imp__less__exp,axiom,
! [X: rat,M: int,N: int] :
( ( ord_less_rat @ one_one_rat @ X )
=> ( ( ord_less_rat @ ( power_int_rat @ X @ M ) @ ( power_int_rat @ X @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ord_less_int @ M @ N ) ) ) ) ).
% power_int_le_imp_less_exp
thf(fact_5688_power__int__minus__mult,axiom,
! [X: real,N: int] :
( ( ( X != zero_zero_real )
| ( N != zero_zero_int ) )
=> ( ( times_times_real @ ( power_int_real @ X @ ( minus_minus_int @ N @ one_one_int ) ) @ X )
= ( power_int_real @ X @ N ) ) ) ).
% power_int_minus_mult
thf(fact_5689_power__int__minus__mult,axiom,
! [X: rat,N: int] :
( ( ( X != zero_zero_rat )
| ( N != zero_zero_int ) )
=> ( ( times_times_rat @ ( power_int_rat @ X @ ( minus_minus_int @ N @ one_one_int ) ) @ X )
= ( power_int_rat @ X @ N ) ) ) ).
% power_int_minus_mult
thf(fact_5690_power__int__add__1_H,axiom,
! [X: real,M: int] :
( ( ( X != zero_zero_real )
| ( M
!= ( uminus_uminus_int @ one_one_int ) ) )
=> ( ( power_int_real @ X @ ( plus_plus_int @ M @ one_one_int ) )
= ( times_times_real @ X @ ( power_int_real @ X @ M ) ) ) ) ).
% power_int_add_1'
thf(fact_5691_power__int__add__1_H,axiom,
! [X: rat,M: int] :
( ( ( X != zero_zero_rat )
| ( M
!= ( uminus_uminus_int @ one_one_int ) ) )
=> ( ( power_int_rat @ X @ ( plus_plus_int @ M @ one_one_int ) )
= ( times_times_rat @ X @ ( power_int_rat @ X @ M ) ) ) ) ).
% power_int_add_1'
thf(fact_5692_power__int__add__1,axiom,
! [X: real,M: int] :
( ( ( X != zero_zero_real )
| ( M
!= ( uminus_uminus_int @ one_one_int ) ) )
=> ( ( power_int_real @ X @ ( plus_plus_int @ M @ one_one_int ) )
= ( times_times_real @ ( power_int_real @ X @ M ) @ X ) ) ) ).
% power_int_add_1
thf(fact_5693_power__int__add__1,axiom,
! [X: rat,M: int] :
( ( ( X != zero_zero_rat )
| ( M
!= ( uminus_uminus_int @ one_one_int ) ) )
=> ( ( power_int_rat @ X @ ( plus_plus_int @ M @ one_one_int ) )
= ( times_times_rat @ ( power_int_rat @ X @ M ) @ X ) ) ) ).
% power_int_add_1
thf(fact_5694_Cauchy__altdef,axiom,
( topolo4055970368930404560y_real
= ( ^ [F2: nat > real] :
! [E3: real] :
( ( ord_less_real @ zero_zero_real @ E3 )
=> ? [M7: nat] :
! [M4: nat] :
( ( ord_less_eq_nat @ M7 @ M4 )
=> ! [N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( ord_less_real @ ( real_V975177566351809787t_real @ ( F2 @ M4 ) @ ( F2 @ N4 ) ) @ E3 ) ) ) ) ) ) ).
% Cauchy_altdef
thf(fact_5695_CauchyI_H,axiom,
! [X9: nat > real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [M8: nat] :
! [M2: nat] :
( ( ord_less_eq_nat @ M8 @ M2 )
=> ! [N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_real @ ( real_V975177566351809787t_real @ ( X9 @ M2 ) @ ( X9 @ N2 ) ) @ E ) ) ) )
=> ( topolo4055970368930404560y_real @ X9 ) ) ).
% CauchyI'
thf(fact_5696_dist__of__nat,axiom,
! [M: nat,N: nat] :
( ( real_V975177566351809787t_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ring_1_of_int_real @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% dist_of_nat
thf(fact_5697_le__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: real,C: real] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_5698_le__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: rat,C: rat] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_5699_divide__le__eq__numeral_I1_J,axiom,
! [B: real,C: real,W2: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_5700_divide__le__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W2: num] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_5701_divide__less__eq__numeral_I2_J,axiom,
! [B: real,C: real,W2: num] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_5702_divide__less__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W2: num] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_5703_less__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: real,C: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_5704_less__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: rat,C: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_5705_norm__inverse__le__norm,axiom,
! [R2: real,X: real] :
( ( ord_less_eq_real @ R2 @ ( real_V7735802525324610683m_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_5706_norm__inverse__le__norm,axiom,
! [R2: real,X: complex] :
( ( ord_less_eq_real @ R2 @ ( real_V1022390504157884413omplex @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_5707_Cauchy__iff,axiom,
( topolo6517432010174082258omplex
= ( ^ [X8: nat > complex] :
! [E3: real] :
( ( ord_less_real @ zero_zero_real @ E3 )
=> ? [M7: nat] :
! [M4: nat] :
( ( ord_less_eq_nat @ M7 @ M4 )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M7 @ N4 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M4 ) @ ( X8 @ N4 ) ) ) @ E3 ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_5708_Cauchy__iff,axiom,
( topolo4055970368930404560y_real
= ( ^ [X8: nat > real] :
! [E3: real] :
( ( ord_less_real @ zero_zero_real @ E3 )
=> ? [M7: nat] :
! [M4: nat] :
( ( ord_less_eq_nat @ M7 @ M4 )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M7 @ N4 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M4 ) @ ( X8 @ N4 ) ) ) @ E3 ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_5709_CauchyI,axiom,
! [X9: nat > complex] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [M8: nat] :
! [M2: nat] :
( ( ord_less_eq_nat @ M8 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq_nat @ M8 @ N2 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X9 @ M2 ) @ ( X9 @ N2 ) ) ) @ E ) ) ) )
=> ( topolo6517432010174082258omplex @ X9 ) ) ).
% CauchyI
thf(fact_5710_CauchyI,axiom,
! [X9: nat > real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [M8: nat] :
! [M2: nat] :
( ( ord_less_eq_nat @ M8 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq_nat @ M8 @ N2 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X9 @ M2 ) @ ( X9 @ N2 ) ) ) @ E ) ) ) )
=> ( topolo4055970368930404560y_real @ X9 ) ) ).
% CauchyI
thf(fact_5711_CauchyD,axiom,
! [X9: nat > complex,E2: real] :
( ( topolo6517432010174082258omplex @ X9 )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [M9: nat] :
! [M5: nat] :
( ( ord_less_eq_nat @ M9 @ M5 )
=> ! [N6: nat] :
( ( ord_less_eq_nat @ M9 @ N6 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X9 @ M5 ) @ ( X9 @ N6 ) ) ) @ E2 ) ) ) ) ) ).
% CauchyD
thf(fact_5712_CauchyD,axiom,
! [X9: nat > real,E2: real] :
( ( topolo4055970368930404560y_real @ X9 )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [M9: nat] :
! [M5: nat] :
( ( ord_less_eq_nat @ M9 @ M5 )
=> ! [N6: nat] :
( ( ord_less_eq_nat @ M9 @ N6 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X9 @ M5 ) @ ( X9 @ N6 ) ) ) @ E2 ) ) ) ) ) ).
% CauchyD
thf(fact_5713_power__int__def,axiom,
( power_int_real
= ( ^ [X4: real,N4: int] : ( if_real @ ( ord_less_eq_int @ zero_zero_int @ N4 ) @ ( power_power_real @ X4 @ ( nat2 @ N4 ) ) @ ( power_power_real @ ( inverse_inverse_real @ X4 ) @ ( nat2 @ ( uminus_uminus_int @ N4 ) ) ) ) ) ) ).
% power_int_def
thf(fact_5714_power__int__def,axiom,
( power_int_complex
= ( ^ [X4: complex,N4: int] : ( if_complex @ ( ord_less_eq_int @ zero_zero_int @ N4 ) @ ( power_power_complex @ X4 @ ( nat2 @ N4 ) ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X4 ) @ ( nat2 @ ( uminus_uminus_int @ N4 ) ) ) ) ) ) ).
% power_int_def
thf(fact_5715_power__int__def,axiom,
( power_int_rat
= ( ^ [X4: rat,N4: int] : ( if_rat @ ( ord_less_eq_int @ zero_zero_int @ N4 ) @ ( power_power_rat @ X4 @ ( nat2 @ N4 ) ) @ ( power_power_rat @ ( inverse_inverse_rat @ X4 ) @ ( nat2 @ ( uminus_uminus_int @ N4 ) ) ) ) ) ) ).
% power_int_def
thf(fact_5716_power__int__numeral__neg__numeral,axiom,
! [M: num,N: num] :
( ( power_int_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( inverse_inverse_real @ ( numeral_numeral_real @ ( pow @ M @ N ) ) ) ) ).
% power_int_numeral_neg_numeral
thf(fact_5717_power__int__numeral__neg__numeral,axiom,
! [M: num,N: num] :
( ( power_int_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( inverse_inverse_rat @ ( numeral_numeral_rat @ ( pow @ M @ N ) ) ) ) ).
% power_int_numeral_neg_numeral
thf(fact_5718_enat__ord__number_I2_J,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_5719_lemma__termdiff3,axiom,
! [H: real,Z2: real,K5: real,N: nat] :
( ( H != zero_zero_real )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ K5 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z2 @ H ) ) @ K5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H ) @ N ) @ ( power_power_real @ Z2 @ N ) ) @ H ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_5720_lemma__termdiff3,axiom,
! [H: complex,Z2: complex,K5: real,N: nat] :
( ( H != zero_zero_complex )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ K5 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z2 @ H ) ) @ K5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H ) @ N ) @ ( power_power_complex @ Z2 @ N ) ) @ H ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_5721_complex__mod__minus__le__complex__mod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% complex_mod_minus_le_complex_mod
thf(fact_5722_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_5723_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_5724_ceiling__log__nat__eq__powr__iff,axiom,
! [B: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( archim7802044766580827645g_real @ ( log2 @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
= ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_5725_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_5726_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_5727_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_5728_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_5729_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_5730_verit__eq__simplify_I8_J,axiom,
! [X2: num,Y2: num] :
( ( ( bit0 @ X2 )
= ( bit0 @ Y2 ) )
= ( X2 = Y2 ) ) ).
% verit_eq_simplify(8)
thf(fact_5731_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_5732_member__bound,axiom,
! [Tree: vEBT_VEBT,X: nat,N: nat] :
( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_invar_vebt @ Tree @ N )
=> ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% member_bound
thf(fact_5733_bit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H2: nat,L3: nat,D3: nat] : ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D3 ) ) @ L3 ) ) ) ).
% bit_concat_def
thf(fact_5734_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera6690914467698888265omplex @ N )
= one_one_complex )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_5735_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_5736_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_rat @ N )
= one_one_rat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_5737_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_5738_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_5739_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_complex
= ( numera6690914467698888265omplex @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_5740_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_5741_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_rat
= ( numeral_numeral_rat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_5742_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_5743_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_5744_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_5745_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_5746_of__real__0,axiom,
( ( real_V1803761363581548252l_real @ zero_zero_real )
= zero_zero_real ) ).
% of_real_0
thf(fact_5747_of__real__0,axiom,
( ( real_V4546457046886955230omplex @ zero_zero_real )
= zero_zero_complex ) ).
% of_real_0
thf(fact_5748_of__real__1,axiom,
( ( real_V1803761363581548252l_real @ one_one_real )
= one_one_real ) ).
% of_real_1
thf(fact_5749_of__real__1,axiom,
( ( real_V4546457046886955230omplex @ one_one_real )
= one_one_complex ) ).
% of_real_1
thf(fact_5750_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_5751_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_5752_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_5753_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_5754_of__real__minus,axiom,
! [X: real] :
( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( real_V1803761363581548252l_real @ X ) ) ) ).
% of_real_minus
thf(fact_5755_of__real__minus,axiom,
! [X: real] :
( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ X ) )
= ( uminus1482373934393186551omplex @ ( real_V4546457046886955230omplex @ X ) ) ) ).
% of_real_minus
thf(fact_5756_minus__of__real__eq__of__real__iff,axiom,
! [X: real,Y: real] :
( ( ( uminus_uminus_real @ ( real_V1803761363581548252l_real @ X ) )
= ( real_V1803761363581548252l_real @ Y ) )
= ( ( uminus_uminus_real @ X )
= Y ) ) ).
% minus_of_real_eq_of_real_iff
thf(fact_5757_minus__of__real__eq__of__real__iff,axiom,
! [X: real,Y: real] :
( ( ( uminus1482373934393186551omplex @ ( real_V4546457046886955230omplex @ X ) )
= ( real_V4546457046886955230omplex @ Y ) )
= ( ( uminus_uminus_real @ X )
= Y ) ) ).
% minus_of_real_eq_of_real_iff
thf(fact_5758_of__real__eq__minus__of__real__iff,axiom,
! [X: real,Y: real] :
( ( ( real_V1803761363581548252l_real @ X )
= ( uminus_uminus_real @ ( real_V1803761363581548252l_real @ Y ) ) )
= ( X
= ( uminus_uminus_real @ Y ) ) ) ).
% of_real_eq_minus_of_real_iff
thf(fact_5759_of__real__eq__minus__of__real__iff,axiom,
! [X: real,Y: real] :
( ( ( real_V4546457046886955230omplex @ X )
= ( uminus1482373934393186551omplex @ ( real_V4546457046886955230omplex @ Y ) ) )
= ( X
= ( uminus_uminus_real @ Y ) ) ) ).
% of_real_eq_minus_of_real_iff
thf(fact_5760_of__real__diff,axiom,
! [X: real,Y: real] :
( ( real_V1803761363581548252l_real @ ( minus_minus_real @ X @ Y ) )
= ( minus_minus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% of_real_diff
thf(fact_5761_of__real__diff,axiom,
! [X: real,Y: real] :
( ( real_V4546457046886955230omplex @ ( minus_minus_real @ X @ Y ) )
= ( minus_minus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% of_real_diff
thf(fact_5762_of__real__of__nat__eq,axiom,
! [N: nat] :
( ( real_V4546457046886955230omplex @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri8010041392384452111omplex @ N ) ) ).
% of_real_of_nat_eq
thf(fact_5763_of__real__of__nat__eq,axiom,
! [N: nat] :
( ( real_V1803761363581548252l_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% of_real_of_nat_eq
thf(fact_5764_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_5765_of__real__inverse,axiom,
! [X: real] :
( ( real_V1803761363581548252l_real @ ( inverse_inverse_real @ X ) )
= ( inverse_inverse_real @ ( real_V1803761363581548252l_real @ X ) ) ) ).
% of_real_inverse
thf(fact_5766_of__real__inverse,axiom,
! [X: real] :
( ( real_V4546457046886955230omplex @ ( inverse_inverse_real @ X ) )
= ( invers8013647133539491842omplex @ ( real_V4546457046886955230omplex @ X ) ) ) ).
% of_real_inverse
thf(fact_5767_of__real__of__int__eq,axiom,
! [Z2: int] :
( ( real_V1803761363581548252l_real @ ( ring_1_of_int_real @ Z2 ) )
= ( ring_1_of_int_real @ Z2 ) ) ).
% of_real_of_int_eq
thf(fact_5768_of__real__of__int__eq,axiom,
! [Z2: int] :
( ( real_V4546457046886955230omplex @ ( ring_1_of_int_real @ Z2 ) )
= ( ring_17405671764205052669omplex @ Z2 ) ) ).
% of_real_of_int_eq
thf(fact_5769_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ one_one_int ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_5770_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
= ( uminus_uminus_real @ one_one_real ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_5771_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_5772_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_5773_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
= ( uminus_uminus_rat @ one_one_rat ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_5774_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_5775_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_real @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_5776_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus1482373934393186551omplex @ one_one_complex )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_5777_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_5778_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_rat @ one_one_rat )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_5779_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_5780_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_5781_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_5782_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_5783_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_5784_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_5785_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_5786_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_5787_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_5788_one__add__one,axiom,
( ( plus_plus_complex @ one_one_complex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_5789_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_5790_one__add__one,axiom,
( ( plus_plus_rat @ one_one_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_5791_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_5792_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_5793_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_5794_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_5795_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_5796_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_5797_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_5798_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_5799_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_5800_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_5801_one__mod__two__eq__one,axiom,
( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= one_one_Code_natural ) ).
% one_mod_two_eq_one
thf(fact_5802_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_5803_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_5804_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_5805_bits__one__mod__two__eq__one,axiom,
( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= one_one_Code_natural ) ).
% bits_one_mod_two_eq_one
thf(fact_5806_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_5807_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_5808_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_5809_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_5810_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_5811_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_5812_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_5813_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_5814_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_5815_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_5816_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_5817_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_5818_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_5819_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_5820_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_5821_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_5822_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_5823_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_5824_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
= ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_5825_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_5826_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_5827_of__real__neg__numeral,axiom,
! [W2: num] :
( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ).
% of_real_neg_numeral
thf(fact_5828_of__real__neg__numeral,axiom,
! [W2: num] :
( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ).
% of_real_neg_numeral
thf(fact_5829_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_5830_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_5831_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_5832_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_5833_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_5834_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_5835_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_5836_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_5837_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_5838_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_5839_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_5840_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_5841_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_5842_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_5843_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_5844_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_5845_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_5846_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_5847_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_5848_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_5849_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_5850_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_5851_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_5852_inrange,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% inrange
thf(fact_5853_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_5854_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_5855_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_5856_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_5857_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_5858_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_5859_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_5860_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_5861_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_5862_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_5863_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_5864_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_5865_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_5866_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_5867_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_5868_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_5869_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_5870_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_5871_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_5872_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_5873_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_5874_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_5875_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_5876_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_5877_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_5878_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_5879_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_5880_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_5881_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_5882_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_5883_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_5884_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_5885_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_5886_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_5887_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_5888_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_5889_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_5890_not__mod__2__eq__0__eq__1,axiom,
! [A: code_natural] :
( ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
!= zero_z2226904508553997617atural )
= ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= one_one_Code_natural ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_5891_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_5892_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_5893_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_5894_not__mod__2__eq__1__eq__0,axiom,
! [A: code_natural] :
( ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
!= one_one_Code_natural )
= ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= zero_z2226904508553997617atural ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_5895_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_5896_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_5897_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_5898_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_5899_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: int,N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5900_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5901_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5902_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: code_integer,N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5903_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5904_not__mod2__eq__Suc__0__eq__0,axiom,
! [N: nat] :
( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= ( suc @ zero_zero_nat ) )
= ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% not_mod2_eq_Suc_0_eq_0
thf(fact_5905_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_5906_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_5907_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_5908_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_5909_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_5910_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_5911_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_5912_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_5913_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_5914_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_5915_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_5916_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_5917_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_5918_power__minus1__even,axiom,
! [N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_int ) ).
% power_minus1_even
thf(fact_5919_power__minus1__even,axiom,
! [N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_real ) ).
% power_minus1_even
thf(fact_5920_power__minus1__even,axiom,
! [N: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_complex ) ).
% power_minus1_even
thf(fact_5921_power__minus1__even,axiom,
! [N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_Code_integer ) ).
% power_minus1_even
thf(fact_5922_power__minus1__even,axiom,
! [N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_rat ) ).
% power_minus1_even
thf(fact_5923_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_5924_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_5925_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_5926_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_5927_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_5928_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_5929_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_5930_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_5931__C3_Ohyps_C_I2_J,axiom,
( ( size_s6755466524823107622T_VEBT @ treeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% "3.hyps"(2)
thf(fact_5932_verit__eq__simplify_I10_J,axiom,
! [X2: num] :
( one
!= ( bit0 @ X2 ) ) ).
% verit_eq_simplify(10)
thf(fact_5933_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one )
=> ( ! [X3: num] :
( ( P @ X3 )
=> ( P @ ( inc @ X3 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_5934_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one )
= X ) ).
% pow.simps(1)
thf(fact_5935_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_5936_add__One,axiom,
! [X: num] :
( ( plus_plus_num @ X @ one )
= ( inc @ X ) ) ).
% add_One
thf(fact_5937_add__inc,axiom,
! [X: num,Y: num] :
( ( plus_plus_num @ X @ ( inc @ Y ) )
= ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).
% add_inc
thf(fact_5938_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_5939_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_5940_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_5941_zero__power2,axiom,
( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_rat ) ).
% zero_power2
thf(fact_5942_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_5943_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real ) ).
% zero_power2
thf(fact_5944_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_5945_zero__power2,axiom,
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_complex ) ).
% zero_power2
thf(fact_5946_one__power2,axiom,
( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_rat ) ).
% one_power2
thf(fact_5947_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_5948_one__power2,axiom,
( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real ) ).
% one_power2
thf(fact_5949_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_5950_one__power2,axiom,
( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_complex ) ).
% one_power2
thf(fact_5951_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_5952_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_5953_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_5954_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_5955_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_5956_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_5957_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_5958_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_5959_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_5960_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_5961_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_5962_double__not__eq__Suc__double,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% double_not_eq_Suc_double
thf(fact_5963_Suc__double__not__eq__double,axiom,
! [M: nat,N: nat] :
( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% Suc_double_not_eq_double
thf(fact_5964_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_5965_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_5966_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_5967_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_5968_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_5969_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_5970_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_5971_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_5972_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_5973_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_5974_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_5975_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_5976_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_5977_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_5978_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_5979_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_5980_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_5981_mult__2,axiom,
! [Z2: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_complex @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_5982_mult__2,axiom,
! [Z2: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_real @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_5983_mult__2,axiom,
! [Z2: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_rat @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_5984_mult__2,axiom,
! [Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_5985_mult__2,axiom,
! [Z2: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_int @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_5986_mult__2__right,axiom,
! [Z2: complex] :
( ( times_times_complex @ Z2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
= ( plus_plus_complex @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_5987_mult__2__right,axiom,
! [Z2: real] :
( ( times_times_real @ Z2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( plus_plus_real @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_5988_mult__2__right,axiom,
! [Z2: rat] :
( ( times_times_rat @ Z2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
= ( plus_plus_rat @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_5989_mult__2__right,axiom,
! [Z2: nat] :
( ( times_times_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_5990_mult__2__right,axiom,
! [Z2: int] :
( ( times_times_int @ Z2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_5991_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_5992_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_5993_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_5994_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_5995_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_5996_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_5997_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_5998_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_5999_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_6000_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_6001_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_6002_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_6003_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_6004_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_6005_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_6006_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_6007_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_6008_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_6009_abs__sqrt__wlog,axiom,
! [P: code_integer > code_integer > $o,X: code_integer] :
( ! [X3: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
=> ( P @ X3 @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_6010_abs__sqrt__wlog,axiom,
! [P: real > real > $o,X: real] :
( ! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( P @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_6011_abs__sqrt__wlog,axiom,
! [P: rat > rat > $o,X: rat] :
( ! [X3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
=> ( P @ X3 @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_6012_abs__sqrt__wlog,axiom,
! [P: int > int > $o,X: int] :
( ! [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( P @ X3 @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_6013_nat__2,axiom,
( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% nat_2
thf(fact_6014_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_6015_two__realpow__ge__one,axiom,
! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% two_realpow_ge_one
thf(fact_6016_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_6017_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_6018_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_6019_not__exp__less__eq__0__int,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% not_exp_less_eq_0_int
thf(fact_6020_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_6021_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_6022_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_6023_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_6024_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_6025_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_6026_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_6027_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_6028_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_6029_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_6030_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_6031_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_6032_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_6033_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_6034_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_6035_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_6036_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_6037_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_6038_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_6039_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_6040_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_6041_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_6042_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_6043_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_6044_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_6045_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_6046_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_6047_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_6048_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_6049_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_6050_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_6051_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_6052_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_6053_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_6054_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_6055_of__nat__less__two__power,axiom,
! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% of_nat_less_two_power
thf(fact_6056_of__nat__less__two__power,axiom,
! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% of_nat_less_two_power
thf(fact_6057_of__nat__less__two__power,axiom,
! [N: nat] : ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ).
% of_nat_less_two_power
thf(fact_6058_of__nat__less__two__power,axiom,
! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% of_nat_less_two_power
thf(fact_6059_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_right
thf(fact_6060_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_right
thf(fact_6061_exp__add__not__zero__imp__left,axiom,
! [M: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_left
thf(fact_6062_exp__add__not__zero__imp__left,axiom,
! [M: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_left
thf(fact_6063_zero__le__even__power_H,axiom,
! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_6064_zero__le__even__power_H,axiom,
! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_6065_zero__le__even__power_H,axiom,
! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_6066_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
!= zero_zero_nat ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_6067_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
!= zero_zero_int ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_6068_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_6069_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_6070_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_6071_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_6072_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_6073_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_6074_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_6075_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_6076_div__exp__eq,axiom,
! [A: nat,M: nat,N: nat] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% div_exp_eq
thf(fact_6077_div__exp__eq,axiom,
! [A: int,M: nat,N: nat] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% div_exp_eq
thf(fact_6078_minus__power__mult__self,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
= ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_6079_minus__power__mult__self,axiom,
! [A: real,N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
= ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_6080_minus__power__mult__self,axiom,
! [A: complex,N: nat] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
= ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_6081_minus__power__mult__self,axiom,
! [A: code_integer,N: nat] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
= ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_6082_minus__power__mult__self,axiom,
! [A: rat,N: nat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
= ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_6083_power__odd__eq,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_6084_power__odd__eq,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_6085_power__odd__eq,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_6086_power__odd__eq,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_6087_power__odd__eq,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_6088_nat__bit__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_6089_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_6090_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_6091_div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_6092_Suc__n__div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_6093_numeral__Bit0,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% numeral_Bit0
thf(fact_6094_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_Bit0
thf(fact_6095_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit0 @ N ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% numeral_Bit0
thf(fact_6096_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_6097_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_Bit0
thf(fact_6098_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_6099_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_6100_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_6101_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_6102_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_6103_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_6104_minus__1__div__exp__eq__int,axiom,
! [N: nat] :
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% minus_1_div_exp_eq_int
thf(fact_6105_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_6106_mult__numeral__1,axiom,
! [A: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_6107_mult__numeral__1,axiom,
! [A: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_6108_mult__numeral__1,axiom,
! [A: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_6109_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_6110_mult__numeral__1,axiom,
! [A: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_6111_mult__numeral__1__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_6112_mult__numeral__1__right,axiom,
! [A: real] :
( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_6113_mult__numeral__1__right,axiom,
! [A: rat] :
( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_6114_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_6115_mult__numeral__1__right,axiom,
! [A: int] :
( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_6116_numeral__One,axiom,
( ( numera6690914467698888265omplex @ one )
= one_one_complex ) ).
% numeral_One
thf(fact_6117_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_6118_numeral__One,axiom,
( ( numeral_numeral_rat @ one )
= one_one_rat ) ).
% numeral_One
thf(fact_6119_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_6120_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_6121_divide__numeral__1,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_6122_divide__numeral__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_6123_divide__numeral__1,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_6124_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_6125_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_6126_inverse__numeral__1,axiom,
( ( inverse_inverse_real @ ( numeral_numeral_real @ one ) )
= ( numeral_numeral_real @ one ) ) ).
% inverse_numeral_1
thf(fact_6127_inverse__numeral__1,axiom,
( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ one ) )
= ( numera6690914467698888265omplex @ one ) ) ).
% inverse_numeral_1
thf(fact_6128_inverse__numeral__1,axiom,
( ( inverse_inverse_rat @ ( numeral_numeral_rat @ one ) )
= ( numeral_numeral_rat @ one ) ) ).
% inverse_numeral_1
thf(fact_6129_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_6130_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_6131_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_6132_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_6133_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_6134_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_6135_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_6136_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_6137_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_6138_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_6139_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_6140_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_6141_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_6142_bits__stable__imp__add__self,axiom,
! [A: code_natural] :
( ( ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= A )
=> ( ( plus_p4538020629002901425atural @ A @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
= zero_z2226904508553997617atural ) ) ).
% bits_stable_imp_add_self
thf(fact_6143_odd__0__le__power__imp__0__le,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_6144_odd__0__le__power__imp__0__le,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_6145_odd__0__le__power__imp__0__le,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_6146_odd__power__less__zero,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% odd_power_less_zero
thf(fact_6147_odd__power__less__zero,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% odd_power_less_zero
thf(fact_6148_odd__power__less__zero,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% odd_power_less_zero
thf(fact_6149_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% power_minus1_odd
thf(fact_6150_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% power_minus1_odd
thf(fact_6151_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% power_minus1_odd
thf(fact_6152_power__minus1__odd,axiom,
! [N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% power_minus1_odd
thf(fact_6153_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% power_minus1_odd
thf(fact_6154_div__exp__mod__exp__eq,axiom,
! [A: int,N: nat,M: nat] :
( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
= ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_6155_div__exp__mod__exp__eq,axiom,
! [A: nat,N: nat,M: nat] :
( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
= ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_6156_div__exp__mod__exp__eq,axiom,
! [A: code_integer,N: nat,M: nat] :
( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
= ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_6157_div__exp__mod__exp__eq,axiom,
! [A: code_natural,N: nat,M: nat] :
( ( modulo8411746178871703098atural @ ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) )
= ( divide5121882707175180666atural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_6158_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 )
=> ? [N2: 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 ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_6159_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 )
=> ? [N2: 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 ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_6160_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_6161_exp__bound__half,axiom,
! [Z2: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% exp_bound_half
thf(fact_6162_exp__bound__half,axiom,
! [Z2: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% exp_bound_half
thf(fact_6163_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ zero_zero_int )
=> ( ( P @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2 != zero_zero_int )
=> ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_6164_less__log2__of__power,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% less_log2_of_power
thf(fact_6165_arsinh__def,axiom,
( arsinh_real
= ( ^ [X4: real] : ( ln_ln_real @ ( plus_plus_real @ X4 @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( 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_6166_nonzero__of__real__inverse,axiom,
! [X: real] :
( ( X != zero_zero_real )
=> ( ( real_V1803761363581548252l_real @ ( inverse_inverse_real @ X ) )
= ( inverse_inverse_real @ ( real_V1803761363581548252l_real @ X ) ) ) ) ).
% nonzero_of_real_inverse
thf(fact_6167_nonzero__of__real__inverse,axiom,
! [X: real] :
( ( X != zero_zero_real )
=> ( ( real_V4546457046886955230omplex @ ( inverse_inverse_real @ X ) )
= ( invers8013647133539491842omplex @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).
% nonzero_of_real_inverse
thf(fact_6168_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_6169_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_6170_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_6171_arcosh__def,axiom,
( arcosh_real
= ( ^ [X4: real] : ( ln_ln_real @ ( plus_plus_real @ X4 @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X4 @ ( 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_6172_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_6173_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_6174_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A: code_integer] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_6175_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A: code_natural] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
= ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_6176_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_6177_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_6178_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q4: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
= zero_zero_int )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q4 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(2)
thf(fact_6179_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q4: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
= zero_zero_nat )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q4 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(2)
thf(fact_6180_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q4: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
= zero_z3403309356797280102nteger )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q4 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(2)
thf(fact_6181_cosh__field__def,axiom,
( cosh_real
= ( ^ [Z3: real] : ( divide_divide_real @ ( plus_plus_real @ ( exp_real @ Z3 ) @ ( exp_real @ ( uminus_uminus_real @ Z3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% cosh_field_def
thf(fact_6182_cosh__field__def,axiom,
( cosh_complex
= ( ^ [Z3: complex] : ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( exp_complex @ Z3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z3 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% cosh_field_def
thf(fact_6183_log2__of__power__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% log2_of_power_less
thf(fact_6184_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_6185_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_6186_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_6187_round__def,axiom,
( archim8280529875227126926d_real
= ( ^ [X4: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% round_def
thf(fact_6188_round__def,axiom,
( archim7778729529865785530nd_rat
= ( ^ [X4: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% round_def
thf(fact_6189_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_6190_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_6191_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_6192_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_6193_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_6194_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_6195_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_6196_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_6197_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_6198_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_6199_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_6200_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_6201_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_6202_uminus__numeral__One,axiom,
( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% uminus_numeral_One
thf(fact_6203_uminus__numeral__One,axiom,
( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% uminus_numeral_One
thf(fact_6204_uminus__numeral__One,axiom,
( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% uminus_numeral_One
thf(fact_6205_uminus__numeral__One,axiom,
( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% uminus_numeral_One
thf(fact_6206_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_6207_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
= zero_zero_int ) ).
% cong_exp_iff_simps(1)
thf(fact_6208_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
= zero_zero_nat ) ).
% cong_exp_iff_simps(1)
thf(fact_6209_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
= zero_z3403309356797280102nteger ) ).
% cong_exp_iff_simps(1)
thf(fact_6210_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_6211_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_6212_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_6213_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_6214_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_6215_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_6216_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_6217_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_6218_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_6219_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_6220_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_6221_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_6222_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_6223_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_6224_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_6225_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_6226_log2__of__power__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_eq_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% log2_of_power_le
thf(fact_6227_exp__bound__lemma,axiom,
! [Z2: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z2 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ) ).
% exp_bound_lemma
thf(fact_6228_exp__bound__lemma,axiom,
! [Z2: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z2 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ) ).
% exp_bound_lemma
thf(fact_6229_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_6230_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_6231_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_6232_artanh__def,axiom,
( artanh_real
= ( ^ [X4: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( minus_minus_real @ one_one_real @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% artanh_def
thf(fact_6233_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_6234_numeral__inc,axiom,
! [X: num] :
( ( numera6690914467698888265omplex @ ( inc @ X ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% numeral_inc
thf(fact_6235_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_real @ ( inc @ X ) )
= ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% numeral_inc
thf(fact_6236_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_rat @ ( inc @ X ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% numeral_inc
thf(fact_6237_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_nat @ ( inc @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% numeral_inc
thf(fact_6238_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_int @ ( inc @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% numeral_inc
thf(fact_6239_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_6240_floor__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( archim6058952711729229775r_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
= ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% floor_log2_div2
thf(fact_6241_fact__double,axiom,
! [N: nat] :
( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s2602460028002588243omplex @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% fact_double
thf(fact_6242_fact__double,axiom,
! [N: nat] :
( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% fact_double
thf(fact_6243_fact__double,axiom,
! [N: nat] :
( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_real @ ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s7457072308508201937r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% fact_double
thf(fact_6244_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_6245_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_6246_round__unique_H,axiom,
! [X: real,N: int] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( archim8280529875227126926d_real @ X )
= N ) ) ).
% round_unique'
thf(fact_6247_round__unique_H,axiom,
! [X: rat,N: int] :
( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
=> ( ( archim7778729529865785530nd_rat @ X )
= N ) ) ).
% round_unique'
thf(fact_6248_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_6249_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_6250_round__altdef,axiom,
( archim8280529875227126926d_real
= ( ^ [X4: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X4 ) ) @ ( archim7802044766580827645g_real @ X4 ) @ ( archim6058952711729229775r_real @ X4 ) ) ) ) ).
% round_altdef
thf(fact_6251_round__altdef,axiom,
( archim7778729529865785530nd_rat
= ( ^ [X4: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X4 ) ) @ ( archim2889992004027027881ng_rat @ X4 ) @ ( archim3151403230148437115or_rat @ X4 ) ) ) ) ).
% round_altdef
thf(fact_6252_tanh__real__altdef,axiom,
( tanh_real
= ( ^ [X4: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ) ) ).
% tanh_real_altdef
thf(fact_6253_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_6254_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_6255_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_6256_pochhammer__double,axiom,
! [Z2: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s2602460028002588243omplex @ Z2 @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% pochhammer_double
thf(fact_6257_pochhammer__double,axiom,
! [Z2: real,N: nat] :
( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s7457072308508201937r_real @ Z2 @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% pochhammer_double
thf(fact_6258_pochhammer__double,axiom,
! [Z2: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% pochhammer_double
thf(fact_6259_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_6260_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_6261_norm__of__real__diff,axiom,
! [B: real,A: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( real_V1803761363581548252l_real @ B ) @ ( real_V1803761363581548252l_real @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% norm_of_real_diff
thf(fact_6262_norm__of__real__diff,axiom,
! [B: real,A: real] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( real_V4546457046886955230omplex @ B ) @ ( real_V4546457046886955230omplex @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% norm_of_real_diff
thf(fact_6263_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_6264_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_6265_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_6266_floor__log__nat__eq__if,axiom,
! [B: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
=> ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( archim6058952711729229775r_real @ ( log2 @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
= ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_6267_floor__log__nat__eq__powr__iff,axiom,
! [B: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( archim6058952711729229775r_real @ ( log2 @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
= ( semiri1314217659103216013at_int @ N ) )
= ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_6268_ceiling__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( archim7802044766580827645g_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
= ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% ceiling_log2_div2
thf(fact_6269_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_6270_ceiling__log__nat__eq__if,axiom,
! [B: nat,N: nat,K: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
=> ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( archim7802044766580827645g_real @ ( log2 @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_6271_set__n__deg__not__0,axiom,
! [TreeList2: list_VEBT_VEBT,N: nat,M: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% set_n_deg_not_0
thf(fact_6272_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_6273_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_6274_low__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
= X ) ) ).
% low_inv
thf(fact_6275_high__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
= Y ) ) ).
% high_inv
thf(fact_6276_high__bound__aux,axiom,
! [Ma: nat,N: nat,M: nat] :
( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
=> ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% high_bound_aux
thf(fact_6277_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_6278_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_6279_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_6280_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X4: nat,N4: nat] : ( divide_divide_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% high_def
thf(fact_6281_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X4: nat,N4: nat] : ( modulo_modulo_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% low_def
thf(fact_6282_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_6283_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_6284_unset__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% unset_bit_negative_int_iff
thf(fact_6285_set__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% set_bit_negative_int_iff
thf(fact_6286_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_6287_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_6288_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_6289_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_6290_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_6291_length__induct,axiom,
! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
( ! [Xs2: list_VEBT_VEBT] :
( ! [Ys: list_VEBT_VEBT] :
( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_6292_length__induct,axiom,
! [P: list_o > $o,Xs: list_o] :
( ! [Xs2: list_o] :
( ! [Ys: list_o] :
( ( ord_less_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_6293_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_6294_length__induct,axiom,
! [P: list_int > $o,Xs: list_int] :
( ! [Xs2: list_int] :
( ! [Ys: list_int] :
( ( ord_less_nat @ ( size_size_list_int @ Ys ) @ ( size_size_list_int @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_6295_unset__bit__nat__def,axiom,
( bit_se4205575877204974255it_nat
= ( ^ [M4: nat,N4: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M4 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% unset_bit_nat_def
thf(fact_6296_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_6297_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_6298_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_6299_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_6300_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_6301_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_6302_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_6303_card__length,axiom,
! [Xs: list_complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( set_complex2 @ Xs ) ) @ ( size_s3451745648224563538omplex @ Xs ) ) ).
% card_length
thf(fact_6304_card__length,axiom,
! [Xs: list_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( set_list_nat2 @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ).
% card_length
thf(fact_6305_card__length,axiom,
! [Xs: list_set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( set_set_nat2 @ Xs ) ) @ ( size_s3254054031482475050et_nat @ Xs ) ) ).
% card_length
thf(fact_6306_card__length,axiom,
! [Xs: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( finite7802652506058667612T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% card_length
thf(fact_6307_card__length,axiom,
! [Xs: list_o] : ( ord_less_eq_nat @ ( finite_card_o @ ( set_o2 @ Xs ) ) @ ( size_size_list_o @ Xs ) ) ).
% card_length
thf(fact_6308_card__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_length
thf(fact_6309_card__length,axiom,
! [Xs: list_int] : ( ord_less_eq_nat @ ( finite_card_int @ ( set_int2 @ Xs ) ) @ ( size_size_list_int @ Xs ) ) ).
% card_length
thf(fact_6310_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_6311_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_6312_invar__vebt_Ointros_I2_J,axiom,
! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus_nat @ N @ M ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_6313_invar__vebt_Ointros_I3_J,axiom,
! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus_nat @ N @ M ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_6314_unset__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_6315_unset__bit__Suc,axiom,
! [N: nat,A: code_natural] :
( ( bit_se7083795435491715335atural @ ( suc @ N ) @ A )
= ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se7083795435491715335atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_6316_unset__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_6317_unset__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_6318_set__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_6319_set__bit__Suc,axiom,
! [N: nat,A: code_natural] :
( ( bit_se1617098188084679374atural @ ( suc @ N ) @ A )
= ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se1617098188084679374atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_6320_set__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_6321_set__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_6322_both__member__options__ding,axiom,
! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% both_member_options_ding
thf(fact_6323_flip__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_6324_flip__bit__Suc,axiom,
! [N: nat,A: code_natural] :
( ( bit_se168947363167071951atural @ ( suc @ N ) @ A )
= ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se168947363167071951atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_6325_flip__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_6326_flip__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_6327_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_6328_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_6329_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_6330_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_6331_dbl__simps_I4_J,axiom,
( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_6332_dbl__simps_I4_J,axiom,
( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_6333_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_6334_log__base__10__eq1,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log2 @ ( 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_6335_inthall,axiom,
! [Xs: list_complex,P: complex > $o,N: nat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( P @ ( nth_complex @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_6336_inthall,axiom,
! [Xs: list_real,P: real > $o,N: nat] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( P @ ( nth_real @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_6337_inthall,axiom,
! [Xs: list_set_nat,P: set_nat > $o,N: nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( P @ ( nth_set_nat @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_6338_inthall,axiom,
! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_6339_inthall,axiom,
! [Xs: list_o,P: $o > $o,N: nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ ( set_o2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_6340_inthall,axiom,
! [Xs: list_nat,P: nat > $o,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_6341_inthall,axiom,
! [Xs: list_int,P: int > $o,N: nat] :
( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_6342_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_6343_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_6344_flip__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% flip_bit_nonnegative_int_iff
thf(fact_6345_flip__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% flip_bit_negative_int_iff
thf(fact_6346_signed__take__bit__of__0,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
= zero_zero_int ) ).
% signed_take_bit_of_0
thf(fact_6347_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_real @ zero_zero_real )
= zero_zero_real ) ).
% dbl_simps(2)
thf(fact_6348_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% dbl_simps(2)
thf(fact_6349_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_6350_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_6351_signed__take__bit__of__minus__1,axiom,
! [N: nat] :
( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% signed_take_bit_of_minus_1
thf(fact_6352_signed__take__bit__of__minus__1,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% signed_take_bit_of_minus_1
thf(fact_6353_signed__take__bit__Suc__1,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
= one_one_int ) ).
% signed_take_bit_Suc_1
thf(fact_6354_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_6355_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
= ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_6356_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_6357_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_6358_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_6359_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_6360_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_6361_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
= ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_6362_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_6363_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_6364_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
= ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_6365_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_6366_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_6367_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_6368_signed__take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_6369_dbl__simps_I3_J,axiom,
( ( neg_nu7009210354673126013omplex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_6370_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_6371_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_6372_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_6373_dbl__inc__simps_I3_J,axiom,
( ( neg_nu8557863876264182079omplex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_6374_dbl__inc__simps_I3_J,axiom,
( ( neg_nu8295874005876285629c_real @ one_one_real )
= ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_6375_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_6376_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5851722552734809277nc_int @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_6377_div__Suc__eq__div__add3,axiom,
! [M: nat,N: nat] :
( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_6378_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_6379_signed__take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_6380_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_6381_mod__Suc__eq__mod__add3,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% mod_Suc_eq_mod_add3
thf(fact_6382_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_6383_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_6384_dbl__dec__simps_I4_J,axiom,
( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_6385_dbl__dec__simps_I4_J,axiom,
( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_6386_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_6387_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_6388_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_6389_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_6390_zmod__numeral__Bit1,axiom,
! [V: num,W2: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W2 ) ) ) @ one_one_int ) ) ).
% zmod_numeral_Bit1
thf(fact_6391_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_6392_verit__eq__simplify_I14_J,axiom,
! [X2: num,X32: num] :
( ( bit0 @ X2 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_6393_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_6394_signed__take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% signed_take_bit_minus
thf(fact_6395_signed__take__bit__add,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L ) ) ) ).
% signed_take_bit_add
thf(fact_6396_signed__take__bit__diff,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% signed_take_bit_diff
thf(fact_6397_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_6398_nth__equalityI,axiom,
! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_s6755466524823107622T_VEBT @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( nth_VEBT_VEBT @ Xs @ I2 )
= ( nth_VEBT_VEBT @ Ys2 @ I2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_6399_nth__equalityI,axiom,
! [Xs: list_o,Ys2: list_o] :
( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
=> ( ( nth_o @ Xs @ I2 )
= ( nth_o @ Ys2 @ I2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_6400_nth__equalityI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys2 @ I2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_6401_nth__equalityI,axiom,
! [Xs: list_int,Ys2: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ Xs @ I2 )
= ( nth_int @ Ys2 @ I2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_6402_Skolem__list__nth,axiom,
! [K: nat,P: nat > vEBT_VEBT > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X8: vEBT_VEBT] : ( P @ I3 @ X8 ) ) )
= ( ? [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_6403_Skolem__list__nth,axiom,
! [K: nat,P: nat > $o > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X8: $o] : ( P @ I3 @ X8 ) ) )
= ( ? [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_6404_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X8: nat] : ( P @ I3 @ X8 ) ) )
= ( ? [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_6405_Skolem__list__nth,axiom,
! [K: nat,P: nat > int > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X8: int] : ( P @ I3 @ X8 ) ) )
= ( ? [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_6406_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_VEBT_VEBT,Z: list_VEBT_VEBT] : Y4 = Z )
= ( ^ [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_6407_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_o,Z: list_o] : Y4 = Z )
= ( ^ [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_6408_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : Y4 = Z )
= ( ^ [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_6409_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_int,Z: list_int] : Y4 = Z )
= ( ^ [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_6410_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X23: num] :
( Y
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_6411_dbl__def,axiom,
( neg_numeral_dbl_real
= ( ^ [X4: real] : ( plus_plus_real @ X4 @ X4 ) ) ) ).
% dbl_def
thf(fact_6412_dbl__def,axiom,
( neg_numeral_dbl_rat
= ( ^ [X4: rat] : ( plus_plus_rat @ X4 @ X4 ) ) ) ).
% dbl_def
thf(fact_6413_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X4: int] : ( plus_plus_int @ X4 @ X4 ) ) ) ).
% dbl_def
thf(fact_6414_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_6415_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_6416_nth__mem,axiom,
! [N: nat,Xs: list_complex] :
( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( member_complex @ ( nth_complex @ Xs @ N ) @ ( set_complex2 @ Xs ) ) ) ).
% nth_mem
thf(fact_6417_nth__mem,axiom,
! [N: nat,Xs: list_real] :
( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( member_real @ ( nth_real @ Xs @ N ) @ ( set_real2 @ Xs ) ) ) ).
% nth_mem
thf(fact_6418_nth__mem,axiom,
! [N: nat,Xs: list_set_nat] :
( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( member_set_nat @ ( nth_set_nat @ Xs @ N ) @ ( set_set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_6419_nth__mem,axiom,
! [N: nat,Xs: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% nth_mem
thf(fact_6420_nth__mem,axiom,
! [N: nat,Xs: list_o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( member_o @ ( nth_o @ Xs @ N ) @ ( set_o2 @ Xs ) ) ) ).
% nth_mem
thf(fact_6421_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_6422_nth__mem,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).
% nth_mem
thf(fact_6423_list__ball__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_6424_list__ball__nth,axiom,
! [N: nat,Xs: list_o,P: $o > $o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ ( set_o2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_6425_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_6426_list__ball__nth,axiom,
! [N: nat,Xs: list_int,P: int > $o] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_6427_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_6428_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_6429_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_6430_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_6431_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_6432_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_6433_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_6434_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_6435_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_6436_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_6437_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_6438_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_6439_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_6440_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_6441_all__set__conv__all__nth,axiom,
! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_6442_all__set__conv__all__nth,axiom,
! [Xs: list_o,P: $o > $o] :
( ( ! [X4: $o] :
( ( member_o @ X4 @ ( set_o2 @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
=> ( P @ ( nth_o @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_6443_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_6444_all__set__conv__all__nth,axiom,
! [Xs: list_int,P: int > $o] :
( ( ! [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( P @ ( nth_int @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_6445_numeral__Bit1,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% numeral_Bit1
thf(fact_6446_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit1 @ N ) )
= ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% numeral_Bit1
thf(fact_6447_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit1 @ N ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% numeral_Bit1
thf(fact_6448_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_Bit1
thf(fact_6449_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_Bit1
thf(fact_6450_eval__nat__numeral_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_6451_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_6452_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_6453_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_6454_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_6455_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_6456_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q4: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
!= zero_zero_int ) ).
% cong_exp_iff_simps(3)
thf(fact_6457_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q4: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
!= zero_zero_nat ) ).
% cong_exp_iff_simps(3)
thf(fact_6458_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q4: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
!= zero_z3403309356797280102nteger ) ).
% cong_exp_iff_simps(3)
thf(fact_6459_numeral__3__eq__3,axiom,
( ( numeral_numeral_nat @ ( bit1 @ one ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% numeral_3_eq_3
thf(fact_6460_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% Suc3_eq_add_3
thf(fact_6461_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_6462_VEBT_Osize_I4_J,axiom,
! [X21: $o,X22: $o] :
( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
= zero_zero_nat ) ).
% VEBT.size(4)
thf(fact_6463_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q4: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q4 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(11)
thf(fact_6464_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q4: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q4 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(11)
thf(fact_6465_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q4: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q4 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(11)
thf(fact_6466_cong__exp__iff__simps_I7_J,axiom,
! [Q4: num,N: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q4 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(7)
thf(fact_6467_cong__exp__iff__simps_I7_J,axiom,
! [Q4: num,N: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q4 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(7)
thf(fact_6468_cong__exp__iff__simps_I7_J,axiom,
! [Q4: num,N: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q4 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(7)
thf(fact_6469_Suc__div__eq__add3__div,axiom,
! [M: nat,N: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
= ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% Suc_div_eq_add3_div
thf(fact_6470_Suc__mod__eq__add3__mod,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
= ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% Suc_mod_eq_add3_mod
thf(fact_6471_signed__take__bit__int__less__exp,axiom,
! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% signed_take_bit_int_less_exp
thf(fact_6472_exp__le,axiom,
ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% exp_le
thf(fact_6473_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_6474_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
= ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_6475_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_6476_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_6477_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
= ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_6478_signed__take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
= ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_6479_signed__take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
=> ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% signed_take_bit_int_less_eq
thf(fact_6480_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_ri631733984087533419it_int @ N @ K )
= K )
= ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
& ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_6481_signed__take__bit__int__eq__self,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
=> ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_ri631733984087533419it_int @ N @ K )
= K ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_6482_num_Osize_I5_J,axiom,
! [X2: num] :
( ( size_size_num @ ( bit0 @ X2 ) )
= ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(5)
thf(fact_6483_signed__take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_6484_log__base__10__eq2,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log2 @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
= ( times_times_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% log_base_10_eq2
thf(fact_6485_signed__take__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_6486_signed__take__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_6487_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N4: nat,TreeList3: list_VEBT_VEBT,X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X4 @ N4 ) ) @ ( vEBT_VEBT_low @ X4 @ N4 ) ) ) ) ).
% in_children_def
thf(fact_6488_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_6489_central__binomial__lower__bound,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).
% central_binomial_lower_bound
thf(fact_6490_concat__bit__Suc,axiom,
! [N: nat,K: int,L: int] :
( ( bit_concat_bit @ ( suc @ N ) @ 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 @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_6491_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_6492_option_Osize_I3_J,axiom,
( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_6493_option_Osize_I3_J,axiom,
( ( size_size_option_num @ none_num )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_6494_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_6495_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= one_one_nat ) ).
% binomial_n_n
thf(fact_6496_concat__bit__0,axiom,
! [K: int,L: int] :
( ( bit_concat_bit @ zero_zero_nat @ K @ L )
= L ) ).
% concat_bit_0
thf(fact_6497_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% binomial_0_Suc
thf(fact_6498_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ zero_zero_nat ) )
= N ) ).
% binomial_1
thf(fact_6499_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( binomial @ N @ K )
= zero_zero_nat )
= ( ord_less_nat @ N @ K ) ) ).
% binomial_eq_0_iff
thf(fact_6500_binomial__Suc__Suc,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_6501_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one )
= zero_zero_nat ) ).
% pred_numeral_simps(1)
thf(fact_6502_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ zero_zero_nat )
= one_one_nat ) ).
% binomial_n_0
thf(fact_6503_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral_nat @ K )
= ( suc @ N ) )
= ( ( pred_numeral @ K )
= N ) ) ).
% eq_numeral_Suc
thf(fact_6504_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( ( suc @ N )
= ( numeral_numeral_nat @ K ) )
= ( N
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_6505_concat__bit__nonnegative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
= ( ord_less_eq_int @ zero_zero_int @ L ) ) ).
% concat_bit_nonnegative_iff
thf(fact_6506_concat__bit__negative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
= ( ord_less_int @ L @ zero_zero_int ) ) ).
% concat_bit_negative_iff
thf(fact_6507_pred__numeral__inc,axiom,
! [K: num] :
( ( pred_numeral @ ( inc @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% pred_numeral_inc
thf(fact_6508_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
= ( ord_less_eq_nat @ K @ N ) ) ).
% zero_less_binomial_iff
thf(fact_6509_pred__numeral__simps_I3_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit1 @ K ) )
= ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% pred_numeral_simps(3)
thf(fact_6510_less__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
= ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% less_numeral_Suc
thf(fact_6511_less__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% less_Suc_numeral
thf(fact_6512_le__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
= ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% le_numeral_Suc
thf(fact_6513_le__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% le_Suc_numeral
thf(fact_6514_diff__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% diff_Suc_numeral
thf(fact_6515_diff__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
= ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% diff_numeral_Suc
thf(fact_6516_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_6517_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_6518_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ one_one_nat )
= N ) ).
% choose_one
thf(fact_6519_concat__bit__assoc,axiom,
! [N: nat,K: int,M: nat,L: int,R2: int] :
( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
= ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R2 ) ) ).
% concat_bit_assoc
thf(fact_6520_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( binomial @ N @ K )
= zero_zero_nat ) ) ).
% binomial_eq_0
thf(fact_6521_Suc__times__binomial__eq,axiom,
! [N: nat,K: nat] :
( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
= ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% Suc_times_binomial_eq
thf(fact_6522_Suc__times__binomial,axiom,
! [K: nat,N: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
= ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% Suc_times_binomial
thf(fact_6523_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_6524_choose__mult__lemma,axiom,
! [M: nat,R2: nat,K: nat] :
( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
= ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).
% choose_mult_lemma
thf(fact_6525_binomial__gbinomial,axiom,
! [N: nat,K: nat] :
( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
= ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K ) ) ).
% binomial_gbinomial
thf(fact_6526_binomial__gbinomial,axiom,
! [N: nat,K: nat] :
( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
= ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K ) ) ).
% binomial_gbinomial
thf(fact_6527_numeral__eq__Suc,axiom,
( numeral_numeral_nat
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_6528_zero__less__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% zero_less_binomial
thf(fact_6529_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_6530_choose__mult,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
= ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_6531_binomial__Suc__Suc__eq__times,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_6532_binomial__absorb__comp,axiom,
! [N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_6533_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).
% pred_numeral_def
thf(fact_6534_binomial__absorption,axiom,
! [K: nat,N: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% binomial_absorption
thf(fact_6535_binomial__fact__lemma,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% binomial_fact_lemma
thf(fact_6536_binomial__ge__n__over__k__pow__k,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_6537_binomial__ge__n__over__k__pow__k,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_6538_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( binomial @ N @ K )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_6539_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_6540_binomial__altdef__nat,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_6541_binomial__strict__mono,axiom,
! [K: nat,K6: nat,N: nat] :
( ( ord_less_nat @ K @ K6 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
=> ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% binomial_strict_mono
thf(fact_6542_binomial__strict__antimono,axiom,
! [K: nat,K6: nat,N: nat] :
( ( ord_less_nat @ K @ K6 )
=> ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
=> ( ( ord_less_eq_nat @ K6 @ N )
=> ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_6543_binomial__less__binomial__Suc,axiom,
! [K: nat,N: nat] :
( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_less_binomial_Suc
thf(fact_6544_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( binomial @ N @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_6545_fact__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
= ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% fact_binomial
thf(fact_6546_fact__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
= ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% fact_binomial
thf(fact_6547_fact__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
= ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% fact_binomial
thf(fact_6548_binomial__fact,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
= ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% binomial_fact
thf(fact_6549_binomial__fact,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
= ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% binomial_fact
thf(fact_6550_binomial__fact,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
= ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% binomial_fact
thf(fact_6551_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_6552_arctan__one,axiom,
( ( arctan @ one_one_real )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% arctan_one
thf(fact_6553_choose__two,axiom,
! [N: nat] :
( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% choose_two
thf(fact_6554_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_6555_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_6556_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_6557_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_6558_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_6559_binomial__code,axiom,
( binomial
= ( ^ [N4: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N4 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N4 @ ( minus_minus_nat @ N4 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N4 @ K3 ) @ one_one_nat ) @ N4 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_6560_sin__cos__npi,axiom,
! [N: nat] :
( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% sin_cos_npi
thf(fact_6561_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_6562_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_6563_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_6564_even__succ__mod__exp,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_6565_even__succ__mod__exp,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_6566_even__succ__mod__exp,axiom,
! [A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_6567_even__succ__mod__exp,axiom,
! [A: code_natural,N: nat] :
( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
= ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_6568_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_6569_int__dvd__int__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% int_dvd_int_iff
thf(fact_6570_dvd__0__left__iff,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
= ( A = zero_zero_real ) ) ).
% dvd_0_left_iff
thf(fact_6571_dvd__0__left__iff,axiom,
! [A: rat] :
( ( dvd_dvd_rat @ zero_zero_rat @ A )
= ( A = zero_zero_rat ) ) ).
% dvd_0_left_iff
thf(fact_6572_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_6573_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_6574_dvd__0__right,axiom,
! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% dvd_0_right
thf(fact_6575_dvd__0__right,axiom,
! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% dvd_0_right
thf(fact_6576_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_6577_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_6578_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_6579_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_6580_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_6581_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_6582_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_6583_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_6584_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_6585_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_6586_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_6587_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_6588_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_6589_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_6590_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_6591_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_6592_minus__dvd__iff,axiom,
! [X: complex,Y: complex] :
( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X ) @ Y )
= ( dvd_dvd_complex @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_6593_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_6594_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_6595_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_6596_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_6597_dvd__minus__iff,axiom,
! [X: complex,Y: complex] :
( ( dvd_dvd_complex @ X @ ( uminus1482373934393186551omplex @ Y ) )
= ( dvd_dvd_complex @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_6598_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_6599_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_6600_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_6601_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_6602_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_6603_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_6604_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_6605_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_6606_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_6607_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_6608_take__bit__of__0,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
= zero_zero_int ) ).
% take_bit_of_0
thf(fact_6609_take__bit__of__0,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% take_bit_of_0
thf(fact_6610_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_6611_sin__zero,axiom,
( ( sin_real @ zero_zero_real )
= zero_zero_real ) ).
% sin_zero
thf(fact_6612_cos__minus,axiom,
! [X: real] :
( ( cos_real @ ( uminus_uminus_real @ X ) )
= ( cos_real @ X ) ) ).
% cos_minus
thf(fact_6613_cos__minus,axiom,
! [X: complex] :
( ( cos_complex @ ( uminus1482373934393186551omplex @ X ) )
= ( cos_complex @ X ) ) ).
% cos_minus
thf(fact_6614_sin__minus,axiom,
! [X: real] :
( ( sin_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% sin_minus
thf(fact_6615_sin__minus,axiom,
! [X: complex] :
( ( sin_complex @ ( uminus1482373934393186551omplex @ X ) )
= ( uminus1482373934393186551omplex @ ( sin_complex @ X ) ) ) ).
% sin_minus
thf(fact_6616_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
= ( ( abs_abs_int @ X )
= one_one_int ) ) ).
% zdvd1_eq
thf(fact_6617_cot__zero,axiom,
( ( cot_real @ zero_zero_real )
= zero_zero_real ) ).
% cot_zero
thf(fact_6618_cot__minus,axiom,
! [X: real] :
( ( cot_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( cot_real @ X ) ) ) ).
% cot_minus
thf(fact_6619_cot__minus,axiom,
! [X: complex] :
( ( cot_complex @ ( uminus1482373934393186551omplex @ X ) )
= ( uminus1482373934393186551omplex @ ( cot_complex @ X ) ) ) ).
% cot_minus
thf(fact_6620_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_6621_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_6622_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_6623_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_6624_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_6625_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_6626_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_6627_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_6628_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_6629_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_6630_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_6631_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_6632_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_6633_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_6634_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_6635_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_6636_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_6637_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_6638_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_6639_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_6640_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_6641_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_6642_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_6643_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_6644_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_6645_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_6646_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_6647_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_6648_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_6649_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_6650_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_6651_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_6652_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_6653_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_6654_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_6655_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_6656_dvd__imp__mod__0,axiom,
! [A: code_natural,B: code_natural] :
( ( dvd_dvd_Code_natural @ A @ B )
=> ( ( modulo8411746178871703098atural @ B @ A )
= zero_z2226904508553997617atural ) ) ).
% dvd_imp_mod_0
thf(fact_6657_cos__zero,axiom,
( ( cos_complex @ zero_zero_complex )
= one_one_complex ) ).
% cos_zero
thf(fact_6658_cos__zero,axiom,
( ( cos_real @ zero_zero_real )
= one_one_real ) ).
% cos_zero
thf(fact_6659_take__bit__0,axiom,
! [A: int] :
( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
= zero_zero_int ) ).
% take_bit_0
thf(fact_6660_take__bit__0,axiom,
! [A: nat] :
( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% take_bit_0
thf(fact_6661_take__bit__Suc__1,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
= one_one_int ) ).
% take_bit_Suc_1
thf(fact_6662_take__bit__Suc__1,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
= one_one_nat ) ).
% take_bit_Suc_1
thf(fact_6663_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_6664_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_6665_dvd__nat__abs__iff,axiom,
! [N: nat,K: int] :
( ( dvd_dvd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
= ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% dvd_nat_abs_iff
thf(fact_6666_nat__abs__dvd__iff,axiom,
! [K: int,N: nat] :
( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
= ( dvd_dvd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_abs_dvd_iff
thf(fact_6667_sgn__mult__dvd__iff,axiom,
! [R2: int,L: int,K: int] :
( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K )
= ( ( dvd_dvd_int @ L @ K )
& ( ( R2 = zero_zero_int )
=> ( K = zero_zero_int ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_6668_mult__sgn__dvd__iff,axiom,
! [L: int,R2: int,K: int] :
( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K )
= ( ( dvd_dvd_int @ L @ K )
& ( ( R2 = zero_zero_int )
=> ( K = zero_zero_int ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_6669_dvd__sgn__mult__iff,axiom,
! [L: int,R2: int,K: int] :
( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
= ( ( dvd_dvd_int @ L @ K )
| ( R2 = zero_zero_int ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_6670_dvd__mult__sgn__iff,axiom,
! [L: int,K: int,R2: int] :
( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
= ( ( dvd_dvd_int @ L @ K )
| ( R2 = zero_zero_int ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_6671_concat__bit__of__zero__2,axiom,
! [N: nat,K: int] :
( ( bit_concat_bit @ N @ K @ zero_zero_int )
= ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% concat_bit_of_zero_2
thf(fact_6672_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_6673_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_6674_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_6675_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_6676_even__Suc__Suc__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_Suc_Suc_iff
thf(fact_6677_even__Suc,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% even_Suc
thf(fact_6678_pow__divides__pow__iff,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_6679_pow__divides__pow__iff,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_6680_take__bit__of__1__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
= zero_zero_int )
= ( N = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_6681_take__bit__of__1__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
= zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_6682_sin__of__real__pi,axiom,
( ( sin_real @ ( real_V1803761363581548252l_real @ pi ) )
= zero_zero_real ) ).
% sin_of_real_pi
thf(fact_6683_sin__of__real__pi,axiom,
( ( sin_complex @ ( real_V4546457046886955230omplex @ pi ) )
= zero_zero_complex ) ).
% sin_of_real_pi
thf(fact_6684_cos__pi,axiom,
( ( cos_real @ pi )
= ( uminus_uminus_real @ one_one_real ) ) ).
% cos_pi
thf(fact_6685_of__nat__nat__take__bit__eq,axiom,
! [N: nat,K: int] :
( ( semiri8010041392384452111omplex @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
= ( ring_17405671764205052669omplex @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% of_nat_nat_take_bit_eq
thf(fact_6686_of__nat__nat__take__bit__eq,axiom,
! [N: nat,K: int] :
( ( semiri1314217659103216013at_int @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
= ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% of_nat_nat_take_bit_eq
thf(fact_6687_of__nat__nat__take__bit__eq,axiom,
! [N: nat,K: int] :
( ( semiri5074537144036343181t_real @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
= ( ring_1_of_int_real @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% of_nat_nat_take_bit_eq
thf(fact_6688_of__nat__nat__take__bit__eq,axiom,
! [N: nat,K: int] :
( ( semiri4939895301339042750nteger @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
= ( ring_18347121197199848620nteger @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% of_nat_nat_take_bit_eq
thf(fact_6689_of__nat__nat__take__bit__eq,axiom,
! [N: nat,K: int] :
( ( semiri681578069525770553at_rat @ ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
= ( ring_1_of_int_rat @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% of_nat_nat_take_bit_eq
thf(fact_6690_cos__periodic__pi2,axiom,
! [X: real] :
( ( cos_real @ ( plus_plus_real @ pi @ X ) )
= ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% cos_periodic_pi2
thf(fact_6691_cos__periodic__pi,axiom,
! [X: real] :
( ( cos_real @ ( plus_plus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% cos_periodic_pi
thf(fact_6692_sin__periodic__pi2,axiom,
! [X: real] :
( ( sin_real @ ( plus_plus_real @ pi @ X ) )
= ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% sin_periodic_pi2
thf(fact_6693_sin__periodic__pi,axiom,
! [X: real] :
( ( sin_real @ ( plus_plus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% sin_periodic_pi
thf(fact_6694_cos__minus__pi,axiom,
! [X: real] :
( ( cos_real @ ( minus_minus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% cos_minus_pi
thf(fact_6695_cos__pi__minus,axiom,
! [X: real] :
( ( cos_real @ ( minus_minus_real @ pi @ X ) )
= ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% cos_pi_minus
thf(fact_6696_sin__minus__pi,axiom,
! [X: real] :
( ( sin_real @ ( minus_minus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% sin_minus_pi
thf(fact_6697_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_6698_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_6699_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_6700_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_6701_power__minus__odd,axiom,
! [N: nat,A: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_6702_power__minus__odd,axiom,
! [N: nat,A: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_6703_power__minus__odd,axiom,
! [N: nat,A: complex] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_6704_power__minus__odd,axiom,
! [N: nat,A: code_integer] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_6705_power__minus__odd,axiom,
! [N: nat,A: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_6706_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( power_power_int @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_6707_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( power_power_real @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_6708_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: complex] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( power_power_complex @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_6709_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: code_integer] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_6710_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( power_power_rat @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_6711_odd__Suc__div__two,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% odd_Suc_div_two
thf(fact_6712_even__Suc__div__two,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_Suc_div_two
thf(fact_6713_power__int__minus__left__odd,axiom,
! [N: int,A: real] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( power_int_real @ ( uminus_uminus_real @ A ) @ N )
= ( uminus_uminus_real @ ( power_int_real @ A @ N ) ) ) ) ).
% power_int_minus_left_odd
thf(fact_6714_power__int__minus__left__odd,axiom,
! [N: int,A: complex] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( power_int_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( uminus1482373934393186551omplex @ ( power_int_complex @ A @ N ) ) ) ) ).
% power_int_minus_left_odd
thf(fact_6715_power__int__minus__left__odd,axiom,
! [N: int,A: rat] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( power_int_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( uminus_uminus_rat @ ( power_int_rat @ A @ N ) ) ) ) ).
% power_int_minus_left_odd
thf(fact_6716_power__int__minus__left__even,axiom,
! [N: int,A: real] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( power_int_real @ ( uminus_uminus_real @ A ) @ N )
= ( power_int_real @ A @ N ) ) ) ).
% power_int_minus_left_even
thf(fact_6717_power__int__minus__left__even,axiom,
! [N: int,A: complex] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( power_int_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( power_int_complex @ A @ N ) ) ) ).
% power_int_minus_left_even
thf(fact_6718_power__int__minus__left__even,axiom,
! [N: int,A: rat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( power_int_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( power_int_rat @ A @ N ) ) ) ).
% power_int_minus_left_even
thf(fact_6719_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_6720_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_6721_cos__of__real__pi,axiom,
( ( cos_real @ ( real_V1803761363581548252l_real @ pi ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% cos_of_real_pi
thf(fact_6722_cos__of__real__pi,axiom,
( ( cos_complex @ ( real_V4546457046886955230omplex @ pi ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% cos_of_real_pi
thf(fact_6723_sin__npi__int,axiom,
! [N: int] :
( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
= zero_zero_real ) ).
% sin_npi_int
thf(fact_6724_zero__le__power__eq__numeral,axiom,
! [A: real,W2: num] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_6725_zero__le__power__eq__numeral,axiom,
! [A: rat,W2: num] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_6726_zero__le__power__eq__numeral,axiom,
! [A: int,W2: num] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_6727_power__less__zero__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% power_less_zero_eq
thf(fact_6728_power__less__zero__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% power_less_zero_eq
thf(fact_6729_power__less__zero__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq
thf(fact_6730_power__less__zero__eq__numeral,axiom,
! [A: real,W2: num] :
( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_6731_power__less__zero__eq__numeral,axiom,
! [A: rat,W2: num] :
( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_6732_power__less__zero__eq__numeral,axiom,
! [A: int,W2: num] :
( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_6733_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_6734_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_6735_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_6736_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% neg_one_odd_power
thf(fact_6737_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
= ( uminus_uminus_real @ one_one_real ) ) ) ).
% neg_one_odd_power
thf(fact_6738_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% neg_one_odd_power
thf(fact_6739_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% neg_one_odd_power
thf(fact_6740_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
= ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% neg_one_odd_power
thf(fact_6741_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= one_one_int ) ) ).
% neg_one_even_power
thf(fact_6742_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
= one_one_real ) ) ).
% neg_one_even_power
thf(fact_6743_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
= one_one_complex ) ) ).
% neg_one_even_power
thf(fact_6744_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
= one_one_Code_integer ) ) ).
% neg_one_even_power
thf(fact_6745_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
= one_one_rat ) ) ).
% neg_one_even_power
thf(fact_6746_even__of__nat,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_of_nat
thf(fact_6747_even__of__nat,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_of_nat
thf(fact_6748_even__of__nat,axiom,
! [N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_of_nat
thf(fact_6749_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% odd_Suc_minus_one
thf(fact_6750_even__take__bit__eq,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
= ( ( N = zero_zero_nat )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_take_bit_eq
thf(fact_6751_even__take__bit__eq,axiom,
! [N: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
= ( ( N = zero_zero_nat )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_take_bit_eq
thf(fact_6752_even__diff__nat,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% even_diff_nat
thf(fact_6753_zero__less__power__eq__numeral,axiom,
! [A: real,W2: num] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( ( numeral_numeral_nat @ W2 )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A != zero_zero_real ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_6754_zero__less__power__eq__numeral,axiom,
! [A: rat,W2: num] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( ( numeral_numeral_nat @ W2 )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A != zero_zero_rat ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_6755_zero__less__power__eq__numeral,axiom,
! [A: int,W2: num] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( ( numeral_numeral_nat @ W2 )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_6756_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_6757_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_6758_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_6759_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_6760_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_6761_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_6762_even__power,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_6763_even__power,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_6764_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_6765_take__bit__Suc__0,axiom,
! [A: code_natural] :
( ( bit_se569199155075624693atural @ ( suc @ zero_zero_nat ) @ A )
= ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_0
thf(fact_6766_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_6767_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_6768_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_6769_sin__pi__half,axiom,
( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= one_one_real ) ).
% sin_pi_half
thf(fact_6770_cos__two__pi,axiom,
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
= one_one_real ) ).
% cos_two_pi
thf(fact_6771_cos__npi,axiom,
! [N: nat] :
( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% cos_npi
thf(fact_6772_cos__npi2,axiom,
! [N: nat] :
( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% cos_npi2
thf(fact_6773_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_6774_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_6775_power__le__zero__eq__numeral,axiom,
! [A: real,W2: num] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_real @ A @ zero_zero_real ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_6776_power__le__zero__eq__numeral,axiom,
! [A: rat,W2: num] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_rat @ A @ zero_zero_rat ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A = zero_zero_rat ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_6777_power__le__zero__eq__numeral,axiom,
! [A: int,W2: num] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_6778_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_6779_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_6780_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_6781_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_6782_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_6783_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_6784_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_6785_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_6786_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_6787_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_6788_cos__2npi,axiom,
! [N: nat] :
( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
= one_one_real ) ).
% cos_2npi
thf(fact_6789_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_6790_sin__int__2pin,axiom,
! [N: int] :
( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
= zero_zero_real ) ).
% sin_int_2pin
thf(fact_6791_cos__int__2pin,axiom,
! [N: int] :
( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
= one_one_real ) ).
% cos_int_2pin
thf(fact_6792_cos__npi__int,axiom,
! [N: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
= one_one_real ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% cos_npi_int
thf(fact_6793_even__succ__div__exp,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_6794_even__succ__div__exp,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_6795_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_6796_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% of_nat_dvd_iff
thf(fact_6797_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% of_nat_dvd_iff
thf(fact_6798_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% of_nat_dvd_iff
thf(fact_6799_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_6800_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_6801_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_6802_dvd__refl,axiom,
! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% dvd_refl
thf(fact_6803_take__bit__of__nat,axiom,
! [N: nat,M: nat] :
( ( bit_se1745604003318907178nteger @ N @ ( semiri4939895301339042750nteger @ M ) )
= ( semiri4939895301339042750nteger @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% take_bit_of_nat
thf(fact_6804_take__bit__of__nat,axiom,
! [N: nat,M: nat] :
( ( bit_se2923211474154528505it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
= ( semiri1314217659103216013at_int @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% take_bit_of_nat
thf(fact_6805_take__bit__of__nat,axiom,
! [N: nat,M: nat] :
( ( bit_se2925701944663578781it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
= ( semiri1316708129612266289at_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% take_bit_of_nat
thf(fact_6806_take__bit__of__int,axiom,
! [N: nat,K: int] :
( ( bit_se1745604003318907178nteger @ N @ ( ring_18347121197199848620nteger @ K ) )
= ( ring_18347121197199848620nteger @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% take_bit_of_int
thf(fact_6807_take__bit__of__int,axiom,
! [N: nat,K: int] :
( ( bit_se2923211474154528505it_int @ N @ ( ring_1_of_int_int @ K ) )
= ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% take_bit_of_int
thf(fact_6808_sin__diff,axiom,
! [X: real,Y: real] :
( ( sin_real @ ( minus_minus_real @ X @ Y ) )
= ( minus_minus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% sin_diff
thf(fact_6809_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_6810_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_6811_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_6812_take__bit__add,axiom,
! [N: nat,A: int,B: int] :
( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) )
= ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ A @ B ) ) ) ).
% take_bit_add
thf(fact_6813_take__bit__add,axiom,
! [N: nat,A: nat,B: nat] :
( ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) )
= ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ A @ B ) ) ) ).
% take_bit_add
thf(fact_6814_dvd__0__left,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
=> ( A = zero_zero_real ) ) ).
% dvd_0_left
thf(fact_6815_dvd__0__left,axiom,
! [A: rat] :
( ( dvd_dvd_rat @ zero_zero_rat @ A )
=> ( A = zero_zero_rat ) ) ).
% dvd_0_left
thf(fact_6816_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_6817_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_6818_dvd__field__iff,axiom,
( dvd_dvd_real
= ( ^ [A3: real,B3: real] :
( ( A3 = zero_zero_real )
=> ( B3 = zero_zero_real ) ) ) ) ).
% dvd_field_iff
thf(fact_6819_dvd__field__iff,axiom,
( dvd_dvd_rat
= ( ^ [A3: rat,B3: rat] :
( ( A3 = zero_zero_rat )
=> ( B3 = zero_zero_rat ) ) ) ) ).
% dvd_field_iff
thf(fact_6820_dvd__triv__right,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% dvd_triv_right
thf(fact_6821_dvd__triv__right,axiom,
! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_6822_dvd__triv__right,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_6823_dvd__triv__right,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% dvd_triv_right
thf(fact_6824_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_6825_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_6826_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_6827_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_6828_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_6829_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_6830_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_6831_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_6832_dvd__triv__left,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% dvd_triv_left
thf(fact_6833_dvd__triv__left,axiom,
! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_6834_dvd__triv__left,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_6835_dvd__triv__left,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% dvd_triv_left
thf(fact_6836_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_6837_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_6838_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_6839_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_6840_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_6841_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_6842_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_6843_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_6844_division__decomp,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
=> ? [B6: nat,C4: nat] :
( ( A
= ( times_times_nat @ B6 @ C4 ) )
& ( dvd_dvd_nat @ B6 @ B )
& ( dvd_dvd_nat @ C4 @ C ) ) ) ).
% division_decomp
thf(fact_6845_division__decomp,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
=> ? [B6: int,C4: int] :
( ( A
= ( times_times_int @ B6 @ C4 ) )
& ( dvd_dvd_int @ B6 @ B )
& ( dvd_dvd_int @ C4 @ C ) ) ) ).
% division_decomp
thf(fact_6846_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_6847_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_6848_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_6849_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_6850_dvd__productE,axiom,
! [P5: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ P5 @ ( times_times_nat @ A @ B ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( P5
= ( times_times_nat @ X3 @ Y3 ) )
=> ( ( dvd_dvd_nat @ X3 @ A )
=> ~ ( dvd_dvd_nat @ Y3 @ B ) ) ) ) ).
% dvd_productE
thf(fact_6851_dvd__productE,axiom,
! [P5: int,A: int,B: int] :
( ( dvd_dvd_int @ P5 @ ( times_times_int @ A @ B ) )
=> ~ ! [X3: int,Y3: int] :
( ( P5
= ( times_times_int @ X3 @ Y3 ) )
=> ( ( dvd_dvd_int @ X3 @ A )
=> ~ ( dvd_dvd_int @ Y3 @ B ) ) ) ) ).
% dvd_productE
thf(fact_6852_dvd__def,axiom,
( dvd_dvd_real
= ( ^ [B3: real,A3: real] :
? [K3: real] :
( A3
= ( times_times_real @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_6853_dvd__def,axiom,
( dvd_dvd_rat
= ( ^ [B3: rat,A3: rat] :
? [K3: rat] :
( A3
= ( times_times_rat @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_6854_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B3: nat,A3: nat] :
? [K3: nat] :
( A3
= ( times_times_nat @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_6855_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B3: int,A3: int] :
? [K3: int] :
( A3
= ( times_times_int @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_6856_dvdI,axiom,
! [A: real,B: real,K: real] :
( ( A
= ( times_times_real @ B @ K ) )
=> ( dvd_dvd_real @ B @ A ) ) ).
% dvdI
thf(fact_6857_dvdI,axiom,
! [A: rat,B: rat,K: rat] :
( ( A
= ( times_times_rat @ B @ K ) )
=> ( dvd_dvd_rat @ B @ A ) ) ).
% dvdI
thf(fact_6858_dvdI,axiom,
! [A: nat,B: nat,K: nat] :
( ( A
= ( times_times_nat @ B @ K ) )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% dvdI
thf(fact_6859_dvdI,axiom,
! [A: int,B: int,K: int] :
( ( A
= ( times_times_int @ B @ K ) )
=> ( dvd_dvd_int @ B @ A ) ) ).
% dvdI
thf(fact_6860_dvdE,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ~ ! [K2: real] :
( A
!= ( times_times_real @ B @ K2 ) ) ) ).
% dvdE
thf(fact_6861_dvdE,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ~ ! [K2: rat] :
( A
!= ( times_times_rat @ B @ K2 ) ) ) ).
% dvdE
thf(fact_6862_dvdE,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ~ ! [K2: nat] :
( A
!= ( times_times_nat @ B @ K2 ) ) ) ).
% dvdE
thf(fact_6863_dvdE,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ~ ! [K2: int] :
( A
!= ( times_times_int @ B @ K2 ) ) ) ).
% dvdE
thf(fact_6864_take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
= ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% take_bit_minus
thf(fact_6865_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_6866_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_6867_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_6868_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_6869_one__dvd,axiom,
! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% one_dvd
thf(fact_6870_one__dvd,axiom,
! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% one_dvd
thf(fact_6871_one__dvd,axiom,
! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% one_dvd
thf(fact_6872_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_6873_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_6874_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_6875_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_6876_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_6877_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_6878_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_6879_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_6880_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_6881_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_6882_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_6883_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_6884_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_6885_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_6886_dvd__diff,axiom,
! [X: real,Y: real,Z2: real] :
( ( dvd_dvd_real @ X @ Y )
=> ( ( dvd_dvd_real @ X @ Z2 )
=> ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z2 ) ) ) ) ).
% dvd_diff
thf(fact_6887_dvd__diff,axiom,
! [X: rat,Y: rat,Z2: rat] :
( ( dvd_dvd_rat @ X @ Y )
=> ( ( dvd_dvd_rat @ X @ Z2 )
=> ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y @ Z2 ) ) ) ) ).
% dvd_diff
thf(fact_6888_dvd__diff,axiom,
! [X: int,Y: int,Z2: int] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( dvd_dvd_int @ X @ Z2 )
=> ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z2 ) ) ) ) ).
% dvd_diff
thf(fact_6889_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_6890_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_6891_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_6892_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_6893_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_6894_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_6895_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_6896_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_6897_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_6898_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_6899_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_6900_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_6901_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_6902_take__bit__diff,axiom,
! [N: nat,K: int,L: int] :
( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
= ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% take_bit_diff
thf(fact_6903_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_6904_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_6905_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_6906_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_6907_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_6908_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_6909_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_6910_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_6911_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_6912_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_6913_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_6914_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_6915_dvd__mod__imp__dvd,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( ( dvd_dvd_Code_natural @ C @ ( modulo8411746178871703098atural @ A @ B ) )
=> ( ( dvd_dvd_Code_natural @ C @ B )
=> ( dvd_dvd_Code_natural @ C @ A ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_6916_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_6917_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_6918_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_6919_dvd__mod__iff,axiom,
! [C: code_natural,B: code_natural,A: code_natural] :
( ( dvd_dvd_Code_natural @ C @ B )
=> ( ( dvd_dvd_Code_natural @ C @ ( modulo8411746178871703098atural @ A @ B ) )
= ( dvd_dvd_Code_natural @ C @ A ) ) ) ).
% dvd_mod_iff
thf(fact_6920_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_6921_uminus__dvd__conv_I2_J,axiom,
( dvd_dvd_int
= ( ^ [D3: int,T2: int] : ( dvd_dvd_int @ D3 @ ( uminus_uminus_int @ T2 ) ) ) ) ).
% uminus_dvd_conv(2)
thf(fact_6922_uminus__dvd__conv_I1_J,axiom,
( dvd_dvd_int
= ( ^ [D3: int] : ( dvd_dvd_int @ ( uminus_uminus_int @ D3 ) ) ) ) ).
% uminus_dvd_conv(1)
thf(fact_6923_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_6924_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_6925_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_6926_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_6927_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_6928_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_6929_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_6930_sincos__principal__value,axiom,
! [X: real] :
? [Y3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y3 )
& ( ord_less_eq_real @ Y3 @ pi )
& ( ( sin_real @ Y3 )
= ( sin_real @ X ) )
& ( ( cos_real @ Y3 )
= ( cos_real @ X ) ) ) ).
% sincos_principal_value
thf(fact_6931_sin__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).
% sin_le_one
thf(fact_6932_take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% take_bit_int_less_eq_self_iff
thf(fact_6933_take__bit__nonnegative,axiom,
! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% take_bit_nonnegative
thf(fact_6934_take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% take_bit_int_greater_self_iff
thf(fact_6935_not__take__bit__negative,axiom,
! [N: nat,K: int] :
~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% not_take_bit_negative
thf(fact_6936_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [N: nat,A: int,B: int] :
( ( ( bit_ri631733984087533419it_int @ N @ A )
= ( bit_ri631733984087533419it_int @ N @ B ) )
= ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
= ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B ) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
thf(fact_6937_cos__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).
% cos_le_one
thf(fact_6938_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_6939_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_6940_pinf_I9_J,axiom,
! [D: real,S: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).
% pinf(9)
thf(fact_6941_pinf_I9_J,axiom,
! [D: rat,S: rat] :
? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z4 @ X5 )
=> ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).
% pinf(9)
thf(fact_6942_pinf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% pinf(9)
thf(fact_6943_pinf_I9_J,axiom,
! [D: int,S: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% pinf(9)
thf(fact_6944_pinf_I10_J,axiom,
! [D: real,S: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_6945_pinf_I10_J,axiom,
! [D: rat,S: rat] :
? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z4 @ X5 )
=> ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_6946_pinf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_6947_pinf_I10_J,axiom,
! [D: int,S: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_6948_minf_I9_J,axiom,
! [D: real,S: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).
% minf(9)
thf(fact_6949_minf_I9_J,axiom,
! [D: rat,S: rat] :
? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z4 )
=> ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).
% minf(9)
thf(fact_6950_minf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% minf(9)
thf(fact_6951_minf_I9_J,axiom,
! [D: int,S: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% minf(9)
thf(fact_6952_minf_I10_J,axiom,
! [D: real,S: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_6953_minf_I10_J,axiom,
! [D: rat,S: rat] :
? [Z4: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z4 )
=> ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_6954_minf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_6955_minf_I10_J,axiom,
! [D: int,S: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_6956_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_6957_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_6958_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_6959_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_6960_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_6961_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_6962_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_6963_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_6964_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_6965_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_6966_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_6967_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_6968_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_6969_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_6970_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_6971_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_6972_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_6973_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_6974_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_6975_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_6976_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_6977_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_6978_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_6979_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_6980_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_6981_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_6982_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_6983_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_6984_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_6985_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_6986_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_6987_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_6988_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_6989_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_6990_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_6991_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_6992_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_6993_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_6994_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_6995_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_6996_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_6997_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_6998_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_6999_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_7000_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_7001_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_7002_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_7003_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_7004_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_7005_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_7006_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_7007_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_7008_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_7009_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_7010_mod__0__imp__dvd,axiom,
! [A: code_natural,B: code_natural] :
( ( ( modulo8411746178871703098atural @ A @ B )
= zero_z2226904508553997617atural )
=> ( dvd_dvd_Code_natural @ B @ A ) ) ).
% mod_0_imp_dvd
thf(fact_7011_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_int
= ( ^ [A3: int,B3: int] :
( ( modulo_modulo_int @ B3 @ A3 )
= zero_zero_int ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_7012_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_nat
= ( ^ [A3: nat,B3: nat] :
( ( modulo_modulo_nat @ B3 @ A3 )
= zero_zero_nat ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_7013_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_Code_integer
= ( ^ [A3: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger @ B3 @ A3 )
= zero_z3403309356797280102nteger ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_7014_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_Code_natural
= ( ^ [A3: code_natural,B3: code_natural] :
( ( modulo8411746178871703098atural @ B3 @ A3 )
= zero_z2226904508553997617atural ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_7015_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_7016_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_7017_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_7018_mod__eq__0__iff__dvd,axiom,
! [A: code_natural,B: code_natural] :
( ( ( modulo8411746178871703098atural @ A @ B )
= zero_z2226904508553997617atural )
= ( dvd_dvd_Code_natural @ B @ A ) ) ).
% mod_eq_0_iff_dvd
thf(fact_7019_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_7020_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_7021_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_7022_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_7023_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_7024_dvd__minus__mod,axiom,
! [B: code_natural,A: code_natural] : ( dvd_dvd_Code_natural @ B @ ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) ) ) ).
% dvd_minus_mod
thf(fact_7025_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_7026_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_7027_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_7028_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M @ N )
=> ( ( dvd_dvd_int @ N @ M )
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_7029_zdvd__not__zless,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N )
=> ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% zdvd_not_zless
thf(fact_7030_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( dvd_dvd_nat @ M @ N )
= ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_7031_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_7032_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_7033_zdvd__mono,axiom,
! [K: int,M: int,T: int] :
( ( K != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T )
= ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% zdvd_mono
thf(fact_7034_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_7035_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 ) ) )
=> ? [X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D @ A )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
& ( ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y3 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_7036_bezout__add__nat,axiom,
! [A: nat,B: nat] :
? [D4: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D4 ) )
| ( ( times_times_nat @ B @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D4 ) ) ) ) ).
% bezout_add_nat
thf(fact_7037_bezout1__nat,axiom,
! [A: nat,B: nat] :
? [D4: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
= D4 )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
= D4 ) ) ) ).
% bezout1_nat
thf(fact_7038_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
= ( dvd_dvd_int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_7039_zdvd__period,axiom,
! [A: int,D: int,X: int,T: int,C: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
= ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% zdvd_period
thf(fact_7040_take__bit__eq__0__iff,axiom,
! [N: nat,A: int] :
( ( ( bit_se2923211474154528505it_int @ N @ A )
= zero_zero_int )
= ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% take_bit_eq_0_iff
thf(fact_7041_take__bit__eq__0__iff,axiom,
! [N: nat,A: nat] :
( ( ( bit_se2925701944663578781it_nat @ N @ A )
= zero_zero_nat )
= ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% take_bit_eq_0_iff
thf(fact_7042_cos__int__times__real,axiom,
! [M: int,X: real] :
( ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
= ( real_V1803761363581548252l_real @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% cos_int_times_real
thf(fact_7043_cos__int__times__real,axiom,
! [M: int,X: real] :
( ( cos_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
= ( real_V4546457046886955230omplex @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% cos_int_times_real
thf(fact_7044_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_7045_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_7046_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_7047_sin__int__times__real,axiom,
! [M: int,X: real] :
( ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
= ( real_V1803761363581548252l_real @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% sin_int_times_real
thf(fact_7048_sin__int__times__real,axiom,
! [M: int,X: real] :
( ( sin_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
= ( real_V4546457046886955230omplex @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).
% sin_int_times_real
thf(fact_7049_nat__dvd__iff,axiom,
! [Z2: int,M: nat] :
( ( dvd_dvd_nat @ ( nat2 @ Z2 ) @ M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( dvd_dvd_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_dvd_iff
thf(fact_7050_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_7051_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_7052_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_7053_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_7054_take__bit__signed__take__bit,axiom,
! [M: nat,N: nat,A: int] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N @ A ) )
= ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).
% take_bit_signed_take_bit
thf(fact_7055_sin__x__ge__neg__x,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).
% sin_x_ge_neg_x
thf(fact_7056_unit__dvdE,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [C2: nat] :
( B
!= ( times_times_nat @ A @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_7057_unit__dvdE,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [C2: int] :
( B
!= ( times_times_int @ A @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_7058_unity__coeff__ex,axiom,
! [P: real > $o,L: real] :
( ( ? [X4: real] : ( P @ ( times_times_real @ L @ X4 ) ) )
= ( ? [X4: real] :
( ( dvd_dvd_real @ L @ ( plus_plus_real @ X4 @ zero_zero_real ) )
& ( P @ X4 ) ) ) ) ).
% unity_coeff_ex
thf(fact_7059_unity__coeff__ex,axiom,
! [P: rat > $o,L: rat] :
( ( ? [X4: rat] : ( P @ ( times_times_rat @ L @ X4 ) ) )
= ( ? [X4: rat] :
( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X4 @ zero_zero_rat ) )
& ( P @ X4 ) ) ) ) ).
% unity_coeff_ex
thf(fact_7060_unity__coeff__ex,axiom,
! [P: nat > $o,L: nat] :
( ( ? [X4: nat] : ( P @ ( times_times_nat @ L @ X4 ) ) )
= ( ? [X4: nat] :
( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X4 @ zero_zero_nat ) )
& ( P @ X4 ) ) ) ) ).
% unity_coeff_ex
thf(fact_7061_unity__coeff__ex,axiom,
! [P: int > $o,L: int] :
( ( ? [X4: int] : ( P @ ( times_times_int @ L @ X4 ) ) )
= ( ? [X4: int] :
( ( dvd_dvd_int @ L @ ( plus_plus_int @ X4 @ zero_zero_int ) )
& ( P @ X4 ) ) ) ) ).
% unity_coeff_ex
thf(fact_7062_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_7063_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_7064_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_7065_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_7066_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_7067_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_7068_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_7069_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_7070_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_7071_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_7072_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_7073_inf__period_I4_J,axiom,
! [D: real,D6: real,T: real] :
( ( dvd_dvd_real @ D @ D6 )
=> ! [X5: real,K4: real] :
( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D6 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_7074_inf__period_I4_J,axiom,
! [D: rat,D6: rat,T: rat] :
( ( dvd_dvd_rat @ D @ D6 )
=> ! [X5: rat,K4: rat] :
( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D6 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_7075_inf__period_I4_J,axiom,
! [D: int,D6: int,T: int] :
( ( dvd_dvd_int @ D @ D6 )
=> ! [X5: int,K4: int] :
( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D6 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_7076_inf__period_I3_J,axiom,
! [D: real,D6: real,T: real] :
( ( dvd_dvd_real @ D @ D6 )
=> ! [X5: real,K4: real] :
( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D6 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_7077_inf__period_I3_J,axiom,
! [D: rat,D6: rat,T: rat] :
( ( dvd_dvd_rat @ D @ D6 )
=> ! [X5: rat,K4: rat] :
( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D6 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_7078_inf__period_I3_J,axiom,
! [D: int,D6: int,T: int] :
( ( dvd_dvd_int @ D @ D6 )
=> ! [X5: int,K4: int] :
( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D6 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_7079_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_7080_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_7081_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_7082_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_7083_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_7084_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_7085_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_7086_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_7087_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_7088_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_7089_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_7090_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_7091_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_7092_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_7093_take__bit__decr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
!= zero_zero_int )
=> ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
= ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% take_bit_decr_eq
thf(fact_7094_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_7095_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_7096_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_7097_unit__imp__mod__eq__0,axiom,
! [B: code_natural,A: code_natural] :
( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( ( modulo8411746178871703098atural @ A @ B )
= zero_z2226904508553997617atural ) ) ).
% unit_imp_mod_eq_0
thf(fact_7098_is__unit__power__iff,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_7099_is__unit__power__iff,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_7100_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_7101_sin__times__sin,axiom,
! [W2: complex,Z2: complex] :
( ( times_times_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% sin_times_sin
thf(fact_7102_sin__times__sin,axiom,
! [W2: real,Z2: real] :
( ( times_times_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% sin_times_sin
thf(fact_7103_sin__times__cos,axiom,
! [W2: complex,Z2: complex] :
( ( times_times_complex @ ( sin_complex @ W2 ) @ ( cos_complex @ Z2 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% sin_times_cos
thf(fact_7104_sin__times__cos,axiom,
! [W2: real,Z2: real] :
( ( times_times_real @ ( sin_real @ W2 ) @ ( cos_real @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% sin_times_cos
thf(fact_7105_cos__times__sin,axiom,
! [W2: complex,Z2: complex] :
( ( times_times_complex @ ( cos_complex @ W2 ) @ ( sin_complex @ Z2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% cos_times_sin
thf(fact_7106_cos__times__sin,axiom,
! [W2: real,Z2: real] :
( ( times_times_real @ ( cos_real @ W2 ) @ ( sin_real @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% cos_times_sin
thf(fact_7107_sin__plus__sin,axiom,
! [W2: complex,Z2: complex] :
( ( plus_plus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_7108_sin__plus__sin,axiom,
! [W2: real,Z2: real] :
( ( plus_plus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_7109_sin__diff__sin,axiom,
! [W2: complex,Z2: complex] :
( ( minus_minus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_7110_sin__diff__sin,axiom,
! [W2: real,Z2: real] :
( ( minus_minus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_7111_cos__diff__cos,axiom,
! [W2: complex,Z2: complex] :
( ( minus_minus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z2 @ W2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_7112_cos__diff__cos,axiom,
! [W2: real,Z2: real] :
( ( minus_minus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z2 @ W2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_7113_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_7114_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_7115_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_7116_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7117_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7118_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7119_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7120_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_7121_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_7122_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D4: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D4 @ A )
& ( dvd_dvd_nat @ D4 @ B )
& ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D4 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_7123_sin__cos__eq,axiom,
( sin_real
= ( ^ [X4: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ).
% sin_cos_eq
thf(fact_7124_sin__cos__eq,axiom,
( sin_complex
= ( ^ [X4: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ).
% sin_cos_eq
thf(fact_7125_cos__sin__eq,axiom,
( cos_real
= ( ^ [X4: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ).
% cos_sin_eq
thf(fact_7126_cos__sin__eq,axiom,
( cos_complex
= ( ^ [X4: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ).
% cos_sin_eq
thf(fact_7127_zdvd__imp__le,axiom,
! [Z2: int,N: int] :
( ( dvd_dvd_int @ Z2 @ N )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ Z2 @ N ) ) ) ).
% zdvd_imp_le
thf(fact_7128_mod__greater__zero__iff__not__dvd,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
= ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_7129_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_7130_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M: nat,Q4: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( ( modulo_modulo_nat @ M @ Q4 )
= ( modulo_modulo_nat @ N @ Q4 ) )
= ( dvd_dvd_nat @ Q4 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_7131_of__int__divide__in__Ints,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( member_Code_integer @ ( divide6298287555418463151nteger @ ( ring_18347121197199848620nteger @ A ) @ ( ring_18347121197199848620nteger @ B ) ) @ ring_11222124179247155820nteger ) ) ).
% of_int_divide_in_Ints
thf(fact_7132_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_7133_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_7134_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_7135_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_7136_real__of__nat__div,axiom,
! [D: nat,N: nat] :
( ( dvd_dvd_nat @ D @ N )
=> ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% real_of_nat_div
thf(fact_7137_real__of__int__div,axiom,
! [D: int,N: int] :
( ( dvd_dvd_int @ D @ N )
=> ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
= ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% real_of_int_div
thf(fact_7138_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_7139_dvd__fact,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% dvd_fact
thf(fact_7140_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_7141_cos__double__sin,axiom,
! [W2: complex] :
( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W2 ) )
= ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_double_sin
thf(fact_7142_cos__double__sin,axiom,
! [W2: real] :
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W2 ) )
= ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_double_sin
thf(fact_7143_stable__imp__take__bit__eq,axiom,
! [A: int,N: nat] :
( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A )
=> ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se2923211474154528505it_int @ N @ A )
= zero_zero_int ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se2923211474154528505it_int @ N @ A )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_7144_stable__imp__take__bit__eq,axiom,
! [A: nat,N: nat] :
( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se2925701944663578781it_nat @ N @ A )
= zero_zero_nat ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se2925701944663578781it_nat @ N @ A )
= ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_7145_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_7146_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_7147_even__zero,axiom,
dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% even_zero
thf(fact_7148_even__zero,axiom,
dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% even_zero
thf(fact_7149_is__unitE,axiom,
! [A: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [B4: nat] :
( ( B4 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B4 @ one_one_nat )
=> ( ( ( divide_divide_nat @ one_one_nat @ A )
= B4 )
=> ( ( ( divide_divide_nat @ one_one_nat @ B4 )
= A )
=> ( ( ( times_times_nat @ A @ B4 )
= one_one_nat )
=> ( ( divide_divide_nat @ C @ A )
!= ( times_times_nat @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_7150_is__unitE,axiom,
! [A: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [B4: int] :
( ( B4 != zero_zero_int )
=> ( ( dvd_dvd_int @ B4 @ one_one_int )
=> ( ( ( divide_divide_int @ one_one_int @ A )
= B4 )
=> ( ( ( divide_divide_int @ one_one_int @ B4 )
= A )
=> ( ( ( times_times_int @ A @ B4 )
= one_one_int )
=> ( ( divide_divide_int @ C @ A )
!= ( times_times_int @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_7151_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_7152_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_7153_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_7154_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_7155_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_7156_odd__one,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% odd_one
thf(fact_7157_odd__one,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% odd_one
thf(fact_7158_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_7159_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_7160_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_7161_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_7162_cos__monotone__minus__pi__0_H,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0'
thf(fact_7163_dvd__power__iff,axiom,
! [X: int,M: nat,N: nat] :
( ( X != zero_zero_int )
=> ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
= ( ( dvd_dvd_int @ X @ one_one_int )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_7164_dvd__power__iff,axiom,
! [X: nat,M: nat,N: nat] :
( ( X != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
= ( ( dvd_dvd_nat @ X @ one_one_nat )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_7165_dvd__power,axiom,
! [N: nat,X: rat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_rat ) )
=> ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).
% dvd_power
thf(fact_7166_dvd__power,axiom,
! [N: nat,X: int] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_int ) )
=> ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% dvd_power
thf(fact_7167_dvd__power,axiom,
! [N: nat,X: real] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_real ) )
=> ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).
% dvd_power
thf(fact_7168_dvd__power,axiom,
! [N: nat,X: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_nat ) )
=> ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% dvd_power
thf(fact_7169_dvd__power,axiom,
! [N: nat,X: complex] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_complex ) )
=> ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).
% dvd_power
thf(fact_7170_sin__zero__iff__int2,axiom,
! [X: real] :
( ( ( sin_real @ X )
= zero_zero_real )
= ( ? [I3: int] :
( X
= ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ pi ) ) ) ) ).
% sin_zero_iff_int2
thf(fact_7171_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_7172_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% choose_dvd
thf(fact_7173_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% choose_dvd
thf(fact_7174_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% choose_dvd
thf(fact_7175_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% choose_dvd
thf(fact_7176_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_7177_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_7178_dvd__minus__add,axiom,
! [Q4: nat,N: nat,R2: nat,M: nat] :
( ( ord_less_eq_nat @ Q4 @ N )
=> ( ( ord_less_eq_nat @ Q4 @ ( times_times_nat @ R2 @ M ) )
=> ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q4 ) )
= ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q4 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_7179_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 )
=> ? [T3: real] :
( ( ord_less_eq_real @ zero_zero_real @ T3 )
& ( ord_less_eq_real @ T3 @ pi )
& ( X
= ( cos_real @ T3 ) )
& ( Y
= ( sin_real @ T3 ) ) ) ) ) ).
% sincos_total_pi
thf(fact_7180_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_7181_mod__nat__eqI,axiom,
! [R2: nat,N: nat,M: nat] :
( ( ord_less_nat @ R2 @ N )
=> ( ( ord_less_eq_nat @ R2 @ M )
=> ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
=> ( ( modulo_modulo_nat @ M @ N )
= R2 ) ) ) ) ).
% mod_nat_eqI
thf(fact_7182_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_7183_zdvd__mult__cancel1,axiom,
! [M: int,N: int] :
( ( M != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
= ( ( abs_abs_int @ N )
= one_one_int ) ) ) ).
% zdvd_mult_cancel1
thf(fact_7184_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_7185_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_7186_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_7187_aset_I10_J,axiom,
! [D: int,D6: int,A2: set_int,T: int] :
( ( dvd_dvd_int @ D @ D6 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
=> ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D6 ) @ T ) ) ) ) ) ).
% aset(10)
thf(fact_7188_aset_I9_J,axiom,
! [D: int,D6: int,A2: set_int,T: int] :
( ( dvd_dvd_int @ D @ D6 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
=> ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D6 ) @ T ) ) ) ) ) ).
% aset(9)
thf(fact_7189_bset_I10_J,axiom,
! [D: int,D6: int,B2: set_int,T: int] :
( ( dvd_dvd_int @ D @ D6 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
=> ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D6 ) @ T ) ) ) ) ) ).
% bset(10)
thf(fact_7190_bset_I9_J,axiom,
! [D: int,D6: int,B2: set_int,T: int] :
( ( dvd_dvd_int @ D @ D6 )
=> ! [X5: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
=> ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D6 ) @ T ) ) ) ) ) ).
% bset(9)
thf(fact_7191_take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_7192_take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_7193_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_7194_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_7195_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_7196_take__bit__int__less__exp,axiom,
! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% take_bit_int_less_exp
thf(fact_7197_fold__atLeastAtMost__nat_Osimps,axiom,
( set_fo2584398358068434914at_nat
= ( ^ [F2: nat > nat > nat,A3: nat,B3: nat,Acc: nat] : ( if_nat @ ( ord_less_nat @ B3 @ A3 ) @ Acc @ ( set_fo2584398358068434914at_nat @ F2 @ ( plus_plus_nat @ A3 @ one_one_nat ) @ B3 @ ( F2 @ A3 @ Acc ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_7198_fold__atLeastAtMost__nat_Oelims,axiom,
! [X: nat > nat > nat,Xa3: nat,Xb3: nat,Xc: nat,Y: nat] :
( ( ( set_fo2584398358068434914at_nat @ X @ Xa3 @ Xb3 @ Xc )
= Y )
=> ( ( ( ord_less_nat @ Xb3 @ Xa3 )
=> ( Y = Xc ) )
& ( ~ ( ord_less_nat @ Xb3 @ Xa3 )
=> ( Y
= ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa3 @ one_one_nat ) @ Xb3 @ ( X @ Xa3 @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_7199_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_7200_cos__total,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ? [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
& ( ord_less_eq_real @ X3 @ pi )
& ( ( cos_real @ X3 )
= Y )
& ! [Y6: real] :
( ( ( ord_less_eq_real @ zero_zero_real @ Y6 )
& ( ord_less_eq_real @ Y6 @ pi )
& ( ( cos_real @ Y6 )
= Y ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% cos_total
thf(fact_7201_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_7202_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_7203_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_7204_even__iff__mod__2__eq__zero,axiom,
! [A: code_natural] :
( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
= ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= zero_z2226904508553997617atural ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_7205_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_7206_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_7207_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_7208_odd__iff__mod__2__eq__one,axiom,
! [A: code_natural] :
( ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) )
= ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= one_one_Code_natural ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_7209_uminus__power__if,axiom,
! [N: nat,A: int] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( power_power_int @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_7210_uminus__power__if,axiom,
! [N: nat,A: real] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( power_power_real @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_7211_uminus__power__if,axiom,
! [N: nat,A: complex] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( power_power_complex @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_7212_uminus__power__if,axiom,
! [N: nat,A: code_integer] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( power_8256067586552552935nteger @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_7213_uminus__power__if,axiom,
! [N: nat,A: rat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( power_power_rat @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_7214_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% odd_pos
thf(fact_7215_power__int__minus__left,axiom,
! [N: int,A: real] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( power_int_real @ ( uminus_uminus_real @ A ) @ N )
= ( power_int_real @ A @ N ) ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( power_int_real @ ( uminus_uminus_real @ A ) @ N )
= ( uminus_uminus_real @ ( power_int_real @ A @ N ) ) ) ) ) ).
% power_int_minus_left
thf(fact_7216_power__int__minus__left,axiom,
! [N: int,A: complex] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( power_int_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( power_int_complex @ A @ N ) ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( power_int_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( uminus1482373934393186551omplex @ ( power_int_complex @ A @ N ) ) ) ) ) ).
% power_int_minus_left
thf(fact_7217_power__int__minus__left,axiom,
! [N: int,A: rat] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( power_int_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( power_int_rat @ A @ N ) ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( power_int_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( uminus_uminus_rat @ ( power_int_rat @ A @ N ) ) ) ) ) ).
% power_int_minus_left
thf(fact_7218_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_7219_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_7220_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_7221_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_7222_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_7223_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_7224_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_7225_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 )
=> ? [T3: real] :
( ( ord_less_eq_real @ zero_zero_real @ T3 )
& ( ord_less_eq_real @ T3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( X
= ( cos_real @ T3 ) )
& ( Y
= ( sin_real @ T3 ) ) ) ) ) ) ).
% sincos_total_pi_half
thf(fact_7226_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 )
=> ? [T3: real] :
( ( ord_less_eq_real @ zero_zero_real @ T3 )
& ( ord_less_eq_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
& ( X
= ( cos_real @ T3 ) )
& ( Y
= ( sin_real @ T3 ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_7227_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_7228_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_7229_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 )
=> ~ ! [T3: real] :
( ( ord_less_eq_real @ zero_zero_real @ T3 )
=> ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( ( X
= ( cos_real @ T3 ) )
=> ( Y
!= ( sin_real @ T3 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_7230_take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_7231_take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
= ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_7232_take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% take_bit_int_less_self_iff
thf(fact_7233_cos__integer__2pi,axiom,
! [N: real] :
( ( member_real @ N @ ring_1_Ints_real )
=> ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
= one_one_real ) ) ).
% cos_integer_2pi
thf(fact_7234_oddE,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ! [B4: nat] :
( A
!= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) @ one_one_nat ) ) ) ).
% oddE
thf(fact_7235_oddE,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ! [B4: int] :
( A
!= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) @ one_one_int ) ) ) ).
% oddE
thf(fact_7236_sin__pi__divide__n__ge__0,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% sin_pi_divide_n_ge_0
thf(fact_7237_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_7238_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_7239_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_7240_mod2__eq__if,axiom,
! [A: code_natural] :
( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= zero_z2226904508553997617atural ) )
& ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= one_one_Code_natural ) ) ) ).
% mod2_eq_if
thf(fact_7241_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_7242_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_7243_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_7244_parity__cases,axiom,
! [A: code_natural] :
( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
!= zero_z2226904508553997617atural ) )
=> ~ ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
!= one_one_Code_natural ) ) ) ).
% parity_cases
thf(fact_7245_zero__le__power__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_7246_zero__le__power__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_7247_zero__le__power__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_7248_zero__le__odd__power,axiom,
! [N: nat,A: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% zero_le_odd_power
thf(fact_7249_zero__le__odd__power,axiom,
! [N: nat,A: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% zero_le_odd_power
thf(fact_7250_zero__le__odd__power,axiom,
! [N: nat,A: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% zero_le_odd_power
thf(fact_7251_zero__le__even__power,axiom,
! [N: nat,A: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_7252_zero__le__even__power,axiom,
! [N: nat,A: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_7253_zero__le__even__power,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_7254_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= one_one_int ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% minus_one_power_iff
thf(fact_7255_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
= one_one_real ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% minus_one_power_iff
thf(fact_7256_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
= one_one_complex ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% minus_one_power_iff
thf(fact_7257_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
= one_one_Code_integer ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% minus_one_power_iff
thf(fact_7258_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
= one_one_rat ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
= ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% minus_one_power_iff
thf(fact_7259_central__binomial__odd,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% central_binomial_odd
thf(fact_7260_cos__times__cos,axiom,
! [W2: complex,Z2: complex] :
( ( times_times_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% cos_times_cos
thf(fact_7261_cos__times__cos,axiom,
! [W2: real,Z2: real] :
( ( times_times_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% cos_times_cos
thf(fact_7262_cos__plus__cos,axiom,
! [W2: complex,Z2: complex] :
( ( plus_plus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_7263_cos__plus__cos,axiom,
! [W2: real,Z2: real] :
( ( plus_plus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_7264_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_7265_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_7266_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_7267_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_7268_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_7269_take__bit__int__eq__self,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_se2923211474154528505it_int @ N @ K )
= K ) ) ) ).
% take_bit_int_eq_self
thf(fact_7270_take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
= K )
= ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_7271_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_7272_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_7273_take__bit__incr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
!= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
=> ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
= ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% take_bit_incr_eq
thf(fact_7274_cos__one__2pi__int,axiom,
! [X: real] :
( ( ( cos_real @ X )
= one_one_real )
= ( ? [X4: int] :
( X
= ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% cos_one_2pi_int
thf(fact_7275_zero__less__power__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_real ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_7276_zero__less__power__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_rat ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_7277_zero__less__power__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_7278_cos__double__cos,axiom,
! [W2: complex] :
( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W2 ) )
= ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% cos_double_cos
thf(fact_7279_cos__double__cos,axiom,
! [W2: real] :
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W2 ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).
% cos_double_cos
thf(fact_7280_take__bit__Suc__minus__1__eq,axiom,
! [N: nat] :
( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_7281_take__bit__Suc__minus__1__eq,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_7282_take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_Suc_bit1
thf(fact_7283_take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% take_bit_Suc_bit1
thf(fact_7284_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_7285_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_7286_take__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% take_bit_Suc
thf(fact_7287_take__bit__Suc,axiom,
! [N: nat,A: code_natural] :
( ( bit_se569199155075624693atural @ ( suc @ N ) @ A )
= ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).
% take_bit_Suc
thf(fact_7288_take__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( 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_7289_take__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( 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_7290_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_7291_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_7292_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_7293_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_7294_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_7295_sin__total,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ? [X3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
& ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ X3 )
= Y )
& ! [Y6: real] :
( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y6 )
& ( ord_less_eq_real @ Y6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ Y6 )
= Y ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% sin_total
thf(fact_7296_take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_7297_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_7298_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_7299_take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_int @ K @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% take_bit_int_greater_eq
thf(fact_7300_even__mask__div__iff_H,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% even_mask_div_iff'
thf(fact_7301_even__mask__div__iff_H,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% even_mask_div_iff'
thf(fact_7302_cos__one__2pi,axiom,
! [X: real] :
( ( ( cos_real @ X )
= one_one_real )
= ( ? [X4: nat] :
( X
= ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
| ? [X4: nat] :
( X
= ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% cos_one_2pi
thf(fact_7303_signed__take__bit__eq__take__bit__shift,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N4: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ ( plus_plus_int @ K3 @ ( 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_7304_power__le__zero__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_real @ A @ zero_zero_real ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_7305_power__le__zero__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_rat @ A @ zero_zero_rat ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_rat ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_7306_power__le__zero__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_7307_even__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( suc @ zero_zero_nat ) )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_mod_4_div_2
thf(fact_7308_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_7309_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_7310_take__bit__minus__small__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_7311_even__mask__div__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= zero_zero_nat )
| ( ord_less_eq_nat @ M @ N ) ) ) ).
% even_mask_div_iff
thf(fact_7312_even__mask__div__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
= zero_zero_int )
| ( ord_less_eq_nat @ M @ N ) ) ) ).
% even_mask_div_iff
thf(fact_7313_odd__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( numeral_numeral_nat @ ( bit1 @ one ) ) )
=> ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% odd_mod_4_div_2
thf(fact_7314_Bernoulli__inequality__even,axiom,
! [N: nat,X: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% Bernoulli_inequality_even
thf(fact_7315_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_7316_even__mult__exp__div__exp__iff,axiom,
! [A: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M )
| ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= zero_zero_nat )
| ( ( ord_less_eq_nat @ M @ N )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_7317_even__mult__exp__div__exp__iff,axiom,
! [A: int,M: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M )
| ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
= zero_zero_int )
| ( ( ord_less_eq_nat @ M @ N )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_7318_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_7319_take__bit__rec,axiom,
( bit_se569199155075624693atural
= ( ^ [N4: nat,A3: code_natural] : ( if_Code_natural @ ( N4 = zero_zero_nat ) @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide5121882707175180666atural @ A3 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A3 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_7320_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_7321_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_7322_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_7323_fact__code,axiom,
( semiri3624122377584611663nteger
= ( ^ [N4: nat] : ( semiri4939895301339042750nteger @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_7324_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_7325_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_7326_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_7327_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_7328_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_7329_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_7330_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_7331_complex__unimodular__polar,axiom,
! [Z2: complex] :
( ( ( real_V1022390504157884413omplex @ Z2 )
= one_one_real )
=> ~ ! [T3: real] :
( ( ord_less_eq_real @ zero_zero_real @ T3 )
=> ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( Z2
!= ( complex2 @ ( cos_real @ T3 ) @ ( sin_real @ T3 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_7332_arccos__cos__eq__abs__2pi,axiom,
! [Theta: real] :
~ ! [K2: int] :
( ( arccos @ ( cos_real @ Theta ) )
!= ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).
% arccos_cos_eq_abs_2pi
thf(fact_7333_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_7334_tan__zero,axiom,
( ( tan_real @ zero_zero_real )
= zero_zero_real ) ).
% tan_zero
thf(fact_7335_tan__minus,axiom,
! [X: real] :
( ( tan_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( tan_real @ X ) ) ) ).
% tan_minus
thf(fact_7336_tan__minus,axiom,
! [X: complex] :
( ( tan_complex @ ( uminus1482373934393186551omplex @ X ) )
= ( uminus1482373934393186551omplex @ ( tan_complex @ X ) ) ) ).
% tan_minus
thf(fact_7337_arccos__1,axiom,
( ( arccos @ one_one_real )
= zero_zero_real ) ).
% arccos_1
thf(fact_7338_sin__coeff__0,axiom,
( ( sin_coeff @ zero_zero_nat )
= zero_zero_real ) ).
% sin_coeff_0
thf(fact_7339_cos__coeff__0,axiom,
( ( cos_coeff @ zero_zero_nat )
= one_one_real ) ).
% cos_coeff_0
thf(fact_7340_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_7341_arccos__minus__1,axiom,
( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
= pi ) ).
% arccos_minus_1
thf(fact_7342_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_7343_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_7344_norm__cos__sin,axiom,
! [T: real] :
( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
= one_one_real ) ).
% norm_cos_sin
thf(fact_7345_arcsin__1,axiom,
( ( arcsin @ one_one_real )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% arcsin_1
thf(fact_7346_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_7347_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_7348_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_7349_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( A != B ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_7350_gcd__nat_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( dvd_dvd_nat @ A @ B ) ) ).
% gcd_nat.strict_implies_order
thf(fact_7351_gcd__nat_Ostrict__iff__order,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
= ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_7352_gcd__nat_Oorder__iff__strict,axiom,
( dvd_dvd_nat
= ( ^ [A3: nat,B3: nat] :
( ( ( dvd_dvd_nat @ A3 @ B3 )
& ( A3 != B3 ) )
| ( A3 = B3 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_7353_gcd__nat_Ostrict__iff__not,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
= ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_7354_gcd__nat_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_7355_gcd__nat_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_7356_gcd__nat_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_7357_gcd__nat_Oantisym,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( A = B ) ) ) ).
% gcd_nat.antisym
thf(fact_7358_gcd__nat_Oirrefl,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ A @ A )
& ( A != A ) ) ).
% gcd_nat.irrefl
thf(fact_7359_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ B3 )
& ( dvd_dvd_nat @ B3 @ A3 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_7360_gcd__nat_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% gcd_nat.trans
thf(fact_7361_gcd__nat_Orefl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% gcd_nat.refl
thf(fact_7362_gcd__nat_Oasym,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ~ ( ( dvd_dvd_nat @ B @ A )
& ( B != A ) ) ) ).
% gcd_nat.asym
thf(fact_7363_complex__minus,axiom,
! [A: real,B: real] :
( ( uminus1482373934393186551omplex @ ( complex2 @ A @ B ) )
= ( complex2 @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% complex_minus
thf(fact_7364_one__complex_Ocode,axiom,
( one_one_complex
= ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% one_complex.code
thf(fact_7365_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_7366_sin__coeff__Suc,axiom,
! [N: nat] :
( ( sin_coeff @ ( suc @ N ) )
= ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% sin_coeff_Suc
thf(fact_7367_cot__altdef,axiom,
( cot_real
= ( ^ [X4: real] : ( inverse_inverse_real @ ( tan_real @ X4 ) ) ) ) ).
% cot_altdef
thf(fact_7368_cot__altdef,axiom,
( cot_complex
= ( ^ [X4: complex] : ( invers8013647133539491842omplex @ ( tan_complex @ X4 ) ) ) ) ).
% cot_altdef
thf(fact_7369_tan__altdef,axiom,
( tan_real
= ( ^ [X4: real] : ( inverse_inverse_real @ ( cot_real @ X4 ) ) ) ) ).
% tan_altdef
thf(fact_7370_tan__altdef,axiom,
( tan_complex
= ( ^ [X4: complex] : ( invers8013647133539491842omplex @ ( cot_complex @ X4 ) ) ) ) ).
% tan_altdef
thf(fact_7371_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_7372_Complex__eq__neg__numeral,axiom,
! [A: real,B: real,W2: num] :
( ( ( complex2 @ A @ B )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= ( ( A
= ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
& ( B = zero_zero_real ) ) ) ).
% Complex_eq_neg_numeral
thf(fact_7373_cos__coeff__Suc,axiom,
! [N: nat] :
( ( cos_coeff @ ( suc @ N ) )
= ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% cos_coeff_Suc
thf(fact_7374_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_7375_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_7376_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_7377_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_7378_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_7379_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_7380_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_7381_take__bit__nat__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% take_bit_nat_eq
thf(fact_7382_nat__take__bit__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
= ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% nat_take_bit_eq
thf(fact_7383_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_7384_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_7385_take__bit__nat__eq__self__iff,axiom,
! [N: nat,M: nat] :
( ( ( bit_se2925701944663578781it_nat @ N @ M )
= M )
= ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_nat_eq_self_iff
thf(fact_7386_take__bit__nat__less__exp,axiom,
! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% take_bit_nat_less_exp
thf(fact_7387_take__bit__nat__eq__self,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_se2925701944663578781it_nat @ N @ M )
= M ) ) ).
% take_bit_nat_eq_self
thf(fact_7388_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_7389_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_7390_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_7391_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_7392_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_7393_take__bit__nat__less__self__iff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% take_bit_nat_less_self_iff
thf(fact_7394_tan__45,axiom,
( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
= one_one_real ) ).
% tan_45
thf(fact_7395_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_7396_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_7397_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_7398_arccos__cos2,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
=> ( ( arccos @ ( cos_real @ X ) )
= ( uminus_uminus_real @ X ) ) ) ) ).
% arccos_cos2
thf(fact_7399_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_7400_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_7401_tan__total,axiom,
! [Y: 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 )
= Y )
& ! [Y6: real] :
( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y6 )
& ( ord_less_real @ Y6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ Y6 )
= Y ) )
=> ( Y6 = X3 ) ) ) ).
% tan_total
thf(fact_7402_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_7403_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_7404_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_7405_lemma__tan__total1,axiom,
! [Y: 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 )
= Y ) ) ).
% lemma_tan_total1
thf(fact_7406_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_7407_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_7408_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_7409_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_7410_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_7411_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_7412_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_7413_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_7414_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_7415_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_7416_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_7417_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_7418_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_7419_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_7420_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_7421_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_7422_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_7423_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_7424_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_7425_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_7426_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_7427_tan__total__pi4,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ? [Z4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z4 )
& ( ord_less_real @ Z4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
& ( ( tan_real @ Z4 )
= X ) ) ) ).
% tan_total_pi4
thf(fact_7428_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_7429_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_7430_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_7431_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_7432_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_7433_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_7434_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_7435_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_7436_tan__half,axiom,
( tan_complex
= ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) ) @ one_one_complex ) ) ) ) ).
% tan_half
thf(fact_7437_tan__half,axiom,
( tan_real
= ( ^ [X4: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) @ one_one_real ) ) ) ) ).
% tan_half
thf(fact_7438_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_7439_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_7440_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_7441_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_7442_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_7443_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_7444_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_7445_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_7446_set__decode__Suc,axiom,
! [N: nat,X: nat] :
( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
= ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_7447_nat__of__bool,axiom,
! [P: $o] :
( ( nat2 @ ( zero_n2684676970156552555ol_int @ P ) )
= ( zero_n2687167440665602831ol_nat @ P ) ) ).
% nat_of_bool
thf(fact_7448_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_7449_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_7450_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_7451_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n3304061248610475627l_real @ P )
= zero_zero_real )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_7452_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n2052037380579107095ol_rat @ P )
= zero_zero_rat )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_7453_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n2684676970156552555ol_int @ P )
= zero_zero_int )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_7454_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P )
= zero_zero_nat )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_7455_of__bool__eq_I1_J,axiom,
( ( zero_n3304061248610475627l_real @ $false )
= zero_zero_real ) ).
% of_bool_eq(1)
thf(fact_7456_of__bool__eq_I1_J,axiom,
( ( zero_n2052037380579107095ol_rat @ $false )
= zero_zero_rat ) ).
% of_bool_eq(1)
thf(fact_7457_of__bool__eq_I1_J,axiom,
( ( zero_n2684676970156552555ol_int @ $false )
= zero_zero_int ) ).
% of_bool_eq(1)
thf(fact_7458_of__bool__eq_I1_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $false )
= zero_zero_nat ) ).
% of_bool_eq(1)
thf(fact_7459_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_7460_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_7461_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_7462_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_7463_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n1201886186963655149omplex @ P )
= one_one_complex )
= P ) ).
% of_bool_eq_1_iff
thf(fact_7464_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n3304061248610475627l_real @ P )
= one_one_real )
= P ) ).
% of_bool_eq_1_iff
thf(fact_7465_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2052037380579107095ol_rat @ P )
= one_one_rat )
= P ) ).
% of_bool_eq_1_iff
thf(fact_7466_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2684676970156552555ol_int @ P )
= one_one_int )
= P ) ).
% of_bool_eq_1_iff
thf(fact_7467_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P )
= one_one_nat )
= P ) ).
% of_bool_eq_1_iff
thf(fact_7468_of__bool__eq_I2_J,axiom,
( ( zero_n1201886186963655149omplex @ $true )
= one_one_complex ) ).
% of_bool_eq(2)
thf(fact_7469_of__bool__eq_I2_J,axiom,
( ( zero_n3304061248610475627l_real @ $true )
= one_one_real ) ).
% of_bool_eq(2)
thf(fact_7470_of__bool__eq_I2_J,axiom,
( ( zero_n2052037380579107095ol_rat @ $true )
= one_one_rat ) ).
% of_bool_eq(2)
thf(fact_7471_of__bool__eq_I2_J,axiom,
( ( zero_n2684676970156552555ol_int @ $true )
= one_one_int ) ).
% of_bool_eq(2)
thf(fact_7472_of__bool__eq_I2_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $true )
= one_one_nat ) ).
% of_bool_eq(2)
thf(fact_7473_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n3304061248610475627l_real @ P ) ) ).
% of_nat_of_bool
thf(fact_7474_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n356916108424825756nteger @ P ) ) ).
% of_nat_of_bool
thf(fact_7475_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n2052037380579107095ol_rat @ P ) ) ).
% of_nat_of_bool
thf(fact_7476_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n2684676970156552555ol_int @ P ) ) ).
% of_nat_of_bool
thf(fact_7477_of__nat__of__bool,axiom,
! [P: $o] :
( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( zero_n2687167440665602831ol_nat @ P ) ) ).
% of_nat_of_bool
thf(fact_7478_real__sqrt__eq__1__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= one_one_real )
= ( X = one_one_real ) ) ).
% real_sqrt_eq_1_iff
thf(fact_7479_real__sqrt__one,axiom,
( ( sqrt @ one_one_real )
= one_one_real ) ).
% real_sqrt_one
thf(fact_7480_abs__bool__eq,axiom,
! [P: $o] :
( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
= ( zero_n3304061248610475627l_real @ P ) ) ).
% abs_bool_eq
thf(fact_7481_abs__bool__eq,axiom,
! [P: $o] :
( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
= ( zero_n356916108424825756nteger @ P ) ) ).
% abs_bool_eq
thf(fact_7482_abs__bool__eq,axiom,
! [P: $o] :
( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
= ( zero_n2052037380579107095ol_rat @ P ) ) ).
% abs_bool_eq
thf(fact_7483_abs__bool__eq,axiom,
! [P: $o] :
( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
= ( zero_n2684676970156552555ol_int @ P ) ) ).
% abs_bool_eq
thf(fact_7484_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_7485_of__int__of__bool,axiom,
! [P: $o] :
( ( ring_18347121197199848620nteger @ ( zero_n2684676970156552555ol_int @ P ) )
= ( zero_n356916108424825756nteger @ P ) ) ).
% of_int_of_bool
thf(fact_7486_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_7487_of__int__of__bool,axiom,
! [P: $o] :
( ( ring_17405671764205052669omplex @ ( zero_n2684676970156552555ol_int @ P ) )
= ( zero_n1201886186963655149omplex @ P ) ) ).
% of_int_of_bool
thf(fact_7488_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_7489_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_7490_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_7491_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_7492_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_7493_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_7494_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_7495_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_7496_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_7497_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_7498_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_7499_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_7500_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_7501_Suc__0__mod__eq,axiom,
! [N: nat] :
( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
= ( zero_n2687167440665602831ol_nat
@ ( N
!= ( suc @ zero_zero_nat ) ) ) ) ).
% Suc_0_mod_eq
thf(fact_7502_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_7503_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_7504_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_7505_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_7506_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_7507_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_7508_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_7509_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_7510_sgn__abs,axiom,
! [A: complex] :
( ( abs_abs_complex @ ( sgn_sgn_complex @ A ) )
= ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).
% sgn_abs
thf(fact_7511_sgn__abs,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
= ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% sgn_abs
thf(fact_7512_sgn__abs,axiom,
! [A: real] :
( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
= ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).
% sgn_abs
thf(fact_7513_sgn__abs,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
= ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% sgn_abs
thf(fact_7514_sgn__abs,axiom,
! [A: int] :
( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
= ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% sgn_abs
thf(fact_7515_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_7516_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_7517_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_7518_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_7519_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_7520_take__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% take_bit_of_Suc_0
thf(fact_7521_take__bit__of__1,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ N @ one_one_int )
= ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% take_bit_of_1
thf(fact_7522_take__bit__of__1,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% take_bit_of_1
thf(fact_7523_sgn__of__nat,axiom,
! [N: nat] :
( ( sgn_sgn_real @ ( semiri5074537144036343181t_real @ N ) )
= ( zero_n3304061248610475627l_real @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% sgn_of_nat
thf(fact_7524_sgn__of__nat,axiom,
! [N: nat] :
( ( sgn_sgn_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
= ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% sgn_of_nat
thf(fact_7525_sgn__of__nat,axiom,
! [N: nat] :
( ( sgn_sgn_rat @ ( semiri681578069525770553at_rat @ N ) )
= ( zero_n2052037380579107095ol_rat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% sgn_of_nat
thf(fact_7526_sgn__of__nat,axiom,
! [N: nat] :
( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N ) )
= ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% sgn_of_nat
thf(fact_7527_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_7528_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_7529_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_7530_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_7531_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_7532_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_7533_take__bit__of__exp,axiom,
! [M: nat,N: nat] :
( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_of_exp
thf(fact_7534_take__bit__of__exp,axiom,
! [M: nat,N: nat] :
( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_of_exp
thf(fact_7535_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_7536_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
= ( zero_n8403883297036319079atural @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_7537_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_7538_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_7539_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_7540_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_7541_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_7542_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_7543_real__sqrt__minus,axiom,
! [X: real] :
( ( sqrt @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).
% real_sqrt_minus
thf(fact_7544_of__bool__eq__iff,axiom,
! [P5: $o,Q4: $o] :
( ( ( zero_n2684676970156552555ol_int @ P5 )
= ( zero_n2684676970156552555ol_int @ Q4 ) )
= ( P5 = Q4 ) ) ).
% of_bool_eq_iff
thf(fact_7545_of__bool__eq__iff,axiom,
! [P5: $o,Q4: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P5 )
= ( zero_n2687167440665602831ol_nat @ Q4 ) )
= ( P5 = Q4 ) ) ).
% of_bool_eq_iff
thf(fact_7546_divide__complex__def,axiom,
( divide1717551699836669952omplex
= ( ^ [X4: complex,Y5: complex] : ( times_times_complex @ X4 @ ( invers8013647133539491842omplex @ Y5 ) ) ) ) ).
% divide_complex_def
thf(fact_7547_real__sqrt__inverse,axiom,
! [X: real] :
( ( sqrt @ ( inverse_inverse_real @ X ) )
= ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).
% real_sqrt_inverse
thf(fact_7548_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_7549_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_7550_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_7551_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_7552_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_7553_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_7554_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_7555_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_7556_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_7557_split__of__bool__asm,axiom,
! [P: complex > $o,P5: $o] :
( ( P @ ( zero_n1201886186963655149omplex @ P5 ) )
= ( ~ ( ( P5
& ~ ( P @ one_one_complex ) )
| ( ~ P5
& ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_7558_split__of__bool__asm,axiom,
! [P: real > $o,P5: $o] :
( ( P @ ( zero_n3304061248610475627l_real @ P5 ) )
= ( ~ ( ( P5
& ~ ( P @ one_one_real ) )
| ( ~ P5
& ~ ( P @ zero_zero_real ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_7559_split__of__bool__asm,axiom,
! [P: rat > $o,P5: $o] :
( ( P @ ( zero_n2052037380579107095ol_rat @ P5 ) )
= ( ~ ( ( P5
& ~ ( P @ one_one_rat ) )
| ( ~ P5
& ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_7560_split__of__bool__asm,axiom,
! [P: int > $o,P5: $o] :
( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
= ( ~ ( ( P5
& ~ ( P @ one_one_int ) )
| ( ~ P5
& ~ ( P @ zero_zero_int ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_7561_split__of__bool__asm,axiom,
! [P: nat > $o,P5: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
= ( ~ ( ( P5
& ~ ( P @ one_one_nat ) )
| ( ~ P5
& ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_7562_split__of__bool,axiom,
! [P: complex > $o,P5: $o] :
( ( P @ ( zero_n1201886186963655149omplex @ P5 ) )
= ( ( P5
=> ( P @ one_one_complex ) )
& ( ~ P5
=> ( P @ zero_zero_complex ) ) ) ) ).
% split_of_bool
thf(fact_7563_split__of__bool,axiom,
! [P: real > $o,P5: $o] :
( ( P @ ( zero_n3304061248610475627l_real @ P5 ) )
= ( ( P5
=> ( P @ one_one_real ) )
& ( ~ P5
=> ( P @ zero_zero_real ) ) ) ) ).
% split_of_bool
thf(fact_7564_split__of__bool,axiom,
! [P: rat > $o,P5: $o] :
( ( P @ ( zero_n2052037380579107095ol_rat @ P5 ) )
= ( ( P5
=> ( P @ one_one_rat ) )
& ( ~ P5
=> ( P @ zero_zero_rat ) ) ) ) ).
% split_of_bool
thf(fact_7565_split__of__bool,axiom,
! [P: int > $o,P5: $o] :
( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
= ( ( P5
=> ( P @ one_one_int ) )
& ( ~ P5
=> ( P @ zero_zero_int ) ) ) ) ).
% split_of_bool
thf(fact_7566_split__of__bool,axiom,
! [P: nat > $o,P5: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
= ( ( P5
=> ( P @ one_one_nat ) )
& ( ~ P5
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_of_bool
thf(fact_7567_of__bool__def,axiom,
( zero_n1201886186963655149omplex
= ( ^ [P6: $o] : ( if_complex @ P6 @ one_one_complex @ zero_zero_complex ) ) ) ).
% of_bool_def
thf(fact_7568_of__bool__def,axiom,
( zero_n3304061248610475627l_real
= ( ^ [P6: $o] : ( if_real @ P6 @ one_one_real @ zero_zero_real ) ) ) ).
% of_bool_def
thf(fact_7569_of__bool__def,axiom,
( zero_n2052037380579107095ol_rat
= ( ^ [P6: $o] : ( if_rat @ P6 @ one_one_rat @ zero_zero_rat ) ) ) ).
% of_bool_def
thf(fact_7570_of__bool__def,axiom,
( zero_n2684676970156552555ol_int
= ( ^ [P6: $o] : ( if_int @ P6 @ one_one_int @ zero_zero_int ) ) ) ).
% of_bool_def
thf(fact_7571_of__bool__def,axiom,
( zero_n2687167440665602831ol_nat
= ( ^ [P6: $o] : ( if_nat @ P6 @ one_one_nat @ zero_zero_nat ) ) ) ).
% of_bool_def
thf(fact_7572_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_7573_subset__decode__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% subset_decode_imp_le
thf(fact_7574_bits__induct,axiom,
! [P: int > $o,A: int] :
( ! [A4: int] :
( ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A4 )
=> ( P @ A4 ) )
=> ( ! [A4: int,B4: $o] :
( ( P @ A4 )
=> ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A4 )
=> ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
=> ( P @ A ) ) ) ).
% bits_induct
thf(fact_7575_bits__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [A4: nat] :
( ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A4 )
=> ( P @ A4 ) )
=> ( ! [A4: nat,B4: $o] :
( ( P @ A4 )
=> ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A4 )
=> ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
=> ( P @ A ) ) ) ).
% bits_induct
thf(fact_7576_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_7577_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_7578_exp__mod__exp,axiom,
! [M: nat,N: nat] :
( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% exp_mod_exp
thf(fact_7579_exp__mod__exp,axiom,
! [M: nat,N: nat] :
( ( modulo8411746178871703098atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
= ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_nat @ M @ N ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) ) ).
% exp_mod_exp
thf(fact_7580_exp__mod__exp,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% exp_mod_exp
thf(fact_7581_exp__mod__exp,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% exp_mod_exp
thf(fact_7582_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_7583_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_7584_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_7585_arsinh__real__def,axiom,
( arsinh_real
= ( ^ [X4: real] : ( ln_ln_real @ ( plus_plus_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% arsinh_real_def
thf(fact_7586_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_7587_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_7588_exp__div__exp__eq,axiom,
! [M: nat,N: nat] :
( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int
@ ( zero_n2684676970156552555ol_int
@ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
!= zero_zero_int )
& ( ord_less_eq_nat @ N @ M ) ) )
@ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% exp_div_exp_eq
thf(fact_7589_exp__div__exp__eq,axiom,
! [M: nat,N: nat] :
( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= zero_zero_nat )
& ( ord_less_eq_nat @ N @ M ) ) )
@ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% exp_div_exp_eq
thf(fact_7590_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_7591_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_7592_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_7593_arctan__half,axiom,
( arctan
= ( ^ [X4: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X4 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% arctan_half
thf(fact_7594_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_7595_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_7596_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_7597_divide__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N = 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 @ N ) ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
& ( ( ( 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 @ N ) ) )
= ( uminus_uminus_int
@ ( semiri1314217659103216013at_int
@ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_7598_modulo__int__def,axiom,
( modulo_modulo_int
= ( ^ [K3: int,L3: int] :
( if_int @ ( L3 = zero_zero_int ) @ K3
@ ( if_int
@ ( ( sgn_sgn_int @ K3 )
= ( sgn_sgn_int @ L3 ) )
@ ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) )
@ ( times_times_int @ ( sgn_sgn_int @ L3 )
@ ( minus_minus_int
@ ( times_times_int @ ( abs_abs_int @ L3 )
@ ( zero_n2684676970156552555ol_int
@ ~ ( dvd_dvd_int @ L3 @ K3 ) ) )
@ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) ) ) ) ) ) ) ).
% modulo_int_def
thf(fact_7599_modulo__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
& ( ~ ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N = 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 @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
& ( ( ( 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 @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ L )
@ ( minus_minus_int
@ ( semiri1314217659103216013at_int
@ ( times_times_nat @ N
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
@ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_7600_divide__int__def,axiom,
( divide_divide_int
= ( ^ [K3: int,L3: int] :
( if_int @ ( L3 = zero_zero_int ) @ zero_zero_int
@ ( if_int
@ ( ( sgn_sgn_int @ K3 )
= ( sgn_sgn_int @ L3 ) )
@ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) )
@ ( uminus_uminus_int
@ ( semiri1314217659103216013at_int
@ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_int @ L3 @ K3 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_7601_cis__multiple__2pi,axiom,
! [N: real] :
( ( member_real @ N @ ring_1_Ints_real )
=> ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
= one_one_complex ) ) ).
% cis_multiple_2pi
thf(fact_7602_buildup__gives__empty,axiom,
! [N: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
= bot_bot_set_nat ) ).
% buildup_gives_empty
thf(fact_7603_mask__numeral,axiom,
! [N: num] :
( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).
% mask_numeral
thf(fact_7604_mask__numeral,axiom,
! [N: num] :
( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).
% mask_numeral
thf(fact_7605_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_7606_invar__vebt_Ocases,axiom,
! [A1: vEBT_VEBT,A22: nat] :
( ( vEBT_invar_vebt @ A1 @ A22 )
=> ( ( ? [A4: $o,B4: $o] :
( A1
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( A22
!= ( suc @ zero_zero_nat ) ) )
=> ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M2 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( M2 = N2 )
=> ( ( Deg2
= ( plus_plus_nat @ N2 @ M2 ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M2 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( M2
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus_nat @ N2 @ M2 ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M2 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( M2 = N2 )
=> ( ( Deg2
= ( plus_plus_nat @ N2 @ M2 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi = Ma2 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
=> ( ( 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 ) ) @ M2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
=> ( ( ord_less_nat @ Mi @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M2 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( M2
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus_nat @ N2 @ M2 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi = Ma2 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
=> ( ( 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 ) ) @ M2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
=> ( ( ord_less_nat @ Mi @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_7607_mask__nat__positive__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% mask_nat_positive_iff
thf(fact_7608_mi__ma__2__deg,axiom,
! [Mi2: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( ord_less_eq_nat @ Mi2 @ Ma )
& ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% mi_ma_2_deg
thf(fact_7609_atLeastatMost__empty,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( set_or633870826150836451st_rat @ A @ B )
= bot_bot_set_rat ) ) ).
% atLeastatMost_empty
thf(fact_7610_atLeastatMost__empty,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ( set_or7049704709247886629st_num @ A @ B )
= bot_bot_set_num ) ) ).
% atLeastatMost_empty
thf(fact_7611_atLeastatMost__empty,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( set_or1266510415728281911st_int @ A @ B )
= bot_bot_set_int ) ) ).
% atLeastatMost_empty
thf(fact_7612_atLeastatMost__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_7613_atLeastatMost__empty,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( set_or1222579329274155063t_real @ A @ B )
= bot_bot_set_real ) ) ).
% atLeastatMost_empty
thf(fact_7614_mask__0,axiom,
( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% mask_0
thf(fact_7615_mask__0,axiom,
( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
= zero_zero_int ) ).
% mask_0
thf(fact_7616_mask__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2002935070580805687sk_nat @ N )
= zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_7617_mask__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2000444600071755411sk_int @ N )
= zero_zero_int )
= ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_7618_norm__cis,axiom,
! [A: real] :
( ( real_V1022390504157884413omplex @ ( cis @ A ) )
= one_one_real ) ).
% norm_cis
thf(fact_7619_cis__zero,axiom,
( ( cis @ zero_zero_real )
= one_one_complex ) ).
% cis_zero
thf(fact_7620_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi2: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq_nat @ one_one_nat @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
| ( X = Mi2 )
| ( X = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_7621_set__decode__zero,axiom,
( ( nat_set_decode @ zero_zero_nat )
= bot_bot_set_nat ) ).
% set_decode_zero
thf(fact_7622_cis__inverse,axiom,
! [A: real] :
( ( invers8013647133539491842omplex @ ( cis @ A ) )
= ( cis @ ( uminus_uminus_real @ A ) ) ) ).
% cis_inverse
thf(fact_7623_member__inv,axiom,
! [Mi2: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
& ( ( X = Mi2 )
| ( X = Ma )
| ( ( ord_less_nat @ X @ Ma )
& ( ord_less_nat @ Mi2 @ X )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
& ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% member_inv
thf(fact_7624_both__member__options__from__chilf__to__complete__tree,axiom,
! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi2: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( ( ord_less_eq_nat @ one_one_nat @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_7625_mask__Suc__0,axiom,
( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% mask_Suc_0
thf(fact_7626_mask__Suc__0,axiom,
( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
% mask_Suc_0
thf(fact_7627_take__bit__minus__one__eq__mask,axiom,
! [N: nat] :
( ( bit_se1745604003318907178nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( bit_se2119862282449309892nteger @ N ) ) ).
% take_bit_minus_one_eq_mask
thf(fact_7628_take__bit__minus__one__eq__mask,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( bit_se2000444600071755411sk_int @ N ) ) ).
% take_bit_minus_one_eq_mask
thf(fact_7629_cis__pi,axiom,
( ( cis @ pi )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% cis_pi
thf(fact_7630_cis__2pi,axiom,
( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
= one_one_complex ) ).
% cis_2pi
thf(fact_7631_of__int__mask__eq,axiom,
! [N: nat] :
( ( ring_18347121197199848620nteger @ ( bit_se2000444600071755411sk_int @ N ) )
= ( bit_se2119862282449309892nteger @ N ) ) ).
% of_int_mask_eq
thf(fact_7632_of__int__mask__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( bit_se2000444600071755411sk_int @ N ) )
= ( bit_se2000444600071755411sk_int @ N ) ) ).
% of_int_mask_eq
thf(fact_7633_of__nat__mask__eq,axiom,
! [N: nat] :
( ( semiri4939895301339042750nteger @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( bit_se2119862282449309892nteger @ N ) ) ).
% of_nat_mask_eq
thf(fact_7634_of__nat__mask__eq,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( bit_se2002935070580805687sk_nat @ N ) ) ).
% of_nat_mask_eq
thf(fact_7635_of__nat__mask__eq,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( bit_se2000444600071755411sk_int @ N ) ) ).
% of_nat_mask_eq
thf(fact_7636_nat__mask__eq,axiom,
! [N: nat] :
( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
= ( bit_se2002935070580805687sk_nat @ N ) ) ).
% nat_mask_eq
thf(fact_7637_prod__decode__aux_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [K2: nat,M2: nat] :
( X
!= ( product_Pair_nat_nat @ K2 @ M2 ) ) ).
% prod_decode_aux.cases
thf(fact_7638_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_7639_bot_Onot__eq__extremum,axiom,
! [A: set_int] :
( ( A != bot_bot_set_int )
= ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% bot.not_eq_extremum
thf(fact_7640_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_7641_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_7642_bot_Oextremum__strict,axiom,
! [A: set_int] :
~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% bot.extremum_strict
thf(fact_7643_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_7644_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi2: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ zero_zero_nat @ Va2 @ Vb ) @ X )
= ( ( X = Mi2 )
| ( X = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_7645_diff__shunt__var,axiom,
! [X: set_int,Y: set_int] :
( ( ( minus_minus_set_int @ X @ Y )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_7646_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_7647_subset__Compl__self__eq,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
= ( A2 = bot_bot_set_int ) ) ).
% subset_Compl_self_eq
thf(fact_7648_subset__Compl__self__eq,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_7649_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% mask_nonnegative_int
thf(fact_7650_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% not_mask_negative_int
thf(fact_7651_VEBT__internal_OminNull_Osimps_I5_J,axiom,
! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) ) ).
% VEBT_internal.minNull.simps(5)
thf(fact_7652_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_7653_VEBT__internal_OminNull_Ocases,axiom,
! [X: vEBT_VEBT] :
( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.cases
thf(fact_7654_less__mask,axiom,
! [N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% less_mask
thf(fact_7655_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% VEBT_internal.minNull.elims(3)
thf(fact_7656_option_Osize_I4_J,axiom,
! [X2: product_prod_nat_nat] :
( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X2 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_7657_option_Osize_I4_J,axiom,
! [X2: num] :
( ( size_size_option_num @ ( some_num @ X2 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_7658_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_7659_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y )
=> ( ( ? [Uv2: $o] :
( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> Y )
=> ( ( ? [Uu2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> Y )
=> ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ Y )
=> ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> Y ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_7660_take__bit__eq__mask__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
= ( bit_se2000444600071755411sk_int @ N ) )
= ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
= zero_zero_int ) ) ).
% take_bit_eq_mask_iff
thf(fact_7661_num_Osize__gen_I1_J,axiom,
( ( size_num @ one )
= zero_zero_nat ) ).
% num.size_gen(1)
thf(fact_7662_Suc__mask__eq__exp,axiom,
! [N: nat] :
( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% Suc_mask_eq_exp
thf(fact_7663_mask__nat__less__exp,axiom,
! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% mask_nat_less_exp
thf(fact_7664_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% semiring_bit_operations_class.even_mask_iff
thf(fact_7665_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) )
= ( N = zero_zero_nat ) ) ).
% semiring_bit_operations_class.even_mask_iff
thf(fact_7666_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_7667_mask__half__int,axiom,
! [N: nat] :
( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% mask_half_int
thf(fact_7668_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_7669_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_7670_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_7671_invar__vebt_Ointros_I4_J,axiom,
! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi2: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus_nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi2 = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) ) )
=> ( ( ord_less_eq_nat @ Mi2 @ Ma )
=> ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( ( Mi2 != Ma )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less_nat @ Mi2 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_7672_invar__vebt_Ointros_I5_J,axiom,
! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi2: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus_nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi2 = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) ) )
=> ( ( ord_less_eq_nat @ Mi2 @ Ma )
=> ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( ( Mi2 != Ma )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less_nat @ Mi2 @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_7673_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
= ( bit_se2000444600071755411sk_int @ N ) )
= ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_7674_num_Osize__gen_I2_J,axiom,
! [X2: num] :
( ( size_num @ ( bit0 @ X2 ) )
= ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(2)
thf(fact_7675_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A12: vEBT_VEBT,A23: nat] :
( ( ? [A3: $o,B3: $o] :
( A12
= ( vEBT_Leaf @ A3 @ B3 ) )
& ( A23
= ( suc @ zero_zero_nat ) ) )
| ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList3 @ Summary3 ) )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X4 @ N4 ) )
& ( vEBT_invar_vebt @ Summary3 @ N4 )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
& ( A23
= ( plus_plus_nat @ N4 @ N4 ) )
& ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
| ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList3 @ Summary3 ) )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X4 @ N4 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
& ( A23
= ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) )
& ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
| ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X4 @ N4 ) )
& ( vEBT_invar_vebt @ Summary3 @ N4 )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( 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 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
& ( ( Mi3 != 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 @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N4 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N4 ) ) )
=> ( ( ord_less_nat @ Mi3 @ X4 )
& ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) )
| ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X4 @ N4 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( 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 ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
& ( ( Mi3 != 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 @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N4 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N4 ) ) )
=> ( ( ord_less_nat @ Mi3 @ X4 )
& ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_7676_divmod__step__eq,axiom,
! [L: num,R2: nat,Q4: nat] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
=> ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
=> ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
= ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% divmod_step_eq
thf(fact_7677_divmod__step__eq,axiom,
! [L: num,R2: int,Q4: int] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
=> ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
= ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
=> ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
= ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% divmod_step_eq
thf(fact_7678_divmod__step__eq,axiom,
! [L: num,R2: code_integer,Q4: code_integer] :
( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
=> ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
= ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
=> ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
= ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% divmod_step_eq
thf(fact_7679_card_Oempty,axiom,
( ( finite_card_complex @ bot_bot_set_complex )
= zero_zero_nat ) ).
% card.empty
thf(fact_7680_card_Oempty,axiom,
( ( finite_card_list_nat @ bot_bot_set_list_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_7681_card_Oempty,axiom,
( ( finite_card_set_nat @ bot_bot_set_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_7682_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_7683_card_Oempty,axiom,
( ( finite_card_int @ bot_bot_set_int )
= zero_zero_nat ) ).
% card.empty
thf(fact_7684_divides__aux__eq,axiom,
! [Q4: code_integer,R2: code_integer] :
( ( unique5706413561485394159nteger @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
= ( R2 = zero_z3403309356797280102nteger ) ) ).
% divides_aux_eq
thf(fact_7685_divides__aux__eq,axiom,
! [Q4: nat,R2: nat] :
( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
= ( R2 = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_7686_divides__aux__eq,axiom,
! [Q4: int,R2: int] :
( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q4 @ R2 ) )
= ( R2 = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_7687_product__nth,axiom,
! [N: nat,Xs: list_Code_integer,Ys2: list_Code_integer] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s3445333598471063425nteger @ Xs ) @ ( size_s3445333598471063425nteger @ Ys2 ) ) )
=> ( ( nth_Pr2304437835452373666nteger @ ( produc8792966785426426881nteger @ Xs @ Ys2 ) @ N )
= ( produc1086072967326762835nteger @ ( nth_Code_integer @ Xs @ ( divide_divide_nat @ N @ ( size_s3445333598471063425nteger @ Ys2 ) ) ) @ ( nth_Code_integer @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s3445333598471063425nteger @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_7688_product__nth,axiom,
! [N: nat,Xs: list_Code_integer,Ys2: list_o] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s3445333598471063425nteger @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
=> ( ( nth_Pr8522763379788166057eger_o @ ( produc3607205314601156340eger_o @ Xs @ Ys2 ) @ N )
= ( produc6677183202524767010eger_o @ ( nth_Code_integer @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_7689_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
=> ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys2 ) @ N )
= ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_7690_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_o] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
=> ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys2 ) @ N )
= ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_7691_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
=> ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys2 ) @ N )
= ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_7692_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
=> ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys2 ) @ N )
= ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_7693_product__nth,axiom,
! [N: nat,Xs: list_o,Ys2: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
=> ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys2 ) @ N )
= ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_7694_product__nth,axiom,
! [N: nat,Xs: list_o,Ys2: list_o] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
=> ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs @ Ys2 ) @ N )
= ( product_Pair_o_o @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_7695_product__nth,axiom,
! [N: nat,Xs: list_o,Ys2: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
=> ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs @ Ys2 ) @ N )
= ( product_Pair_o_nat @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_7696_product__nth,axiom,
! [N: nat,Xs: list_o,Ys2: list_int] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
=> ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs @ Ys2 ) @ N )
= ( product_Pair_o_int @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_7697_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_7698_norm__ii,axiom,
( ( real_V1022390504157884413omplex @ imaginary_unit )
= one_one_real ) ).
% norm_ii
thf(fact_7699_complex__i__mult__minus,axiom,
! [X: complex] :
( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X ) )
= ( uminus1482373934393186551omplex @ X ) ) ).
% complex_i_mult_minus
thf(fact_7700_inverse__i,axiom,
( ( invers8013647133539491842omplex @ imaginary_unit )
= ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% inverse_i
thf(fact_7701_divide__i,axiom,
! [X: complex] :
( ( divide1717551699836669952omplex @ X @ imaginary_unit )
= ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).
% divide_i
thf(fact_7702_i__squared,axiom,
( ( times_times_complex @ imaginary_unit @ imaginary_unit )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% i_squared
thf(fact_7703_exp__pi__i_H,axiom,
( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% exp_pi_i'
thf(fact_7704_exp__pi__i,axiom,
( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% exp_pi_i
thf(fact_7705_divide__numeral__i,axiom,
! [Z2: complex,N: num] :
( ( divide1717551699836669952omplex @ Z2 @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
= ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z2 ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% divide_numeral_i
thf(fact_7706_power2__i,axiom,
( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% power2_i
thf(fact_7707_i__even__power,axiom,
! [N: nat] :
( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% i_even_power
thf(fact_7708_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_7709_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_7710_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_7711_complex__i__not__one,axiom,
imaginary_unit != one_one_complex ).
% complex_i_not_one
thf(fact_7712_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,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Deg3 ) ) ) ).
% VEBT_internal.valid'.cases
thf(fact_7713_i__times__eq__iff,axiom,
! [W2: complex,Z2: complex] :
( ( ( times_times_complex @ imaginary_unit @ W2 )
= Z2 )
= ( W2
= ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z2 ) ) ) ) ).
% i_times_eq_iff
thf(fact_7714_complex__i__not__neg__numeral,axiom,
! [W2: num] :
( imaginary_unit
!= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ).
% complex_i_not_neg_numeral
thf(fact_7715_imaginary__unit_Ocode,axiom,
( imaginary_unit
= ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% imaginary_unit.code
thf(fact_7716_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_7717_Complex__mult__i,axiom,
! [A: real,B: real] :
( ( times_times_complex @ ( complex2 @ A @ B ) @ imaginary_unit )
= ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% Complex_mult_i
thf(fact_7718_i__mult__Complex,axiom,
! [A: real,B: real] :
( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B ) )
= ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% i_mult_Complex
thf(fact_7719_VEBT__internal_Onaive__member_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A4: $o,B4: $o,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X3 ) )
=> ( ! [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,V3: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList @ S2 ) @ X3 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_7720_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_7721_vebt__member_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A4: $o,B4: $o,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X3 ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X3 ) )
=> ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X3 ) )
=> ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X3 ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% vebt_member.cases
thf(fact_7722_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,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X3 ) )
=> ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList @ Vc2 ) @ X3 ) )
=> ~ ! [V3: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList @ Vd ) @ X3 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_7723_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_7724_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_7725_divmod__algorithm__code_I7_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_num @ M @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_7726_divmod__algorithm__code_I7_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_num @ M @ N )
=> ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M @ N )
=> ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_7727_divmod__algorithm__code_I7_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_num @ M @ N )
=> ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M @ N )
=> ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_7728_divmod__algorithm__code_I8_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_num @ M @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less_num @ M @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_7729_divmod__algorithm__code_I8_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_num @ M @ N )
=> ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less_num @ M @ N )
=> ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_7730_divmod__algorithm__code_I8_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_num @ M @ N )
=> ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less_num @ M @ N )
=> ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_7731_neg__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q4: int,R2: int] :
( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_7732_csqrt__eq__1,axiom,
! [Z2: complex] :
( ( ( csqrt @ Z2 )
= one_one_complex )
= ( Z2 = one_one_complex ) ) ).
% csqrt_eq_1
thf(fact_7733_csqrt__1,axiom,
( ( csqrt @ one_one_complex )
= one_one_complex ) ).
% csqrt_1
thf(fact_7734_divmod__algorithm__code_I2_J,axiom,
! [M: num] :
( ( unique3479559517661332726nteger @ M @ one )
= ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% divmod_algorithm_code(2)
thf(fact_7735_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_7736_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_7737_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_7738_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_7739_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_7740_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_7741_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_7742_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_7743_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_7744_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q4: int] :
( ( L != zero_zero_int )
=> ( ( K
= ( times_times_int @ Q4 @ L ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ zero_zero_int ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_7745_zminus1__lemma,axiom,
! [A: int,B: int,Q4: int,R2: int] :
( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
=> ( ( B != zero_zero_int )
=> ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q4 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q4 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_7746_Arg__bounded,axiom,
! [Z2: complex] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z2 ) )
& ( ord_less_eq_real @ ( arg @ Z2 ) @ pi ) ) ).
% Arg_bounded
thf(fact_7747_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q4: int,R2: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
= ( ( K
= ( plus_plus_int @ ( times_times_int @ L @ Q4 ) @ R2 ) )
& ( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
& ( ord_less_int @ R2 @ L ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ L )
=> ( ( ( ord_less_int @ L @ zero_zero_int )
=> ( ( ord_less_int @ L @ R2 )
& ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
& ( ~ ( ord_less_int @ L @ zero_zero_int )
=> ( Q4 = zero_zero_int ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_7748_eucl__rel__int__remainderI,axiom,
! [R2: int,L: int,K: int,Q4: int] :
( ( ( sgn_sgn_int @ R2 )
= ( sgn_sgn_int @ L ) )
=> ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
=> ( ( K
= ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R2 ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_7749_cis__Arg__unique,axiom,
! [Z2: complex,X: real] :
( ( ( sgn_sgn_complex @ Z2 )
= ( cis @ X ) )
=> ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ( arg @ Z2 )
= X ) ) ) ) ).
% cis_Arg_unique
thf(fact_7750_Arg__correct,axiom,
! [Z2: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( ( sgn_sgn_complex @ Z2 )
= ( cis @ ( arg @ Z2 ) ) )
& ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z2 ) )
& ( ord_less_eq_real @ ( arg @ Z2 ) @ pi ) ) ) ).
% Arg_correct
thf(fact_7751_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A12: int,A23: int,A32: product_prod_int_int] :
( ? [K3: int] :
( ( A12 = K3 )
& ( A23 = zero_zero_int )
& ( A32
= ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
| ? [L3: int,K3: int,Q6: int] :
( ( A12 = K3 )
& ( A23 = L3 )
& ( A32
= ( product_Pair_int_int @ Q6 @ zero_zero_int ) )
& ( L3 != zero_zero_int )
& ( K3
= ( times_times_int @ Q6 @ L3 ) ) )
| ? [R5: int,L3: int,K3: int,Q6: int] :
( ( A12 = K3 )
& ( A23 = L3 )
& ( A32
= ( product_Pair_int_int @ Q6 @ R5 ) )
& ( ( sgn_sgn_int @ R5 )
= ( sgn_sgn_int @ L3 ) )
& ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L3 ) )
& ( K3
= ( plus_plus_int @ ( times_times_int @ Q6 @ L3 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_7752_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A33: product_prod_int_int] :
( ( eucl_rel_int @ A1 @ A22 @ A33 )
=> ( ( ( A22 = zero_zero_int )
=> ( A33
!= ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
=> ( ! [Q2: int] :
( ( A33
= ( product_Pair_int_int @ Q2 @ zero_zero_int ) )
=> ( ( A22 != zero_zero_int )
=> ( A1
!= ( times_times_int @ Q2 @ A22 ) ) ) )
=> ~ ! [R4: int,Q2: int] :
( ( A33
= ( product_Pair_int_int @ Q2 @ R4 ) )
=> ( ( ( sgn_sgn_int @ R4 )
= ( sgn_sgn_int @ A22 ) )
=> ( ( ord_less_int @ ( abs_abs_int @ R4 ) @ ( abs_abs_int @ A22 ) )
=> ( A1
!= ( plus_plus_int @ ( times_times_int @ Q2 @ A22 ) @ R4 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_7753_divmod__divmod__step,axiom,
( unique5055182867167087721od_nat
= ( ^ [M4: num,N4: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M4 @ N4 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M4 ) ) @ ( unique5026877609467782581ep_nat @ N4 @ ( unique5055182867167087721od_nat @ M4 @ ( bit0 @ N4 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_7754_divmod__divmod__step,axiom,
( unique5052692396658037445od_int
= ( ^ [M4: num,N4: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M4 @ N4 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M4 ) ) @ ( unique5024387138958732305ep_int @ N4 @ ( unique5052692396658037445od_int @ M4 @ ( bit0 @ N4 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_7755_divmod__divmod__step,axiom,
( unique3479559517661332726nteger
= ( ^ [M4: num,N4: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M4 @ N4 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M4 ) ) @ ( unique4921790084139445826nteger @ N4 @ ( unique3479559517661332726nteger @ M4 @ ( bit0 @ N4 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_7756_pos__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q4: int,R2: int] :
( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_7757_one__div__minus__numeral,axiom,
! [N: num] :
( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% one_div_minus_numeral
thf(fact_7758_minus__one__div__numeral,axiom,
! [N: num] :
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% minus_one_div_numeral
thf(fact_7759_Divides_Oadjust__div__eq,axiom,
! [Q4: int,R2: int] :
( ( adjust_div @ ( product_Pair_int_int @ Q4 @ R2 ) )
= ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_7760_numeral__div__minus__numeral,axiom,
! [M: num,N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% numeral_div_minus_numeral
thf(fact_7761_minus__numeral__div__numeral,axiom,
! [M: num,N: num] :
( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% minus_numeral_div_numeral
thf(fact_7762_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_7763_option_Osize__gen_I2_J,axiom,
! [X: product_prod_nat_nat > nat,X2: product_prod_nat_nat] :
( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X2 ) )
= ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_7764_option_Osize__gen_I2_J,axiom,
! [X: num > nat,X2: num] :
( ( size_option_num @ X @ ( some_num @ X2 ) )
= ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_7765_signed__take__bit__eq__take__bit__minus,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N4: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_7766_and__int__unfold,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K3: int,L3: int] :
( if_int
@ ( ( K3 = zero_zero_int )
| ( L3 = zero_zero_int ) )
@ zero_zero_int
@ ( if_int
@ ( K3
= ( uminus_uminus_int @ one_one_int ) )
@ L3
@ ( if_int
@ ( L3
= ( uminus_uminus_int @ one_one_int ) )
@ K3
@ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_7767_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_7768_bit_Oconj__zero__right,axiom,
! [X: int] :
( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
= zero_zero_int ) ).
% bit.conj_zero_right
thf(fact_7769_bit_Oconj__zero__left,axiom,
! [X: int] :
( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
= zero_zero_int ) ).
% bit.conj_zero_left
thf(fact_7770_zero__and__eq,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% zero_and_eq
thf(fact_7771_zero__and__eq,axiom,
! [A: nat] :
( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_and_eq
thf(fact_7772_and__zero__eq,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% and_zero_eq
thf(fact_7773_and__zero__eq,axiom,
! [A: nat] :
( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% and_zero_eq
thf(fact_7774_bit__0__eq,axiom,
( ( bit_se1146084159140164899it_int @ zero_zero_int )
= bot_bot_nat_o ) ).
% bit_0_eq
thf(fact_7775_bit__0__eq,axiom,
( ( bit_se1148574629649215175it_nat @ zero_zero_nat )
= bot_bot_nat_o ) ).
% bit_0_eq
thf(fact_7776_sinh__0,axiom,
( ( sinh_real @ zero_zero_real )
= zero_zero_real ) ).
% sinh_0
thf(fact_7777_sinh__minus,axiom,
! [X: real] :
( ( sinh_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( sinh_real @ X ) ) ) ).
% sinh_minus
thf(fact_7778_sinh__minus,axiom,
! [X: complex] :
( ( sinh_complex @ ( uminus1482373934393186551omplex @ X ) )
= ( uminus1482373934393186551omplex @ ( sinh_complex @ X ) ) ) ).
% sinh_minus
thf(fact_7779_bit_Oconj__one__right,axiom,
! [X: code_integer] :
( ( bit_se3949692690581998587nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= X ) ).
% bit.conj_one_right
thf(fact_7780_bit_Oconj__one__right,axiom,
! [X: int] :
( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
= X ) ).
% bit.conj_one_right
thf(fact_7781_and_Oright__neutral,axiom,
! [A: code_integer] :
( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= A ) ).
% and.right_neutral
thf(fact_7782_and_Oright__neutral,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= A ) ).
% and.right_neutral
thf(fact_7783_and_Oleft__neutral,axiom,
! [A: code_integer] :
( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
= A ) ).
% and.left_neutral
thf(fact_7784_and_Oleft__neutral,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
= A ) ).
% and.left_neutral
thf(fact_7785_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_7786_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_7787_dbl__dec__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
= ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).
% dbl_dec_simps(5)
thf(fact_7788_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_7789_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_7790_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_7791_bit__numeral__Bit0__Suc__iff,axiom,
! [M: num,N: nat] :
( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N ) )
= ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% bit_numeral_Bit0_Suc_iff
thf(fact_7792_bit__numeral__Bit0__Suc__iff,axiom,
! [M: num,N: nat] :
( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N ) )
= ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% bit_numeral_Bit0_Suc_iff
thf(fact_7793_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_7794_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_7795_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_7796_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_7797_bit__numeral__Bit1__Suc__iff,axiom,
! [M: num,N: nat] :
( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N ) )
= ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% bit_numeral_Bit1_Suc_iff
thf(fact_7798_bit__numeral__Bit1__Suc__iff,axiom,
! [M: num,N: nat] :
( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N ) )
= ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% bit_numeral_Bit1_Suc_iff
thf(fact_7799_pred__numeral__simps_I2_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit0 @ K ) )
= ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% pred_numeral_simps(2)
thf(fact_7800_signed__take__bit__nonnegative__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
= ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% signed_take_bit_nonnegative_iff
thf(fact_7801_signed__take__bit__negative__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
= ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% signed_take_bit_negative_iff
thf(fact_7802_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_7803_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_7804_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_7805_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_7806_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W2: num,N: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) ) @ ( suc @ N ) )
= ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ N ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_7807_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W2: num,N: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W2 ) ) ) @ ( suc @ N ) )
= ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) @ N ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_7808_and__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= one_one_int ) ).
% and_minus_numerals(2)
thf(fact_7809_and__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
= one_one_int ) ).
% and_minus_numerals(6)
thf(fact_7810_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_7811_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_7812_and__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
= zero_zero_int ) ).
% and_minus_numerals(5)
thf(fact_7813_and__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= zero_zero_int ) ).
% and_minus_numerals(1)
thf(fact_7814_bit__minus__numeral__int_I1_J,axiom,
! [W2: num,N: num] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) ) @ ( numeral_numeral_nat @ N ) )
= ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ ( pred_numeral @ N ) ) ) ).
% bit_minus_numeral_int(1)
thf(fact_7815_bit__minus__numeral__int_I2_J,axiom,
! [W2: num,N: num] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W2 ) ) ) @ ( numeral_numeral_nat @ N ) )
= ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_minus_numeral_int(2)
thf(fact_7816_bit__mod__2__iff,axiom,
! [A: code_integer,N: nat] :
( ( bit_se9216721137139052372nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N )
= ( ( N = zero_zero_nat )
& ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% bit_mod_2_iff
thf(fact_7817_bit__mod__2__iff,axiom,
! [A: code_natural,N: nat] :
( ( bit_se8040316288895769887atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ N )
= ( ( N = zero_zero_nat )
& ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ).
% bit_mod_2_iff
thf(fact_7818_bit__mod__2__iff,axiom,
! [A: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N )
= ( ( N = zero_zero_nat )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% bit_mod_2_iff
thf(fact_7819_bit__mod__2__iff,axiom,
! [A: nat,N: nat] :
( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
= ( ( N = zero_zero_nat )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% bit_mod_2_iff
thf(fact_7820_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_7821_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_7822_of__int__and__eq,axiom,
! [K: int,L: int] :
( ( ring_18347121197199848620nteger @ ( bit_se725231765392027082nd_int @ K @ L ) )
= ( bit_se3949692690581998587nteger @ ( ring_18347121197199848620nteger @ K ) @ ( ring_18347121197199848620nteger @ L ) ) ) ).
% of_int_and_eq
thf(fact_7823_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_7824_bit__of__nat__iff__bit,axiom,
! [M: nat,N: nat] :
( ( bit_se9216721137139052372nteger @ ( semiri4939895301339042750nteger @ M ) @ N )
= ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% bit_of_nat_iff_bit
thf(fact_7825_bit__of__nat__iff__bit,axiom,
! [M: nat,N: nat] :
( ( bit_se1146084159140164899it_int @ ( semiri1314217659103216013at_int @ M ) @ N )
= ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% bit_of_nat_iff_bit
thf(fact_7826_bit__of__nat__iff__bit,axiom,
! [M: nat,N: nat] :
( ( bit_se1148574629649215175it_nat @ ( semiri1316708129612266289at_nat @ M ) @ N )
= ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% bit_of_nat_iff_bit
thf(fact_7827_of__nat__and__eq,axiom,
! [M: nat,N: nat] :
( ( semiri4939895301339042750nteger @ ( bit_se727722235901077358nd_nat @ M @ N ) )
= ( bit_se3949692690581998587nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% of_nat_and_eq
thf(fact_7828_of__nat__and__eq,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se727722235901077358nd_nat @ M @ N ) )
= ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_and_eq
thf(fact_7829_of__nat__and__eq,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se727722235901077358nd_nat @ M @ N ) )
= ( bit_se727722235901077358nd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_and_eq
thf(fact_7830_bit__disjunctive__add__iff,axiom,
! [A: int,B: int,N: nat] :
( ! [N2: nat] :
( ~ ( bit_se1146084159140164899it_int @ A @ N2 )
| ~ ( bit_se1146084159140164899it_int @ B @ N2 ) )
=> ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B ) @ N )
= ( ( bit_se1146084159140164899it_int @ A @ N )
| ( bit_se1146084159140164899it_int @ B @ N ) ) ) ) ).
% bit_disjunctive_add_iff
thf(fact_7831_bit__disjunctive__add__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ! [N2: nat] :
( ~ ( bit_se1148574629649215175it_nat @ A @ N2 )
| ~ ( bit_se1148574629649215175it_nat @ B @ N2 ) )
=> ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B ) @ N )
= ( ( bit_se1148574629649215175it_nat @ A @ N )
| ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ) ).
% bit_disjunctive_add_iff
thf(fact_7832_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_7833_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_7834_not__bit__1__Suc,axiom,
! [N: nat] :
~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N ) ) ).
% not_bit_1_Suc
thf(fact_7835_not__bit__1__Suc,axiom,
! [N: nat] :
~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N ) ) ).
% not_bit_1_Suc
thf(fact_7836_bit__1__iff,axiom,
! [N: nat] :
( ( bit_se1146084159140164899it_int @ one_one_int @ N )
= ( N = zero_zero_nat ) ) ).
% bit_1_iff
thf(fact_7837_bit__1__iff,axiom,
! [N: nat] :
( ( bit_se1148574629649215175it_nat @ one_one_nat @ N )
= ( N = zero_zero_nat ) ) ).
% bit_1_iff
thf(fact_7838_bit__numeral__simps_I1_J,axiom,
! [N: num] :
~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N ) ) ).
% bit_numeral_simps(1)
thf(fact_7839_bit__numeral__simps_I1_J,axiom,
! [N: num] :
~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% bit_numeral_simps(1)
thf(fact_7840_bit__take__bit__iff,axiom,
! [M: nat,A: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N )
= ( ( ord_less_nat @ N @ M )
& ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% bit_take_bit_iff
thf(fact_7841_bit__take__bit__iff,axiom,
! [M: nat,A: nat,N: nat] :
( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N )
= ( ( ord_less_nat @ N @ M )
& ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% bit_take_bit_iff
thf(fact_7842_bit__of__bool__iff,axiom,
! [B: $o,N: nat] :
( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B ) @ N )
= ( B
& ( N = zero_zero_nat ) ) ) ).
% bit_of_bool_iff
thf(fact_7843_bit__of__bool__iff,axiom,
! [B: $o,N: nat] :
( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ N )
= ( B
& ( N = zero_zero_nat ) ) ) ).
% bit_of_bool_iff
thf(fact_7844_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_7845_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_7846_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_7847_AND__upper1_H,axiom,
! [Y: int,Z2: int,Ya: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z2 ) ) ) ).
% AND_upper1'
thf(fact_7848_AND__upper2_H,axiom,
! [Y: int,Z2: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z2 ) ) ) ).
% AND_upper2'
thf(fact_7849_and__exp__eq__0__iff__not__bit,axiom,
! [A: int,N: nat] :
( ( ( bit_se725231765392027082nd_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= zero_zero_int )
= ( ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% and_exp_eq_0_iff_not_bit
thf(fact_7850_and__exp__eq__0__iff__not__bit,axiom,
! [A: nat,N: nat] :
( ( ( bit_se727722235901077358nd_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= zero_zero_nat )
= ( ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% and_exp_eq_0_iff_not_bit
thf(fact_7851_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_7852_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_7853_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_7854_AND__upper1_H_H,axiom,
! [Y: int,Z2: int,Ya: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z2 ) ) ) ).
% AND_upper1''
thf(fact_7855_AND__upper2_H_H,axiom,
! [Y: int,Z2: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z2 ) ) ) ).
% AND_upper2''
thf(fact_7856_bit__not__int__iff_H,axiom,
! [K: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
= ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% bit_not_int_iff'
thf(fact_7857_eval__nat__numeral_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% eval_nat_numeral(2)
thf(fact_7858_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_7859_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_7860_sinh__diff,axiom,
! [X: real,Y: real] :
( ( sinh_real @ ( minus_minus_real @ X @ Y ) )
= ( minus_minus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% sinh_diff
thf(fact_7861_cosh__diff,axiom,
! [X: real,Y: real] :
( ( cosh_real @ ( minus_minus_real @ X @ Y ) )
= ( minus_minus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).
% cosh_diff
thf(fact_7862_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_7863_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus_num @ ( bitM @ N ) @ one )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_7864_cosh__plus__sinh,axiom,
! [X: complex] :
( ( plus_plus_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ X ) )
= ( exp_complex @ X ) ) ).
% cosh_plus_sinh
thf(fact_7865_cosh__plus__sinh,axiom,
! [X: real] :
( ( plus_plus_real @ ( cosh_real @ X ) @ ( sinh_real @ X ) )
= ( exp_real @ X ) ) ).
% cosh_plus_sinh
thf(fact_7866_sinh__plus__cosh,axiom,
! [X: complex] :
( ( plus_plus_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) )
= ( exp_complex @ X ) ) ).
% sinh_plus_cosh
thf(fact_7867_sinh__plus__cosh,axiom,
! [X: real] :
( ( plus_plus_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) )
= ( exp_real @ X ) ) ).
% sinh_plus_cosh
thf(fact_7868_bit__imp__take__bit__positive,axiom,
! [N: nat,M: nat,K: int] :
( ( ord_less_nat @ N @ M )
=> ( ( bit_se1146084159140164899it_int @ K @ N )
=> ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_7869_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
= ( ( ( ord_less_nat @ N @ M )
& ( bit_se1146084159140164899it_int @ K @ N ) )
| ( ( ord_less_eq_nat @ M @ N )
& ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_7870_signed__take__bit__eq__concat__bit,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N4: nat,K3: int] : ( bit_concat_bit @ N4 @ K3 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) ) ) ) ) ).
% signed_take_bit_eq_concat_bit
thf(fact_7871_exp__eq__0__imp__not__bit,axiom,
! [N: nat,A: int] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
= zero_zero_int )
=> ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ).
% exp_eq_0_imp_not_bit
thf(fact_7872_exp__eq__0__imp__not__bit,axiom,
! [N: nat,A: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= zero_zero_nat )
=> ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ).
% exp_eq_0_imp_not_bit
thf(fact_7873_bit__Suc,axiom,
! [A: int,N: nat] :
( ( bit_se1146084159140164899it_int @ A @ ( suc @ N ) )
= ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N ) ) ).
% bit_Suc
thf(fact_7874_bit__Suc,axiom,
! [A: nat,N: nat] :
( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N ) )
= ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ) ).
% bit_Suc
thf(fact_7875_one__and__eq,axiom,
! [A: code_integer] :
( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
= ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% one_and_eq
thf(fact_7876_one__and__eq,axiom,
! [A: code_natural] :
( ( bit_se2773287842338716102atural @ one_one_Code_natural @ A )
= ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% one_and_eq
thf(fact_7877_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_7878_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_7879_and__one__eq,axiom,
! [A: code_integer] :
( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
= ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% and_one_eq
thf(fact_7880_and__one__eq,axiom,
! [A: code_natural] :
( ( bit_se2773287842338716102atural @ A @ one_one_Code_natural )
= ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% and_one_eq
thf(fact_7881_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_7882_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_7883_numeral__BitM,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bitM @ N ) )
= ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N ) ) @ one_one_complex ) ) ).
% numeral_BitM
thf(fact_7884_numeral__BitM,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bitM @ N ) )
= ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N ) ) @ one_one_real ) ) ).
% numeral_BitM
thf(fact_7885_numeral__BitM,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bitM @ N ) )
= ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).
% numeral_BitM
thf(fact_7886_numeral__BitM,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bitM @ N ) )
= ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).
% numeral_BitM
thf(fact_7887_int__bit__bound,axiom,
! [K: int] :
~ ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ N2 @ M5 )
=> ( ( bit_se1146084159140164899it_int @ K @ M5 )
= ( bit_se1146084159140164899it_int @ K @ N2 ) ) )
=> ~ ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ) ) ).
% int_bit_bound
thf(fact_7888_sinh__minus__cosh,axiom,
! [X: real] :
( ( minus_minus_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) )
= ( uminus_uminus_real @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ).
% sinh_minus_cosh
thf(fact_7889_sinh__minus__cosh,axiom,
! [X: complex] :
( ( minus_minus_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) )
= ( uminus1482373934393186551omplex @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ).
% sinh_minus_cosh
thf(fact_7890_cosh__minus__sinh,axiom,
! [X: real] :
( ( minus_minus_real @ ( cosh_real @ X ) @ ( sinh_real @ X ) )
= ( exp_real @ ( uminus_uminus_real @ X ) ) ) ).
% cosh_minus_sinh
thf(fact_7891_cosh__minus__sinh,axiom,
! [X: complex] :
( ( minus_minus_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ X ) )
= ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ).
% cosh_minus_sinh
thf(fact_7892_even__bit__succ__iff,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N )
= ( ( bit_se1146084159140164899it_int @ A @ N )
| ( N = zero_zero_nat ) ) ) ) ).
% even_bit_succ_iff
thf(fact_7893_even__bit__succ__iff,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N )
= ( ( bit_se1148574629649215175it_nat @ A @ N )
| ( N = zero_zero_nat ) ) ) ) ).
% even_bit_succ_iff
thf(fact_7894_odd__bit__iff__bit__pred,axiom,
! [A: int,N: nat] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se1146084159140164899it_int @ A @ N )
= ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N )
| ( N = zero_zero_nat ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_7895_odd__bit__iff__bit__pred,axiom,
! [A: nat,N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se1148574629649215175it_nat @ A @ N )
= ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N )
| ( N = zero_zero_nat ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_7896_bit__sum__mult__2__cases,axiom,
! [A: int,B: int,N: 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 ) ) @ N )
= ( ( ( N = zero_zero_nat )
=> ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
& ( ( N != zero_zero_nat )
=> ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_7897_bit__sum__mult__2__cases,axiom,
! [A: nat,B: nat,N: 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 ) ) @ N )
= ( ( ( N = zero_zero_nat )
=> ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
& ( ( N != zero_zero_nat )
=> ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_7898_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_7899_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_7900_and__int__rec,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K3: int,L3: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% and_int_rec
thf(fact_7901_sinh__field__def,axiom,
( sinh_real
= ( ^ [Z3: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ Z3 ) @ ( exp_real @ ( uminus_uminus_real @ Z3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% sinh_field_def
thf(fact_7902_sinh__field__def,axiom,
( sinh_complex
= ( ^ [Z3: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ Z3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z3 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% sinh_field_def
thf(fact_7903_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_7904_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_7905_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_7906_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_7907_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_7908_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_7909_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_7910_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_7911_set__bit__eq,axiom,
( bit_se7879613467334960850it_int
= ( ^ [N4: nat,K3: int] :
( plus_plus_int @ K3
@ ( times_times_int
@ ( zero_n2684676970156552555ol_int
@ ~ ( bit_se1146084159140164899it_int @ K3 @ N4 ) )
@ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% set_bit_eq
thf(fact_7912_unset__bit__eq,axiom,
( bit_se4203085406695923979it_int
= ( ^ [N4: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% unset_bit_eq
thf(fact_7913_take__bit__Suc__from__most,axiom,
! [N: nat,K: int] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
= ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% take_bit_Suc_from_most
thf(fact_7914_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_7915_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_7916_and__int_Oelims,axiom,
! [X: int,Xa3: int,Y: int] :
( ( ( bit_se725231765392027082nd_int @ X @ Xa3 )
= Y )
=> ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa3 @ ( 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 ) ) @ Xa3 ) ) ) ) ) )
& ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa3 @ ( 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 ) ) @ Xa3 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_7917_and__int_Osimps,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K3: int,L3: int] :
( if_int
@ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
@ ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) ) )
@ ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_7918_xor__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
= ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% xor_Suc_0_eq
thf(fact_7919_Suc__0__xor__eq,axiom,
! [N: nat] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% Suc_0_xor_eq
thf(fact_7920_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_7921_bit_Oxor__self,axiom,
! [X: int] :
( ( bit_se6526347334894502574or_int @ X @ X )
= zero_zero_int ) ).
% bit.xor_self
thf(fact_7922_xor__self__eq,axiom,
! [A: nat] :
( ( bit_se6528837805403552850or_nat @ A @ A )
= zero_zero_nat ) ).
% xor_self_eq
thf(fact_7923_xor__self__eq,axiom,
! [A: int] :
( ( bit_se6526347334894502574or_int @ A @ A )
= zero_zero_int ) ).
% xor_self_eq
thf(fact_7924_xor_Oleft__neutral,axiom,
! [A: nat] :
( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
= A ) ).
% xor.left_neutral
thf(fact_7925_xor_Oleft__neutral,axiom,
! [A: int] :
( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
= A ) ).
% xor.left_neutral
thf(fact_7926_xor_Oright__neutral,axiom,
! [A: nat] :
( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
= A ) ).
% xor.right_neutral
thf(fact_7927_xor_Oright__neutral,axiom,
! [A: int] :
( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
= A ) ).
% xor.right_neutral
thf(fact_7928_subset__Compl__singleton,axiom,
! [A2: set_complex,B: complex] :
( ( ord_le211207098394363844omplex @ A2 @ ( uminus8566677241136511917omplex @ ( insert_complex @ B @ bot_bot_set_complex ) ) )
= ( ~ ( member_complex @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_7929_subset__Compl__singleton,axiom,
! [A2: set_real,B: real] :
( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
= ( ~ ( member_real @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_7930_subset__Compl__singleton,axiom,
! [A2: set_set_nat,B: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
= ( ~ ( member_set_nat @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_7931_subset__Compl__singleton,axiom,
! [A2: set_int,B: int] :
( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
= ( ~ ( member_int @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_7932_subset__Compl__singleton,axiom,
! [A2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_7933_card__Diff__insert,axiom,
! [A: real,A2: set_real,B2: set_real] :
( ( member_real @ A @ A2 )
=> ( ~ ( member_real @ A @ B2 )
=> ( ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_real @ ( minus_minus_set_real @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_7934_card__Diff__insert,axiom,
! [A: complex,A2: set_complex,B2: set_complex] :
( ( member_complex @ A @ A2 )
=> ( ~ ( member_complex @ A @ B2 )
=> ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_7935_card__Diff__insert,axiom,
! [A: list_nat,A2: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ A @ A2 )
=> ( ~ ( member_list_nat @ A @ B2 )
=> ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_7936_card__Diff__insert,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ~ ( member_set_nat @ A @ B2 )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_7937_card__Diff__insert,axiom,
! [A: int,A2: set_int,B2: set_int] :
( ( member_int @ A @ A2 )
=> ( ~ ( member_int @ A @ B2 )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_7938_card__Diff__insert,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ A @ B2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_7939_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_7940_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_7941_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_7942_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_7943_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_7944_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_7945_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_7946_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_7947_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_7948_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_7949_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_7950_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_7951_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_7952_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_7953_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_7954_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_7955_Suc__0__and__eq,axiom,
! [N: nat] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
= ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Suc_0_and_eq
thf(fact_7956_and__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
= ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% and_Suc_0_eq
thf(fact_7957_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_7958_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_7959_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_7960_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_7961_of__nat__xor__eq,axiom,
! [M: nat,N: nat] :
( ( semiri4939895301339042750nteger @ ( bit_se6528837805403552850or_nat @ M @ N ) )
= ( bit_se3222712562003087583nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% of_nat_xor_eq
thf(fact_7962_of__nat__xor__eq,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se6528837805403552850or_nat @ M @ N ) )
= ( bit_se6528837805403552850or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_xor_eq
thf(fact_7963_of__nat__xor__eq,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se6528837805403552850or_nat @ M @ N ) )
= ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_xor_eq
thf(fact_7964_of__int__xor__eq,axiom,
! [K: int,L: int] :
( ( ring_18347121197199848620nteger @ ( bit_se6526347334894502574or_int @ K @ L ) )
= ( bit_se3222712562003087583nteger @ ( ring_18347121197199848620nteger @ K ) @ ( ring_18347121197199848620nteger @ L ) ) ) ).
% of_int_xor_eq
thf(fact_7965_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_7966_bit__Suc__0__iff,axiom,
! [N: nat] :
( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
= ( N = zero_zero_nat ) ) ).
% bit_Suc_0_iff
thf(fact_7967_not__bit__Suc__0__Suc,axiom,
! [N: nat] :
~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% not_bit_Suc_0_Suc
thf(fact_7968_Compl__insert,axiom,
! [X: int,A2: set_int] :
( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A2 ) )
= ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).
% Compl_insert
thf(fact_7969_Compl__insert,axiom,
! [X: nat,A2: set_nat] :
( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A2 ) )
= ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% Compl_insert
thf(fact_7970_not__bit__Suc__0__numeral,axiom,
! [N: num] :
~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% not_bit_Suc_0_numeral
thf(fact_7971_and__nat__def,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M4: nat,N4: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% and_nat_def
thf(fact_7972_card__1__singletonE,axiom,
! [A2: set_complex] :
( ( ( finite_card_complex @ A2 )
= one_one_nat )
=> ~ ! [X3: complex] :
( A2
!= ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ).
% card_1_singletonE
thf(fact_7973_card__1__singletonE,axiom,
! [A2: set_list_nat] :
( ( ( finite_card_list_nat @ A2 )
= one_one_nat )
=> ~ ! [X3: list_nat] :
( A2
!= ( insert_list_nat @ X3 @ bot_bot_set_list_nat ) ) ) ).
% card_1_singletonE
thf(fact_7974_card__1__singletonE,axiom,
! [A2: set_set_nat] :
( ( ( finite_card_set_nat @ A2 )
= one_one_nat )
=> ~ ! [X3: set_nat] :
( A2
!= ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_7975_card__1__singletonE,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= one_one_nat )
=> ~ ! [X3: nat] :
( A2
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_7976_card__1__singletonE,axiom,
! [A2: set_int] :
( ( ( finite_card_int @ A2 )
= one_one_nat )
=> ~ ! [X3: int] :
( A2
!= ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).
% card_1_singletonE
thf(fact_7977_bit__nat__iff,axiom,
! [K: int,N: nat] :
( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
= ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% bit_nat_iff
thf(fact_7978_card__1__singleton__iff,axiom,
! [A2: set_complex] :
( ( ( finite_card_complex @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X4: complex] :
( A2
= ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) ) ).
% card_1_singleton_iff
thf(fact_7979_card__1__singleton__iff,axiom,
! [A2: set_list_nat] :
( ( ( finite_card_list_nat @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X4: list_nat] :
( A2
= ( insert_list_nat @ X4 @ bot_bot_set_list_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_7980_card__1__singleton__iff,axiom,
! [A2: set_set_nat] :
( ( ( finite_card_set_nat @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X4: set_nat] :
( A2
= ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_7981_card__1__singleton__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X4: nat] :
( A2
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_7982_card__1__singleton__iff,axiom,
! [A2: set_int] :
( ( ( finite_card_int @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X4: int] :
( A2
= ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ).
% card_1_singleton_iff
thf(fact_7983_card__eq__SucD,axiom,
! [A2: set_real,K: nat] :
( ( ( finite_card_real @ A2 )
= ( suc @ K ) )
=> ? [B4: real,B7: set_real] :
( ( A2
= ( insert_real @ B4 @ B7 ) )
& ~ ( member_real @ B4 @ B7 )
& ( ( finite_card_real @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_real ) ) ) ) ).
% card_eq_SucD
thf(fact_7984_card__eq__SucD,axiom,
! [A2: set_complex,K: nat] :
( ( ( finite_card_complex @ A2 )
= ( suc @ K ) )
=> ? [B4: complex,B7: set_complex] :
( ( A2
= ( insert_complex @ B4 @ B7 ) )
& ~ ( member_complex @ B4 @ B7 )
& ( ( finite_card_complex @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_complex ) ) ) ) ).
% card_eq_SucD
thf(fact_7985_card__eq__SucD,axiom,
! [A2: set_list_nat,K: nat] :
( ( ( finite_card_list_nat @ A2 )
= ( suc @ K ) )
=> ? [B4: list_nat,B7: set_list_nat] :
( ( A2
= ( insert_list_nat @ B4 @ B7 ) )
& ~ ( member_list_nat @ B4 @ B7 )
& ( ( finite_card_list_nat @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_list_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_7986_card__eq__SucD,axiom,
! [A2: set_set_nat,K: nat] :
( ( ( finite_card_set_nat @ A2 )
= ( suc @ K ) )
=> ? [B4: set_nat,B7: set_set_nat] :
( ( A2
= ( insert_set_nat @ B4 @ B7 ) )
& ~ ( member_set_nat @ B4 @ B7 )
& ( ( finite_card_set_nat @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_set_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_7987_card__eq__SucD,axiom,
! [A2: set_nat,K: nat] :
( ( ( finite_card_nat @ A2 )
= ( suc @ K ) )
=> ? [B4: nat,B7: set_nat] :
( ( A2
= ( insert_nat @ B4 @ B7 ) )
& ~ ( member_nat @ B4 @ B7 )
& ( ( finite_card_nat @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_7988_card__eq__SucD,axiom,
! [A2: set_int,K: nat] :
( ( ( finite_card_int @ A2 )
= ( suc @ K ) )
=> ? [B4: int,B7: set_int] :
( ( A2
= ( insert_int @ B4 @ B7 ) )
& ~ ( member_int @ B4 @ B7 )
& ( ( finite_card_int @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_int ) ) ) ) ).
% card_eq_SucD
thf(fact_7989_card__Suc__eq,axiom,
! [A2: set_real,K: nat] :
( ( ( finite_card_real @ A2 )
= ( suc @ K ) )
= ( ? [B3: real,B8: set_real] :
( ( A2
= ( insert_real @ B3 @ B8 ) )
& ~ ( member_real @ B3 @ B8 )
& ( ( finite_card_real @ B8 )
= K )
& ( ( K = zero_zero_nat )
=> ( B8 = bot_bot_set_real ) ) ) ) ) ).
% card_Suc_eq
thf(fact_7990_card__Suc__eq,axiom,
! [A2: set_complex,K: nat] :
( ( ( finite_card_complex @ A2 )
= ( suc @ K ) )
= ( ? [B3: complex,B8: set_complex] :
( ( A2
= ( insert_complex @ B3 @ B8 ) )
& ~ ( member_complex @ B3 @ B8 )
& ( ( finite_card_complex @ B8 )
= K )
& ( ( K = zero_zero_nat )
=> ( B8 = bot_bot_set_complex ) ) ) ) ) ).
% card_Suc_eq
thf(fact_7991_card__Suc__eq,axiom,
! [A2: set_list_nat,K: nat] :
( ( ( finite_card_list_nat @ A2 )
= ( suc @ K ) )
= ( ? [B3: list_nat,B8: set_list_nat] :
( ( A2
= ( insert_list_nat @ B3 @ B8 ) )
& ~ ( member_list_nat @ B3 @ B8 )
& ( ( finite_card_list_nat @ B8 )
= K )
& ( ( K = zero_zero_nat )
=> ( B8 = bot_bot_set_list_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_7992_card__Suc__eq,axiom,
! [A2: set_set_nat,K: nat] :
( ( ( finite_card_set_nat @ A2 )
= ( suc @ K ) )
= ( ? [B3: set_nat,B8: set_set_nat] :
( ( A2
= ( insert_set_nat @ B3 @ B8 ) )
& ~ ( member_set_nat @ B3 @ B8 )
& ( ( finite_card_set_nat @ B8 )
= K )
& ( ( K = zero_zero_nat )
=> ( B8 = bot_bot_set_set_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_7993_card__Suc__eq,axiom,
! [A2: set_nat,K: nat] :
( ( ( finite_card_nat @ A2 )
= ( suc @ K ) )
= ( ? [B3: nat,B8: set_nat] :
( ( A2
= ( insert_nat @ B3 @ B8 ) )
& ~ ( member_nat @ B3 @ B8 )
& ( ( finite_card_nat @ B8 )
= K )
& ( ( K = zero_zero_nat )
=> ( B8 = bot_bot_set_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_7994_card__Suc__eq,axiom,
! [A2: set_int,K: nat] :
( ( ( finite_card_int @ A2 )
= ( suc @ K ) )
= ( ? [B3: int,B8: set_int] :
( ( A2
= ( insert_int @ B3 @ B8 ) )
& ~ ( member_int @ B3 @ B8 )
& ( ( finite_card_int @ B8 )
= K )
& ( ( K = zero_zero_nat )
=> ( B8 = bot_bot_set_int ) ) ) ) ) ).
% card_Suc_eq
thf(fact_7995_card__Diff__singleton__if,axiom,
! [X: real,A2: set_real] :
( ( ( member_real @ X @ A2 )
=> ( ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) )
= ( minus_minus_nat @ ( finite_card_real @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_real @ X @ A2 )
=> ( ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) )
= ( finite_card_real @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_7996_card__Diff__singleton__if,axiom,
! [X: complex,A2: set_complex] :
( ( ( member_complex @ X @ A2 )
=> ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) )
= ( minus_minus_nat @ ( finite_card_complex @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_complex @ X @ A2 )
=> ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) )
= ( finite_card_complex @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_7997_card__Diff__singleton__if,axiom,
! [X: list_nat,A2: set_list_nat] :
( ( ( member_list_nat @ X @ A2 )
=> ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) )
= ( minus_minus_nat @ ( finite_card_list_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_list_nat @ X @ A2 )
=> ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) )
= ( finite_card_list_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_7998_card__Diff__singleton__if,axiom,
! [X: set_nat,A2: set_set_nat] :
( ( ( member_set_nat @ X @ A2 )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_set_nat @ X @ A2 )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) )
= ( finite_card_set_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_7999_card__Diff__singleton__if,axiom,
! [X: int,A2: set_int] :
( ( ( member_int @ X @ A2 )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) )
= ( minus_minus_nat @ ( finite_card_int @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_int @ X @ A2 )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) )
= ( finite_card_int @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_8000_card__Diff__singleton__if,axiom,
! [X: nat,A2: set_nat] :
( ( ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_8001_card__Diff__singleton,axiom,
! [X: real,A2: set_real] :
( ( member_real @ X @ A2 )
=> ( ( finite_card_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) )
= ( minus_minus_nat @ ( finite_card_real @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_8002_card__Diff__singleton,axiom,
! [X: complex,A2: set_complex] :
( ( member_complex @ X @ A2 )
=> ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) )
= ( minus_minus_nat @ ( finite_card_complex @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_8003_card__Diff__singleton,axiom,
! [X: list_nat,A2: set_list_nat] :
( ( member_list_nat @ X @ A2 )
=> ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) )
= ( minus_minus_nat @ ( finite_card_list_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_8004_card__Diff__singleton,axiom,
! [X: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_8005_card__Diff__singleton,axiom,
! [X: int,A2: set_int] :
( ( member_int @ X @ A2 )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) )
= ( minus_minus_nat @ ( finite_card_int @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_8006_card__Diff__singleton,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_8007_atLeastAtMostPlus1__int__conv,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
=> ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
= ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_8008_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_8009_card__insert__le__m1,axiom,
! [N: nat,Y: set_complex,X: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_complex @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_complex @ ( insert_complex @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_8010_card__insert__le__m1,axiom,
! [N: nat,Y: set_list_nat,X: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ ( insert_list_nat @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_8011_card__insert__le__m1,axiom,
! [N: nat,Y: set_set_nat,X: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( insert_set_nat @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_8012_card__insert__le__m1,axiom,
! [N: nat,Y: set_nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_8013_card__insert__le__m1,axiom,
! [N: nat,Y: set_int,X: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( insert_int @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_8014_sinh__zero__iff,axiom,
! [X: real] :
( ( ( sinh_real @ X )
= zero_zero_real )
= ( member_real @ ( exp_real @ X ) @ ( insert_real @ one_one_real @ ( insert_real @ ( uminus_uminus_real @ one_one_real ) @ bot_bot_set_real ) ) ) ) ).
% sinh_zero_iff
thf(fact_8015_sinh__zero__iff,axiom,
! [X: complex] :
( ( ( sinh_complex @ X )
= zero_zero_complex )
= ( member_complex @ ( exp_complex @ X ) @ ( insert_complex @ one_one_complex @ ( insert_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ bot_bot_set_complex ) ) ) ) ).
% sinh_zero_iff
thf(fact_8016_and__nat__unfold,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M4: nat,N4: nat] :
( if_nat
@ ( ( M4 = zero_zero_nat )
| ( N4 = zero_zero_nat ) )
@ zero_zero_nat
@ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M4 @ ( 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 @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_8017_and__nat__rec,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M4: nat,N4: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_8018_xor__nat__unfold,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M4: nat,N4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M4 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M4 @ ( 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 @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_8019_xor__nat__rec,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M4: nat,N4: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_8020_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_8021_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_8022_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_8023_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_8024_bit__horner__sum__bit__iff,axiom,
! [Bs: list_o,N: nat] :
( ( bit_se1148574629649215175it_nat @ ( groups9119017779487936845_o_nat @ zero_n2687167440665602831ol_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Bs ) @ N )
= ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
& ( nth_o @ Bs @ N ) ) ) ).
% bit_horner_sum_bit_iff
thf(fact_8025_bit__horner__sum__bit__iff,axiom,
! [Bs: list_o,N: nat] :
( ( bit_se1146084159140164899it_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ N )
= ( ( ord_less_nat @ N @ ( size_size_list_o @ Bs ) )
& ( nth_o @ Bs @ N ) ) ) ).
% bit_horner_sum_bit_iff
thf(fact_8026_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_8027_and__int_Opelims,axiom,
! [X: int,Xa3: int,Y: int] :
( ( ( bit_se725231765392027082nd_int @ X @ Xa3 )
= Y )
=> ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa3 ) )
=> ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa3 @ ( 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 ) ) @ Xa3 ) ) ) ) ) )
& ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa3 @ ( 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 ) ) @ Xa3 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa3 ) ) ) ) ) ).
% and_int.pelims
thf(fact_8028_push__bit__numeral__minus__1,axiom,
! [N: num] :
( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_8029_push__bit__numeral__minus__1,axiom,
! [N: num] :
( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_8030_vebt__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa3 )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( ( Xa3 = zero_zero_nat )
=> A4 )
& ( ( Xa3 != zero_zero_nat )
=> ( ( ( Xa3 = one_one_nat )
=> B4 )
& ( Xa3 = one_one_nat ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ( ( Xa3 != Mi )
=> ( ( Xa3 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa3 @ Mi )
& ( ~ ( ord_less_nat @ Xa3 @ Mi )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_8031_vebt__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa3 )
= Y )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
= ( ~ ( ( ( Xa3 = zero_zero_nat )
=> A4 )
& ( ( Xa3 != zero_zero_nat )
=> ( ( ( Xa3 = one_one_nat )
=> B4 )
& ( Xa3 = one_one_nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> Y )
=> ( ( ? [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> Y )
=> ( ( ? [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> Y )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ( Y
= ( ~ ( ( Xa3 != Mi )
=> ( ( Xa3 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa3 @ Mi )
& ( ~ ( ord_less_nat @ Xa3 @ Mi )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_8032_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).
% set_vebt'_def
thf(fact_8033_push__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% push_bit_nonnegative_int_iff
thf(fact_8034_push__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% push_bit_negative_int_iff
thf(fact_8035_push__bit__of__0,axiom,
! [N: nat] :
( ( bit_se545348938243370406it_int @ N @ zero_zero_int )
= zero_zero_int ) ).
% push_bit_of_0
thf(fact_8036_push__bit__of__0,axiom,
! [N: nat] :
( ( bit_se547839408752420682it_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% push_bit_of_0
thf(fact_8037_push__bit__eq__0__iff,axiom,
! [N: nat,A: int] :
( ( ( bit_se545348938243370406it_int @ N @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% push_bit_eq_0_iff
thf(fact_8038_push__bit__eq__0__iff,axiom,
! [N: nat,A: nat] :
( ( ( bit_se547839408752420682it_nat @ N @ A )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% push_bit_eq_0_iff
thf(fact_8039_push__bit__push__bit,axiom,
! [M: nat,N: nat,A: int] :
( ( bit_se545348938243370406it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
= ( bit_se545348938243370406it_int @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% push_bit_push_bit
thf(fact_8040_push__bit__push__bit,axiom,
! [M: nat,N: nat,A: nat] :
( ( bit_se547839408752420682it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
= ( bit_se547839408752420682it_nat @ ( plus_plus_nat @ M @ N ) @ A ) ) ).
% push_bit_push_bit
thf(fact_8041_concat__bit__of__zero__1,axiom,
! [N: nat,L: int] :
( ( bit_concat_bit @ N @ zero_zero_int @ L )
= ( bit_se545348938243370406it_int @ N @ L ) ) ).
% concat_bit_of_zero_1
thf(fact_8042_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_8043_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_8044_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_8045_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_8046_push__bit__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( numeral_numeral_int @ K ) )
= ( bit_se545348938243370406it_int @ N @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).
% push_bit_Suc_numeral
thf(fact_8047_push__bit__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( bit_se547839408752420682it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% push_bit_Suc_numeral
thf(fact_8048_push__bit__Suc__minus__numeral,axiom,
! [N: nat,K: num] :
( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_Suc_minus_numeral
thf(fact_8049_push__bit__Suc__minus__numeral,axiom,
! [N: nat,K: num] :
( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_Suc_minus_numeral
thf(fact_8050_push__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se545348938243370406it_int @ ( suc @ N ) @ A )
= ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% push_bit_Suc
thf(fact_8051_push__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ A )
= ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% push_bit_Suc
thf(fact_8052_push__bit__of__1,axiom,
! [N: nat] :
( ( bit_se545348938243370406it_int @ N @ one_one_int )
= ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% push_bit_of_1
thf(fact_8053_push__bit__of__1,axiom,
! [N: nat] :
( ( bit_se547839408752420682it_nat @ N @ one_one_nat )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% push_bit_of_1
thf(fact_8054_push__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% push_bit_of_Suc_0
thf(fact_8055_even__push__bit__iff,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se545348938243370406it_int @ N @ A ) )
= ( ( N != zero_zero_nat )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_push_bit_iff
thf(fact_8056_even__push__bit__iff,axiom,
! [N: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se547839408752420682it_nat @ N @ A ) )
= ( ( N != zero_zero_nat )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_push_bit_iff
thf(fact_8057_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_8058_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_8059_flip__bit__int__def,axiom,
( bit_se2159334234014336723it_int
= ( ^ [N4: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).
% flip_bit_int_def
thf(fact_8060_lambda__zero,axiom,
( ( ^ [H2: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_8061_lambda__zero,axiom,
( ( ^ [H2: rat] : zero_zero_rat )
= ( times_times_rat @ zero_zero_rat ) ) ).
% lambda_zero
thf(fact_8062_lambda__zero,axiom,
( ( ^ [H2: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_8063_lambda__zero,axiom,
( ( ^ [H2: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_8064_Compl__eq,axiom,
( uminus8566677241136511917omplex
= ( ^ [A6: set_complex] :
( collect_complex
@ ^ [X4: complex] :
~ ( member_complex @ X4 @ A6 ) ) ) ) ).
% Compl_eq
thf(fact_8065_Compl__eq,axiom,
( uminus612125837232591019t_real
= ( ^ [A6: set_real] :
( collect_real
@ ^ [X4: real] :
~ ( member_real @ X4 @ A6 ) ) ) ) ).
% Compl_eq
thf(fact_8066_Compl__eq,axiom,
( uminus3195874150345416415st_nat
= ( ^ [A6: set_list_nat] :
( collect_list_nat
@ ^ [X4: list_nat] :
~ ( member_list_nat @ X4 @ A6 ) ) ) ) ).
% Compl_eq
thf(fact_8067_Compl__eq,axiom,
( uminus613421341184616069et_nat
= ( ^ [A6: set_set_nat] :
( collect_set_nat
@ ^ [X4: set_nat] :
~ ( member_set_nat @ X4 @ A6 ) ) ) ) ).
% Compl_eq
thf(fact_8068_Compl__eq,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A6: set_nat] :
( collect_nat
@ ^ [X4: nat] :
~ ( member_nat @ X4 @ A6 ) ) ) ) ).
% Compl_eq
thf(fact_8069_Compl__eq,axiom,
( uminus1532241313380277803et_int
= ( ^ [A6: set_int] :
( collect_int
@ ^ [X4: int] :
~ ( member_int @ X4 @ A6 ) ) ) ) ).
% Compl_eq
thf(fact_8070_Collect__neg__eq,axiom,
! [P: real > $o] :
( ( collect_real
@ ^ [X4: real] :
~ ( P @ X4 ) )
= ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ).
% Collect_neg_eq
thf(fact_8071_Collect__neg__eq,axiom,
! [P: list_nat > $o] :
( ( collect_list_nat
@ ^ [X4: list_nat] :
~ ( P @ X4 ) )
= ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) ) ).
% Collect_neg_eq
thf(fact_8072_Collect__neg__eq,axiom,
! [P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X4: set_nat] :
~ ( P @ X4 ) )
= ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).
% Collect_neg_eq
thf(fact_8073_Collect__neg__eq,axiom,
! [P: nat > $o] :
( ( collect_nat
@ ^ [X4: nat] :
~ ( P @ X4 ) )
= ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).
% Collect_neg_eq
thf(fact_8074_Collect__neg__eq,axiom,
! [P: int > $o] :
( ( collect_int
@ ^ [X4: int] :
~ ( P @ X4 ) )
= ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).
% Collect_neg_eq
thf(fact_8075_uminus__set__def,axiom,
( uminus8566677241136511917omplex
= ( ^ [A6: set_complex] :
( collect_complex
@ ( uminus1680532995456772888plex_o
@ ^ [X4: complex] : ( member_complex @ X4 @ A6 ) ) ) ) ) ).
% uminus_set_def
thf(fact_8076_uminus__set__def,axiom,
( uminus612125837232591019t_real
= ( ^ [A6: set_real] :
( collect_real
@ ( uminus_uminus_real_o
@ ^ [X4: real] : ( member_real @ X4 @ A6 ) ) ) ) ) ).
% uminus_set_def
thf(fact_8077_uminus__set__def,axiom,
( uminus3195874150345416415st_nat
= ( ^ [A6: set_list_nat] :
( collect_list_nat
@ ( uminus5770388063884162150_nat_o
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A6 ) ) ) ) ) ).
% uminus_set_def
thf(fact_8078_uminus__set__def,axiom,
( uminus613421341184616069et_nat
= ( ^ [A6: set_set_nat] :
( collect_set_nat
@ ( uminus6401447641752708672_nat_o
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A6 ) ) ) ) ) ).
% uminus_set_def
thf(fact_8079_uminus__set__def,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A6: set_nat] :
( collect_nat
@ ( uminus_uminus_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A6 ) ) ) ) ) ).
% uminus_set_def
thf(fact_8080_uminus__set__def,axiom,
( uminus1532241313380277803et_int
= ( ^ [A6: set_int] :
( collect_int
@ ( uminus_uminus_int_o
@ ^ [X4: int] : ( member_int @ X4 @ A6 ) ) ) ) ) ).
% uminus_set_def
thf(fact_8081_push__bit__of__int,axiom,
! [N: nat,K: int] :
( ( bit_se7788150548672797655nteger @ N @ ( ring_18347121197199848620nteger @ K ) )
= ( ring_18347121197199848620nteger @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% push_bit_of_int
thf(fact_8082_push__bit__of__int,axiom,
! [N: nat,K: int] :
( ( bit_se545348938243370406it_int @ N @ ( ring_1_of_int_int @ K ) )
= ( ring_1_of_int_int @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% push_bit_of_int
thf(fact_8083_push__bit__of__nat,axiom,
! [N: nat,M: nat] :
( ( bit_se7788150548672797655nteger @ N @ ( semiri4939895301339042750nteger @ M ) )
= ( semiri4939895301339042750nteger @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% push_bit_of_nat
thf(fact_8084_push__bit__of__nat,axiom,
! [N: nat,M: nat] :
( ( bit_se545348938243370406it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
= ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% push_bit_of_nat
thf(fact_8085_push__bit__of__nat,axiom,
! [N: nat,M: nat] :
( ( bit_se547839408752420682it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
= ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ N @ M ) ) ) ).
% push_bit_of_nat
thf(fact_8086_of__nat__push__bit,axiom,
! [M: nat,N: nat] :
( ( semiri4939895301339042750nteger @ ( bit_se547839408752420682it_nat @ M @ N ) )
= ( bit_se7788150548672797655nteger @ M @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% of_nat_push_bit
thf(fact_8087_of__nat__push__bit,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ M @ N ) )
= ( bit_se545348938243370406it_int @ M @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_push_bit
thf(fact_8088_of__nat__push__bit,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ M @ N ) )
= ( bit_se547839408752420682it_nat @ M @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_push_bit
thf(fact_8089_push__bit__nat__eq,axiom,
! [N: nat,K: int] :
( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% push_bit_nat_eq
thf(fact_8090_strict__subset__divisors__dvd,axiom,
! [A: real,B: real] :
( ( ord_less_set_real
@ ( collect_real
@ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
@ ( collect_real
@ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
= ( ( dvd_dvd_real @ A @ B )
& ~ ( dvd_dvd_real @ B @ A ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_8091_strict__subset__divisors__dvd,axiom,
! [A: nat,B: nat] :
( ( ord_less_set_nat
@ ( collect_nat
@ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
@ ( collect_nat
@ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
= ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_8092_strict__subset__divisors__dvd,axiom,
! [A: int,B: int] :
( ( ord_less_set_int
@ ( collect_int
@ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
@ ( collect_int
@ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
= ( ( dvd_dvd_int @ A @ B )
& ~ ( dvd_dvd_int @ B @ A ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_8093_subset__divisors__dvd,axiom,
! [A: real,B: real] :
( ( ord_less_eq_set_real
@ ( collect_real
@ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
@ ( collect_real
@ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_8094_subset__divisors__dvd,axiom,
! [A: int,B: int] :
( ( ord_less_eq_set_int
@ ( collect_int
@ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
@ ( collect_int
@ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_8095_subset__divisors__dvd,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
@ ( collect_nat
@ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_8096_push__bit__minus,axiom,
! [N: nat,A: code_integer] :
( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ A ) )
= ( uminus1351360451143612070nteger @ ( bit_se7788150548672797655nteger @ N @ A ) ) ) ).
% push_bit_minus
thf(fact_8097_push__bit__minus,axiom,
! [N: nat,A: int] :
( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ A ) )
= ( uminus_uminus_int @ ( bit_se545348938243370406it_int @ N @ A ) ) ) ).
% push_bit_minus
thf(fact_8098_lambda__one,axiom,
( ( ^ [X4: complex] : X4 )
= ( times_times_complex @ one_one_complex ) ) ).
% lambda_one
thf(fact_8099_lambda__one,axiom,
( ( ^ [X4: real] : X4 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_8100_lambda__one,axiom,
( ( ^ [X4: rat] : X4 )
= ( times_times_rat @ one_one_rat ) ) ).
% lambda_one
thf(fact_8101_lambda__one,axiom,
( ( ^ [X4: nat] : X4 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_8102_lambda__one,axiom,
( ( ^ [X4: int] : X4 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_8103_push__bit__add,axiom,
! [N: nat,A: int,B: int] :
( ( bit_se545348938243370406it_int @ N @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( bit_se545348938243370406it_int @ N @ B ) ) ) ).
% push_bit_add
thf(fact_8104_push__bit__add,axiom,
! [N: nat,A: nat,B: nat] :
( ( bit_se547839408752420682it_nat @ N @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( bit_se547839408752420682it_nat @ N @ B ) ) ) ).
% push_bit_add
thf(fact_8105_numeral__code_I2_J,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% numeral_code(2)
thf(fact_8106_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_code(2)
thf(fact_8107_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit0 @ N ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% numeral_code(2)
thf(fact_8108_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_code(2)
thf(fact_8109_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_code(2)
thf(fact_8110_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_less_as_int
thf(fact_8111_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_leq_as_int
thf(fact_8112_numeral__code_I3_J,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% numeral_code(3)
thf(fact_8113_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit1 @ N ) )
= ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% numeral_code(3)
thf(fact_8114_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit1 @ N ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% numeral_code(3)
thf(fact_8115_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_code(3)
thf(fact_8116_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_code(3)
thf(fact_8117_nat__plus__as__int,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_8118_nat__times__as__int,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_8119_nat__minus__as__int,axiom,
( minus_minus_nat
= ( ^ [A3: nat,B3: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_8120_card__less__Suc2,axiom,
! [M10: set_nat,I: nat] :
( ~ ( member_nat @ zero_zero_nat @ M10 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M10 )
& ( ord_less_nat @ K3 @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M10 )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_8121_card__less__Suc,axiom,
! [M10: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M10 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M10 )
& ( ord_less_nat @ K3 @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M10 )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_8122_card__less,axiom,
! [M10: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M10 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M10 )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_8123_nat__div__as__int,axiom,
( divide_divide_nat
= ( ^ [A3: nat,B3: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_8124_nat__mod__as__int,axiom,
( modulo_modulo_nat
= ( ^ [A3: nat,B3: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_mod_as_int
thf(fact_8125_card__nth__roots,axiom,
! [C: complex,N: nat] :
( ( C != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= C ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_8126_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= one_one_complex ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_8127_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_8128_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_8129_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_8130_card__roots__unity,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ord_less_eq_nat
@ ( finite_card_real
@ ( collect_real
@ ^ [Z3: real] :
( ( power_power_real @ Z3 @ N )
= one_one_real ) ) )
@ N ) ) ).
% card_roots_unity
thf(fact_8131_card__roots__unity,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ord_less_eq_nat
@ ( finite_card_complex
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= one_one_complex ) ) )
@ N ) ) ).
% card_roots_unity
thf(fact_8132_push__bit__take__bit,axiom,
! [M: nat,N: nat,A: int] :
( ( bit_se545348938243370406it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
= ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ M @ A ) ) ) ).
% push_bit_take_bit
thf(fact_8133_push__bit__take__bit,axiom,
! [M: nat,N: nat,A: nat] :
( ( bit_se547839408752420682it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) )
= ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N ) @ ( bit_se547839408752420682it_nat @ M @ A ) ) ) ).
% push_bit_take_bit
thf(fact_8134_take__bit__push__bit,axiom,
! [M: nat,N: nat,A: int] :
( ( bit_se2923211474154528505it_int @ M @ ( bit_se545348938243370406it_int @ N @ A ) )
= ( bit_se545348938243370406it_int @ N @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% take_bit_push_bit
thf(fact_8135_take__bit__push__bit,axiom,
! [M: nat,N: nat,A: nat] :
( ( bit_se2925701944663578781it_nat @ M @ ( bit_se547839408752420682it_nat @ N @ A ) )
= ( bit_se547839408752420682it_nat @ N @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ M @ N ) @ A ) ) ) ).
% take_bit_push_bit
thf(fact_8136_flip__bit__nat__def,axiom,
( bit_se2161824704523386999it_nat
= ( ^ [M4: nat,N4: nat] : ( bit_se6528837805403552850or_nat @ N4 @ ( bit_se547839408752420682it_nat @ M4 @ one_one_nat ) ) ) ) ).
% flip_bit_nat_def
thf(fact_8137_diff__nat__eq__if,axiom,
! [Z5: int,Z2: int] :
( ( ( ord_less_int @ Z5 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) )
= ( nat2 @ Z2 ) ) )
& ( ~ ( ord_less_int @ Z5 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) )
= ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z2 @ Z5 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z2 @ Z5 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_8138_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_8139_atLeastAtMost__insertL,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_8140_atLeastAtMostSuc__conv,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_8141_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( set_or1269000886237332187st_nat @ M @ N )
= ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_8142_bit__push__bit__iff__int,axiom,
! [M: nat,K: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
= ( ( ord_less_eq_nat @ M @ N )
& ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_8143_xor__nat__def,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M4: nat,N4: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% xor_nat_def
thf(fact_8144_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X4: nat] :
( collect_nat
@ ^ [N4: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_8145_bit__push__bit__iff__nat,axiom,
! [M: nat,Q4: nat,N: nat] :
( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q4 ) @ N )
= ( ( ord_less_eq_nat @ M @ N )
& ( bit_se1148574629649215175it_nat @ Q4 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_8146_concat__bit__eq,axiom,
( bit_concat_bit
= ( ^ [N4: nat,K3: int,L3: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N4 @ K3 ) @ ( bit_se545348938243370406it_int @ N4 @ L3 ) ) ) ) ).
% concat_bit_eq
thf(fact_8147_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_8148_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_8149_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_8150_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_8151_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_8152_pochhammer__code,axiom,
( comm_s8582702949713902594nteger
= ( ^ [A3: code_integer,N4: nat] :
( if_Code_integer @ ( N4 = zero_zero_nat ) @ one_one_Code_integer
@ ( set_fo1084959871951514735nteger
@ ^ [O: nat] : ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A3 @ ( semiri4939895301339042750nteger @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N4 @ one_one_nat )
@ one_one_Code_integer ) ) ) ) ).
% pochhammer_code
thf(fact_8153_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_8154_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_8155_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_8156_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_8157_gbinomial__code,axiom,
( gbinomial_complex
= ( ^ [A3: complex,K3: nat] :
( if_complex @ ( K3 = zero_zero_nat ) @ one_one_complex
@ ( divide1717551699836669952omplex
@ ( set_fo1517530859248394432omplex
@ ^ [L3: nat] : ( times_times_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ L3 ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ K3 @ one_one_nat )
@ one_one_complex )
@ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).
% gbinomial_code
thf(fact_8158_gbinomial__code,axiom,
( gbinomial_rat
= ( ^ [A3: rat,K3: nat] :
( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
@ ( divide_divide_rat
@ ( set_fo1949268297981939178at_rat
@ ^ [L3: nat] : ( times_times_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ L3 ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ K3 @ one_one_nat )
@ one_one_rat )
@ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).
% gbinomial_code
thf(fact_8159_gbinomial__code,axiom,
( gbinomial_real
= ( ^ [A3: real,K3: nat] :
( if_real @ ( K3 = zero_zero_nat ) @ one_one_real
@ ( divide_divide_real
@ ( set_fo3111899725591712190t_real
@ ^ [L3: nat] : ( times_times_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ L3 ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ K3 @ one_one_nat )
@ one_one_real )
@ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).
% gbinomial_code
thf(fact_8160_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X )
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_8161_push__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% push_bit_minus_one
thf(fact_8162_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd2 ) @ X )
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_8163_set__decode__plus__power__2,axiom,
! [N: nat,Z2: nat] :
( ~ ( member_nat @ N @ ( nat_set_decode @ Z2 ) )
=> ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z2 ) )
= ( insert_nat @ N @ ( nat_set_decode @ Z2 ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_8164_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi2: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X )
= ( ( X = Mi2 )
| ( X = Ma )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_8165_vebt__member_Osimps_I5_J,axiom,
! [Mi2: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X )
= ( ( X != Mi2 )
=> ( ( X != Ma )
=> ( ~ ( ord_less_nat @ X @ Mi2 )
& ( ~ ( ord_less_nat @ X @ Mi2 )
=> ( ~ ( 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 @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_8166_XOR__upper,axiom,
! [X: int,N: nat,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_8167_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa3 )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( ( Xa3 = zero_zero_nat )
=> A4 )
& ( ( Xa3 != zero_zero_nat )
=> ( ( ( Xa3 = one_one_nat )
=> B4 )
& ( Xa3 = 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,V3: nat,TreeList: list_VEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList @ S2 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_8168_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa3 )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ( ( ( Xa3 = zero_zero_nat )
=> A4 )
& ( ( Xa3 != zero_zero_nat )
=> ( ( ( Xa3 = one_one_nat )
=> B4 )
& ( Xa3 = one_one_nat ) ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList: list_VEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList @ S2 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_8169_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa3 )
= Y )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
= ( ~ ( ( ( Xa3 = zero_zero_nat )
=> A4 )
& ( ( Xa3 != zero_zero_nat )
=> ( ( ( Xa3 = one_one_nat )
=> B4 )
& ( Xa3 = 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,V3: nat,TreeList: list_VEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList @ S2 ) )
=> ( Y
= ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_8170_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa3 )
=> ( ! [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 ) )
=> ~ ( ( Xa3 = Mi )
| ( Xa3 = Ma2 ) ) )
=> ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList @ Vc2 ) )
=> ~ ( ( Xa3 = Mi )
| ( Xa3 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) )
=> ~ ! [V3: nat,TreeList: list_VEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList @ Vd ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_8171_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 ) )
=> ( ! [K2: int,L2: int] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L2 ) )
=> ( ( ~ ( ( 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 ) ) ) )
=> ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
=> ( P @ K2 @ L2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_8172_xor__int__rec,axiom,
( bit_se6526347334894502574or_int
= ( ^ [K3: int,L3: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% xor_int_rec
thf(fact_8173_vebt__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ( vEBT_vebt_member @ X @ Xa3 )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ( ( ( Xa3 = zero_zero_nat )
=> A4 )
& ( ( Xa3 != zero_zero_nat )
=> ( ( ( Xa3 = one_one_nat )
=> B4 )
& ( Xa3 = one_one_nat ) ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ~ ( ( Xa3 != Mi )
=> ( ( Xa3 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa3 @ Mi )
& ( ~ ( ord_less_nat @ Xa3 @ Mi )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_8174_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa3 )
= 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
= ( ~ ( ( Xa3 = Mi )
| ( Xa3 = Ma2 ) ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList @ Vc2 ) )
=> ( Y
= ( ~ ( ( Xa3 = Mi )
| ( Xa3 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList: list_VEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList @ Vd ) )
=> ( Y
= ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_8175_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa3 )
=> ( ! [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 ) )
=> ( ( Xa3 = Mi )
| ( Xa3 = Ma2 ) ) )
=> ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList @ Vc2 ) )
=> ( ( Xa3 = Mi )
| ( Xa3 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) )
=> ~ ! [V3: nat,TreeList: list_VEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList @ Vd ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_8176_of__int__code__if,axiom,
( ring_1_of_int_int
= ( ^ [K3: int] :
( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
@ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_int
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_8177_of__int__code__if,axiom,
( ring_1_of_int_real
= ( ^ [K3: int] :
( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
@ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_real
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_8178_of__int__code__if,axiom,
( ring_17405671764205052669omplex
= ( ^ [K3: int] :
( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
@ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_complex
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_8179_of__int__code__if,axiom,
( ring_18347121197199848620nteger
= ( ^ [K3: int] :
( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
@ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_Code_integer
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_8180_of__int__code__if,axiom,
( ring_1_of_int_rat
= ( ^ [K3: int] :
( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
@ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_rat
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_8181_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_8182_pochhammer__times__pochhammer__half,axiom,
! [Z2: complex,N: nat] :
( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
= ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_8183_pochhammer__times__pochhammer__half,axiom,
! [Z2: real,N: nat] :
( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
= ( groups129246275422532515t_real
@ ^ [K3: nat] : ( plus_plus_real @ Z2 @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_8184_pochhammer__times__pochhammer__half,axiom,
! [Z2: rat,N: nat] :
( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
= ( groups73079841787564623at_rat
@ ^ [K3: nat] : ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_8185_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_8186_arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( arctan @ X )
= ( suminf_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_8187_Ints__prod,axiom,
! [A2: set_complex,F: complex > int] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( member_int @ ( F @ X3 ) @ ring_1_Ints_int ) )
=> ( member_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ring_1_Ints_int ) ) ).
% Ints_prod
thf(fact_8188_Ints__prod,axiom,
! [A2: set_real,F: real > int] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_int @ ( F @ X3 ) @ ring_1_Ints_int ) )
=> ( member_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ring_1_Ints_int ) ) ).
% Ints_prod
thf(fact_8189_Ints__prod,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( member_real @ ( F @ X3 ) @ ring_1_Ints_real ) )
=> ( member_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ring_1_Ints_real ) ) ).
% Ints_prod
thf(fact_8190_Ints__prod,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ ( F @ X3 ) @ ring_1_Ints_real ) )
=> ( member_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ring_1_Ints_real ) ) ).
% Ints_prod
thf(fact_8191_Ints__prod,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_real @ ( F @ X3 ) @ ring_1_Ints_real ) )
=> ( member_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ring_1_Ints_real ) ) ).
% Ints_prod
thf(fact_8192_Ints__prod,axiom,
! [A2: set_int,F: int > real] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( member_real @ ( F @ X3 ) @ ring_1_Ints_real ) )
=> ( member_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ring_1_Ints_real ) ) ).
% Ints_prod
thf(fact_8193_Ints__prod,axiom,
! [A2: set_complex,F: complex > complex] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( member_complex @ ( F @ X3 ) @ ring_1_Ints_complex ) )
=> ( member_complex @ ( groups3708469109370488835omplex @ F @ A2 ) @ ring_1_Ints_complex ) ) ).
% Ints_prod
thf(fact_8194_Ints__prod,axiom,
! [A2: set_real,F: real > complex] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_complex @ ( F @ X3 ) @ ring_1_Ints_complex ) )
=> ( member_complex @ ( groups713298508707869441omplex @ F @ A2 ) @ ring_1_Ints_complex ) ) ).
% Ints_prod
thf(fact_8195_Ints__prod,axiom,
! [A2: set_nat,F: nat > complex] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_complex @ ( F @ X3 ) @ ring_1_Ints_complex ) )
=> ( member_complex @ ( groups6464643781859351333omplex @ F @ A2 ) @ ring_1_Ints_complex ) ) ).
% Ints_prod
thf(fact_8196_Ints__prod,axiom,
! [A2: set_int,F: int > complex] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( member_complex @ ( F @ X3 ) @ ring_1_Ints_complex ) )
=> ( member_complex @ ( groups7440179247065528705omplex @ F @ A2 ) @ ring_1_Ints_complex ) ) ).
% Ints_prod
thf(fact_8197_of__nat__prod,axiom,
! [F: int > nat,A2: set_int] :
( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
= ( groups1705073143266064639nt_int
@ ^ [X4: int] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
@ A2 ) ) ).
% of_nat_prod
thf(fact_8198_of__nat__prod,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ F @ A2 ) )
= ( groups129246275422532515t_real
@ ^ [X4: nat] : ( semiri5074537144036343181t_real @ ( F @ X4 ) )
@ A2 ) ) ).
% of_nat_prod
thf(fact_8199_of__nat__prod,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri4939895301339042750nteger @ ( groups708209901874060359at_nat @ F @ A2 ) )
= ( groups3455450783089532116nteger
@ ^ [X4: nat] : ( semiri4939895301339042750nteger @ ( F @ X4 ) )
@ A2 ) ) ).
% of_nat_prod
thf(fact_8200_of__nat__prod,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri681578069525770553at_rat @ ( groups708209901874060359at_nat @ F @ A2 ) )
= ( groups73079841787564623at_rat
@ ^ [X4: nat] : ( semiri681578069525770553at_rat @ ( F @ X4 ) )
@ A2 ) ) ).
% of_nat_prod
thf(fact_8201_of__nat__prod,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ F @ A2 ) )
= ( groups708209901874060359at_nat
@ ^ [X4: nat] : ( semiri1316708129612266289at_nat @ ( F @ X4 ) )
@ A2 ) ) ).
% of_nat_prod
thf(fact_8202_of__nat__prod,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A2 ) )
= ( groups705719431365010083at_int
@ ^ [X4: nat] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
@ A2 ) ) ).
% of_nat_prod
thf(fact_8203_of__int__prod,axiom,
! [F: nat > int,A2: set_nat] :
( ( ring_1_of_int_real @ ( groups705719431365010083at_int @ F @ A2 ) )
= ( groups129246275422532515t_real
@ ^ [X4: nat] : ( ring_1_of_int_real @ ( F @ X4 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_8204_of__int__prod,axiom,
! [F: nat > int,A2: set_nat] :
( ( ring_18347121197199848620nteger @ ( groups705719431365010083at_int @ F @ A2 ) )
= ( groups3455450783089532116nteger
@ ^ [X4: nat] : ( ring_18347121197199848620nteger @ ( F @ X4 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_8205_of__int__prod,axiom,
! [F: nat > int,A2: set_nat] :
( ( ring_1_of_int_rat @ ( groups705719431365010083at_int @ F @ A2 ) )
= ( groups73079841787564623at_rat
@ ^ [X4: nat] : ( ring_1_of_int_rat @ ( F @ X4 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_8206_of__int__prod,axiom,
! [F: nat > int,A2: set_nat] :
( ( ring_17405671764205052669omplex @ ( groups705719431365010083at_int @ F @ A2 ) )
= ( groups6464643781859351333omplex
@ ^ [X4: nat] : ( ring_17405671764205052669omplex @ ( F @ X4 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_8207_of__int__prod,axiom,
! [F: nat > int,A2: set_nat] :
( ( ring_1_of_int_int @ ( groups705719431365010083at_int @ F @ A2 ) )
= ( groups705719431365010083at_int
@ ^ [X4: nat] : ( ring_1_of_int_int @ ( F @ X4 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_8208_of__int__prod,axiom,
! [F: int > int,A2: set_int] :
( ( ring_1_of_int_real @ ( groups1705073143266064639nt_int @ F @ A2 ) )
= ( groups2316167850115554303t_real
@ ^ [X4: int] : ( ring_1_of_int_real @ ( F @ X4 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_8209_of__int__prod,axiom,
! [F: int > int,A2: set_int] :
( ( ring_18347121197199848620nteger @ ( groups1705073143266064639nt_int @ F @ A2 ) )
= ( groups3827104343326376752nteger
@ ^ [X4: int] : ( ring_18347121197199848620nteger @ ( F @ X4 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_8210_of__int__prod,axiom,
! [F: int > int,A2: set_int] :
( ( ring_1_of_int_rat @ ( groups1705073143266064639nt_int @ F @ A2 ) )
= ( groups1072433553688619179nt_rat
@ ^ [X4: int] : ( ring_1_of_int_rat @ ( F @ X4 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_8211_of__int__prod,axiom,
! [F: int > int,A2: set_int] :
( ( ring_17405671764205052669omplex @ ( groups1705073143266064639nt_int @ F @ A2 ) )
= ( groups7440179247065528705omplex
@ ^ [X4: int] : ( ring_17405671764205052669omplex @ ( F @ X4 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_8212_of__int__prod,axiom,
! [F: int > int,A2: set_int] :
( ( ring_1_of_int_int @ ( groups1705073143266064639nt_int @ F @ A2 ) )
= ( groups1705073143266064639nt_int
@ ^ [X4: int] : ( ring_1_of_int_int @ ( F @ X4 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_8213_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_8214_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_8215_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > complex] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_complex ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_8216_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_8217_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > rat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_rat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_8218_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_8219_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > int] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_int ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_8220_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_8221_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > int,M: nat,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_8222_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > nat,M: nat,K: nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_8223_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > int,M: nat,K: nat,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_8224_prod_OatLeastAtMost__rev,axiom,
! [G: nat > nat,N: nat,M: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_8225_prod_OatLeastAtMost__rev,axiom,
! [G: nat > int,N: nat,M: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_8226_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_8227_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_8228_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_8229_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_8230_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ ( suc @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_8231_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ ( suc @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_8232_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ ( suc @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_8233_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ ( suc @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_8234_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_8235_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_8236_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_8237_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_8238_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ M )
@ ( groups129246275422532515t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_8239_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ M )
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_8240_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ M )
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_8241_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ M )
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_8242_fact__prod,axiom,
( semiri1406184849735516958ct_int
= ( ^ [N4: nat] :
( semiri1314217659103216013at_int
@ ( groups708209901874060359at_nat
@ ^ [X4: nat] : X4
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).
% fact_prod
thf(fact_8243_fact__prod,axiom,
( semiri3624122377584611663nteger
= ( ^ [N4: nat] :
( semiri4939895301339042750nteger
@ ( groups708209901874060359at_nat
@ ^ [X4: nat] : X4
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).
% fact_prod
thf(fact_8244_fact__prod,axiom,
( semiri773545260158071498ct_rat
= ( ^ [N4: nat] :
( semiri681578069525770553at_rat
@ ( groups708209901874060359at_nat
@ ^ [X4: nat] : X4
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).
% fact_prod
thf(fact_8245_fact__prod,axiom,
( semiri1408675320244567234ct_nat
= ( ^ [N4: nat] :
( semiri1316708129612266289at_nat
@ ( groups708209901874060359at_nat
@ ^ [X4: nat] : X4
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).
% fact_prod
thf(fact_8246_fact__prod,axiom,
( semiri2265585572941072030t_real
= ( ^ [N4: nat] :
( semiri5074537144036343181t_real
@ ( groups708209901874060359at_nat
@ ^ [X4: nat] : X4
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).
% fact_prod
thf(fact_8247_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_8248_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_8249_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_8250_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_8251_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_8252_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > real,P5: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P5 ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P5 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_8253_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > rat,P5: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P5 ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P5 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_8254_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > nat,P5: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P5 ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P5 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_8255_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > int,P5: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P5 ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P5 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_8256_fact__eq__fact__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1408675320244567234ct_nat @ M )
= ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
@ ( groups708209901874060359at_nat
@ ^ [X4: nat] : X4
@ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_8257_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_8258_pochhammer__Suc__prod,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( groups129246275422532515t_real
@ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8259_pochhammer__Suc__prod,axiom,
! [A: code_integer,N: nat] :
( ( comm_s8582702949713902594nteger @ A @ ( suc @ N ) )
= ( groups3455450783089532116nteger
@ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8260_pochhammer__Suc__prod,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8261_pochhammer__Suc__prod,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8262_pochhammer__Suc__prod,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8263_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_8264_pochhammer__prod__rev,axiom,
( comm_s8582702949713902594nteger
= ( ^ [A3: code_integer,N4: nat] :
( groups3455450783089532116nteger
@ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A3 @ ( semiri4939895301339042750nteger @ ( minus_minus_nat @ N4 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_8265_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_8266_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_8267_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_8268_fact__div__fact,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
= ( groups708209901874060359at_nat
@ ^ [X4: nat] : X4
@ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_8269_prod_Oin__pairs,axiom,
! [G: nat > real,M: nat,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups129246275422532515t_real
@ ^ [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 @ N ) ) ) ).
% prod.in_pairs
thf(fact_8270_prod_Oin__pairs,axiom,
! [G: nat > rat,M: nat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups73079841787564623at_rat
@ ^ [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 @ N ) ) ) ).
% prod.in_pairs
thf(fact_8271_prod_Oin__pairs,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [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 @ N ) ) ) ).
% prod.in_pairs
thf(fact_8272_prod_Oin__pairs,axiom,
! [G: nat > int,M: nat,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [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 @ N ) ) ) ).
% prod.in_pairs
thf(fact_8273_pochhammer__Suc__prod__rev,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( groups129246275422532515t_real
@ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8274_pochhammer__Suc__prod__rev,axiom,
! [A: code_integer,N: nat] :
( ( comm_s8582702949713902594nteger @ A @ ( suc @ N ) )
= ( groups3455450783089532116nteger
@ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8275_pochhammer__Suc__prod__rev,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8276_pochhammer__Suc__prod__rev,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8277_pochhammer__Suc__prod__rev,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8278_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_8279_gbinomial__Suc,axiom,
! [A: code_integer,K: nat] :
( ( gbinom8545251970709558553nteger @ A @ ( suc @ K ) )
= ( divide6298287555418463151nteger
@ ( groups3455450783089532116nteger
@ ^ [I3: nat] : ( minus_8373710615458151222nteger @ A @ ( semiri4939895301339042750nteger @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri3624122377584611663nteger @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_8280_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_8281_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_8282_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_8283_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_8284_pi__series,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( suminf_real
@ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% pi_series
thf(fact_8285_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_8286_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_8287_prod_Oempty,axiom,
! [G: nat > complex] :
( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
= one_one_complex ) ).
% prod.empty
thf(fact_8288_prod_Oempty,axiom,
! [G: nat > real] :
( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
= one_one_real ) ).
% prod.empty
thf(fact_8289_prod_Oempty,axiom,
! [G: nat > rat] :
( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
= one_one_rat ) ).
% prod.empty
thf(fact_8290_prod_Oempty,axiom,
! [G: int > complex] :
( ( groups7440179247065528705omplex @ G @ bot_bot_set_int )
= one_one_complex ) ).
% prod.empty
thf(fact_8291_prod_Oempty,axiom,
! [G: int > real] :
( ( groups2316167850115554303t_real @ G @ bot_bot_set_int )
= one_one_real ) ).
% prod.empty
thf(fact_8292_prod_Oempty,axiom,
! [G: int > rat] :
( ( groups1072433553688619179nt_rat @ G @ bot_bot_set_int )
= one_one_rat ) ).
% prod.empty
thf(fact_8293_prod_Oempty,axiom,
! [G: int > nat] :
( ( groups1707563613775114915nt_nat @ G @ bot_bot_set_int )
= one_one_nat ) ).
% prod.empty
thf(fact_8294_prod_Oempty,axiom,
! [G: nat > nat] :
( ( groups708209901874060359at_nat @ G @ bot_bot_set_nat )
= one_one_nat ) ).
% prod.empty
thf(fact_8295_prod_Oempty,axiom,
! [G: nat > int] :
( ( groups705719431365010083at_int @ G @ bot_bot_set_nat )
= one_one_int ) ).
% prod.empty
thf(fact_8296_prod_Oempty,axiom,
! [G: int > int] :
( ( groups1705073143266064639nt_int @ G @ bot_bot_set_int )
= one_one_int ) ).
% prod.empty
thf(fact_8297_prod__le__power,axiom,
! [A2: set_real,F: real > real,N: real,K: nat] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_eq_real @ ( F @ I2 ) @ N ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ K )
=> ( ( ord_less_eq_real @ one_one_real @ N )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( power_power_real @ N @ K ) ) ) ) ) ).
% prod_le_power
thf(fact_8298_prod__le__power,axiom,
! [A2: set_complex,F: complex > real,N: real,K: nat] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_eq_real @ ( F @ I2 ) @ N ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ K )
=> ( ( ord_less_eq_real @ one_one_real @ N )
=> ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( power_power_real @ N @ K ) ) ) ) ) ).
% prod_le_power
thf(fact_8299_prod__le__power,axiom,
! [A2: set_nat,F: nat > real,N: real,K: nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_eq_real @ ( F @ I2 ) @ N ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ K )
=> ( ( ord_less_eq_real @ one_one_real @ N )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( power_power_real @ N @ K ) ) ) ) ) ).
% prod_le_power
thf(fact_8300_prod__le__power,axiom,
! [A2: set_int,F: int > real,N: real,K: nat] :
( ! [I2: int] :
( ( member_int @ I2 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_eq_real @ ( F @ I2 ) @ N ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ K )
=> ( ( ord_less_eq_real @ one_one_real @ N )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( power_power_real @ N @ K ) ) ) ) ) ).
% prod_le_power
thf(fact_8301_prod__le__power,axiom,
! [A2: set_real,F: real > rat,N: rat,K: nat] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_eq_rat @ ( F @ I2 ) @ N ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ K )
=> ( ( ord_less_eq_rat @ one_one_rat @ N )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( power_power_rat @ N @ K ) ) ) ) ) ).
% prod_le_power
thf(fact_8302_prod__le__power,axiom,
! [A2: set_complex,F: complex > rat,N: rat,K: nat] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_eq_rat @ ( F @ I2 ) @ N ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ K )
=> ( ( ord_less_eq_rat @ one_one_rat @ N )
=> ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( power_power_rat @ N @ K ) ) ) ) ) ).
% prod_le_power
thf(fact_8303_prod__le__power,axiom,
! [A2: set_nat,F: nat > rat,N: rat,K: nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_eq_rat @ ( F @ I2 ) @ N ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ K )
=> ( ( ord_less_eq_rat @ one_one_rat @ N )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( power_power_rat @ N @ K ) ) ) ) ) ).
% prod_le_power
thf(fact_8304_prod__le__power,axiom,
! [A2: set_int,F: int > rat,N: rat,K: nat] :
( ! [I2: int] :
( ( member_int @ I2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_eq_rat @ ( F @ I2 ) @ N ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ K )
=> ( ( ord_less_eq_rat @ one_one_rat @ N )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( power_power_rat @ N @ K ) ) ) ) ) ).
% prod_le_power
thf(fact_8305_prod__le__power,axiom,
! [A2: set_real,F: real > nat,N: nat,K: nat] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
& ( ord_less_eq_nat @ ( F @ I2 ) @ N ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ K )
=> ( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( power_power_nat @ N @ K ) ) ) ) ) ).
% prod_le_power
thf(fact_8306_prod__le__power,axiom,
! [A2: set_complex,F: complex > nat,N: nat,K: nat] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
& ( ord_less_eq_nat @ ( F @ I2 ) @ N ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ K )
=> ( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ord_less_eq_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( power_power_nat @ N @ K ) ) ) ) ) ).
% prod_le_power
thf(fact_8307_suminf__zero,axiom,
( ( suminf_real
@ ^ [N4: nat] : zero_zero_real )
= zero_zero_real ) ).
% suminf_zero
thf(fact_8308_suminf__zero,axiom,
( ( suminf_nat
@ ^ [N4: nat] : zero_zero_nat )
= zero_zero_nat ) ).
% suminf_zero
thf(fact_8309_suminf__zero,axiom,
( ( suminf_int
@ ^ [N4: nat] : zero_zero_int )
= zero_zero_int ) ).
% suminf_zero
thf(fact_8310_prod_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups708209901874060359at_nat
@ ^ [Uu3: nat] : one_one_nat
@ A2 )
= one_one_nat ) ).
% prod.neutral_const
thf(fact_8311_prod_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups705719431365010083at_int
@ ^ [Uu3: nat] : one_one_int
@ A2 )
= one_one_int ) ).
% prod.neutral_const
thf(fact_8312_prod_Oneutral__const,axiom,
! [A2: set_int] :
( ( groups1705073143266064639nt_int
@ ^ [Uu3: int] : one_one_int
@ A2 )
= one_one_int ) ).
% prod.neutral_const
thf(fact_8313_int__prod,axiom,
! [F: int > nat,A2: set_int] :
( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
= ( groups1705073143266064639nt_int
@ ^ [X4: int] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
@ A2 ) ) ).
% int_prod
thf(fact_8314_int__prod,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A2 ) )
= ( groups705719431365010083at_int
@ ^ [X4: nat] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
@ A2 ) ) ).
% int_prod
thf(fact_8315_prod__int__eq,axiom,
! [I: nat,J: nat] :
( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
= ( groups1705073143266064639nt_int
@ ^ [X4: int] : X4
@ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).
% prod_int_eq
thf(fact_8316_prod__int__plus__eq,axiom,
! [I: nat,J: nat] :
( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
= ( groups1705073143266064639nt_int
@ ^ [X4: int] : X4
@ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_8317_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > complex,A2: set_complex] :
( ( ( groups3708469109370488835omplex @ G @ A2 )
!= one_one_complex )
=> ~ ! [A4: complex] :
( ( member_complex @ A4 @ A2 )
=> ( ( G @ A4 )
= one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8318_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: real > complex,A2: set_real] :
( ( ( groups713298508707869441omplex @ G @ A2 )
!= one_one_complex )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8319_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > complex,A2: set_nat] :
( ( ( groups6464643781859351333omplex @ G @ A2 )
!= one_one_complex )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( G @ A4 )
= one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8320_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: int > complex,A2: set_int] :
( ( ( groups7440179247065528705omplex @ G @ A2 )
!= one_one_complex )
=> ~ ! [A4: int] :
( ( member_int @ A4 @ A2 )
=> ( ( G @ A4 )
= one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8321_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > real,A2: set_complex] :
( ( ( groups766887009212190081x_real @ G @ A2 )
!= one_one_real )
=> ~ ! [A4: complex] :
( ( member_complex @ A4 @ A2 )
=> ( ( G @ A4 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8322_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A2: set_real] :
( ( ( groups1681761925125756287l_real @ G @ A2 )
!= one_one_real )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8323_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A2: set_nat] :
( ( ( groups129246275422532515t_real @ G @ A2 )
!= one_one_real )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( G @ A4 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8324_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: int > real,A2: set_int] :
( ( ( groups2316167850115554303t_real @ G @ A2 )
!= one_one_real )
=> ~ ! [A4: int] :
( ( member_int @ A4 @ A2 )
=> ( ( G @ A4 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8325_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > rat,A2: set_complex] :
( ( ( groups225925009352817453ex_rat @ G @ A2 )
!= one_one_rat )
=> ~ ! [A4: complex] :
( ( member_complex @ A4 @ A2 )
=> ( ( G @ A4 )
= one_one_rat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8326_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: real > rat,A2: set_real] :
( ( ( groups4061424788464935467al_rat @ G @ A2 )
!= one_one_rat )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= one_one_rat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_8327_prod_Oneutral,axiom,
! [A2: set_nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ A2 )
= one_one_nat ) ) ).
% prod.neutral
thf(fact_8328_prod_Oneutral,axiom,
! [A2: set_nat,G: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ( groups705719431365010083at_int @ G @ A2 )
= one_one_int ) ) ).
% prod.neutral
thf(fact_8329_prod_Oneutral,axiom,
! [A2: set_int,G: int > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ( groups1705073143266064639nt_int @ G @ A2 )
= one_one_int ) ) ).
% prod.neutral
thf(fact_8330_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_8331_prod__mono,axiom,
! [A2: set_real,F: real > real,G: real > real] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8332_prod__mono,axiom,
! [A2: set_nat,F: nat > real,G: nat > real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8333_prod__mono,axiom,
! [A2: set_int,F: int > real,G: int > real] :
( ! [I2: int] :
( ( member_int @ I2 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8334_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_8335_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_8336_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_8337_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_8338_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_8339_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_8340_prod__nonneg,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% prod_nonneg
thf(fact_8341_prod__nonneg,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% prod_nonneg
thf(fact_8342_prod__nonneg,axiom,
! [A2: set_int,F: int > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).
% prod_nonneg
thf(fact_8343_prod__pos,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% prod_pos
thf(fact_8344_prod__pos,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% prod_pos
thf(fact_8345_prod__pos,axiom,
! [A2: set_int,F: int > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).
% prod_pos
thf(fact_8346_prod__ge__1,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8347_prod__ge__1,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8348_prod__ge__1,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8349_prod__ge__1,axiom,
! [A2: set_int,F: int > real] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8350_prod__ge__1,axiom,
! [A2: set_complex,F: complex > rat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8351_prod__ge__1,axiom,
! [A2: set_real,F: real > rat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8352_prod__ge__1,axiom,
! [A2: set_nat,F: nat > rat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8353_prod__ge__1,axiom,
! [A2: set_int,F: int > rat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8354_prod__ge__1,axiom,
! [A2: set_complex,F: complex > nat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8355_prod__ge__1,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8356_prod__le__1,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
& ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_8357_prod__le__1,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
& ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_8358_prod__le__1,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
& ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_8359_prod__le__1,axiom,
! [A2: set_int,F: int > real] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
& ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_8360_prod__le__1,axiom,
! [A2: set_complex,F: complex > rat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
& ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_8361_prod__le__1,axiom,
! [A2: set_real,F: real > rat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
& ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_8362_prod__le__1,axiom,
! [A2: set_nat,F: nat > rat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
& ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_8363_prod__le__1,axiom,
! [A2: set_int,F: int > rat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
& ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_8364_prod__le__1,axiom,
! [A2: set_complex,F: complex > nat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
& ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_8365_prod__le__1,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
& ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_8366_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_8367_bij__betw__nth__root__unity,axiom,
! [C: complex,N: nat] :
( ( C != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= one_one_complex ) )
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= C ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_8368_summable__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( summable_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_8369_xor__int__unfold,axiom,
( bit_se6526347334894502574or_int
= ( ^ [K3: int,L3: int] :
( if_int
@ ( K3
= ( uminus_uminus_int @ one_one_int ) )
@ ( bit_ri7919022796975470100ot_int @ L3 )
@ ( if_int
@ ( L3
= ( uminus_uminus_int @ one_one_int ) )
@ ( bit_ri7919022796975470100ot_int @ K3 )
@ ( if_int @ ( K3 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_8370_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_8371_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_8372_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_8373_summable__zero,axiom,
( summable_real
@ ^ [N4: nat] : zero_zero_real ) ).
% summable_zero
thf(fact_8374_summable__zero,axiom,
( summable_nat
@ ^ [N4: nat] : zero_zero_nat ) ).
% summable_zero
thf(fact_8375_summable__zero,axiom,
( summable_int
@ ^ [N4: nat] : zero_zero_int ) ).
% summable_zero
thf(fact_8376_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_8377_bit_Oconj__cancel__right,axiom,
! [X: int] :
( ( bit_se725231765392027082nd_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
= zero_zero_int ) ).
% bit.conj_cancel_right
thf(fact_8378_bit_Oconj__cancel__left,axiom,
! [X: int] :
( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
= zero_zero_int ) ).
% bit.conj_cancel_left
thf(fact_8379_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_8380_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_8381_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_8382_bit_Ocompl__one,axiom,
( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% bit.compl_one
thf(fact_8383_bit_Ocompl__one,axiom,
( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% bit.compl_one
thf(fact_8384_bit_Ocompl__zero,axiom,
( ( bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% bit.compl_zero
thf(fact_8385_bit_Ocompl__zero,axiom,
( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% bit.compl_zero
thf(fact_8386_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_8387_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_8388_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_8389_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_8390_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_8391_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_8392_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_8393_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_8394_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_8395_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_8396_minus__not__numeral__eq,axiom,
! [N: num] :
( ( uminus1351360451143612070nteger @ ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ).
% minus_not_numeral_eq
thf(fact_8397_minus__not__numeral__eq,axiom,
! [N: num] :
( ( uminus_uminus_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( inc @ N ) ) ) ).
% minus_not_numeral_eq
thf(fact_8398_push__bit__minus__one__eq__not__mask,axiom,
! [N: nat] :
( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% push_bit_minus_one_eq_not_mask
thf(fact_8399_push__bit__minus__one__eq__not__mask,axiom,
! [N: nat] :
( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% push_bit_minus_one_eq_not_mask
thf(fact_8400_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_8401_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_8402_not__one__eq,axiom,
( ( bit_ri7632146776885996613nteger @ one_one_Code_integer )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% not_one_eq
thf(fact_8403_not__one__eq,axiom,
( ( bit_ri7919022796975470100ot_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% not_one_eq
thf(fact_8404_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_8405_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_8406_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_8407_summable__diff,axiom,
! [F: nat > real,G: nat > real] :
( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ).
% summable_diff
thf(fact_8408_summable__const__iff,axiom,
! [C: real] :
( ( summable_real
@ ^ [Uu3: nat] : C )
= ( C = zero_zero_real ) ) ).
% summable_const_iff
thf(fact_8409_of__int__not__eq,axiom,
! [K: int] :
( ( ring_18347121197199848620nteger @ ( bit_ri7919022796975470100ot_int @ K ) )
= ( bit_ri7632146776885996613nteger @ ( ring_18347121197199848620nteger @ K ) ) ) ).
% of_int_not_eq
thf(fact_8410_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_8411_summable__minus__iff,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N4: nat] : ( uminus_uminus_real @ ( F @ N4 ) ) )
= ( summable_real @ F ) ) ).
% summable_minus_iff
thf(fact_8412_summable__minus__iff,axiom,
! [F: nat > complex] :
( ( summable_complex
@ ^ [N4: nat] : ( uminus1482373934393186551omplex @ ( F @ N4 ) ) )
= ( summable_complex @ F ) ) ).
% summable_minus_iff
thf(fact_8413_summable__minus,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( summable_real
@ ^ [N4: nat] : ( uminus_uminus_real @ ( F @ N4 ) ) ) ) ).
% summable_minus
thf(fact_8414_summable__minus,axiom,
! [F: nat > complex] :
( ( summable_complex @ F )
=> ( summable_complex
@ ^ [N4: nat] : ( uminus1482373934393186551omplex @ ( F @ N4 ) ) ) ) ).
% summable_minus
thf(fact_8415_summable__Suc__iff,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N4: nat] : ( F @ ( suc @ N4 ) ) )
= ( summable_real @ F ) ) ).
% summable_Suc_iff
thf(fact_8416_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_8417_of__int__not__numeral,axiom,
! [K: num] :
( ( ring_18347121197199848620nteger @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) )
= ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% of_int_not_numeral
thf(fact_8418_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_8419_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_8420_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_8421_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_8422_summable__zero__power,axiom,
summable_int @ ( power_power_int @ zero_zero_int ) ).
% summable_zero_power
thf(fact_8423_summable__zero__power,axiom,
summable_real @ ( power_power_real @ zero_zero_real ) ).
% summable_zero_power
thf(fact_8424_summable__zero__power,axiom,
summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% summable_zero_power
thf(fact_8425_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_8426_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_8427_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_8428_suminf__diff,axiom,
! [F: nat > real,G: nat > real] :
( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( ( minus_minus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
= ( suminf_real
@ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).
% suminf_diff
thf(fact_8429_suminf__minus,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N4: nat] : ( uminus_uminus_real @ ( F @ N4 ) ) )
= ( uminus_uminus_real @ ( suminf_real @ F ) ) ) ) ).
% suminf_minus
thf(fact_8430_suminf__minus,axiom,
! [F: nat > complex] :
( ( summable_complex @ F )
=> ( ( suminf_complex
@ ^ [N4: nat] : ( uminus1482373934393186551omplex @ ( F @ N4 ) ) )
= ( uminus1482373934393186551omplex @ ( suminf_complex @ F ) ) ) ) ).
% suminf_minus
thf(fact_8431_suminf__eq__zero__iff,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
=> ( ( ( suminf_real @ F )
= zero_zero_real )
= ( ! [N4: nat] :
( ( F @ N4 )
= zero_zero_real ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_8432_suminf__eq__zero__iff,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
=> ( ( ( suminf_nat @ F )
= zero_zero_nat )
= ( ! [N4: nat] :
( ( F @ N4 )
= zero_zero_nat ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_8433_suminf__eq__zero__iff,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
=> ( ( ( suminf_int @ F )
= zero_zero_int )
= ( ! [N4: nat] :
( ( F @ N4 )
= zero_zero_int ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_8434_suminf__nonneg,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_8435_suminf__nonneg,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_8436_suminf__nonneg,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_8437_suminf__pos,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N2 ) )
=> ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% suminf_pos
thf(fact_8438_suminf__pos,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% suminf_pos
thf(fact_8439_suminf__pos,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N2 ) )
=> ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% suminf_pos
thf(fact_8440_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_8441_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_8442_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_8443_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_8444_not__eq__complement,axiom,
( bit_ri7632146776885996613nteger
= ( ^ [A3: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A3 ) @ one_one_Code_integer ) ) ) ).
% not_eq_complement
thf(fact_8445_not__eq__complement,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [A3: int] : ( minus_minus_int @ ( uminus_uminus_int @ A3 ) @ one_one_int ) ) ) ).
% not_eq_complement
thf(fact_8446_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_8447_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_8448_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_8449_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_8450_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_8451_summable__powser__split__head,axiom,
! [F: nat > complex,Z2: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z2 @ N4 ) ) )
= ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z2 @ N4 ) ) ) ) ).
% summable_powser_split_head
thf(fact_8452_summable__powser__split__head,axiom,
! [F: nat > real,Z2: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z2 @ N4 ) ) )
= ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z2 @ N4 ) ) ) ) ).
% summable_powser_split_head
thf(fact_8453_powser__split__head_I3_J,axiom,
! [F: nat > complex,Z2: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z2 @ N4 ) ) )
=> ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z2 @ N4 ) ) ) ) ).
% powser_split_head(3)
thf(fact_8454_powser__split__head_I3_J,axiom,
! [F: nat > real,Z2: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z2 @ N4 ) ) )
=> ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z2 @ N4 ) ) ) ) ).
% powser_split_head(3)
thf(fact_8455_not__int__def,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).
% not_int_def
thf(fact_8456_summable__powser__ignore__initial__segment,axiom,
! [F: nat > complex,M: nat,Z2: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N4 @ M ) ) @ ( power_power_complex @ Z2 @ N4 ) ) )
= ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z2 @ N4 ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_8457_summable__powser__ignore__initial__segment,axiom,
! [F: nat > real,M: nat,Z2: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N4 @ M ) ) @ ( power_power_real @ Z2 @ N4 ) ) )
= ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z2 @ N4 ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_8458_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_8459_disjunctive__diff,axiom,
! [B: int,A: int] :
( ! [N2: nat] :
( ( bit_se1146084159140164899it_int @ B @ N2 )
=> ( bit_se1146084159140164899it_int @ A @ N2 ) )
=> ( ( minus_minus_int @ A @ B )
= ( bit_se725231765392027082nd_int @ A @ ( bit_ri7919022796975470100ot_int @ B ) ) ) ) ).
% disjunctive_diff
thf(fact_8460_take__bit__not__eq__mask__diff,axiom,
! [N: nat,A: int] :
( ( bit_se2923211474154528505it_int @ N @ ( bit_ri7919022796975470100ot_int @ A ) )
= ( minus_minus_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ).
% take_bit_not_eq_mask_diff
thf(fact_8461_minus__numeral__inc__eq,axiom,
! [N: num] :
( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) )
= ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% minus_numeral_inc_eq
thf(fact_8462_minus__numeral__inc__eq,axiom,
! [N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ).
% minus_numeral_inc_eq
thf(fact_8463_suminf__pos__iff,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
= ( ? [I3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_8464_suminf__pos__iff,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
= ( ? [I3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_8465_suminf__pos__iff,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
= ( ? [I3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_8466_suminf__pos2,axiom,
! [F: nat > real,I: nat] :
( ( summable_real @ F )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
=> ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_8467_suminf__pos2,axiom,
! [F: nat > nat,I: nat] :
( ( summable_nat @ F )
=> ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_8468_suminf__pos2,axiom,
! [F: nat > int,I: nat] :
( ( summable_int @ F )
=> ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
=> ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_8469_and__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= one_one_int ) ).
% and_not_numerals(2)
thf(fact_8470_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_8471_not__numeral__Bit0__eq,axiom,
! [N: num] :
( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ) ).
% not_numeral_Bit0_eq
thf(fact_8472_not__numeral__Bit0__eq,axiom,
! [N: num] :
( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% not_numeral_Bit0_eq
thf(fact_8473_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_8474_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_8475_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_8476_summable__geometric,axiom,
! [C: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
=> ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% summable_geometric
thf(fact_8477_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_8478_summable__exp,axiom,
! [X: complex] :
( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri5044797733671781792omplex @ N4 ) ) @ ( power_power_complex @ X @ N4 ) ) ) ).
% summable_exp
thf(fact_8479_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_8480_bit__minus__int__iff,axiom,
! [K: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
= ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% bit_minus_int_iff
thf(fact_8481_not__numeral__BitM__eq,axiom,
! [N: num] :
( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bitM @ N ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ) ).
% not_numeral_BitM_eq
thf(fact_8482_not__numeral__BitM__eq,axiom,
! [N: num] :
( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bitM @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% not_numeral_BitM_eq
thf(fact_8483_take__bit__not__mask__eq__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) )
= zero_zero_int ) ) ).
% take_bit_not_mask_eq_0
thf(fact_8484_push__bit__mask__eq,axiom,
! [M: nat,N: nat] :
( ( bit_se545348938243370406it_int @ M @ ( bit_se2000444600071755411sk_int @ N ) )
= ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ N @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ M ) ) ) ) ).
% push_bit_mask_eq
thf(fact_8485_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_8486_unset__bit__int__def,axiom,
( bit_se4203085406695923979it_int
= ( ^ [N4: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ) ).
% unset_bit_int_def
thf(fact_8487_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_8488_and__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= zero_zero_int ) ).
% and_not_numerals(3)
thf(fact_8489_powser__split__head_I1_J,axiom,
! [F: nat > complex,Z2: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z2 @ N4 ) ) )
=> ( ( suminf_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z2 @ N4 ) ) )
= ( plus_plus_complex @ ( F @ zero_zero_nat )
@ ( times_times_complex
@ ( suminf_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z2 @ N4 ) ) )
@ Z2 ) ) ) ) ).
% powser_split_head(1)
thf(fact_8490_powser__split__head_I1_J,axiom,
! [F: nat > real,Z2: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z2 @ N4 ) ) )
=> ( ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z2 @ N4 ) ) )
= ( plus_plus_real @ ( F @ zero_zero_nat )
@ ( times_times_real
@ ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z2 @ N4 ) ) )
@ Z2 ) ) ) ) ).
% powser_split_head(1)
thf(fact_8491_powser__split__head_I2_J,axiom,
! [F: nat > complex,Z2: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z2 @ N4 ) ) )
=> ( ( times_times_complex
@ ( suminf_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z2 @ N4 ) ) )
@ Z2 )
= ( minus_minus_complex
@ ( suminf_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z2 @ N4 ) ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% powser_split_head(2)
thf(fact_8492_powser__split__head_I2_J,axiom,
! [F: nat > real,Z2: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z2 @ N4 ) ) )
=> ( ( times_times_real
@ ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z2 @ N4 ) ) )
@ Z2 )
= ( minus_minus_real
@ ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z2 @ N4 ) ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% powser_split_head(2)
thf(fact_8493_suminf__exist__split,axiom,
! [R2: real,F: nat > real] :
( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ( summable_real @ F )
=> ? [N7: nat] :
! [N6: nat] :
( ( ord_less_eq_nat @ N7 @ N6 )
=> ( ord_less_real
@ ( real_V7735802525324610683m_real
@ ( suminf_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N6 ) ) ) )
@ R2 ) ) ) ) ).
% suminf_exist_split
thf(fact_8494_suminf__exist__split,axiom,
! [R2: real,F: nat > complex] :
( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ( summable_complex @ F )
=> ? [N7: nat] :
! [N6: nat] :
( ( ord_less_eq_nat @ N7 @ N6 )
=> ( ord_less_real
@ ( real_V1022390504157884413omplex
@ ( suminf_complex
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N6 ) ) ) )
@ R2 ) ) ) ) ).
% suminf_exist_split
thf(fact_8495_summable__power__series,axiom,
! [F: nat > real,Z2: 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 @ Z2 )
=> ( ( ord_less_real @ Z2 @ one_one_real )
=> ( summable_real
@ ^ [I3: nat] : ( times_times_real @ ( F @ I3 ) @ ( power_power_real @ Z2 @ I3 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_8496_bit__not__iff__eq,axiom,
! [A: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ A ) @ N )
= ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int )
& ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% bit_not_iff_eq
thf(fact_8497_minus__exp__eq__not__mask,axiom,
! [N: nat] :
( ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% minus_exp_eq_not_mask
thf(fact_8498_minus__exp__eq__not__mask,axiom,
! [N: nat] :
( ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% minus_exp_eq_not_mask
thf(fact_8499_summable__ratio__test,axiom,
! [C: real,N5: nat,F: nat > real] :
( ( ord_less_real @ C @ one_one_real )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ N5 @ N2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N2 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) ) ) )
=> ( summable_real @ F ) ) ) ).
% summable_ratio_test
thf(fact_8500_summable__ratio__test,axiom,
! [C: real,N5: nat,F: nat > complex] :
( ( ord_less_real @ C @ one_one_real )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ N5 @ N2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N2 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) )
=> ( summable_complex @ F ) ) ) ).
% summable_ratio_test
thf(fact_8501_and__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_8502_not__int__rec,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [K3: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% not_int_rec
thf(fact_8503_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_8504_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_8505_sum__gp,axiom,
! [N: nat,M: nat,X: complex] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_complex ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_8506_sum__gp,axiom,
! [N: nat,M: nat,X: rat] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_rat ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_8507_sum__gp,axiom,
! [N: nat,M: nat,X: real] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_8508_vebt__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa3 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa3 ) )
=> ( ( ( Xa3 = zero_zero_nat )
=> A4 )
& ( ( Xa3 != zero_zero_nat )
=> ( ( ( Xa3 = one_one_nat )
=> B4 )
& ( Xa3 = 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 ) @ Xa3 ) ) )
=> ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa3 ) ) )
=> ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa3 ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa3 ) )
=> ( ( Xa3 != Mi )
=> ( ( Xa3 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa3 @ Mi )
& ( ~ ( ord_less_nat @ Xa3 @ Mi )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_8509_intind,axiom,
! [I: nat,N: nat,P: nat > $o,X: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( P @ X )
=> ( P @ ( nth_nat @ ( replicate_nat @ N @ X ) @ I ) ) ) ) ).
% intind
thf(fact_8510_intind,axiom,
! [I: nat,N: nat,P: int > $o,X: int] :
( ( ord_less_nat @ I @ N )
=> ( ( P @ X )
=> ( P @ ( nth_int @ ( replicate_int @ N @ X ) @ I ) ) ) ) ).
% intind
thf(fact_8511_intind,axiom,
! [I: nat,N: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
( ( ord_less_nat @ I @ N )
=> ( ( P @ X )
=> ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I ) ) ) ) ).
% intind
thf(fact_8512_replicate__eq__replicate,axiom,
! [M: nat,X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
( ( ( replicate_VEBT_VEBT @ M @ X )
= ( replicate_VEBT_VEBT @ N @ Y ) )
= ( ( M = N )
& ( ( M != zero_zero_nat )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_8513_Ints__sum,axiom,
! [A2: set_complex,F: complex > int] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( member_int @ ( F @ X3 ) @ ring_1_Ints_int ) )
=> ( member_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ring_1_Ints_int ) ) ).
% Ints_sum
thf(fact_8514_Ints__sum,axiom,
! [A2: set_real,F: real > int] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_int @ ( F @ X3 ) @ ring_1_Ints_int ) )
=> ( member_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ring_1_Ints_int ) ) ).
% Ints_sum
thf(fact_8515_Ints__sum,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_int @ ( F @ X3 ) @ ring_1_Ints_int ) )
=> ( member_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ring_1_Ints_int ) ) ).
% Ints_sum
thf(fact_8516_Ints__sum,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( member_real @ ( F @ X3 ) @ ring_1_Ints_real ) )
=> ( member_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ring_1_Ints_real ) ) ).
% Ints_sum
thf(fact_8517_Ints__sum,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ ( F @ X3 ) @ ring_1_Ints_real ) )
=> ( member_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ring_1_Ints_real ) ) ).
% Ints_sum
thf(fact_8518_Ints__sum,axiom,
! [A2: set_int,F: int > real] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( member_real @ ( F @ X3 ) @ ring_1_Ints_real ) )
=> ( member_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ring_1_Ints_real ) ) ).
% Ints_sum
thf(fact_8519_Ints__sum,axiom,
! [A2: set_real,F: real > complex] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_complex @ ( F @ X3 ) @ ring_1_Ints_complex ) )
=> ( member_complex @ ( groups5754745047067104278omplex @ F @ A2 ) @ ring_1_Ints_complex ) ) ).
% Ints_sum
thf(fact_8520_Ints__sum,axiom,
! [A2: set_nat,F: nat > complex] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_complex @ ( F @ X3 ) @ ring_1_Ints_complex ) )
=> ( member_complex @ ( groups2073611262835488442omplex @ F @ A2 ) @ ring_1_Ints_complex ) ) ).
% Ints_sum
thf(fact_8521_Ints__sum,axiom,
! [A2: set_int,F: int > complex] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( member_complex @ ( F @ X3 ) @ ring_1_Ints_complex ) )
=> ( member_complex @ ( groups3049146728041665814omplex @ F @ A2 ) @ ring_1_Ints_complex ) ) ).
% Ints_sum
thf(fact_8522_Ints__sum,axiom,
! [A2: set_complex,F: complex > complex] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( member_complex @ ( F @ X3 ) @ ring_1_Ints_complex ) )
=> ( member_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ring_1_Ints_complex ) ) ).
% Ints_sum
thf(fact_8523_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu3: nat] : zero_zero_nat
@ A2 )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_8524_sum_Oneutral__const,axiom,
! [A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [Uu3: complex] : zero_zero_complex
@ A2 )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_8525_sum_Oneutral__const,axiom,
! [A2: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [Uu3: int] : zero_zero_int
@ A2 )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_8526_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu3: nat] : zero_zero_real
@ A2 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_8527_of__nat__sum,axiom,
! [F: complex > nat,A2: set_complex] :
( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A2 ) )
= ( groups7754918857620584856omplex
@ ^ [X4: complex] : ( semiri8010041392384452111omplex @ ( F @ X4 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_8528_of__nat__sum,axiom,
! [F: int > nat,A2: set_int] :
( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
= ( groups4538972089207619220nt_int
@ ^ [X4: int] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_8529_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3539618377306564664at_int
@ ^ [X4: nat] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_8530_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri4939895301339042750nteger @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups7501900531339628137nteger
@ ^ [X4: nat] : ( semiri4939895301339042750nteger @ ( F @ X4 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_8531_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri681578069525770553at_rat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups2906978787729119204at_rat
@ ^ [X4: nat] : ( semiri681578069525770553at_rat @ ( F @ X4 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_8532_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( semiri1316708129612266289at_nat @ ( F @ X4 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_8533_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( semiri5074537144036343181t_real @ ( F @ X4 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_8534_of__int__sum,axiom,
! [F: complex > int,A2: set_complex] :
( ( ring_17405671764205052669omplex @ ( groups5690904116761175830ex_int @ F @ A2 ) )
= ( groups7754918857620584856omplex
@ ^ [X4: complex] : ( ring_17405671764205052669omplex @ ( F @ X4 ) )
@ A2 ) ) ).
% of_int_sum
thf(fact_8535_of__int__sum,axiom,
! [F: nat > int,A2: set_nat] :
( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( ring_1_of_int_real @ ( F @ X4 ) )
@ A2 ) ) ).
% of_int_sum
thf(fact_8536_of__int__sum,axiom,
! [F: int > int,A2: set_int] :
( ( ring_1_of_int_real @ ( groups4538972089207619220nt_int @ F @ A2 ) )
= ( groups8778361861064173332t_real
@ ^ [X4: int] : ( ring_1_of_int_real @ ( F @ X4 ) )
@ A2 ) ) ).
% of_int_sum
thf(fact_8537_of__int__sum,axiom,
! [F: int > int,A2: set_int] :
( ( ring_18347121197199848620nteger @ ( groups4538972089207619220nt_int @ F @ A2 ) )
= ( groups7873554091576472773nteger
@ ^ [X4: int] : ( ring_18347121197199848620nteger @ ( F @ X4 ) )
@ A2 ) ) ).
% of_int_sum
thf(fact_8538_of__int__sum,axiom,
! [F: int > int,A2: set_int] :
( ( ring_1_of_int_rat @ ( groups4538972089207619220nt_int @ F @ A2 ) )
= ( groups3906332499630173760nt_rat
@ ^ [X4: int] : ( ring_1_of_int_rat @ ( F @ X4 ) )
@ A2 ) ) ).
% of_int_sum
thf(fact_8539_of__int__sum,axiom,
! [F: int > int,A2: set_int] :
( ( ring_17405671764205052669omplex @ ( groups4538972089207619220nt_int @ F @ A2 ) )
= ( groups3049146728041665814omplex
@ ^ [X4: int] : ( ring_17405671764205052669omplex @ ( F @ X4 ) )
@ A2 ) ) ).
% of_int_sum
thf(fact_8540_of__int__sum,axiom,
! [F: int > int,A2: set_int] :
( ( ring_1_of_int_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
= ( groups4538972089207619220nt_int
@ ^ [X4: int] : ( ring_1_of_int_int @ ( F @ X4 ) )
@ A2 ) ) ).
% of_int_sum
thf(fact_8541_sums__zero,axiom,
( sums_real
@ ^ [N4: nat] : zero_zero_real
@ zero_zero_real ) ).
% sums_zero
thf(fact_8542_sums__zero,axiom,
( sums_nat
@ ^ [N4: nat] : zero_zero_nat
@ zero_zero_nat ) ).
% sums_zero
thf(fact_8543_sums__zero,axiom,
( sums_int
@ ^ [N4: nat] : zero_zero_int
@ zero_zero_int ) ).
% sums_zero
thf(fact_8544_sum_Oempty,axiom,
! [G: nat > rat] :
( ( groups2906978787729119204at_rat @ G @ bot_bot_set_nat )
= zero_zero_rat ) ).
% sum.empty
thf(fact_8545_sum_Oempty,axiom,
! [G: nat > int] :
( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
= zero_zero_int ) ).
% sum.empty
thf(fact_8546_sum_Oempty,axiom,
! [G: int > real] :
( ( groups8778361861064173332t_real @ G @ bot_bot_set_int )
= zero_zero_real ) ).
% sum.empty
thf(fact_8547_sum_Oempty,axiom,
! [G: int > rat] :
( ( groups3906332499630173760nt_rat @ G @ bot_bot_set_int )
= zero_zero_rat ) ).
% sum.empty
thf(fact_8548_sum_Oempty,axiom,
! [G: int > nat] :
( ( groups4541462559716669496nt_nat @ G @ bot_bot_set_int )
= zero_zero_nat ) ).
% sum.empty
thf(fact_8549_sum_Oempty,axiom,
! [G: nat > nat] :
( ( groups3542108847815614940at_nat @ G @ bot_bot_set_nat )
= zero_zero_nat ) ).
% sum.empty
thf(fact_8550_sum_Oempty,axiom,
! [G: complex > complex] :
( ( groups7754918857620584856omplex @ G @ bot_bot_set_complex )
= zero_zero_complex ) ).
% sum.empty
thf(fact_8551_sum_Oempty,axiom,
! [G: int > int] :
( ( groups4538972089207619220nt_int @ G @ bot_bot_set_int )
= zero_zero_int ) ).
% sum.empty
thf(fact_8552_sum_Oempty,axiom,
! [G: nat > real] :
( ( groups6591440286371151544t_real @ G @ bot_bot_set_nat )
= zero_zero_real ) ).
% sum.empty
thf(fact_8553_Ball__set__replicate,axiom,
! [N: nat,A: nat,P: nat > $o] :
( ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
=> ( P @ X4 ) ) )
= ( ( P @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_8554_Ball__set__replicate,axiom,
! [N: nat,A: int,P: int > $o] :
( ( ! [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
=> ( P @ X4 ) ) )
= ( ( P @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_8555_Ball__set__replicate,axiom,
! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
=> ( P @ X4 ) ) )
= ( ( P @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_8556_Bex__set__replicate,axiom,
! [N: nat,A: nat,P: nat > $o] :
( ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
& ( P @ X4 ) ) )
= ( ( P @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_8557_Bex__set__replicate,axiom,
! [N: nat,A: int,P: int > $o] :
( ( ? [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
& ( P @ X4 ) ) )
= ( ( P @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_8558_Bex__set__replicate,axiom,
! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ? [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
& ( P @ X4 ) ) )
= ( ( P @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_8559_in__set__replicate,axiom,
! [X: complex,N: nat,Y: complex] :
( ( member_complex @ X @ ( set_complex2 @ ( replicate_complex @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_8560_in__set__replicate,axiom,
! [X: real,N: nat,Y: real] :
( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_8561_in__set__replicate,axiom,
! [X: set_nat,N: nat,Y: set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_8562_in__set__replicate,axiom,
! [X: nat,N: nat,Y: nat] :
( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_8563_in__set__replicate,axiom,
! [X: int,N: nat,Y: int] :
( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_8564_in__set__replicate,axiom,
! [X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_8565_nth__replicate,axiom,
! [I: nat,N: nat,X: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_8566_nth__replicate,axiom,
! [I: nat,N: nat,X: int] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_int @ ( replicate_int @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_8567_nth__replicate,axiom,
! [I: nat,N: nat,X: vEBT_VEBT] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_8568_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_8569_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_8570_sum__constant,axiom,
! [Y: int,A2: set_complex] :
( ( groups5690904116761175830ex_int
@ ^ [X4: complex] : Y
@ A2 )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_complex @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_8571_sum__constant,axiom,
! [Y: int,A2: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [X4: nat] : Y
@ A2 )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_8572_sum__constant,axiom,
! [Y: real,A2: set_complex] :
( ( groups5808333547571424918x_real
@ ^ [X4: complex] : Y
@ A2 )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_8573_sum__constant,axiom,
! [Y: real,A2: set_int] :
( ( groups8778361861064173332t_real
@ ^ [X4: int] : Y
@ A2 )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_8574_sum__constant,axiom,
! [Y: nat,A2: set_complex] :
( ( groups5693394587270226106ex_nat
@ ^ [X4: complex] : Y
@ A2 )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_complex @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_8575_sum__constant,axiom,
! [Y: nat,A2: set_int] :
( ( groups4541462559716669496nt_nat
@ ^ [X4: int] : Y
@ A2 )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_8576_sum__constant,axiom,
! [Y: code_integer,A2: set_complex] :
( ( groups6621422865394947399nteger
@ ^ [X4: complex] : Y
@ A2 )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_complex @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_8577_sum__constant,axiom,
! [Y: code_integer,A2: set_nat] :
( ( groups7501900531339628137nteger
@ ^ [X4: nat] : Y
@ A2 )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_nat @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_8578_sum__constant,axiom,
! [Y: code_integer,A2: set_int] :
( ( groups7873554091576472773nteger
@ ^ [X4: int] : Y
@ A2 )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_int @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_8579_sum__constant,axiom,
! [Y: rat,A2: set_complex] :
( ( groups5058264527183730370ex_rat
@ ^ [X4: complex] : Y
@ A2 )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_8580_set__replicate,axiom,
! [N: nat,X: vEBT_VEBT] :
( ( N != zero_zero_nat )
=> ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
= ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).
% set_replicate
thf(fact_8581_set__replicate,axiom,
! [N: nat,X: nat] :
( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% set_replicate
thf(fact_8582_set__replicate,axiom,
! [N: nat,X: int] :
( ( N != zero_zero_nat )
=> ( ( set_int2 @ ( replicate_int @ N @ X ) )
= ( insert_int @ X @ bot_bot_set_int ) ) ) ).
% set_replicate
thf(fact_8583_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_8584_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_8585_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > rat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_rat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_8586_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > int] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_8587_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_8588_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_8589_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_8590_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_8591_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_8592_sums__diff,axiom,
! [F: nat > real,A: real,G: nat > real,B: real] :
( ( sums_real @ F @ A )
=> ( ( sums_real @ G @ B )
=> ( sums_real
@ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( G @ N4 ) )
@ ( minus_minus_real @ A @ B ) ) ) ) ).
% sums_diff
thf(fact_8593_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_8594_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_8595_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_8596_sums__0,axiom,
! [F: nat > real] :
( ! [N2: nat] :
( ( F @ N2 )
= zero_zero_real )
=> ( sums_real @ F @ zero_zero_real ) ) ).
% sums_0
thf(fact_8597_sums__0,axiom,
! [F: nat > nat] :
( ! [N2: nat] :
( ( F @ N2 )
= zero_zero_nat )
=> ( sums_nat @ F @ zero_zero_nat ) ) ).
% sums_0
thf(fact_8598_sums__0,axiom,
! [F: nat > int] :
( ! [N2: nat] :
( ( F @ N2 )
= zero_zero_int )
=> ( sums_int @ F @ zero_zero_int ) ) ).
% sums_0
thf(fact_8599_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > real,A2: set_complex] :
( ( ( groups5808333547571424918x_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A4: complex] :
( ( member_complex @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8600_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A2: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8601_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > real,A2: set_int] :
( ( ( groups8778361861064173332t_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A4: int] :
( ( member_int @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8602_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > rat,A2: set_complex] :
( ( ( groups5058264527183730370ex_rat @ G @ A2 )
!= zero_zero_rat )
=> ~ ! [A4: complex] :
( ( member_complex @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8603_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > rat,A2: set_real] :
( ( ( groups1300246762558778688al_rat @ G @ A2 )
!= zero_zero_rat )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8604_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > rat,A2: set_nat] :
( ( ( groups2906978787729119204at_rat @ G @ A2 )
!= zero_zero_rat )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8605_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > rat,A2: set_int] :
( ( ( groups3906332499630173760nt_rat @ G @ A2 )
!= zero_zero_rat )
=> ~ ! [A4: int] :
( ( member_int @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8606_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > nat,A2: set_complex] :
( ( ( groups5693394587270226106ex_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A4: complex] :
( ( member_complex @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8607_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A2: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8608_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > nat,A2: set_int] :
( ( ( groups4541462559716669496nt_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A4: int] :
( ( member_int @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_8609_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_8610_sum_Oneutral,axiom,
! [A2: set_complex,G: complex > complex] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_complex ) )
=> ( ( groups7754918857620584856omplex @ G @ A2 )
= zero_zero_complex ) ) ).
% sum.neutral
thf(fact_8611_sum_Oneutral,axiom,
! [A2: set_int,G: int > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_int ) )
=> ( ( groups4538972089207619220nt_int @ G @ A2 )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_8612_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_8613_sums__minus,axiom,
! [F: nat > real,A: real] :
( ( sums_real @ F @ A )
=> ( sums_real
@ ^ [N4: nat] : ( uminus_uminus_real @ ( F @ N4 ) )
@ ( uminus_uminus_real @ A ) ) ) ).
% sums_minus
thf(fact_8614_sums__minus,axiom,
! [F: nat > complex,A: complex] :
( ( sums_complex @ F @ A )
=> ( sums_complex
@ ^ [N4: nat] : ( uminus1482373934393186551omplex @ ( F @ N4 ) )
@ ( uminus1482373934393186551omplex @ A ) ) ) ).
% sums_minus
thf(fact_8615_sum_Odistrib,axiom,
! [G: nat > nat,H: nat > nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( plus_plus_nat @ ( G @ X4 ) @ ( H @ X4 ) )
@ A2 )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H @ A2 ) ) ) ).
% sum.distrib
thf(fact_8616_sum_Odistrib,axiom,
! [G: complex > complex,H: complex > complex,A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X4: complex] : ( plus_plus_complex @ ( G @ X4 ) @ ( H @ X4 ) )
@ A2 )
= ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H @ A2 ) ) ) ).
% sum.distrib
thf(fact_8617_sum_Odistrib,axiom,
! [G: int > int,H: int > int,A2: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [X4: int] : ( plus_plus_int @ ( G @ X4 ) @ ( H @ X4 ) )
@ A2 )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H @ A2 ) ) ) ).
% sum.distrib
thf(fact_8618_sum_Odistrib,axiom,
! [G: nat > real,H: nat > real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( plus_plus_real @ ( G @ X4 ) @ ( H @ X4 ) )
@ A2 )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H @ A2 ) ) ) ).
% sum.distrib
thf(fact_8619_sum__subtractf,axiom,
! [F: complex > complex,G: complex > complex,A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X4: complex] : ( minus_minus_complex @ ( F @ X4 ) @ ( G @ X4 ) )
@ A2 )
= ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ A2 ) ) ) ).
% sum_subtractf
thf(fact_8620_sum__subtractf,axiom,
! [F: int > int,G: int > int,A2: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [X4: int] : ( minus_minus_int @ ( F @ X4 ) @ ( G @ X4 ) )
@ A2 )
= ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ).
% sum_subtractf
thf(fact_8621_sum__subtractf,axiom,
! [F: nat > real,G: nat > real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( minus_minus_real @ ( F @ X4 ) @ ( G @ X4 ) )
@ A2 )
= ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ).
% sum_subtractf
thf(fact_8622_sum__negf,axiom,
! [F: complex > complex,A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X4: complex] : ( uminus1482373934393186551omplex @ ( F @ X4 ) )
@ A2 )
= ( uminus1482373934393186551omplex @ ( groups7754918857620584856omplex @ F @ A2 ) ) ) ).
% sum_negf
thf(fact_8623_sum__negf,axiom,
! [F: int > int,A2: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [X4: int] : ( uminus_uminus_int @ ( F @ X4 ) )
@ A2 )
= ( uminus_uminus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) ) ) ).
% sum_negf
thf(fact_8624_sum__negf,axiom,
! [F: nat > real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( uminus_uminus_real @ ( F @ X4 ) )
@ A2 )
= ( uminus_uminus_real @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).
% sum_negf
thf(fact_8625_sum__nonneg,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8626_sum__nonneg,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8627_sum__nonneg,axiom,
! [A2: set_int,F: int > real] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8628_sum__nonneg,axiom,
! [A2: set_complex,F: complex > rat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8629_sum__nonneg,axiom,
! [A2: set_real,F: real > rat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8630_sum__nonneg,axiom,
! [A2: set_nat,F: nat > rat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8631_sum__nonneg,axiom,
! [A2: set_int,F: int > rat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8632_sum__nonneg,axiom,
! [A2: set_complex,F: complex > nat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8633_sum__nonneg,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8634_sum__nonneg,axiom,
! [A2: set_int,F: int > nat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_8635_sum__nonpos,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_8636_sum__nonpos,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_8637_sum__nonpos,axiom,
! [A2: set_int,F: int > real] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_8638_sum__nonpos,axiom,
! [A2: set_complex,F: complex > rat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_8639_sum__nonpos,axiom,
! [A2: set_real,F: real > rat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_8640_sum__nonpos,axiom,
! [A2: set_nat,F: nat > rat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_8641_sum__nonpos,axiom,
! [A2: set_int,F: int > rat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_8642_sum__nonpos,axiom,
! [A2: set_complex,F: complex > nat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_8643_sum__nonpos,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_8644_sum__nonpos,axiom,
! [A2: set_int,F: int > nat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_8645_sum__cong__Suc,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ ( suc @ X3 ) @ A2 )
=> ( ( F @ ( suc @ X3 ) )
= ( G @ ( suc @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% sum_cong_Suc
thf(fact_8646_sum__cong__Suc,axiom,
! [A2: set_nat,F: nat > real,G: nat > real] :
( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ ( suc @ X3 ) @ A2 )
=> ( ( F @ ( suc @ X3 ) )
= ( G @ ( suc @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ F @ A2 )
= ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).
% sum_cong_Suc
thf(fact_8647_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_8648_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_8649_replicate__eqI,axiom,
! [Xs: list_complex,N: nat,X: complex] :
( ( ( size_s3451745648224563538omplex @ Xs )
= N )
=> ( ! [Y3: complex] :
( ( member_complex @ Y3 @ ( set_complex2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_complex @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_8650_replicate__eqI,axiom,
! [Xs: list_real,N: nat,X: real] :
( ( ( size_size_list_real @ Xs )
= N )
=> ( ! [Y3: real] :
( ( member_real @ Y3 @ ( set_real2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_real @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_8651_replicate__eqI,axiom,
! [Xs: list_set_nat,N: nat,X: set_nat] :
( ( ( size_s3254054031482475050et_nat @ Xs )
= N )
=> ( ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ ( set_set_nat2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_set_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_8652_replicate__eqI,axiom,
! [Xs: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= N )
=> ( ! [Y3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_VEBT_VEBT @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_8653_replicate__eqI,axiom,
! [Xs: list_o,N: nat,X: $o] :
( ( ( size_size_list_o @ Xs )
= N )
=> ( ! [Y3: $o] :
( ( member_o @ Y3 @ ( set_o2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_o @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_8654_replicate__eqI,axiom,
! [Xs: list_nat,N: nat,X: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_8655_replicate__eqI,axiom,
! [Xs: list_int,N: nat,X: int] :
( ( ( size_size_list_int @ Xs )
= N )
=> ( ! [Y3: int] :
( ( member_int @ Y3 @ ( set_int2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_int @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_8656_replicate__length__same,axiom,
! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( X3 = X ) )
=> ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_8657_replicate__length__same,axiom,
! [Xs: list_o,X: $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ ( set_o2 @ Xs ) )
=> ( X3 = X ) )
=> ( ( replicate_o @ ( size_size_list_o @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_8658_replicate__length__same,axiom,
! [Xs: list_nat,X: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( X3 = X ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_8659_replicate__length__same,axiom,
! [Xs: list_int,X: int] :
( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( X3 = X ) )
=> ( ( replicate_int @ ( size_size_list_int @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_8660_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_8661_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > real,M: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_8662_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > nat,M: nat,K: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_8663_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > real,M: nat,K: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_8664_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_8665_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_8666_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_8667_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_8668_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_8669_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_8670_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_8671_sums__zero__iff__shift,axiom,
! [N: nat,F: nat > real,S: real] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( F @ I2 )
= zero_zero_real ) )
=> ( ( sums_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N ) )
@ S )
= ( sums_real @ F @ S ) ) ) ).
% sums_zero_iff_shift
thf(fact_8672_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_8673_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_8674_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_8675_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_8676_sum_OatLeastAtMost__rev,axiom,
! [G: nat > nat,N: nat,M: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_8677_sum_OatLeastAtMost__rev,axiom,
! [G: nat > real,N: nat,M: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_8678_sum__bounded__above,axiom,
! [A2: set_real,F: real > real,K5: real] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ K5 ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A2 ) ) @ K5 ) ) ) ).
% sum_bounded_above
thf(fact_8679_sum__bounded__above,axiom,
! [A2: set_complex,F: complex > real,K5: real] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ K5 ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A2 ) ) @ K5 ) ) ) ).
% sum_bounded_above
thf(fact_8680_sum__bounded__above,axiom,
! [A2: set_int,F: int > real,K5: real] :
( ! [I2: int] :
( ( member_int @ I2 @ A2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ K5 ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A2 ) ) @ K5 ) ) ) ).
% sum_bounded_above
thf(fact_8681_sum__bounded__above,axiom,
! [A2: set_real,F: real > code_integer,K5: code_integer] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ K5 ) )
=> ( ord_le3102999989581377725nteger @ ( groups7713935264441627589nteger @ F @ A2 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_real @ A2 ) ) @ K5 ) ) ) ).
% sum_bounded_above
thf(fact_8682_sum__bounded__above,axiom,
! [A2: set_complex,F: complex > code_integer,K5: code_integer] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ K5 ) )
=> ( ord_le3102999989581377725nteger @ ( groups6621422865394947399nteger @ F @ A2 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_complex @ A2 ) ) @ K5 ) ) ) ).
% sum_bounded_above
thf(fact_8683_sum__bounded__above,axiom,
! [A2: set_nat,F: nat > code_integer,K5: code_integer] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ K5 ) )
=> ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ A2 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_nat @ A2 ) ) @ K5 ) ) ) ).
% sum_bounded_above
thf(fact_8684_sum__bounded__above,axiom,
! [A2: set_int,F: int > code_integer,K5: code_integer] :
( ! [I2: int] :
( ( member_int @ I2 @ A2 )
=> ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ K5 ) )
=> ( ord_le3102999989581377725nteger @ ( groups7873554091576472773nteger @ F @ A2 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_int @ A2 ) ) @ K5 ) ) ) ).
% sum_bounded_above
thf(fact_8685_sum__bounded__above,axiom,
! [A2: set_real,F: real > rat,K5: rat] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ K5 ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A2 ) ) @ K5 ) ) ) ).
% sum_bounded_above
thf(fact_8686_sum__bounded__above,axiom,
! [A2: set_complex,F: complex > rat,K5: rat] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ K5 ) )
=> ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A2 ) ) @ K5 ) ) ) ).
% sum_bounded_above
thf(fact_8687_sum__bounded__above,axiom,
! [A2: set_nat,F: nat > rat,K5: rat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ K5 ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A2 ) ) @ K5 ) ) ) ).
% sum_bounded_above
thf(fact_8688_sum__bounded__below,axiom,
! [A2: set_real,K5: real,F: real > real] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_eq_real @ K5 @ ( F @ I2 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A2 ) ) @ K5 ) @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_8689_sum__bounded__below,axiom,
! [A2: set_complex,K5: real,F: complex > real] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ord_less_eq_real @ K5 @ ( F @ I2 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A2 ) ) @ K5 ) @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_8690_sum__bounded__below,axiom,
! [A2: set_int,K5: real,F: int > real] :
( ! [I2: int] :
( ( member_int @ I2 @ A2 )
=> ( ord_less_eq_real @ K5 @ ( F @ I2 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A2 ) ) @ K5 ) @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_8691_sum__bounded__below,axiom,
! [A2: set_real,K5: code_integer,F: real > code_integer] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_le3102999989581377725nteger @ K5 @ ( F @ I2 ) ) )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_real @ A2 ) ) @ K5 ) @ ( groups7713935264441627589nteger @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_8692_sum__bounded__below,axiom,
! [A2: set_complex,K5: code_integer,F: complex > code_integer] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ord_le3102999989581377725nteger @ K5 @ ( F @ I2 ) ) )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_complex @ A2 ) ) @ K5 ) @ ( groups6621422865394947399nteger @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_8693_sum__bounded__below,axiom,
! [A2: set_nat,K5: code_integer,F: nat > code_integer] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_le3102999989581377725nteger @ K5 @ ( F @ I2 ) ) )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_nat @ A2 ) ) @ K5 ) @ ( groups7501900531339628137nteger @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_8694_sum__bounded__below,axiom,
! [A2: set_int,K5: code_integer,F: int > code_integer] :
( ! [I2: int] :
( ( member_int @ I2 @ A2 )
=> ( ord_le3102999989581377725nteger @ K5 @ ( F @ I2 ) ) )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_int @ A2 ) ) @ K5 ) @ ( groups7873554091576472773nteger @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_8695_sum__bounded__below,axiom,
! [A2: set_real,K5: rat,F: real > rat] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_eq_rat @ K5 @ ( F @ I2 ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A2 ) ) @ K5 ) @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_8696_sum__bounded__below,axiom,
! [A2: set_complex,K5: rat,F: complex > rat] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ord_less_eq_rat @ K5 @ ( F @ I2 ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A2 ) ) @ K5 ) @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_8697_sum__bounded__below,axiom,
! [A2: set_nat,K5: rat,F: nat > rat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_rat @ K5 @ ( F @ I2 ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A2 ) ) @ K5 ) @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).
% sum_bounded_below
thf(fact_8698_powser__sums__if,axiom,
! [M: nat,Z2: complex] :
( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ ( if_complex @ ( N4 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z2 @ N4 ) )
@ ( power_power_complex @ Z2 @ M ) ) ).
% powser_sums_if
thf(fact_8699_powser__sums__if,axiom,
! [M: nat,Z2: real] :
( sums_real
@ ^ [N4: nat] : ( times_times_real @ ( if_real @ ( N4 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z2 @ N4 ) )
@ ( power_power_real @ Z2 @ M ) ) ).
% powser_sums_if
thf(fact_8700_powser__sums__if,axiom,
! [M: nat,Z2: int] :
( sums_int
@ ^ [N4: nat] : ( times_times_int @ ( if_int @ ( N4 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z2 @ N4 ) )
@ ( power_power_int @ Z2 @ M ) ) ).
% powser_sums_if
thf(fact_8701_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_8702_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_8703_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_8704_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_8705_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_8706_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_8707_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_8708_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_8709_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_8710_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_8711_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_8712_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_8713_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_8714_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_8715_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_8716_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_8717_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_8718_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_8719_set__replicate__Suc,axiom,
! [N: nat,X: vEBT_VEBT] :
( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N ) @ X ) )
= ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ).
% set_replicate_Suc
thf(fact_8720_set__replicate__Suc,axiom,
! [N: nat,X: nat] :
( ( set_nat2 @ ( replicate_nat @ ( suc @ N ) @ X ) )
= ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% set_replicate_Suc
thf(fact_8721_set__replicate__Suc,axiom,
! [N: nat,X: int] :
( ( set_int2 @ ( replicate_int @ ( suc @ N ) @ X ) )
= ( insert_int @ X @ bot_bot_set_int ) ) ).
% set_replicate_Suc
thf(fact_8722_set__replicate__conv__if,axiom,
! [N: nat,X: vEBT_VEBT] :
( ( ( N = zero_zero_nat )
=> ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
= bot_bo8194388402131092736T_VEBT ) )
& ( ( N != zero_zero_nat )
=> ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
= ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).
% set_replicate_conv_if
thf(fact_8723_set__replicate__conv__if,axiom,
! [N: nat,X: nat] :
( ( ( N = zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
= bot_bot_set_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% set_replicate_conv_if
thf(fact_8724_set__replicate__conv__if,axiom,
! [N: nat,X: int] :
( ( ( N = zero_zero_nat )
=> ( ( set_int2 @ ( replicate_int @ N @ X ) )
= bot_bot_set_int ) )
& ( ( N != zero_zero_nat )
=> ( ( set_int2 @ ( replicate_int @ N @ X ) )
= ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% set_replicate_conv_if
thf(fact_8725_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ M )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_8726_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ M )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_8727_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ M )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_8728_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ M )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_8729_sum__Suc__diff,axiom,
! [M: nat,N: nat,F: nat > rat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% sum_Suc_diff
thf(fact_8730_sum__Suc__diff,axiom,
! [M: nat,N: nat,F: nat > int] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% sum_Suc_diff
thf(fact_8731_sum__Suc__diff,axiom,
! [M: nat,N: nat,F: nat > real] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( minus_minus_real @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% sum_Suc_diff
thf(fact_8732_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_8733_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_8734_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_8735_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_8736_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > rat,P5: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P5 ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P5 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_8737_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > int,P5: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P5 ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P5 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_8738_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > nat,P5: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P5 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P5 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_8739_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > real,P5: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P5 ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P5 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_8740_sum__bounded__above__strict,axiom,
! [A2: set_real,F: real > int,K5: int] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_int @ ( F @ I2 ) @ K5 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A2 ) )
=> ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_real @ A2 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_8741_sum__bounded__above__strict,axiom,
! [A2: set_complex,F: complex > int,K5: int] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ord_less_int @ ( F @ I2 ) @ K5 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A2 ) )
=> ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_complex @ A2 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_8742_sum__bounded__above__strict,axiom,
! [A2: set_nat,F: nat > int,K5: int] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_int @ ( F @ I2 ) @ K5 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A2 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_8743_sum__bounded__above__strict,axiom,
! [A2: set_real,F: real > real,K5: real] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_real @ ( F @ I2 ) @ K5 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A2 ) )
=> ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A2 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_8744_sum__bounded__above__strict,axiom,
! [A2: set_complex,F: complex > real,K5: real] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ord_less_real @ ( F @ I2 ) @ K5 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A2 ) )
=> ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A2 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_8745_sum__bounded__above__strict,axiom,
! [A2: set_int,F: int > real,K5: real] :
( ! [I2: int] :
( ( member_int @ I2 @ A2 )
=> ( ord_less_real @ ( F @ I2 ) @ K5 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A2 ) )
=> ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A2 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_8746_sum__bounded__above__strict,axiom,
! [A2: set_real,F: real > nat,K5: nat] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_nat @ ( F @ I2 ) @ K5 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A2 ) )
=> ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_real @ A2 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_8747_sum__bounded__above__strict,axiom,
! [A2: set_complex,F: complex > nat,K5: nat] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ord_less_nat @ ( F @ I2 ) @ K5 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A2 ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_complex @ A2 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_8748_sum__bounded__above__strict,axiom,
! [A2: set_int,F: int > nat,K5: nat] :
( ! [I2: int] :
( ( member_int @ I2 @ A2 )
=> ( ord_less_nat @ ( F @ I2 ) @ K5 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A2 ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A2 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_8749_sum__bounded__above__strict,axiom,
! [A2: set_real,F: real > code_integer,K5: code_integer] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_le6747313008572928689nteger @ ( F @ I2 ) @ K5 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A2 ) )
=> ( ord_le6747313008572928689nteger @ ( groups7713935264441627589nteger @ F @ A2 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_real @ A2 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_8750_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_8751_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_8752_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_8753_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_8754_convex__sum__bound__le,axiom,
! [I5: set_complex,X: complex > real,A: complex > real,B: real,Delta: real] :
( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
=> ( ( ( groups5808333547571424918x_real @ X @ I5 )
= one_one_real )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I5 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real
@ ( groups5808333547571424918x_real
@ ^ [I3: complex] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_8755_convex__sum__bound__le,axiom,
! [I5: set_real,X: real > real,A: real > real,B: real,Delta: real] :
( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
=> ( ( ( groups8097168146408367636l_real @ X @ I5 )
= one_one_real )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I5 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real
@ ( groups8097168146408367636l_real
@ ^ [I3: real] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_8756_convex__sum__bound__le,axiom,
! [I5: set_int,X: int > real,A: int > real,B: real,Delta: real] :
( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
=> ( ( ( groups8778361861064173332t_real @ X @ I5 )
= one_one_real )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I5 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real
@ ( groups8778361861064173332t_real
@ ^ [I3: int] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_8757_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_8758_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_8759_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_8760_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F: nat > rat] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_rat ) ) ) ).
% sum_natinterval_diff
thf(fact_8761_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F: nat > int] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_int ) ) ) ).
% sum_natinterval_diff
thf(fact_8762_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F: nat > real] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_real ) ) ) ).
% sum_natinterval_diff
thf(fact_8763_sum__telescope_H_H,axiom,
! [M: nat,N: nat,F: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% sum_telescope''
thf(fact_8764_sum__telescope_H_H,axiom,
! [M: nat,N: nat,F: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% sum_telescope''
thf(fact_8765_sum__telescope_H_H,axiom,
! [M: nat,N: nat,F: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% sum_telescope''
thf(fact_8766_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_8767_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_8768_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_8769_mask__eq__sum__exp,axiom,
! [N: nat] :
( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
= ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_8770_mask__eq__sum__exp,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_8771_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: complex] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
= ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_8772_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
= ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_8773_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
= ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_8774_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
= ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_8775_sum_Oin__pairs,axiom,
! [G: nat > rat,M: nat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups2906978787729119204at_rat
@ ^ [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 @ N ) ) ) ).
% sum.in_pairs
thf(fact_8776_sum_Oin__pairs,axiom,
! [G: nat > int,M: nat,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [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 @ N ) ) ) ).
% sum.in_pairs
thf(fact_8777_sum_Oin__pairs,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [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 @ N ) ) ) ).
% sum.in_pairs
thf(fact_8778_sum_Oin__pairs,axiom,
! [G: nat > real,M: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [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 @ N ) ) ) ).
% sum.in_pairs
thf(fact_8779_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_8780_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_8781_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_8782_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X4: nat] : X4
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_nat
thf(fact_8783_gbinomial__sum__up__index,axiom,
! [K: nat,N: nat] :
( ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% gbinomial_sum_up_index
thf(fact_8784_gbinomial__sum__up__index,axiom,
! [K: nat,N: nat] :
( ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% gbinomial_sum_up_index
thf(fact_8785_gbinomial__sum__up__index,axiom,
! [K: nat,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% gbinomial_sum_up_index
thf(fact_8786_double__arith__series,axiom,
! [A: complex,D: complex,N: 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 @ N ) ) )
= ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_8787_double__arith__series,axiom,
! [A: int,D: int,N: 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 @ N ) ) )
= ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_8788_double__arith__series,axiom,
! [A: code_integer,D: code_integer,N: nat] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
@ ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_8789_double__arith__series,axiom,
! [A: rat,D: rat,N: 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 @ N ) ) )
= ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_8790_double__arith__series,axiom,
! [A: nat,D: nat,N: 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 @ N ) ) )
= ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_8791_double__arith__series,axiom,
! [A: real,D: real,N: 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 @ N ) ) )
= ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_8792_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% double_gauss_sum
thf(fact_8793_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% double_gauss_sum
thf(fact_8794_double__gauss__sum,axiom,
! [N: nat] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) ) ).
% double_gauss_sum
thf(fact_8795_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% double_gauss_sum
thf(fact_8796_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% double_gauss_sum
thf(fact_8797_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% double_gauss_sum
thf(fact_8798_arith__series__nat,axiom,
! [A: nat,D: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I3 @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% arith_series_nat
thf(fact_8799_Sum__Icc__nat,axiom,
! [M: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X4: nat] : X4
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Sum_Icc_nat
thf(fact_8800_arith__series,axiom,
! [A: int,D: int,N: nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_8801_arith__series,axiom,
! [A: code_integer,D: code_integer,N: nat] :
( ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_8802_arith__series,axiom,
! [A: nat,D: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_8803_gauss__sum,axiom,
! [N: nat] :
( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_8804_gauss__sum,axiom,
! [N: nat] :
( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_8805_gauss__sum,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_8806_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_8807_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_8808_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_8809_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_8810_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_8811_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_8812_sum__gp__offset,axiom,
! [X: complex,M: nat,N: nat] :
( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% sum_gp_offset
thf(fact_8813_sum__gp__offset,axiom,
! [X: rat,M: nat,N: nat] :
( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% sum_gp_offset
thf(fact_8814_sum__gp__offset,axiom,
! [X: real,M: nat,N: nat] :
( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% sum_gp_offset
thf(fact_8815_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_8816_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_8817_gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_8818_gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_8819_gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_8820_gchoose__row__sum__weighted,axiom,
! [R2: complex,M: nat] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ R2 @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
= ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R2 @ ( suc @ M ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_8821_gchoose__row__sum__weighted,axiom,
! [R2: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R2 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
= ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R2 @ ( suc @ M ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_8822_gchoose__row__sum__weighted,axiom,
! [R2: real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
= ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_8823_vebt__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ( vEBT_vebt_member @ X @ Xa3 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa3 ) )
=> ~ ( ( ( Xa3 = zero_zero_nat )
=> A4 )
& ( ( Xa3 != zero_zero_nat )
=> ( ( ( Xa3 = one_one_nat )
=> B4 )
& ( Xa3 = one_one_nat ) ) ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa3 ) )
=> ~ ( ( Xa3 != Mi )
=> ( ( Xa3 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa3 @ Mi )
& ( ~ ( ord_less_nat @ Xa3 @ Mi )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_8824_geometric__deriv__sums,axiom,
! [Z2: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ one_one_real )
=> ( sums_real
@ ^ [N4: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) @ ( power_power_real @ Z2 @ N4 ) )
@ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_8825_geometric__deriv__sums,axiom,
! [Z2: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ one_one_real )
=> ( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N4 ) ) @ ( power_power_complex @ Z2 @ N4 ) )
@ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_8826_vebt__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa3 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( ( ( Xa3 = zero_zero_nat )
=> A4 )
& ( ( Xa3 != zero_zero_nat )
=> ( ( ( Xa3 = one_one_nat )
=> B4 )
& ( Xa3 = one_one_nat ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa3 ) ) ) )
=> ( ! [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 ) @ Xa3 ) ) ) )
=> ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa3 ) ) ) )
=> ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V3 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa3 ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ( ( Y
= ( ( Xa3 != Mi )
=> ( ( Xa3 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa3 @ Mi )
& ( ~ ( ord_less_nat @ Xa3 @ Mi )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_8827_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa3 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa3 ) )
=> ( ( ( Xa3 = zero_zero_nat )
=> A4 )
& ( ( Xa3 != zero_zero_nat )
=> ( ( ( Xa3 = one_one_nat )
=> B4 )
& ( Xa3 = 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 ) @ Xa3 ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList @ S2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList @ S2 ) @ Xa3 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_8828_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa3 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa3 ) )
=> ~ ( ( ( Xa3 = zero_zero_nat )
=> A4 )
& ( ( Xa3 != zero_zero_nat )
=> ( ( ( Xa3 = one_one_nat )
=> B4 )
& ( Xa3 = one_one_nat ) ) ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList @ S2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList @ S2 ) @ Xa3 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_8829_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa3 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( ( ( Xa3 = zero_zero_nat )
=> A4 )
& ( ( Xa3 != zero_zero_nat )
=> ( ( ( Xa3 = one_one_nat )
=> B4 )
& ( Xa3 = one_one_nat ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa3 ) ) ) )
=> ( ! [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 ) @ Xa3 ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList @ S2 ) )
=> ( ( Y
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList @ S2 ) @ Xa3 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_8830_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa3 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa3 ) ) )
=> ( ! [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 ) @ Xa3 ) ) )
=> ( ! [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 ) @ Xa3 ) )
=> ( ( Xa3 = Mi )
| ( Xa3 = Ma2 ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList @ Vc2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList @ Vc2 ) @ Xa3 ) )
=> ( ( Xa3 = Mi )
| ( Xa3 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList @ Vd ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList @ Vd ) @ Xa3 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_8831_int__sum,axiom,
! [F: int > nat,A2: set_int] :
( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
= ( groups4538972089207619220nt_int
@ ^ [X4: int] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
@ A2 ) ) ).
% int_sum
thf(fact_8832_int__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3539618377306564664at_int
@ ^ [X4: nat] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
@ A2 ) ) ).
% int_sum
thf(fact_8833_sum__subtractf__nat,axiom,
! [A2: set_complex,G: complex > nat,F: complex > nat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups5693394587270226106ex_nat
@ ^ [X4: complex] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A2 )
= ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_8834_sum__subtractf__nat,axiom,
! [A2: set_real,G: real > nat,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups1935376822645274424al_nat
@ ^ [X4: real] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A2 )
= ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_8835_sum__subtractf__nat,axiom,
! [A2: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups8294997508430121362at_nat
@ ^ [X4: set_nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A2 )
= ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( groups8294997508430121362at_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_8836_sum__subtractf__nat,axiom,
! [A2: set_int,G: int > nat,F: int > nat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X4: int] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A2 )
= ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_8837_sum__subtractf__nat,axiom,
! [A2: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A2 )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_8838_card__eq__sum,axiom,
( finite_card_complex
= ( groups5693394587270226106ex_nat
@ ^ [X4: complex] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_8839_card__eq__sum,axiom,
( finite_card_list_nat
= ( groups4396056296759096172at_nat
@ ^ [X4: list_nat] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_8840_card__eq__sum,axiom,
( finite_card_set_nat
= ( groups8294997508430121362at_nat
@ ^ [X4: set_nat] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_8841_card__eq__sum,axiom,
( finite_card_int
= ( groups4541462559716669496nt_nat
@ ^ [X4: int] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_8842_card__eq__sum,axiom,
( finite_card_nat
= ( groups3542108847815614940at_nat
@ ^ [X4: nat] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_8843_sum__SucD,axiom,
! [F: nat > nat,A2: set_nat,N: nat] :
( ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( suc @ N ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ).
% sum_SucD
thf(fact_8844_sum__Suc,axiom,
! [F: complex > nat,A2: set_complex] :
( ( groups5693394587270226106ex_nat
@ ^ [X4: complex] : ( suc @ ( F @ X4 ) )
@ A2 )
= ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( finite_card_complex @ A2 ) ) ) ).
% sum_Suc
thf(fact_8845_sum__Suc,axiom,
! [F: list_nat > nat,A2: set_list_nat] :
( ( groups4396056296759096172at_nat
@ ^ [X4: list_nat] : ( suc @ ( F @ X4 ) )
@ A2 )
= ( plus_plus_nat @ ( groups4396056296759096172at_nat @ F @ A2 ) @ ( finite_card_list_nat @ A2 ) ) ) ).
% sum_Suc
thf(fact_8846_sum__Suc,axiom,
! [F: set_nat > nat,A2: set_set_nat] :
( ( groups8294997508430121362at_nat
@ ^ [X4: set_nat] : ( suc @ ( F @ X4 ) )
@ A2 )
= ( plus_plus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( finite_card_set_nat @ A2 ) ) ) ).
% sum_Suc
thf(fact_8847_sum__Suc,axiom,
! [F: int > nat,A2: set_int] :
( ( groups4541462559716669496nt_nat
@ ^ [X4: int] : ( suc @ ( F @ X4 ) )
@ A2 )
= ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( finite_card_int @ A2 ) ) ) ).
% sum_Suc
thf(fact_8848_sum__Suc,axiom,
! [F: nat > nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( suc @ ( F @ X4 ) )
@ A2 )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( finite_card_nat @ A2 ) ) ) ).
% sum_Suc
thf(fact_8849_real__of__card,axiom,
! [A2: set_complex] :
( ( semiri5074537144036343181t_real @ ( finite_card_complex @ A2 ) )
= ( groups5808333547571424918x_real
@ ^ [X4: complex] : one_one_real
@ A2 ) ) ).
% real_of_card
thf(fact_8850_real__of__card,axiom,
! [A2: set_list_nat] :
( ( semiri5074537144036343181t_real @ ( finite_card_list_nat @ A2 ) )
= ( groups8399112307953289288t_real
@ ^ [X4: list_nat] : one_one_real
@ A2 ) ) ).
% real_of_card
thf(fact_8851_real__of__card,axiom,
! [A2: set_set_nat] :
( ( semiri5074537144036343181t_real @ ( finite_card_set_nat @ A2 ) )
= ( groups5107569545109728110t_real
@ ^ [X4: set_nat] : one_one_real
@ A2 ) ) ).
% real_of_card
thf(fact_8852_real__of__card,axiom,
! [A2: set_int] :
( ( semiri5074537144036343181t_real @ ( finite_card_int @ A2 ) )
= ( groups8778361861064173332t_real
@ ^ [X4: int] : one_one_real
@ A2 ) ) ).
% real_of_card
thf(fact_8853_real__of__card,axiom,
! [A2: set_nat] :
( ( semiri5074537144036343181t_real @ ( finite_card_nat @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [X4: nat] : one_one_real
@ A2 ) ) ).
% real_of_card
thf(fact_8854_sum__diff1__nat,axiom,
! [A: set_nat,A2: set_set_nat,F: set_nat > nat] :
( ( ( member_set_nat @ A @ A2 )
=> ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_set_nat @ A @ A2 )
=> ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= ( groups8294997508430121362at_nat @ F @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_8855_sum__diff1__nat,axiom,
! [A: int,A2: set_int,F: int > nat] :
( ( ( member_int @ A @ A2 )
=> ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_int @ A @ A2 )
=> ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_8856_sum__diff1__nat,axiom,
! [A: nat,A2: set_nat,F: nat > nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_8857_sum__nth__roots,axiom,
! [N: nat,C: complex] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X4: complex] : X4
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= C ) ) )
= zero_zero_complex ) ) ).
% sum_nth_roots
thf(fact_8858_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X4: complex] : X4
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= one_one_complex ) ) )
= zero_zero_complex ) ) ).
% sum_roots_unity
thf(fact_8859_Sum__Icc__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ M @ N )
=> ( ( groups4538972089207619220nt_int
@ ^ [X4: int] : X4
@ ( set_or1266510415728281911st_int @ M @ N ) )
= ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_8860_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa3 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
=> ( ! [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 ) @ Xa3 ) )
=> ~ ( ( Xa3 = Mi )
| ( Xa3 = Ma2 ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList @ Vc2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList @ Vc2 ) @ Xa3 ) )
=> ~ ( ( Xa3 = Mi )
| ( Xa3 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList @ Vd ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList @ Vd ) @ Xa3 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_8861_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa3 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa3 ) ) ) )
=> ( ! [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 ) @ Xa3 ) ) ) )
=> ( ! [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
= ( ( Xa3 = Mi )
| ( Xa3 = 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 ) @ Xa3 ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList @ Vc2 ) )
=> ( ( Y
= ( ( Xa3 = Mi )
| ( Xa3 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList @ Vc2 ) @ Xa3 ) ) ) )
=> ~ ! [V3: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList @ Vd ) )
=> ( ( Y
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V3 ) @ TreeList @ Vd ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_8862_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ? [T3: real] :
( ( ord_less_real @ X @ T3 )
& ( ord_less_real @ T3 @ zero_zero_real )
& ( ( cos_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( cos_coeff @ M4 ) @ ( power_power_real @ X @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_8863_Maclaurin__cos__expansion2,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [T3: real] :
( ( ord_less_real @ zero_zero_real @ T3 )
& ( ord_less_real @ T3 @ X )
& ( ( cos_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( cos_coeff @ M4 ) @ ( power_power_real @ X @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_8864_Maclaurin__sin__expansion3,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ? [T3: real] :
( ( ord_less_real @ zero_zero_real @ T3 )
& ( ord_less_real @ T3 @ X )
& ( ( sin_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( sin_coeff @ M4 ) @ ( power_power_real @ X @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_8865_Maclaurin__sin__expansion4,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [T3: real] :
( ( ord_less_real @ zero_zero_real @ T3 )
& ( ord_less_eq_real @ T3 @ X )
& ( ( sin_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( sin_coeff @ M4 ) @ ( power_power_real @ X @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_8866_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_8867_sumr__cos__zero__one,axiom,
! [N: nat] :
( ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( cos_coeff @ M4 ) @ ( power_power_real @ zero_zero_real @ M4 ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= one_one_real ) ).
% sumr_cos_zero_one
thf(fact_8868_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_8869_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_8870_sum__split__even__odd,axiom,
! [F: nat > real,G: nat > real,N: 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 ) ) @ N ) ) )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum_split_even_odd
thf(fact_8871_Maclaurin__sin__bound,axiom,
! [X: real,N: nat] :
( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real @ ( sin_real @ X )
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( sin_coeff @ M4 ) @ ( power_power_real @ X @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) )
@ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).
% Maclaurin_sin_bound
thf(fact_8872_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_8873_Maclaurin__exp__lt,axiom,
! [X: real,N: nat] :
( ( X != zero_zero_real )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [T3: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
& ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
& ( ( exp_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( divide_divide_real @ ( power_power_real @ X @ M4 ) @ ( semiri2265585572941072030t_real @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_8874_Maclaurin__sin__expansion,axiom,
! [X: real,N: nat] :
? [T3: real] :
( ( sin_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( sin_coeff @ M4 ) @ ( power_power_real @ X @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ).
% Maclaurin_sin_expansion
thf(fact_8875_Maclaurin__sin__expansion2,axiom,
! [X: real,N: nat] :
? [T3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
& ( ( sin_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( sin_coeff @ M4 ) @ ( power_power_real @ X @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% Maclaurin_sin_expansion2
thf(fact_8876_Maclaurin__cos__expansion,axiom,
! [X: real,N: nat] :
? [T3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
& ( ( cos_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( cos_coeff @ M4 ) @ ( power_power_real @ X @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% Maclaurin_cos_expansion
thf(fact_8877_bij__betw__roots__unity,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( bij_betw_nat_complex
@ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
@ ( set_ord_lessThan_nat @ N )
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= one_one_complex ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_8878_card__atMost,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_8879_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_8880_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_8881_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_8882_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost_nat @ ( suc @ K ) )
= ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% atMost_Suc
thf(fact_8883_sum__choose__upper,axiom,
! [M: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( binomial @ K3 @ M )
@ ( set_ord_atMost_nat @ N ) )
= ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% sum_choose_upper
thf(fact_8884_sum__choose__lower,axiom,
! [R2: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
@ ( set_ord_atMost_nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_8885_choose__rising__sum_I2_J,axiom,
! [N: nat,M: nat] :
( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
@ ( set_ord_atMost_nat @ M ) )
= ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).
% choose_rising_sum(2)
thf(fact_8886_choose__rising__sum_I1_J,axiom,
! [N: nat,M: nat] :
( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
@ ( set_ord_atMost_nat @ M ) )
= ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% choose_rising_sum(1)
thf(fact_8887_diffs__cos__coeff,axiom,
( ( diffs_real @ cos_coeff )
= ( ^ [N4: nat] : ( uminus_uminus_real @ ( sin_coeff @ N4 ) ) ) ) ).
% diffs_cos_coeff
thf(fact_8888_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_8889_sum__choose__diagonal,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( binomial @ ( suc @ N ) @ M ) ) ) ).
% sum_choose_diagonal
thf(fact_8890_vandermonde,axiom,
! [M: nat,N: nat,R2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R2 ) )
= ( binomial @ ( plus_plus_nat @ M @ N ) @ R2 ) ) ).
% vandermonde
thf(fact_8891_binomial,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial
thf(fact_8892_polynomial__product__nat,axiom,
! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ M @ I2 )
=> ( ( A @ I2 )
= zero_zero_nat ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ 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 @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [R5: nat] :
( times_times_nat
@ ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_nat @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_8893_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_8894_choose__linear__sum,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( times_times_nat @ I3 @ ( binomial @ N @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% choose_linear_sum
thf(fact_8895_complex__sgn__def,axiom,
( sgn_sgn_complex
= ( ^ [X4: complex] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ X4 ) ) @ X4 ) ) ) ).
% complex_sgn_def
thf(fact_8896_Arg__def,axiom,
( arg
= ( ^ [Z3: complex] :
( if_real @ ( Z3 = zero_zero_complex ) @ zero_zero_real
@ ( fChoice_real
@ ^ [A3: real] :
( ( ( sgn_sgn_complex @ Z3 )
= ( cis @ A3 ) )
& ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A3 )
& ( ord_less_eq_real @ A3 @ pi ) ) ) ) ) ) ).
% Arg_def
thf(fact_8897_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% set_vebt_def
thf(fact_8898_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_8899_divmod__step__nat__def,axiom,
( unique5026877609467782581ep_nat
= ( ^ [L3: num] :
( produc2626176000494625587at_nat
@ ^ [Q6: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L3 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L3 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_8900_divmod__step__int__def,axiom,
( unique5024387138958732305ep_int
= ( ^ [L3: num] :
( produc4245557441103728435nt_int
@ ^ [Q6: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L3 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L3 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_8901_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M4: nat,N4: nat] :
( if_Pro6206227464963214023at_nat
@ ( ( N4 = zero_zero_nat )
| ( ord_less_nat @ M4 @ N4 ) )
@ ( product_Pair_nat_nat @ zero_zero_nat @ M4 )
@ ( produc2626176000494625587at_nat
@ ^ [Q6: nat] : ( product_Pair_nat_nat @ ( suc @ Q6 ) )
@ ( divmod_nat @ ( minus_minus_nat @ M4 @ N4 ) @ N4 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_8902_arctan__def,axiom,
( arctan
= ( ^ [Y5: real] :
( the_real
@ ^ [X4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
& ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X4 )
= Y5 ) ) ) ) ) ).
% arctan_def
thf(fact_8903_arcsin__def,axiom,
( arcsin
= ( ^ [Y5: real] :
( the_real
@ ^ [X4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
& ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ X4 )
= Y5 ) ) ) ) ) ).
% arcsin_def
thf(fact_8904_set__encode__def,axiom,
( nat_set_encode
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% set_encode_def
thf(fact_8905_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_8906_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_8907_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_8908_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_8909_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_8910_set__encode__inverse,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( nat_set_decode @ ( nat_set_encode @ A2 ) )
= A2 ) ) ).
% set_encode_inverse
thf(fact_8911_set__decode__inverse,axiom,
! [N: nat] :
( ( nat_set_encode @ ( nat_set_decode @ N ) )
= N ) ).
% set_decode_inverse
thf(fact_8912_set__encode__empty,axiom,
( ( nat_set_encode @ bot_bot_set_nat )
= zero_zero_nat ) ).
% set_encode_empty
thf(fact_8913_finite__nth__roots,axiom,
! [N: nat,C: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= C ) ) ) ) ).
% finite_nth_roots
thf(fact_8914_set__encode__insert,axiom,
! [A2: set_nat,N: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ N @ A2 )
=> ( ( nat_set_encode @ ( insert_nat @ N @ A2 ) )
= ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% set_encode_insert
thf(fact_8915_set__encode__inf,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( nat_set_encode @ A2 )
= zero_zero_nat ) ) ).
% set_encode_inf
thf(fact_8916_set__encode__eq,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( nat_set_encode @ A2 )
= ( nat_set_encode @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% set_encode_eq
thf(fact_8917_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N8: set_nat] :
? [M4: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N8 )
=> ( ord_less_nat @ X4 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_8918_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_8919_finite__set__decode,axiom,
! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).
% finite_set_decode
thf(fact_8920_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_8921_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_8922_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_8923_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_8924_subset__eq__atLeast0__atMost__finite,axiom,
! [N5: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_8925_Sum__Ico__nat,axiom,
! [M: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X4: nat] : X4
@ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Sum_Ico_nat
thf(fact_8926_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_8927_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_8928_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_8929_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
= ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% atLeastLessThan_singleton
thf(fact_8930_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( P @ M4 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less_eq
thf(fact_8931_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M4: nat] :
( ( ord_less_nat @ M4 @ N )
& ( P @ M4 ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_8932_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_8933_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_8934_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_8935_subset__eq__atLeast0__lessThan__finite,axiom,
! [N5: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_8936_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_8937_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_8938_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_8939_atLeastLessThanSuc,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
= ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThanSuc
thf(fact_8940_prod__Suc__fact,axiom,
! [N: nat] :
( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ).
% prod_Suc_fact
thf(fact_8941_prod__Suc__Suc__fact,axiom,
! [N: nat] :
( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ).
% prod_Suc_Suc_fact
thf(fact_8942_subset__eq__atLeast0__lessThan__card,axiom,
! [N5: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N5 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_8943_card__sum__le__nat__sum,axiom,
! [S3: set_nat] :
( ord_less_eq_nat
@ ( groups3542108847815614940at_nat
@ ^ [X4: nat] : X4
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S3 ) ) )
@ ( groups3542108847815614940at_nat
@ ^ [X4: nat] : X4
@ S3 ) ) ).
% card_sum_le_nat_sum
thf(fact_8944_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_8945_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_8946_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A: nat > nat,B: nat > nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I2 ) ) ) )
=> ( ord_less_eq_nat
@ ( times_times_nat @ N
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( B @ I3 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
@ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_8947_infinite__int__iff__unbounded,axiom,
! [S3: set_int] :
( ( ~ ( finite_finite_int @ S3 ) )
= ( ! [M4: int] :
? [N4: int] :
( ( ord_less_int @ M4 @ ( abs_abs_int @ N4 ) )
& ( member_int @ N4 @ S3 ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_8948_card__atLeastLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% card_atLeastLessThan_int
thf(fact_8949_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_8950_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_8951_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
= ( nat2 @ U ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_8952_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_8953_unbounded__k__infinite,axiom,
! [K: nat,S3: set_nat] :
( ! [M2: nat] :
( ( ord_less_nat @ K @ M2 )
=> ? [N6: nat] :
( ( ord_less_nat @ M2 @ N6 )
& ( member_nat @ N6 @ S3 ) ) )
=> ~ ( finite_finite_nat @ S3 ) ) ).
% unbounded_k_infinite
thf(fact_8954_infinite__nat__iff__unbounded,axiom,
! [S3: set_nat] :
( ( ~ ( finite_finite_nat @ S3 ) )
= ( ! [M4: nat] :
? [N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ( member_nat @ N4 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_8955_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D3: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z7: int,Z3: int] :
( ( ord_less_eq_int @ D3 @ Z3 )
& ( ord_less_int @ Z7 @ Z3 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_8956_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D3: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z7: int,Z3: int] :
( ( ord_less_eq_int @ D3 @ Z7 )
& ( ord_less_int @ Z7 @ Z3 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_8957_int__of__nat__def,axiom,
code_T6385005292777649522of_nat = semiri1314217659103216013at_int ).
% int_of_nat_def
thf(fact_8958_divmod__step__integer__def,axiom,
( unique4921790084139445826nteger
= ( ^ [L3: num] :
( produc6916734918728496179nteger
@ ^ [Q6: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L3 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L3 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_8959_or__int__rec,axiom,
( bit_se1409905431419307370or_int
= ( ^ [K3: int,L3: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
| ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% or_int_rec
thf(fact_8960_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_8961_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_8962_or__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(2)
thf(fact_8963_or__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(6)
thf(fact_8964_or__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% or_minus_minus_numerals
thf(fact_8965_and__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% and_minus_minus_numerals
thf(fact_8966_minus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
= ( uminus1351360451143612070nteger @ L ) ) ).
% minus_integer_code(2)
thf(fact_8967_sgn__integer__code,axiom,
( sgn_sgn_Code_integer
= ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% sgn_integer_code
thf(fact_8968_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_8969_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_8970_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_8971_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_8972_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_8973_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_8974_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_8975_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_8976_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_8977_set__bit__int__def,axiom,
( bit_se7879613467334960850it_int
= ( ^ [N4: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).
% set_bit_int_def
thf(fact_8978_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_8979_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% less_eq_nat.simps(2)
thf(fact_8980_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_8981_or__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(2)
thf(fact_8982_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [K3: nat] : K3
@ ( minus_minus_nat @ M @ N ) ) ) ).
% diff_Suc
thf(fact_8983_or__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(3)
thf(fact_8984_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_8985_OR__upper,axiom,
! [X: int,N: nat,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_8986_or__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_8987_or__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_8988_or__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_8989_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K3: int] :
( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
@ ( if_Code_integer
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_8990_or__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(5)
thf(fact_8991_or__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(1)
thf(fact_8992_integer__of__int__eq__of__int,axiom,
code_integer_of_int = ring_18347121197199848620nteger ).
% integer_of_int_eq_of_int
thf(fact_8993_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_8994_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_8995_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_8996_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_8997_or__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(8)
thf(fact_8998_or__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(4)
thf(fact_8999_or__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(7)
thf(fact_9000_or__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(3)
thf(fact_9001_abs__integer__code,axiom,
( abs_abs_Code_integer
= ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).
% abs_integer_code
thf(fact_9002_uminus__integer__code_I1_J,axiom,
( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% uminus_integer_code(1)
thf(fact_9003_uminus__integer_Oabs__eq,axiom,
! [X: int] :
( ( uminus1351360451143612070nteger @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( uminus_uminus_int @ X ) ) ) ).
% uminus_integer.abs_eq
thf(fact_9004_zero__integer__def,axiom,
( zero_z3403309356797280102nteger
= ( code_integer_of_int @ zero_zero_int ) ) ).
% zero_integer_def
thf(fact_9005_less__integer_Oabs__eq,axiom,
! [Xa3: int,X: int] :
( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
= ( ord_less_int @ Xa3 @ X ) ) ).
% less_integer.abs_eq
thf(fact_9006_plus__integer_Oabs__eq,axiom,
! [Xa3: int,X: int] :
( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( plus_plus_int @ Xa3 @ X ) ) ) ).
% plus_integer.abs_eq
thf(fact_9007_one__integer__def,axiom,
( one_one_Code_integer
= ( code_integer_of_int @ one_one_int ) ) ).
% one_integer_def
thf(fact_9008_minus__integer_Oabs__eq,axiom,
! [Xa3: int,X: int] :
( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( minus_minus_int @ Xa3 @ X ) ) ) ).
% minus_integer.abs_eq
thf(fact_9009_set__bit__nat__def,axiom,
( bit_se7882103937844011126it_nat
= ( ^ [M4: nat,N4: nat] : ( bit_se1412395901928357646or_nat @ N4 @ ( bit_se547839408752420682it_nat @ M4 @ one_one_nat ) ) ) ) ).
% set_bit_nat_def
thf(fact_9010_or__nat__def,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M4: nat,N4: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% or_nat_def
thf(fact_9011_int__numeral__or__not__num__neg,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% int_numeral_or_not_num_neg
thf(fact_9012_int__numeral__not__or__num__neg,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% int_numeral_not_or_num_neg
thf(fact_9013_numeral__or__not__num__eq,axiom,
! [M: num,N: num] :
( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
= ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% numeral_or_not_num_eq
thf(fact_9014_floor__real__def,axiom,
( archim6058952711729229775r_real
= ( ^ [X4: real] :
( the_int
@ ^ [Z3: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X4 )
& ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_9015_Suc__0__or__eq,axiom,
! [N: nat] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
= ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% Suc_0_or_eq
thf(fact_9016_or__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
= ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% or_Suc_0_eq
thf(fact_9017_or__nat__rec,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M4: nat,N4: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 )
| ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_9018_or__int__unfold,axiom,
( bit_se1409905431419307370or_int
= ( ^ [K3: int,L3: int] :
( if_int
@ ( ( K3
= ( uminus_uminus_int @ one_one_int ) )
| ( L3
= ( uminus_uminus_int @ one_one_int ) ) )
@ ( uminus_uminus_int @ one_one_int )
@ ( if_int @ ( K3 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_9019_floor__rat__def,axiom,
( archim3151403230148437115or_rat
= ( ^ [X4: rat] :
( the_int
@ ^ [Z3: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X4 )
& ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_9020_or__nat__unfold,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M4: nat,N4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M4 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M4 @ ( 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 @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_9021_integer__of__num_I3_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit1 @ N ) )
= ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).
% integer_of_num(3)
thf(fact_9022_max__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% max_Suc_Suc
thf(fact_9023_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_9024_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_9025_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_9026_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_9027_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_9028_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_9029_max__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% max_numeral_Suc
thf(fact_9030_max__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% max_Suc_numeral
thf(fact_9031_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_9032_nat__add__max__right,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q4 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q4 ) ) ) ).
% nat_add_max_right
thf(fact_9033_nat__add__max__left,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q4 )
= ( ord_max_nat @ ( plus_plus_nat @ M @ Q4 ) @ ( plus_plus_nat @ N @ Q4 ) ) ) ).
% nat_add_max_left
thf(fact_9034_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q4 )
= ( ord_max_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N @ Q4 ) ) ) ).
% nat_mult_max_left
thf(fact_9035_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q4 ) )
= ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q4 ) ) ) ).
% nat_mult_max_right
thf(fact_9036_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_9037_nat__minus__add__max,axiom,
! [N: nat,M: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
= ( ord_max_nat @ N @ M ) ) ).
% nat_minus_add_max
thf(fact_9038_max__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ M @ ( suc @ N ) )
= ( case_nat_nat @ ( suc @ N )
@ ^ [M3: nat] : ( suc @ ( ord_max_nat @ M3 @ N ) )
@ M ) ) ).
% max_Suc2
thf(fact_9039_max__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_max_nat @ ( suc @ N ) @ M )
= ( case_nat_nat @ ( suc @ N )
@ ^ [M3: nat] : ( suc @ ( ord_max_nat @ N @ M3 ) )
@ M ) ) ).
% max_Suc1
thf(fact_9040_integer__of__num__triv_I1_J,axiom,
( ( code_integer_of_num @ one )
= one_one_Code_integer ) ).
% integer_of_num_triv(1)
thf(fact_9041_pred__def,axiom,
( pred
= ( case_nat_nat @ zero_zero_nat
@ ^ [X24: nat] : X24 ) ) ).
% pred_def
thf(fact_9042_rat__inverse__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( inverse_inverse_rat @ P5 ) )
= ( produc4245557441103728435nt_int
@ ^ [A3: int,B3: 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 ) @ B3 ) @ ( abs_abs_int @ A3 ) ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_inverse_code
thf(fact_9043_normalize__negative,axiom,
! [Q4: int,P5: int] :
( ( ord_less_int @ Q4 @ zero_zero_int )
=> ( ( normalize @ ( product_Pair_int_int @ P5 @ Q4 ) )
= ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P5 ) @ ( uminus_uminus_int @ Q4 ) ) ) ) ) ).
% normalize_negative
thf(fact_9044_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% bezw_0
thf(fact_9045_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K3: nat,M4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M4 @ K3 ) @ ( product_Pair_nat_nat @ M4 @ ( minus_minus_nat @ K3 @ M4 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M4 @ ( suc @ K3 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_9046_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_9047_normalize__denom__zero,axiom,
! [P5: int] :
( ( normalize @ ( product_Pair_int_int @ P5 @ zero_zero_int ) )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% normalize_denom_zero
thf(fact_9048_rat__one__code,axiom,
( ( quotient_of @ one_one_rat )
= ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% rat_one_code
thf(fact_9049_rat__zero__code,axiom,
( ( quotient_of @ zero_zero_rat )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% rat_zero_code
thf(fact_9050_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_9051_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_9052_divide__rat__def,axiom,
( divide_divide_rat
= ( ^ [Q6: rat,R5: rat] : ( times_times_rat @ Q6 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% divide_rat_def
thf(fact_9053_diff__rat__def,axiom,
( minus_minus_rat
= ( ^ [Q6: rat,R5: rat] : ( plus_plus_rat @ Q6 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).
% diff_rat_def
thf(fact_9054_quotient__of__div,axiom,
! [R2: rat,N: int,D: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair_int_int @ N @ D ) )
=> ( R2
= ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% quotient_of_div
thf(fact_9055_rat__plus__code,axiom,
! [P5: rat,Q4: rat] :
( ( quotient_of @ ( plus_plus_rat @ P5 @ Q4 ) )
= ( produc4245557441103728435nt_int
@ ^ [A3: int,C3: int] :
( produc4245557441103728435nt_int
@ ^ [B3: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D3 ) @ ( times_times_int @ B3 @ C3 ) ) @ ( times_times_int @ C3 @ D3 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_plus_code
thf(fact_9056_rat__minus__code,axiom,
! [P5: rat,Q4: rat] :
( ( quotient_of @ ( minus_minus_rat @ P5 @ Q4 ) )
= ( produc4245557441103728435nt_int
@ ^ [A3: int,C3: int] :
( produc4245557441103728435nt_int
@ ^ [B3: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A3 @ D3 ) @ ( times_times_int @ B3 @ C3 ) ) @ ( times_times_int @ C3 @ D3 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_minus_code
thf(fact_9057_quotient__of__denom__pos,axiom,
! [R2: rat,P5: int,Q4: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair_int_int @ P5 @ Q4 ) )
=> ( ord_less_int @ zero_zero_int @ Q4 ) ) ).
% quotient_of_denom_pos
thf(fact_9058_rat__uminus__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( uminus_uminus_rat @ P5 ) )
= ( produc4245557441103728435nt_int
@ ^ [A3: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A3 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_uminus_code
thf(fact_9059_normalize__denom__pos,axiom,
! [R2: product_prod_int_int,P5: int,Q4: int] :
( ( ( normalize @ R2 )
= ( product_Pair_int_int @ P5 @ Q4 ) )
=> ( ord_less_int @ zero_zero_int @ Q4 ) ) ).
% normalize_denom_pos
thf(fact_9060_normalize__crossproduct,axiom,
! [Q4: int,S: int,P5: int,R2: int] :
( ( Q4 != zero_zero_int )
=> ( ( S != zero_zero_int )
=> ( ( ( normalize @ ( product_Pair_int_int @ P5 @ Q4 ) )
= ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
=> ( ( times_times_int @ P5 @ S )
= ( times_times_int @ R2 @ Q4 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_9061_rat__less__code,axiom,
( ord_less_rat
= ( ^ [P6: rat,Q6: rat] :
( produc4947309494688390418_int_o
@ ^ [A3: int,C3: int] :
( produc4947309494688390418_int_o
@ ^ [B3: int,D3: int] : ( ord_less_int @ ( times_times_int @ A3 @ D3 ) @ ( times_times_int @ C3 @ B3 ) )
@ ( quotient_of @ Q6 ) )
@ ( quotient_of @ P6 ) ) ) ) ).
% rat_less_code
thf(fact_9062_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa3: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa3 )
= Y )
=> ( ( ( ord_less_eq_nat @ Xa3 @ X )
=> ( Y
= ( product_Pair_nat_nat @ Xa3 @ ( minus_minus_nat @ X @ Xa3 ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa3 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa3 @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_9063_quotient__of__int,axiom,
! [A: int] :
( ( quotient_of @ ( of_int @ A ) )
= ( product_Pair_int_int @ A @ one_one_int ) ) ).
% quotient_of_int
thf(fact_9064_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_9065_finite__enumerate,axiom,
! [S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ? [R4: nat > nat] :
( ( strict1292158309912662752at_nat @ R4 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S3 ) ) )
& ! [N6: nat] :
( ( ord_less_nat @ N6 @ ( finite_card_nat @ S3 ) )
=> ( member_nat @ ( R4 @ N6 ) @ S3 ) ) ) ) ).
% finite_enumerate
thf(fact_9066_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
@ ( produc9125791028180074456eger_o
@ ^ [R5: code_integer,S4: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S4 ) ) @ ( S4 = one_one_Code_integer ) )
@ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_9067_drop__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% drop_bit_nonnegative_int_iff
thf(fact_9068_drop__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% drop_bit_negative_int_iff
thf(fact_9069_drop__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% drop_bit_minus_one
thf(fact_9070_drop__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% drop_bit_Suc_minus_bit0
thf(fact_9071_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_9072_drop__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_Suc_minus_bit1
thf(fact_9073_drop__bit__push__bit__int,axiom,
! [M: nat,N: nat,K: int] :
( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
= ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).
% drop_bit_push_bit_int
thf(fact_9074_Rat_Oof__int__def,axiom,
of_int = ring_1_of_int_rat ).
% Rat.of_int_def
thf(fact_9075_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_9076_drop__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% drop_bit_of_Suc_0
thf(fact_9077_Frct__code__post_I9_J,axiom,
! [Q4: product_prod_int_int] :
( ( uminus_uminus_rat @ ( uminus_uminus_rat @ ( frct @ Q4 ) ) )
= ( frct @ Q4 ) ) ).
% Frct_code_post(9)
thf(fact_9078_drop__bit__nat__eq,axiom,
! [N: nat,K: int] :
( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).
% drop_bit_nat_eq
thf(fact_9079_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_9080_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_9081_Frct__code__post_I8_J,axiom,
! [A: int,B: int] :
( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
= ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% Frct_code_post(8)
thf(fact_9082_Frct__code__post_I7_J,axiom,
! [A: int,B: int] :
( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
= ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% Frct_code_post(7)
thf(fact_9083_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_9084_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_9085_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L3: code_integer] :
( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L3 )
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L3 )
@ ( 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 @ L3 @ S4 ) ) )
@ ( code_divmod_abs @ K3 @ L3 ) ) )
@ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
@ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L3 )
@ ( 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 @ L3 ) @ S4 ) ) )
@ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_9086_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L3: code_integer] :
( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
@ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L3
@ ( if_Pro6119634080678213985nteger
@ ( ( sgn_sgn_Code_integer @ K3 )
= ( sgn_sgn_Code_integer @ L3 ) )
@ ( code_divmod_abs @ K3 @ L3 )
@ ( 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 @ L3 ) @ S4 ) ) )
@ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_9087_xor__minus__numerals_I1_J,axiom,
! [N: num,K: int] :
( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
= ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% xor_minus_numerals(1)
thf(fact_9088_xor__minus__numerals_I2_J,axiom,
! [K: int,N: num] :
( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% xor_minus_numerals(2)
thf(fact_9089_sub__BitM__One__eq,axiom,
! [N: num] :
( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% sub_BitM_One_eq
thf(fact_9090_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_9091_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
@ ^ [X4: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X4 )
@ ^ [X4: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X4 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_9092_Suc__funpow,axiom,
! [N: nat] :
( ( compow_nat_nat @ N @ suc )
= ( plus_plus_nat @ N ) ) ).
% Suc_funpow
thf(fact_9093_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_9094_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_9095_nat__of__integer_Oabs__eq,axiom,
! [X: int] :
( ( code_nat_of_integer @ ( code_integer_of_int @ X ) )
= ( nat2 @ X ) ) ).
% nat_of_integer.abs_eq
thf(fact_9096_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_9097_Code__Numeral_Onegative__def,axiom,
( code_negative
= ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).
% Code_Numeral.negative_def
thf(fact_9098_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_9099_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K3: code_integer] :
( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
@ ( produc1555791787009142072er_nat
@ ^ [L3: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ one_one_nat ) )
@ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_9100_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K3: code_integer] :
( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
@ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
@ ( produc1553301316500091796er_int
@ ^ [L3: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ one_one_int ) )
@ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_9101_int__of__integer__of__nat,axiom,
! [N: nat] :
( ( code_int_of_integer @ ( semiri4939895301339042750nteger @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% int_of_integer_of_nat
thf(fact_9102_of__int__integer__of,axiom,
! [K: code_integer] :
( ( ring_18347121197199848620nteger @ ( code_int_of_integer @ K ) )
= K ) ).
% of_int_integer_of
thf(fact_9103_int__of__integer__of__int,axiom,
! [K: int] :
( ( code_int_of_integer @ ( ring_18347121197199848620nteger @ K ) )
= K ) ).
% int_of_integer_of_int
thf(fact_9104_zero__integer_Orep__eq,axiom,
( ( code_int_of_integer @ zero_z3403309356797280102nteger )
= zero_zero_int ) ).
% zero_integer.rep_eq
thf(fact_9105_plus__integer_Orep__eq,axiom,
! [X: code_integer,Xa3: code_integer] :
( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X @ Xa3 ) )
= ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% plus_integer.rep_eq
thf(fact_9106_uminus__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( uminus1351360451143612070nteger @ X ) )
= ( uminus_uminus_int @ ( code_int_of_integer @ X ) ) ) ).
% uminus_integer.rep_eq
thf(fact_9107_one__integer_Orep__eq,axiom,
( ( code_int_of_integer @ one_one_Code_integer )
= one_one_int ) ).
% one_integer.rep_eq
thf(fact_9108_minus__integer_Orep__eq,axiom,
! [X: code_integer,Xa3: code_integer] :
( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X @ Xa3 ) )
= ( minus_minus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% minus_integer.rep_eq
thf(fact_9109_less__integer_Orep__eq,axiom,
( ord_le6747313008572928689nteger
= ( ^ [X4: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_integer.rep_eq
thf(fact_9110_integer__less__iff,axiom,
( ord_le6747313008572928689nteger
= ( ^ [K3: code_integer,L3: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% integer_less_iff
thf(fact_9111_nat__of__integer_Orep__eq,axiom,
( code_nat_of_integer
= ( ^ [X4: code_integer] : ( nat2 @ ( code_int_of_integer @ X4 ) ) ) ) ).
% nat_of_integer.rep_eq
thf(fact_9112_times__int_Oabs__eq,axiom,
! [Xa3: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( times_times_int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y5 @ U2 ) ) ) )
@ Xa3
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_9113_Gcd__remove0__nat,axiom,
! [M10: set_nat] :
( ( finite_finite_nat @ M10 )
=> ( ( gcd_Gcd_nat @ M10 )
= ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M10 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_9114_Gcd__nat__eq__one,axiom,
! [N5: set_nat] :
( ( member_nat @ one_one_nat @ N5 )
=> ( ( gcd_Gcd_nat @ N5 )
= one_one_nat ) ) ).
% Gcd_nat_eq_one
thf(fact_9115_int_Oabs__induct,axiom,
! [P: int > $o,X: int] :
( ! [Y3: product_prod_nat_nat] : ( P @ ( abs_Integ @ Y3 ) )
=> ( P @ X ) ) ).
% int.abs_induct
thf(fact_9116_Gcd__greatest__nat,axiom,
! [A2: set_nat,A: nat] :
( ! [B4: nat] :
( ( member_nat @ B4 @ A2 )
=> ( dvd_dvd_nat @ A @ B4 ) )
=> ( dvd_dvd_nat @ A @ ( gcd_Gcd_nat @ A2 ) ) ) ).
% Gcd_greatest_nat
thf(fact_9117_Gcd__dvd__nat,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( dvd_dvd_nat @ ( gcd_Gcd_nat @ A2 ) @ A ) ) ).
% Gcd_dvd_nat
thf(fact_9118_eq__Abs__Integ,axiom,
! [Z2: int] :
~ ! [X3: nat,Y3: nat] :
( Z2
!= ( abs_Integ @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ).
% eq_Abs_Integ
thf(fact_9119_nat_Oabs__eq,axiom,
! [X: product_prod_nat_nat] :
( ( nat2 @ ( abs_Integ @ X ) )
= ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).
% nat.abs_eq
thf(fact_9120_zero__int__def,axiom,
( zero_zero_int
= ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% zero_int_def
thf(fact_9121_int__def,axiom,
( semiri1314217659103216013at_int
= ( ^ [N4: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N4 @ zero_zero_nat ) ) ) ) ).
% int_def
thf(fact_9122_uminus__int_Oabs__eq,axiom,
! [X: product_prod_nat_nat] :
( ( uminus_uminus_int @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_9123_one__int__def,axiom,
( one_one_int
= ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% one_int_def
thf(fact_9124_less__int_Oabs__eq,axiom,
! [Xa3: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( ord_less_int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
= ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) )
@ Xa3
@ X ) ) ).
% less_int.abs_eq
thf(fact_9125_less__eq__int_Oabs__eq,axiom,
! [Xa3: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( ord_less_eq_int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
= ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) )
@ Xa3
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_9126_plus__int_Oabs__eq,axiom,
! [Xa3: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( plus_plus_int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) )
@ Xa3
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_9127_minus__int_Oabs__eq,axiom,
! [Xa3: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( minus_minus_int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y5 @ U2 ) ) )
@ Xa3
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_9128_csqrt_Ocode,axiom,
( csqrt
= ( ^ [Z3: complex] :
( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
@ ( times_times_real
@ ( if_real
@ ( ( im @ Z3 )
= zero_zero_real )
@ one_one_real
@ ( sgn_sgn_real @ ( im @ Z3 ) ) )
@ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% csqrt.code
thf(fact_9129_abs__Gcd__eq,axiom,
! [K5: set_int] :
( ( abs_abs_int @ ( gcd_Gcd_int @ K5 ) )
= ( gcd_Gcd_int @ K5 ) ) ).
% abs_Gcd_eq
thf(fact_9130_complex__Im__of__int,axiom,
! [Z2: int] :
( ( im @ ( ring_17405671764205052669omplex @ Z2 ) )
= zero_zero_real ) ).
% complex_Im_of_int
thf(fact_9131_complex__Re__of__int,axiom,
! [Z2: int] :
( ( re @ ( ring_17405671764205052669omplex @ Z2 ) )
= ( ring_1_of_int_real @ Z2 ) ) ).
% complex_Re_of_int
thf(fact_9132_Re__i__times,axiom,
! [Z2: complex] :
( ( re @ ( times_times_complex @ imaginary_unit @ Z2 ) )
= ( uminus_uminus_real @ ( im @ Z2 ) ) ) ).
% Re_i_times
thf(fact_9133_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_9134_Gcd__greatest__int,axiom,
! [A2: set_int,A: int] :
( ! [B4: int] :
( ( member_int @ B4 @ A2 )
=> ( dvd_dvd_int @ A @ B4 ) )
=> ( dvd_dvd_int @ A @ ( gcd_Gcd_int @ A2 ) ) ) ).
% Gcd_greatest_int
thf(fact_9135_Gcd__dvd__int,axiom,
! [A: int,A2: set_int] :
( ( member_int @ A @ A2 )
=> ( dvd_dvd_int @ ( gcd_Gcd_int @ A2 ) @ A ) ) ).
% Gcd_dvd_int
thf(fact_9136_uminus__complex_Ocode,axiom,
( uminus1482373934393186551omplex
= ( ^ [X4: complex] : ( complex2 @ ( uminus_uminus_real @ ( re @ X4 ) ) @ ( uminus_uminus_real @ ( im @ X4 ) ) ) ) ) ).
% uminus_complex.code
thf(fact_9137_complex__is__Int__iff,axiom,
! [Z2: complex] :
( ( member_complex @ Z2 @ ring_1_Ints_complex )
= ( ( ( im @ Z2 )
= zero_zero_real )
& ? [I3: int] :
( ( re @ Z2 )
= ( ring_1_of_int_real @ I3 ) ) ) ) ).
% complex_is_Int_iff
thf(fact_9138_imaginary__unit_Osimps_I2_J,axiom,
( ( im @ imaginary_unit )
= one_one_real ) ).
% imaginary_unit.simps(2)
thf(fact_9139_one__complex_Osimps_I2_J,axiom,
( ( im @ one_one_complex )
= zero_zero_real ) ).
% one_complex.simps(2)
thf(fact_9140_one__complex_Osimps_I1_J,axiom,
( ( re @ one_one_complex )
= one_one_real ) ).
% one_complex.simps(1)
thf(fact_9141_uminus__complex_Osimps_I2_J,axiom,
! [X: complex] :
( ( im @ ( uminus1482373934393186551omplex @ X ) )
= ( uminus_uminus_real @ ( im @ X ) ) ) ).
% uminus_complex.simps(2)
thf(fact_9142_uminus__complex_Osimps_I1_J,axiom,
! [X: complex] :
( ( re @ ( uminus1482373934393186551omplex @ X ) )
= ( uminus_uminus_real @ ( re @ X ) ) ) ).
% uminus_complex.simps(1)
thf(fact_9143_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_9144_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_9145_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_9146_complex__unit__circle,axiom,
! [Z2: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real ) ) ).
% complex_unit_circle
thf(fact_9147_inverse__complex_Ocode,axiom,
( invers8013647133539491842omplex
= ( ^ [X4: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X4 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X4 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% inverse_complex.code
thf(fact_9148_cmod__plus__Re__le__0__iff,axiom,
! [Z2: complex] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ zero_zero_real )
= ( ( re @ Z2 )
= ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).
% cmod_plus_Re_le_0_iff
thf(fact_9149_csqrt_Osimps_I2_J,axiom,
! [Z2: complex] :
( ( im @ ( csqrt @ Z2 ) )
= ( times_times_real
@ ( if_real
@ ( ( im @ Z2 )
= zero_zero_real )
@ one_one_real
@ ( sgn_sgn_real @ ( im @ Z2 ) ) )
@ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% csqrt.simps(2)
thf(fact_9150_Im__Reals__divide,axiom,
! [R2: complex,Z2: complex] :
( ( member_complex @ R2 @ real_V2521375963428798218omplex )
=> ( ( im @ ( divide1717551699836669952omplex @ R2 @ Z2 ) )
= ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R2 ) ) @ ( im @ Z2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% Im_Reals_divide
thf(fact_9151_less__eq__int_Orep__eq,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Xa4: int] :
( produc8739625826339149834_nat_o
@ ^ [Y5: nat,Z3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y5 @ V4 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
@ ( rep_Integ @ X4 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_9152_nat_Orep__eq,axiom,
( nat2
= ( ^ [X4: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X4 ) ) ) ) ).
% nat.rep_eq
thf(fact_9153_less__int_Orep__eq,axiom,
( ord_less_int
= ( ^ [X4: int,Xa4: int] :
( produc8739625826339149834_nat_o
@ ^ [Y5: nat,Z3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y5 @ V4 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
@ ( rep_Integ @ X4 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_9154_uminus__int__def,axiom,
( uminus_uminus_int
= ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 ) ) ) ) ).
% uminus_int_def
thf(fact_9155_prod__encode__def,axiom,
( nat_prod_encode
= ( produc6842872674320459806at_nat
@ ^ [M4: nat,N4: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M4 @ N4 ) ) @ M4 ) ) ) ).
% prod_encode_def
thf(fact_9156_num__of__nat_Osimps_I2_J,axiom,
! [N: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= ( inc @ ( num_of_nat @ N ) ) ) )
& ( ~ ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= one ) ) ) ).
% num_of_nat.simps(2)
thf(fact_9157_prod__encode__eq,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_encode @ X )
= ( nat_prod_encode @ Y ) )
= ( X = Y ) ) ).
% prod_encode_eq
thf(fact_9158_num__of__nat__numeral__eq,axiom,
! [Q4: num] :
( ( num_of_nat @ ( numeral_numeral_nat @ Q4 ) )
= Q4 ) ).
% num_of_nat_numeral_eq
thf(fact_9159_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ zero_zero_nat )
= one ) ).
% num_of_nat.simps(1)
thf(fact_9160_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_9161_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_9162_numeral__num__of__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
= N ) ) ).
% numeral_num_of_nat
thf(fact_9163_num__of__nat__One,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ one_one_nat )
=> ( ( num_of_nat @ N )
= one ) ) ).
% num_of_nat_One
thf(fact_9164_num__of__nat__double,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
= ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% num_of_nat_double
thf(fact_9165_num__of__nat__plus__distrib,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_9166_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_9167_times__int__def,axiom,
( times_times_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y5 @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_9168_minus__int__def,axiom,
( minus_minus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y5 @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_9169_plus__int__def,axiom,
( plus_plus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_9170_num__of__integer_Oabs__eq,axiom,
! [X: int] :
( ( code_num_of_integer @ ( code_integer_of_int @ X ) )
= ( num_of_nat @ ( nat2 @ X ) ) ) ).
% num_of_integer.abs_eq
thf(fact_9171_num__of__integer_Orep__eq,axiom,
( code_num_of_integer
= ( ^ [X4: code_integer] : ( num_of_nat @ ( nat2 @ ( code_int_of_integer @ X4 ) ) ) ) ) ).
% num_of_integer.rep_eq
thf(fact_9172_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K3: code_integer] :
( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
@ ( produc7336495610019696514er_num
@ ^ [L3: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ one ) )
@ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_9173_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_9174_sqr_Osimps_I2_J,axiom,
! [N: num] :
( ( sqr @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% sqr.simps(2)
thf(fact_9175_sqr_Osimps_I1_J,axiom,
( ( sqr @ one )
= one ) ).
% sqr.simps(1)
thf(fact_9176_sqr__conv__mult,axiom,
( sqr
= ( ^ [X4: num] : ( times_times_num @ X4 @ X4 ) ) ) ).
% sqr_conv_mult
thf(fact_9177_pow_Osimps_I2_J,axiom,
! [X: num,Y: num] :
( ( pow @ X @ ( bit0 @ Y ) )
= ( sqr @ ( pow @ X @ Y ) ) ) ).
% pow.simps(2)
thf(fact_9178_sqr_Osimps_I3_J,axiom,
! [N: num] :
( ( sqr @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% sqr.simps(3)
thf(fact_9179_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_9180_bij__betw__Suc,axiom,
! [M10: set_nat,N5: set_nat] :
( ( bij_betw_nat_nat @ suc @ M10 @ N5 )
= ( ( image_nat_nat @ suc @ M10 )
= N5 ) ) ).
% bij_betw_Suc
thf(fact_9181_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_9182_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_9183_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_9184_image__Suc__lessThan,axiom,
! [N: nat] :
( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% image_Suc_lessThan
thf(fact_9185_image__Suc__atMost,axiom,
! [N: nat] :
( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% image_Suc_atMost
thf(fact_9186_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_9187_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_9188_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_9189_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_9190_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_9191_minus__rat__cancel,axiom,
! [A: int,B: int] :
( ( fract @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( fract @ A @ B ) ) ).
% minus_rat_cancel
thf(fact_9192_inverse__rat,axiom,
! [A: int,B: int] :
( ( inverse_inverse_rat @ ( fract @ A @ B ) )
= ( fract @ B @ A ) ) ).
% inverse_rat
thf(fact_9193_Gcd__abs__eq,axiom,
! [K5: set_int] :
( ( gcd_Gcd_int @ ( image_int_int @ abs_abs_int @ K5 ) )
= ( gcd_Gcd_int @ K5 ) ) ).
% Gcd_abs_eq
thf(fact_9194_minus__rat,axiom,
! [A: int,B: int] :
( ( uminus_uminus_rat @ ( fract @ A @ B ) )
= ( fract @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_rat
thf(fact_9195_Gcd__int__eq,axiom,
! [N5: set_nat] :
( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N5 ) )
= ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N5 ) ) ) ).
% Gcd_int_eq
thf(fact_9196_Gcd__nat__abs__eq,axiom,
! [K5: set_int] :
( ( gcd_Gcd_nat
@ ( image_int_nat
@ ^ [K3: int] : ( nat2 @ ( abs_abs_int @ K3 ) )
@ K5 ) )
= ( nat2 @ ( gcd_Gcd_int @ K5 ) ) ) ).
% Gcd_nat_abs_eq
thf(fact_9197_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_9198_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_9199_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_9200_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_9201_sgn__rat,axiom,
! [A: int,B: int] :
( ( sgn_sgn_rat @ ( fract @ A @ B ) )
= ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).
% sgn_rat
thf(fact_9202_eq__rat_I3_J,axiom,
! [A: int,C: int] :
( ( fract @ zero_zero_int @ A )
= ( fract @ zero_zero_int @ C ) ) ).
% eq_rat(3)
thf(fact_9203_eq__rat_I2_J,axiom,
! [A: int] :
( ( fract @ A @ zero_zero_int )
= ( fract @ zero_zero_int @ one_one_int ) ) ).
% eq_rat(2)
thf(fact_9204_Rat__induct__pos,axiom,
! [P: rat > $o,Q4: rat] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ B4 )
=> ( P @ ( fract @ A4 @ B4 ) ) )
=> ( P @ Q4 ) ) ).
% Rat_induct_pos
thf(fact_9205_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_9206_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_9207_Fract__of__nat__eq,axiom,
! [K: nat] :
( ( fract @ ( semiri1314217659103216013at_int @ K ) @ one_one_int )
= ( semiri681578069525770553at_rat @ K ) ) ).
% Fract_of_nat_eq
thf(fact_9208_rat__number__collapse_I1_J,axiom,
! [K: int] :
( ( fract @ zero_zero_int @ K )
= zero_zero_rat ) ).
% rat_number_collapse(1)
thf(fact_9209_rat__number__collapse_I6_J,axiom,
! [K: int] :
( ( fract @ K @ zero_zero_int )
= zero_zero_rat ) ).
% rat_number_collapse(6)
thf(fact_9210_One__rat__def,axiom,
( one_one_rat
= ( fract @ one_one_int @ one_one_int ) ) ).
% One_rat_def
thf(fact_9211_Fract__of__int__eq,axiom,
! [K: int] :
( ( fract @ K @ one_one_int )
= ( ring_1_of_int_rat @ K ) ) ).
% Fract_of_int_eq
thf(fact_9212_Fract__of__int__quotient,axiom,
( fract
= ( ^ [K3: int,L3: int] : ( divide_divide_rat @ ( ring_1_of_int_rat @ K3 ) @ ( ring_1_of_int_rat @ L3 ) ) ) ) ).
% Fract_of_int_quotient
thf(fact_9213_Zero__rat__def,axiom,
( zero_zero_rat
= ( fract @ zero_zero_int @ one_one_int ) ) ).
% Zero_rat_def
thf(fact_9214_image__int__atLeastAtMost,axiom,
! [A: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastAtMost
thf(fact_9215_rat__number__expand_I3_J,axiom,
( numeral_numeral_rat
= ( ^ [K3: num] : ( fract @ ( numeral_numeral_int @ K3 ) @ one_one_int ) ) ) ).
% rat_number_expand(3)
thf(fact_9216_rat__number__collapse_I3_J,axiom,
! [W2: num] :
( ( fract @ ( numeral_numeral_int @ W2 ) @ one_one_int )
= ( numeral_numeral_rat @ W2 ) ) ).
% rat_number_collapse(3)
thf(fact_9217_image__int__atLeastLessThan,axiom,
! [A: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
= ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastLessThan
thf(fact_9218_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_9219_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_9220_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_9221_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_9222_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_9223_Fract__add__one,axiom,
! [N: int,M: int] :
( ( N != zero_zero_int )
=> ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
= ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).
% Fract_add_one
thf(fact_9224_infinite__int__iff__infinite__nat__abs,axiom,
! [S3: set_int] :
( ( ~ ( finite_finite_int @ S3 ) )
= ( ~ ( finite_finite_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ S3 ) ) ) ) ).
% infinite_int_iff_infinite_nat_abs
thf(fact_9225_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image_int_int
@ ^ [X4: int] : ( plus_plus_int @ X4 @ L )
@ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
= ( set_or4662586982721622107an_int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_9226_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_9227_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_9228_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_9229_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_9230_rat__number__collapse_I4_J,axiom,
! [W2: num] :
( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ one_one_int )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ).
% rat_number_collapse(4)
thf(fact_9231_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_9232_Gcd__int__def,axiom,
( gcd_Gcd_int
= ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).
% Gcd_int_def
thf(fact_9233_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_9234_take__bit__numeral__minus__numeral__int,axiom,
! [M: num,N: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( case_option_int_num @ zero_zero_int
@ ^ [Q6: 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 @ Q6 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_9235_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_9236_of__nat__eq__id,axiom,
semiri1316708129612266289at_nat = id_nat ).
% of_nat_eq_id
thf(fact_9237_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_9238_take__bit__num__simps_I2_J,axiom,
! [N: nat] :
( ( bit_take_bit_num @ ( suc @ N ) @ one )
= ( some_num @ one ) ) ).
% take_bit_num_simps(2)
thf(fact_9239_less__int__def,axiom,
( ord_less_int
= ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
@ ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) ) ) ) ).
% less_int_def
thf(fact_9240_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
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_9241_nat__def,axiom,
( nat2
= ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% nat_def
thf(fact_9242_Rat_Opositive__minus,axiom,
! [X: rat] :
( ~ ( positive @ X )
=> ( ( X != zero_zero_rat )
=> ( positive @ ( uminus_uminus_rat @ X ) ) ) ) ).
% Rat.positive_minus
thf(fact_9243_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N4: nat,M4: num] :
( if_option_num
@ ( ( bit_se2925701944663578781it_nat @ N4 @ ( numeral_numeral_nat @ M4 ) )
= zero_zero_nat )
@ none_num
@ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N4 @ ( numeral_numeral_nat @ M4 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_9244_and__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(7)
thf(fact_9245_and__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(3)
thf(fact_9246_and__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(4)
thf(fact_9247_take__bit__num__simps_I4_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% take_bit_num_simps(4)
thf(fact_9248_take__bit__num__simps_I3_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
= ( case_o6005452278849405969um_num @ none_num
@ ^ [Q6: num] : ( some_num @ ( bit0 @ Q6 ) )
@ ( bit_take_bit_num @ N @ M ) ) ) ).
% take_bit_num_simps(3)
thf(fact_9249_and__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(8)
thf(fact_9250_and__not__num__eq__None__iff,axiom,
! [M: num,N: num] :
( ( ( bit_and_not_num @ M @ N )
= none_num )
= ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
= zero_zero_int ) ) ).
% and_not_num_eq_None_iff
thf(fact_9251_int__numeral__not__and__num,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).
% int_numeral_not_and_num
thf(fact_9252_int__numeral__and__not__num,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).
% int_numeral_and_not_num
thf(fact_9253_measure__function__int,axiom,
fun_is_measure_int @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) ).
% measure_function_int
thf(fact_9254_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_9255_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_9256_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% min_Suc_Suc
thf(fact_9257_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_9258_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_9259_card__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_greaterThanAtMost
thf(fact_9260_min__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% min_numeral_Suc
thf(fact_9261_min__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% min_Suc_numeral
thf(fact_9262_min__diff,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% min_diff
thf(fact_9263_nat__mult__min__left,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q4 )
= ( ord_min_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N @ Q4 ) ) ) ).
% nat_mult_min_left
thf(fact_9264_nat__mult__min__right,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q4 ) )
= ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q4 ) ) ) ).
% nat_mult_min_right
thf(fact_9265_inf__nat__def,axiom,
inf_inf_nat = ord_min_nat ).
% inf_nat_def
thf(fact_9266_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_9267_concat__bit__assoc__sym,axiom,
! [M: nat,N: nat,K: int,L: int,R2: int] :
( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L ) @ R2 )
= ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L @ R2 ) ) ) ).
% concat_bit_assoc_sym
thf(fact_9268_take__bit__concat__bit__eq,axiom,
! [M: nat,N: nat,K: int,L: int] :
( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L ) )
= ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L ) ) ) ).
% take_bit_concat_bit_eq
thf(fact_9269_min__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ M @ ( suc @ N ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [M3: nat] : ( suc @ ( ord_min_nat @ M3 @ N ) )
@ M ) ) ).
% min_Suc2
thf(fact_9270_min__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_min_nat @ ( suc @ N ) @ M )
= ( case_nat_nat @ zero_zero_nat
@ ^ [M3: nat] : ( suc @ ( ord_min_nat @ N @ M3 ) )
@ M ) ) ).
% min_Suc1
thf(fact_9271_card__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or6656581121297822940st_int @ L @ U ) )
= ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% card_greaterThanAtMost_int
thf(fact_9272_inf__int__def,axiom,
inf_inf_int = ord_min_int ).
% inf_int_def
thf(fact_9273_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_9274_vanishes__mult__bounded,axiom,
! [X9: nat > rat,Y7: nat > rat] :
( ? [A7: rat] :
( ( ord_less_rat @ zero_zero_rat @ A7 )
& ! [N2: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N2 ) ) @ A7 ) )
=> ( ( vanishes @ Y7 )
=> ( vanishes
@ ^ [N4: nat] : ( times_times_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% vanishes_mult_bounded
thf(fact_9275_vanishes__const,axiom,
! [C: rat] :
( ( vanishes
@ ^ [N4: nat] : C )
= ( C = zero_zero_rat ) ) ).
% vanishes_const
thf(fact_9276_vanishes__minus,axiom,
! [X9: nat > rat] :
( ( vanishes @ X9 )
=> ( vanishes
@ ^ [N4: nat] : ( uminus_uminus_rat @ ( X9 @ N4 ) ) ) ) ).
% vanishes_minus
thf(fact_9277_vanishes__add,axiom,
! [X9: nat > rat,Y7: nat > rat] :
( ( vanishes @ X9 )
=> ( ( vanishes @ Y7 )
=> ( vanishes
@ ^ [N4: nat] : ( plus_plus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% vanishes_add
thf(fact_9278_vanishes__diff,axiom,
! [X9: nat > rat,Y7: nat > rat] :
( ( vanishes @ X9 )
=> ( ( vanishes @ Y7 )
=> ( vanishes
@ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% vanishes_diff
thf(fact_9279_card__length__sum__list__rec,axiom,
! [M: nat,N5: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L3: list_nat] :
( ( ( size_size_list_nat @ L3 )
= M )
& ( ( groups4561878855575611511st_nat @ L3 )
= N5 ) ) ) )
= ( plus_plus_nat
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L3: list_nat] :
( ( ( size_size_list_nat @ L3 )
= ( minus_minus_nat @ M @ one_one_nat ) )
& ( ( groups4561878855575611511st_nat @ L3 )
= N5 ) ) ) )
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L3: list_nat] :
( ( ( size_size_list_nat @ L3 )
= M )
& ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L3 ) @ one_one_nat )
= N5 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_9280_card__length__sum__list,axiom,
! [M: nat,N5: nat] :
( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L3: list_nat] :
( ( ( size_size_list_nat @ L3 )
= M )
& ( ( groups4561878855575611511st_nat @ L3 )
= N5 ) ) ) )
= ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N5 @ M ) @ one_one_nat ) @ N5 ) ) ).
% card_length_sum_list
thf(fact_9281_vanishes__def,axiom,
( vanishes
= ( ^ [X8: nat > rat] :
! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [N4: nat] :
( ( ord_less_eq_nat @ K3 @ N4 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N4 ) ) @ R5 ) ) ) ) ) ).
% vanishes_def
thf(fact_9282_vanishesI,axiom,
! [X9: nat > rat] :
( ! [R4: rat] :
( ( ord_less_rat @ zero_zero_rat @ R4 )
=> ? [K4: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K4 @ N2 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N2 ) ) @ R4 ) ) )
=> ( vanishes @ X9 ) ) ).
% vanishesI
thf(fact_9283_vanishesD,axiom,
! [X9: nat > rat,R2: rat] :
( ( vanishes @ X9 )
=> ( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ? [K2: nat] :
! [N6: nat] :
( ( ord_less_eq_nat @ K2 @ N6 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N6 ) ) @ R2 ) ) ) ) ).
% vanishesD
thf(fact_9284_rcis__inverse,axiom,
! [R2: real,A: real] :
( ( invers8013647133539491842omplex @ ( rcis @ R2 @ A ) )
= ( rcis @ ( divide_divide_real @ one_one_real @ R2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% rcis_inverse
thf(fact_9285_cis__rcis__eq,axiom,
( cis
= ( rcis @ one_one_real ) ) ).
% cis_rcis_eq
thf(fact_9286_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa3 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa3 = one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( Deg2 = Xa3 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( 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 @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [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 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
& ! [X4: nat] :
( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X4 )
=> ( ( ord_less_nat @ Mi3 @ X4 )
& ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_9287_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ( vEBT_VEBT_valid @ X @ Xa3 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa3 != one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
=> ~ ( ( Deg2 = Xa3 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [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 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
& ! [X4: nat] :
( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X4 )
=> ( ( ord_less_nat @ Mi3 @ X4 )
& ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_9288_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg4 )
= ( ( Deg = Deg4 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
& ! [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 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X4: nat] :
( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less_nat @ Mi3 @ X4 )
& ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_9289_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa3 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Y
= ( Xa3 != one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
=> ( Y
= ( ~ ( ( Deg2 = Xa3 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( 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 @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [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 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
& ! [X4: nat] :
( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X4 )
=> ( ( ord_less_nat @ Mi3 @ X4 )
& ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_9290_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa3 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Y
= ( Xa3 = one_one_nat ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa3 ) ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( Y
= ( ( Deg2 = Xa3 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( 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 @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [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 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
& ! [X4: nat] :
( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X4 )
=> ( ( ord_less_nat @ Mi3 @ X4 )
& ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa3 ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_9291_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ( vEBT_VEBT_valid @ X @ Xa3 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa3 ) )
=> ( Xa3 != one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa3 ) )
=> ~ ( ( Deg2 = Xa3 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [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 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
& ! [X4: nat] :
( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X4 )
=> ( ( ord_less_nat @ Mi3 @ X4 )
& ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_9292_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa3: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa3 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa3 ) )
=> ( Xa3 = one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa3 ) )
=> ( ( Deg2 = Xa3 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( 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 @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [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 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
& ! [X4: nat] :
( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X4 )
=> ( ( ord_less_nat @ Mi3 @ X4 )
& ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_9293_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_9294_take__upt,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
=> ( ( take_nat @ M @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% take_upt
thf(fact_9295_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_9296_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_9297_atLeastLessThan__upt,axiom,
( set_or4665077453230672383an_nat
= ( ^ [I3: nat,J3: nat] : ( set_nat2 @ ( upt @ I3 @ J3 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_9298_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_9299_sorted__wrt__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% sorted_wrt_upt
thf(fact_9300_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_9301_atLeastAtMost__upt,axiom,
( set_or1269000886237332187st_nat
= ( ^ [N4: nat,M4: nat] : ( set_nat2 @ ( upt @ N4 @ ( suc @ M4 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_9302_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N4 ) ) ) ) ).
% atLeast_upt
thf(fact_9303_greaterThanAtMost__upt,axiom,
( set_or6659071591806873216st_nat
= ( ^ [N4: nat,M4: nat] : ( set_nat2 @ ( upt @ ( suc @ N4 ) @ ( suc @ M4 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_9304_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map_nat_nat
@ ^ [I3: nat] : ( plus_plus_nat @ I3 @ N )
@ ( upt @ zero_zero_nat @ M ) )
= ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_9305_greaterThanLessThan__upt,axiom,
( set_or5834768355832116004an_nat
= ( ^ [N4: nat,M4: nat] : ( set_nat2 @ ( upt @ ( suc @ N4 ) @ M4 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_9306_atMost__upto,axiom,
( set_ord_atMost_nat
= ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N4 ) ) ) ) ) ).
% atMost_upto
thf(fact_9307_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat
@ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_9308_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_9309_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_9310_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_9311_Rats__no__top__le,axiom,
! [X: real] :
? [X3: real] :
( ( member_real @ X3 @ field_5140801741446780682s_real )
& ( ord_less_eq_real @ X @ X3 ) ) ).
% Rats_no_top_le
thf(fact_9312_Rats__dense__in__real,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [X3: real] :
( ( member_real @ X3 @ field_5140801741446780682s_real )
& ( ord_less_real @ X @ X3 )
& ( ord_less_real @ X3 @ Y ) ) ) ).
% Rats_dense_in_real
thf(fact_9313_Rats__no__bot__less,axiom,
! [X: real] :
? [X3: real] :
( ( member_real @ X3 @ field_5140801741446780682s_real )
& ( ord_less_real @ X3 @ X ) ) ).
% Rats_no_bot_less
thf(fact_9314_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_9315_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_9316_list__encode_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X3: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs2 ) ) ) ).
% list_encode.cases
thf(fact_9317_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q4: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
= ( upt @ M @ Q4 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q4 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_9318_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_9319_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_9320_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_9321_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_9322_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_9323_list__encode_Oelims,axiom,
! [X: list_nat,Y: nat] :
( ( ( nat_list_encode @ X )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != zero_zero_nat ) )
=> ~ ! [X3: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X3 @ Xs2 ) )
=> ( Y
!= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_9324_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_9325_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_9326_list__encode__eq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( nat_list_encode @ X )
= ( nat_list_encode @ Y ) )
= ( X = Y ) ) ).
% list_encode_eq
thf(fact_9327_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_9328_list__encode_Osimps_I1_J,axiom,
( ( nat_list_encode @ nil_nat )
= zero_zero_nat ) ).
% list_encode.simps(1)
thf(fact_9329_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_9330_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_9331_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_9332_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_9333_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_9334_upto_Opelims,axiom,
! [X: int,Xa3: int,Y: list_int] :
( ( ( upto @ X @ Xa3 )
= Y )
=> ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa3 ) )
=> ~ ( ( ( ( ord_less_eq_int @ X @ Xa3 )
=> ( Y
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa3 ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa3 )
=> ( Y = nil_int ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa3 ) ) ) ) ) ).
% upto.pelims
thf(fact_9335_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_9336_upto__rec__numeral_I2_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(2)
thf(fact_9337_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= nil_int )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil
thf(fact_9338_upto__Nil2,axiom,
! [I: int,J: int] :
( ( nil_int
= ( upto @ I @ J ) )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil2
thf(fact_9339_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less_int @ J @ I )
=> ( ( upto @ I @ J )
= nil_int ) ) ).
% upto_empty
thf(fact_9340_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_9341_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_9342_upto__rec__numeral_I1_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(1)
thf(fact_9343_upto__rec__numeral_I4_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(4)
thf(fact_9344_upto__rec__numeral_I3_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(3)
thf(fact_9345_atLeastAtMost__upto,axiom,
( set_or1266510415728281911st_int
= ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ J3 ) ) ) ) ).
% atLeastAtMost_upto
thf(fact_9346_sorted__wrt__upto,axiom,
! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).
% sorted_wrt_upto
thf(fact_9347_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_9348_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_9349_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_9350_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_9351_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_9352_upto_Oelims,axiom,
! [X: int,Xa3: int,Y: list_int] :
( ( ( upto @ X @ Xa3 )
= Y )
=> ( ( ( ord_less_eq_int @ X @ Xa3 )
=> ( Y
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa3 ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa3 )
=> ( Y = nil_int ) ) ) ) ).
% upto.elims
thf(fact_9353_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_9354_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_9355_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_9356_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_9357_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( hd_nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_9358_tl__upt,axiom,
! [M: nat,N: nat] :
( ( tl_nat @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ N ) ) ).
% tl_upt
thf(fact_9359_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 ) ) )
=> ~ ! [X3: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X3 @ Xs2 ) )
=> ( ( Y
= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) )
=> ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_9360_Code__Target__Nat_ONat_Oabs__eq,axiom,
! [X: int] :
( ( code_Target_Nat @ ( code_integer_of_int @ X ) )
= ( nat2 @ X ) ) ).
% Code_Target_Nat.Nat.abs_eq
thf(fact_9361_Code__Target__Nat_ONat_Orep__eq,axiom,
( code_Target_Nat
= ( ^ [X4: code_integer] : ( nat2 @ ( code_int_of_integer @ X4 ) ) ) ) ).
% Code_Target_Nat.Nat.rep_eq
thf(fact_9362_range__mod,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( image_nat_nat
@ ^ [M4: nat] : ( modulo_modulo_nat @ M4 @ N )
@ top_top_set_nat )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% range_mod
thf(fact_9363_mono__Suc,axiom,
order_mono_nat_nat @ suc ).
% mono_Suc
thf(fact_9364_mono__times__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% mono_times_nat
thf(fact_9365_mono__ge2__power__minus__self,axiom,
! [K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( order_mono_nat_nat
@ ^ [M4: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M4 ) @ M4 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_9366_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_9367_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_9368_range__abs__Nats,axiom,
( ( image_int_int @ abs_abs_int @ top_top_set_int )
= semiring_1_Nats_int ) ).
% range_abs_Nats
thf(fact_9369_infinite__UNIV__int,axiom,
~ ( finite_finite_int @ top_top_set_int ) ).
% infinite_UNIV_int
thf(fact_9370_bij__list__encode,axiom,
bij_be8532844293280997160at_nat @ nat_list_encode @ top_top_set_list_nat @ top_top_set_nat ).
% bij_list_encode
thf(fact_9371_surj__list__encode,axiom,
( ( image_list_nat_nat @ nat_list_encode @ top_top_set_list_nat )
= top_top_set_nat ) ).
% surj_list_encode
thf(fact_9372_surj__prod__encode,axiom,
( ( image_2486076414777270412at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat )
= top_top_set_nat ) ).
% surj_prod_encode
thf(fact_9373_bij__prod__encode,axiom,
bij_be5333170631980326235at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat @ top_top_set_nat ).
% bij_prod_encode
thf(fact_9374_int__in__range__abs,axiom,
! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).
% int_in_range_abs
thf(fact_9375_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_9376_root__def,axiom,
( root
= ( ^ [N4: nat,X4: real] :
( if_real @ ( N4 = zero_zero_nat ) @ zero_zero_real
@ ( the_in5290026491893676941l_real @ top_top_set_real
@ ^ [Y5: real] : ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N4 ) )
@ X4 ) ) ) ) ).
% root_def
thf(fact_9377_DERIV__real__root__generic,axiom,
! [N: nat,X: real,D6: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( X != zero_zero_real )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( D6
= ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( D6
= ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( D6
= ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ D6 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_9378_DERIV__even__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_9379_Inf__real__def,axiom,
( comple4887499456419720421f_real
= ( ^ [X8: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X8 ) ) ) ) ) ).
% Inf_real_def
thf(fact_9380_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_9381_Inf__int__def,axiom,
( complete_Inf_Inf_int
= ( ^ [X8: set_int] : ( uminus_uminus_int @ ( complete_Sup_Sup_int @ ( image_int_int @ uminus_uminus_int @ X8 ) ) ) ) ) ).
% Inf_int_def
thf(fact_9382_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_9383_DERIV__fun__pow,axiom,
! [G: real > real,M: real,X: real,N: nat] :
( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X4: real] : ( power_power_real @ ( G @ X4 ) @ N )
@ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_fun_pow
thf(fact_9384_has__real__derivative__powr,axiom,
! [Z2: real,R2: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( has_fi5821293074295781190e_real
@ ^ [Z3: real] : ( powr_real @ Z3 @ R2 )
@ ( times_times_real @ R2 @ ( powr_real @ Z2 @ ( minus_minus_real @ R2 @ one_one_real ) ) )
@ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ).
% has_real_derivative_powr
thf(fact_9385_DERIV__log,axiom,
! [X: real,B: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ ( log2 @ 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_9386_DERIV__fun__powr,axiom,
! [G: real > real,M: real,X: real,R2: real] :
( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X4: real] : ( powr_real @ ( G @ X4 ) @ R2 )
@ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_fun_powr
thf(fact_9387_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_9388_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_9389_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_9390_DERIV__real__sqrt__generic,axiom,
! [X: real,D6: real] :
( ( X != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ X )
=> ( D6
= ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( ( ord_less_real @ X @ zero_zero_real )
=> ( D6
= ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( has_fi5821293074295781190e_real @ sqrt @ D6 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_9391_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_9392_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_9393_DERIV__power__series_H,axiom,
! [R: real,F: nat > real,X0: real] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
=> ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( F @ N4 ) @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ ( power_power_real @ X3 @ N4 ) ) ) )
=> ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
=> ( ( ord_less_real @ zero_zero_real @ R )
=> ( has_fi5821293074295781190e_real
@ ^ [X4: real] :
( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ X4 @ ( 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_9394_DERIV__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_real_root
thf(fact_9395_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_9396_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_9397_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
( ( ( ( Diff @ zero_zero_nat )
= F )
& ! [M2: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
=> ? [T3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ X @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_9398_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M2: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ? [T3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ X @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_9399_DERIV__odd__real__root,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( X != zero_zero_real )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_9400_Maclaurin,axiom,
! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
( ( ord_less_real @ zero_zero_real @ H )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M2: nat,T3: real] :
( ( ( ord_less_nat @ M2 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T3 )
& ( ord_less_eq_real @ T3 @ H ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
=> ? [T3: real] :
( ( ord_less_real @ zero_zero_real @ T3 )
& ( ord_less_real @ T3 @ H )
& ( ( F @ H )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ H @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_9401_Maclaurin2,axiom,
! [H: real,Diff: nat > real > real,F: real > real,N: nat] :
( ( ord_less_real @ zero_zero_real @ H )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M2: nat,T3: real] :
( ( ( ord_less_nat @ M2 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T3 )
& ( ord_less_eq_real @ T3 @ H ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
=> ? [T3: real] :
( ( ord_less_real @ zero_zero_real @ T3 )
& ( ord_less_eq_real @ T3 @ H )
& ( ( F @ H )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ H @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_9402_Maclaurin__minus,axiom,
! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
( ( ord_less_real @ H @ zero_zero_real )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M2: nat,T3: real] :
( ( ( ord_less_nat @ M2 @ N )
& ( ord_less_eq_real @ H @ T3 )
& ( ord_less_eq_real @ T3 @ zero_zero_real ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
=> ? [T3: real] :
( ( ord_less_real @ H @ T3 )
& ( ord_less_real @ T3 @ zero_zero_real )
& ( ( F @ H )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ H @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_9403_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( X != zero_zero_real )
=> ( ! [M2: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ? [T3: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
& ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ X @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_9404_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M2: nat,T3: real] :
( ( ( ord_less_nat @ M2 @ N )
& ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
=> ? [T3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ X @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_9405_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M2: nat,T3: real] :
( ( ( ord_less_nat @ M2 @ N )
& ( ord_less_eq_real @ A @ T3 )
& ( ord_less_eq_real @ T3 @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
=> ( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ X @ B )
=> ( ( X != C )
=> ? [T3: real] :
( ( ( ord_less_real @ X @ C )
=> ( ( ord_less_real @ X @ T3 )
& ( ord_less_real @ T3 @ C ) ) )
& ( ~ ( ord_less_real @ X @ C )
=> ( ( ord_less_real @ C @ T3 )
& ( ord_less_real @ T3 @ X ) ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ C ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_9406_Taylor__up,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M2: nat,T3: real] :
( ( ( ord_less_nat @ M2 @ N )
& ( ord_less_eq_real @ A @ T3 )
& ( ord_less_eq_real @ T3 @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
=> ( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_real @ C @ B )
=> ? [T3: real] :
( ( ord_less_real @ C @ T3 )
& ( ord_less_real @ T3 @ B )
& ( ( F @ B )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ C ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_9407_Taylor__down,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M2: nat,T3: real] :
( ( ( ord_less_nat @ M2 @ N )
& ( ord_less_eq_real @ A @ T3 )
& ( ord_less_eq_real @ T3 @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
=> ( ( ord_less_real @ A @ C )
=> ( ( ord_less_eq_real @ C @ B )
=> ? [T3: real] :
( ( ord_less_real @ A @ T3 )
& ( ord_less_real @ T3 @ C )
& ( ( F @ A )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ C ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_9408_Maclaurin__lemma2,axiom,
! [N: nat,H: real,Diff: nat > real > real,K: nat,B2: real] :
( ! [M2: nat,T3: real] :
( ( ( ord_less_nat @ M2 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T3 )
& ( ord_less_eq_real @ T3 @ H ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
=> ( ( N
= ( suc @ K ) )
=> ! [M5: nat,T4: real] :
( ( ( ord_less_nat @ M5 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T4 )
& ( ord_less_eq_real @ T4 @ H ) )
=> ( has_fi5821293074295781190e_real
@ ^ [U2: real] :
( minus_minus_real @ ( Diff @ M5 @ U2 )
@ ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M5 @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ U2 @ P6 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M5 ) ) )
@ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M5 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M5 ) ) ) ) ) )
@ ( minus_minus_real @ ( Diff @ ( suc @ M5 ) @ T4 )
@ ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M5 ) @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ T4 @ P6 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M5 ) ) ) )
@ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ T4 @ ( minus_minus_nat @ N @ ( suc @ M5 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M5 ) ) ) ) ) ) )
@ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_9409_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( has_fi5821293074295781190e_real
@ ^ [X10: real] :
( suminf_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X10 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
@ ( suminf_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_arctan_series
thf(fact_9410_DERIV__pow,axiom,
! [N: nat,X: real,S: set_real] :
( has_fi5821293074295781190e_real
@ ^ [X4: real] : ( power_power_real @ X4 @ N )
@ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
@ ( topolo2177554685111907308n_real @ X @ S ) ) ).
% DERIV_pow
thf(fact_9411_DERIV__mirror,axiom,
! [F: real > real,Y: real,X: real] :
( ( has_fi5821293074295781190e_real @ F @ Y @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ X ) @ top_top_set_real ) )
= ( has_fi5821293074295781190e_real
@ ^ [X4: real] : ( F @ ( uminus_uminus_real @ X4 ) )
@ ( uminus_uminus_real @ Y )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_mirror
thf(fact_9412_surj__int__encode,axiom,
( ( image_int_nat @ nat_int_encode @ top_top_set_int )
= top_top_set_nat ) ).
% surj_int_encode
thf(fact_9413_int__encode__eq,axiom,
! [X: int,Y: int] :
( ( ( nat_int_encode @ X )
= ( nat_int_encode @ Y ) )
= ( X = Y ) ) ).
% int_encode_eq
thf(fact_9414_bij__int__encode,axiom,
bij_betw_int_nat @ nat_int_encode @ top_top_set_int @ top_top_set_nat ).
% bij_int_encode
thf(fact_9415_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_9416_continuous__floor,axiom,
! [X: real] :
( ~ ( member_real @ X @ ring_1_Ints_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( comp_int_real_real @ ring_1_of_int_real @ archim6058952711729229775r_real ) ) ) ).
% continuous_floor
thf(fact_9417_DERIV__inverse__function,axiom,
! [F: real > real,D6: real,G: real > real,X: real,A: real,B: real] :
( ( has_fi5821293074295781190e_real @ F @ D6 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
=> ( ( D6 != zero_zero_real )
=> ( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ( ! [Y3: real] :
( ( ord_less_real @ A @ Y3 )
=> ( ( ord_less_real @ Y3 @ B )
=> ( ( F @ ( G @ Y3 ) )
= Y3 ) ) )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
=> ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D6 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).
% DERIV_inverse_function
thf(fact_9418_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_9419_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_9420_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_9421_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 ) )
=> ! [N6: 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 ) ) @ N6 ) ) )
@ ( 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 ) ) @ N6 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_9422_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 )
=> ! [N6: 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 ) ) @ N6 ) @ 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 ) ) @ N6 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_9423_summable__Leibniz_H_I5_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
=> ( 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_9424_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_9425_mult__nat__right__at__top,axiom,
! [C: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( filterlim_nat_nat
@ ^ [X4: nat] : ( times_times_nat @ X4 @ C )
@ at_top_nat
@ at_top_nat ) ) ).
% mult_nat_right_at_top
thf(fact_9426_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_9427_nested__sequence__unique,axiom,
! [F: nat > real,G: nat > real] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
=> ( ( filterlim_nat_real
@ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( G @ N4 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat )
=> ? [L2: real] :
( ! [N6: nat] : ( ord_less_eq_real @ ( F @ N6 ) @ L2 )
& ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat )
& ! [N6: nat] : ( ord_less_eq_real @ L2 @ ( G @ N6 ) )
& ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_9428_LIMSEQ__inverse__zero,axiom,
! [X9: nat > real] :
( ! [R4: real] :
? [N9: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N9 @ N2 )
=> ( ord_less_real @ R4 @ ( X9 @ N2 ) ) )
=> ( filterlim_nat_real
@ ^ [N4: nat] : ( inverse_inverse_real @ ( X9 @ N4 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_inverse_zero
thf(fact_9429_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_9430_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_9431_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_9432_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R2: real] :
( filterlim_nat_real
@ ^ [N4: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
@ ( topolo2815343760600316023s_real @ R2 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_9433_increasing__LIMSEQ,axiom,
! [F: nat > real,L: real] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ L )
=> ( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [N6: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N6 ) @ E ) ) )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_9434_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_9435_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_9436_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_9437_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_9438_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_9439_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R2: real] :
( filterlim_nat_real
@ ^ [N4: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) ) )
@ ( topolo2815343760600316023s_real @ R2 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_9440_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_9441_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R2: real] :
( filterlim_nat_real
@ ^ [N4: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) ) ) )
@ ( topolo2815343760600316023s_real @ R2 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_9442_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_9443_summable,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
=> ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( A @ N4 ) ) ) ) ) ) ).
% summable
thf(fact_9444_cos__diff__limit__1,axiom,
! [Theta: nat > real,Theta2: real] :
( ( filterlim_nat_real
@ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat )
=> ~ ! [K2: nat > int] :
~ ( filterlim_nat_real
@ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
@ ( topolo2815343760600316023s_real @ Theta2 )
@ at_top_nat ) ) ).
% cos_diff_limit_1
thf(fact_9445_cos__limit__1,axiom,
! [Theta: nat > real] :
( ( filterlim_nat_real
@ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat )
=> ? [K2: nat > int] :
( filterlim_nat_real
@ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% cos_limit_1
thf(fact_9446_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_9447_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_9448_summable__Leibniz_H_I3_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
=> ( 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_9449_summable__Leibniz_H_I2_J,axiom,
! [A: nat > real,N: nat] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
=> ( 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 ) ) @ N ) ) )
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_9450_sums__alternating__upper__lower,axiom,
! [A: nat > real] :
( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
=> ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ? [L2: real] :
( ! [N6: 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 ) ) @ N6 ) ) )
@ L2 )
& ( 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 @ L2 )
@ at_top_nat )
& ! [N6: nat] :
( ord_less_eq_real @ L2
@ ( 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 ) ) @ N6 ) @ 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 @ L2 )
@ at_top_nat ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_9451_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_9452_summable__Leibniz_H_I4_J,axiom,
! [A: nat > real,N: nat] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
=> ( 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 ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_9453_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_9454_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_9455_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_9456_filterlim__Suc,axiom,
filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% filterlim_Suc
thf(fact_9457_filterlim__real__of__int__at__top,axiom,
filterlim_int_real @ ring_1_of_int_real @ at_top_real @ at_top_int ).
% filterlim_real_of_int_at_top
thf(fact_9458_filterlim__int__sequentially,axiom,
filterlim_nat_int @ semiri1314217659103216013at_int @ at_top_int @ at_top_nat ).
% filterlim_int_sequentially
thf(fact_9459_filterlim__nat__sequentially,axiom,
filterlim_int_nat @ nat2 @ at_top_nat @ at_top_int ).
% filterlim_nat_sequentially
thf(fact_9460_tanh__real__at__top,axiom,
filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).
% tanh_real_at_top
thf(fact_9461_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_9462_tendsto__exp__limit__at__top,axiom,
! [X: real] :
( filterlim_real_real
@ ^ [Y5: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y5 ) ) @ Y5 )
@ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
@ at_top_real ) ).
% tendsto_exp_limit_at_top
thf(fact_9463_tendsto__exp__limit__at__right,axiom,
! [X: real] :
( filterlim_real_real
@ ^ [Y5: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y5 ) ) @ ( divide_divide_real @ one_one_real @ Y5 ) )
@ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% tendsto_exp_limit_at_right
thf(fact_9464_eventually__at__left__to__right,axiom,
! [P: real > $o,A: real] :
( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
= ( eventually_real
@ ^ [X4: real] : ( P @ ( uminus_uminus_real @ X4 ) )
@ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ A ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ A ) ) ) ) ) ).
% eventually_at_left_to_right
thf(fact_9465_eventually__at__right__to__top,axiom,
! [P: real > $o] :
( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
= ( eventually_real
@ ^ [X4: real] : ( P @ ( inverse_inverse_real @ X4 ) )
@ at_top_real ) ) ).
% eventually_at_right_to_top
thf(fact_9466_eventually__at__top__to__right,axiom,
! [P: real > $o] :
( ( eventually_real @ P @ at_top_real )
= ( eventually_real
@ ^ [X4: real] : ( P @ ( inverse_inverse_real @ X4 ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% eventually_at_top_to_right
thf(fact_9467_filterlim__inverse__at__top__right,axiom,
filterlim_real_real @ inverse_inverse_real @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).
% filterlim_inverse_at_top_right
thf(fact_9468_filterlim__inverse__at__right__top,axiom,
filterlim_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) @ at_top_real ).
% filterlim_inverse_at_right_top
thf(fact_9469_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_9470_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_9471_filterlim__pow__at__bot__even,axiom,
! [N: nat,F: real > real,F3: filter_real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( filterlim_real_real
@ ^ [X4: real] : ( power_power_real @ ( F @ X4 ) @ N )
@ at_top_real
@ F3 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_9472_filterlim__uminus__at__bot__at__top,axiom,
filterlim_real_real @ uminus_uminus_real @ at_bot_real @ at_top_real ).
% filterlim_uminus_at_bot_at_top
thf(fact_9473_filterlim__uminus__at__top__at__bot,axiom,
filterlim_real_real @ uminus_uminus_real @ at_top_real @ at_bot_real ).
% filterlim_uminus_at_top_at_bot
thf(fact_9474_greaterThan__0,axiom,
( ( set_or1210151606488870762an_nat @ zero_zero_nat )
= ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% greaterThan_0
thf(fact_9475_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_9476_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_9477_filterlim__inverse__at__bot__neg,axiom,
filterlim_real_real @ inverse_inverse_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5984915006950818249n_real @ zero_zero_real ) ) ).
% filterlim_inverse_at_bot_neg
thf(fact_9478_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_9479_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F: real > real,F3: filter_real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( filterlim_real_real
@ ^ [X4: real] : ( power_power_real @ ( F @ X4 ) @ N )
@ at_bot_real
@ F3 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_9480_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_9481_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_9482_inj__sgn__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( inj_on_real_real
@ ^ [Y5: real] : ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N ) )
@ top_top_set_real ) ) ).
% inj_sgn_power
thf(fact_9483_continuous__on__arcosh_H,axiom,
! [A2: set_real,F: real > real] :
( ( topolo5044208981011980120l_real @ A2 @ F )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
=> ( topolo5044208981011980120l_real @ A2
@ ^ [X4: real] : ( arcosh_real @ ( F @ X4 ) ) ) ) ) ).
% continuous_on_arcosh'
thf(fact_9484_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_9485_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_9486_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_9487_continuous__on__artanh_H,axiom,
! [A2: set_real,F: real > real] :
( ( topolo5044208981011980120l_real @ A2 @ F )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ ( F @ X3 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
=> ( topolo5044208981011980120l_real @ A2
@ ^ [X4: real] : ( artanh_real @ ( F @ X4 ) ) ) ) ) ).
% continuous_on_artanh'
thf(fact_9488_log__inj,axiom,
! [B: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( inj_on_real_real @ ( log2 @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% log_inj
thf(fact_9489_surj__int__decode,axiom,
( ( image_nat_int @ nat_int_decode @ top_top_set_nat )
= top_top_set_int ) ).
% surj_int_decode
thf(fact_9490_int__decode__inverse,axiom,
! [N: nat] :
( ( nat_int_encode @ ( nat_int_decode @ N ) )
= N ) ).
% int_decode_inverse
thf(fact_9491_int__encode__inverse,axiom,
! [X: int] :
( ( nat_int_decode @ ( nat_int_encode @ X ) )
= X ) ).
% int_encode_inverse
thf(fact_9492_inj__list__encode,axiom,
! [A2: set_list_nat] : ( inj_on_list_nat_nat @ nat_list_encode @ A2 ) ).
% inj_list_encode
thf(fact_9493_inj__prod__encode,axiom,
! [A2: set_Pr1261947904930325089at_nat] : ( inj_on2178005380612969504at_nat @ nat_prod_encode @ A2 ) ).
% inj_prod_encode
thf(fact_9494_inj__Suc,axiom,
! [N5: set_nat] : ( inj_on_nat_nat @ suc @ N5 ) ).
% inj_Suc
thf(fact_9495_inj__int__decode,axiom,
! [A2: set_nat] : ( inj_on_nat_int @ nat_int_decode @ A2 ) ).
% inj_int_decode
thf(fact_9496_inj__int__encode,axiom,
! [A2: set_int] : ( inj_on_int_nat @ nat_int_encode @ A2 ) ).
% inj_int_encode
thf(fact_9497_inj__on__diff__nat,axiom,
! [N5: set_nat,K: nat] :
( ! [N2: nat] :
( ( member_nat @ N2 @ N5 )
=> ( ord_less_eq_nat @ K @ N2 ) )
=> ( inj_on_nat_nat
@ ^ [N4: nat] : ( minus_minus_nat @ N4 @ K )
@ N5 ) ) ).
% inj_on_diff_nat
thf(fact_9498_int__decode__eq,axiom,
! [X: nat,Y: nat] :
( ( ( nat_int_decode @ X )
= ( nat_int_decode @ Y ) )
= ( X = Y ) ) ).
% int_decode_eq
thf(fact_9499_inj__on__set__encode,axiom,
inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).
% inj_on_set_encode
thf(fact_9500_bij__int__decode,axiom,
bij_betw_nat_int @ nat_int_decode @ top_top_set_nat @ top_top_set_int ).
% bij_int_decode
thf(fact_9501_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_9502_Gcd__eq__Max,axiom,
! [M10: set_nat] :
( ( finite_finite_nat @ M10 )
=> ( ( M10 != bot_bot_set_nat )
=> ( ~ ( member_nat @ zero_zero_nat @ M10 )
=> ( ( gcd_Gcd_nat @ M10 )
= ( lattic8265883725875713057ax_nat
@ ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [M4: nat] :
( collect_nat
@ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ M4 ) )
@ M10 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_9503_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_9504_sup__nat__def,axiom,
sup_sup_nat = ord_max_nat ).
% sup_nat_def
thf(fact_9505_sup__int__def,axiom,
sup_sup_int = ord_max_int ).
% sup_int_def
thf(fact_9506_card__le__Suc__Max,axiom,
! [S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ S3 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S3 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_9507_Sup__nat__def,axiom,
( complete_Sup_Sup_nat
= ( ^ [X8: set_nat] : ( if_nat @ ( X8 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X8 ) ) ) ) ).
% Sup_nat_def
thf(fact_9508_divide__nat__def,axiom,
( divide_divide_nat
= ( ^ [M4: nat,N4: nat] :
( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat
@ ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N4 ) @ M4 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_9509_Max__divisors__self__int,axiom,
! [N: int] :
( ( N != zero_zero_int )
=> ( ( lattic8263393255366662781ax_int
@ ( collect_int
@ ^ [D3: int] : ( dvd_dvd_int @ D3 @ N ) ) )
= ( abs_abs_int @ N ) ) ) ).
% Max_divisors_self_int
thf(fact_9510_cauchy__def,axiom,
( cauchy
= ( ^ [X8: nat > rat] :
! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [M4: nat] :
( ( ord_less_eq_nat @ K3 @ M4 )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ K3 @ N4 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M4 ) @ ( X8 @ N4 ) ) ) @ R5 ) ) ) ) ) ) ).
% cauchy_def
thf(fact_9511_cauchyI,axiom,
! [X9: nat > rat] :
( ! [R4: rat] :
( ( ord_less_rat @ zero_zero_rat @ R4 )
=> ? [K4: nat] :
! [M2: nat] :
( ( ord_less_eq_nat @ K4 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq_nat @ K4 @ N2 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X9 @ M2 ) @ ( X9 @ N2 ) ) ) @ R4 ) ) ) )
=> ( cauchy @ X9 ) ) ).
% cauchyI
thf(fact_9512_cauchy__add,axiom,
! [X9: nat > rat,Y7: nat > rat] :
( ( cauchy @ X9 )
=> ( ( cauchy @ Y7 )
=> ( cauchy
@ ^ [N4: nat] : ( plus_plus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% cauchy_add
thf(fact_9513_cauchy__mult,axiom,
! [X9: nat > rat,Y7: nat > rat] :
( ( cauchy @ X9 )
=> ( ( cauchy @ Y7 )
=> ( cauchy
@ ^ [N4: nat] : ( times_times_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% cauchy_mult
thf(fact_9514_cauchy__diff,axiom,
! [X9: nat > rat,Y7: nat > rat] :
( ( cauchy @ X9 )
=> ( ( cauchy @ Y7 )
=> ( cauchy
@ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% cauchy_diff
thf(fact_9515_cauchy__const,axiom,
! [X: rat] :
( cauchy
@ ^ [N4: nat] : X ) ).
% cauchy_const
thf(fact_9516_cauchy__minus,axiom,
! [X9: nat > rat] :
( ( cauchy @ X9 )
=> ( cauchy
@ ^ [N4: nat] : ( uminus_uminus_rat @ ( X9 @ N4 ) ) ) ) ).
% cauchy_minus
thf(fact_9517_cauchy__inverse,axiom,
! [X9: nat > rat] :
( ( cauchy @ X9 )
=> ( ~ ( vanishes @ X9 )
=> ( cauchy
@ ^ [N4: nat] : ( inverse_inverse_rat @ ( X9 @ N4 ) ) ) ) ) ).
% cauchy_inverse
thf(fact_9518_cauchy__imp__bounded,axiom,
! [X9: nat > rat] :
( ( cauchy @ X9 )
=> ? [B4: rat] :
( ( ord_less_rat @ zero_zero_rat @ B4 )
& ! [N6: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N6 ) ) @ B4 ) ) ) ).
% cauchy_imp_bounded
thf(fact_9519_vanishes__diff__inverse,axiom,
! [X9: nat > rat,Y7: nat > rat] :
( ( cauchy @ X9 )
=> ( ~ ( vanishes @ X9 )
=> ( ( cauchy @ Y7 )
=> ( ~ ( vanishes @ Y7 )
=> ( ( vanishes
@ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) )
=> ( vanishes
@ ^ [N4: nat] : ( minus_minus_rat @ ( inverse_inverse_rat @ ( X9 @ N4 ) ) @ ( inverse_inverse_rat @ ( Y7 @ N4 ) ) ) ) ) ) ) ) ) ).
% vanishes_diff_inverse
thf(fact_9520_cauchy__not__vanishes__cases,axiom,
! [X9: nat > rat] :
( ( cauchy @ X9 )
=> ( ~ ( vanishes @ X9 )
=> ? [B4: rat] :
( ( ord_less_rat @ zero_zero_rat @ B4 )
& ? [K2: nat] :
( ! [N6: nat] :
( ( ord_less_eq_nat @ K2 @ N6 )
=> ( ord_less_rat @ B4 @ ( uminus_uminus_rat @ ( X9 @ N6 ) ) ) )
| ! [N6: nat] :
( ( ord_less_eq_nat @ K2 @ N6 )
=> ( ord_less_rat @ B4 @ ( X9 @ N6 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes_cases
thf(fact_9521_cauchy__not__vanishes,axiom,
! [X9: nat > rat] :
( ( cauchy @ X9 )
=> ( ~ ( vanishes @ X9 )
=> ? [B4: rat] :
( ( ord_less_rat @ zero_zero_rat @ B4 )
& ? [K2: nat] :
! [N6: nat] :
( ( ord_less_eq_nat @ K2 @ N6 )
=> ( ord_less_rat @ B4 @ ( abs_abs_rat @ ( X9 @ N6 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes
thf(fact_9522_cauchyD,axiom,
! [X9: nat > rat,R2: rat] :
( ( cauchy @ X9 )
=> ( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ? [K2: nat] :
! [M5: nat] :
( ( ord_less_eq_nat @ K2 @ M5 )
=> ! [N6: nat] :
( ( ord_less_eq_nat @ K2 @ N6 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X9 @ M5 ) @ ( X9 @ N6 ) ) ) @ R2 ) ) ) ) ) ).
% cauchyD
thf(fact_9523_le__Real,axiom,
! [X9: nat > rat,Y7: nat > rat] :
( ( cauchy @ X9 )
=> ( ( cauchy @ Y7 )
=> ( ( ord_less_eq_real @ ( real2 @ X9 ) @ ( real2 @ Y7 ) )
= ( ! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [N4: nat] :
( ( ord_less_eq_nat @ K3 @ N4 )
=> ( ord_less_eq_rat @ ( X9 @ N4 ) @ ( plus_plus_rat @ ( Y7 @ N4 ) @ R5 ) ) ) ) ) ) ) ) ).
% le_Real
thf(fact_9524_inverse__Real,axiom,
! [X9: nat > rat] :
( ( cauchy @ X9 )
=> ( ( ( vanishes @ X9 )
=> ( ( inverse_inverse_real @ ( real2 @ X9 ) )
= zero_zero_real ) )
& ( ~ ( vanishes @ X9 )
=> ( ( inverse_inverse_real @ ( real2 @ X9 ) )
= ( real2
@ ^ [N4: nat] : ( inverse_inverse_rat @ ( X9 @ N4 ) ) ) ) ) ) ) ).
% inverse_Real
thf(fact_9525_Real__induct,axiom,
! [P: real > $o,X: real] :
( ! [X15: nat > rat] :
( ( cauchy @ X15 )
=> ( P @ ( real2 @ X15 ) ) )
=> ( P @ X ) ) ).
% Real_induct
thf(fact_9526_zero__real__def,axiom,
( zero_zero_real
= ( real2
@ ^ [N4: nat] : zero_zero_rat ) ) ).
% zero_real_def
thf(fact_9527_one__real__def,axiom,
( one_one_real
= ( real2
@ ^ [N4: nat] : one_one_rat ) ) ).
% one_real_def
thf(fact_9528_of__int__Real,axiom,
( ring_1_of_int_real
= ( ^ [X4: int] :
( real2
@ ^ [N4: nat] : ( ring_1_of_int_rat @ X4 ) ) ) ) ).
% of_int_Real
thf(fact_9529_of__nat__Real,axiom,
( semiri5074537144036343181t_real
= ( ^ [X4: nat] :
( real2
@ ^ [N4: nat] : ( semiri681578069525770553at_rat @ X4 ) ) ) ) ).
% of_nat_Real
thf(fact_9530_minus__Real,axiom,
! [X9: nat > rat] :
( ( cauchy @ X9 )
=> ( ( uminus_uminus_real @ ( real2 @ X9 ) )
= ( real2
@ ^ [N4: nat] : ( uminus_uminus_rat @ ( X9 @ N4 ) ) ) ) ) ).
% minus_Real
thf(fact_9531_add__Real,axiom,
! [X9: nat > rat,Y7: nat > rat] :
( ( cauchy @ X9 )
=> ( ( cauchy @ Y7 )
=> ( ( plus_plus_real @ ( real2 @ X9 ) @ ( real2 @ Y7 ) )
= ( real2
@ ^ [N4: nat] : ( plus_plus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ) ).
% add_Real
thf(fact_9532_mult__Real,axiom,
! [X9: nat > rat,Y7: nat > rat] :
( ( cauchy @ X9 )
=> ( ( cauchy @ Y7 )
=> ( ( times_times_real @ ( real2 @ X9 ) @ ( real2 @ Y7 ) )
= ( real2
@ ^ [N4: nat] : ( times_times_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ) ).
% mult_Real
thf(fact_9533_diff__Real,axiom,
! [X9: nat > rat,Y7: nat > rat] :
( ( cauchy @ X9 )
=> ( ( cauchy @ Y7 )
=> ( ( minus_minus_real @ ( real2 @ X9 ) @ ( real2 @ Y7 ) )
= ( real2
@ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ) ).
% diff_Real
thf(fact_9534_eq__Real,axiom,
! [X9: nat > rat,Y7: nat > rat] :
( ( cauchy @ X9 )
=> ( ( cauchy @ Y7 )
=> ( ( ( real2 @ X9 )
= ( real2 @ Y7 ) )
= ( vanishes
@ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ) ).
% eq_Real
thf(fact_9535_not__positive__Real,axiom,
! [X9: nat > rat] :
( ( cauchy @ X9 )
=> ( ( ~ ( positive2 @ ( real2 @ X9 ) ) )
= ( ! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [N4: nat] :
( ( ord_less_eq_nat @ K3 @ N4 )
=> ( ord_less_eq_rat @ ( X9 @ N4 ) @ R5 ) ) ) ) ) ) ).
% not_positive_Real
thf(fact_9536_positive__Real,axiom,
! [X9: nat > rat] :
( ( cauchy @ X9 )
=> ( ( positive2 @ ( real2 @ X9 ) )
= ( ? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N4: nat] :
( ( ord_less_eq_nat @ K3 @ N4 )
=> ( ord_less_rat @ R5 @ ( X9 @ N4 ) ) ) ) ) ) ) ).
% positive_Real
thf(fact_9537_Real_Opositive__add,axiom,
! [X: real,Y: real] :
( ( positive2 @ X )
=> ( ( positive2 @ Y )
=> ( positive2 @ ( plus_plus_real @ X @ Y ) ) ) ) ).
% Real.positive_add
thf(fact_9538_Real_Opositive__mult,axiom,
! [X: real,Y: real] :
( ( positive2 @ X )
=> ( ( positive2 @ Y )
=> ( positive2 @ ( times_times_real @ X @ Y ) ) ) ) ).
% Real.positive_mult
thf(fact_9539_Real_Opositive__zero,axiom,
~ ( positive2 @ zero_zero_real ) ).
% Real.positive_zero
thf(fact_9540_Real_Opositive__minus,axiom,
! [X: real] :
( ~ ( positive2 @ X )
=> ( ( X != zero_zero_real )
=> ( positive2 @ ( uminus_uminus_real @ X ) ) ) ) ).
% Real.positive_minus
thf(fact_9541_less__real__def,axiom,
( ord_less_real
= ( ^ [X4: real,Y5: real] : ( positive2 @ ( minus_minus_real @ Y5 @ X4 ) ) ) ) ).
% less_real_def
thf(fact_9542_Real_Opositive_Orep__eq,axiom,
( positive2
= ( ^ [X4: real] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N4: nat] :
( ( ord_less_eq_nat @ K3 @ N4 )
=> ( ord_less_rat @ R5 @ ( rep_real @ X4 @ N4 ) ) ) ) ) ) ).
% Real.positive.rep_eq
thf(fact_9543_inverse__real_Oabs__eq,axiom,
! [X: nat > rat] :
( ( realrel @ X @ X )
=> ( ( inverse_inverse_real @ ( real2 @ X ) )
= ( real2
@ ( if_nat_rat @ ( vanishes @ X )
@ ^ [N4: nat] : zero_zero_rat
@ ^ [N4: nat] : ( inverse_inverse_rat @ ( X @ N4 ) ) ) ) ) ) ).
% inverse_real.abs_eq
thf(fact_9544_realrel__refl,axiom,
! [X9: nat > rat] :
( ( cauchy @ X9 )
=> ( realrel @ X9 @ X9 ) ) ).
% realrel_refl
thf(fact_9545_one__real_Orsp,axiom,
( realrel
@ ^ [N4: nat] : one_one_rat
@ ^ [N4: nat] : one_one_rat ) ).
% one_real.rsp
thf(fact_9546_zero__real_Orsp,axiom,
( realrel
@ ^ [N4: nat] : zero_zero_rat
@ ^ [N4: nat] : zero_zero_rat ) ).
% zero_real.rsp
thf(fact_9547_real_Oabs__induct,axiom,
! [P: real > $o,X: real] :
( ! [Y3: nat > rat] :
( ( realrel @ Y3 @ Y3 )
=> ( P @ ( real2 @ Y3 ) ) )
=> ( P @ X ) ) ).
% real.abs_induct
thf(fact_9548_uminus__real_Oabs__eq,axiom,
! [X: nat > rat] :
( ( realrel @ X @ X )
=> ( ( uminus_uminus_real @ ( real2 @ X ) )
= ( real2
@ ^ [N4: nat] : ( uminus_uminus_rat @ ( X @ N4 ) ) ) ) ) ).
% uminus_real.abs_eq
thf(fact_9549_plus__real_Oabs__eq,axiom,
! [Xa3: nat > rat,X: nat > rat] :
( ( realrel @ Xa3 @ Xa3 )
=> ( ( realrel @ X @ X )
=> ( ( plus_plus_real @ ( real2 @ Xa3 ) @ ( real2 @ X ) )
= ( real2
@ ^ [N4: nat] : ( plus_plus_rat @ ( Xa3 @ N4 ) @ ( X @ N4 ) ) ) ) ) ) ).
% plus_real.abs_eq
thf(fact_9550_times__real_Oabs__eq,axiom,
! [Xa3: nat > rat,X: nat > rat] :
( ( realrel @ Xa3 @ Xa3 )
=> ( ( realrel @ X @ X )
=> ( ( times_times_real @ ( real2 @ Xa3 ) @ ( real2 @ X ) )
= ( real2
@ ^ [N4: nat] : ( times_times_rat @ ( Xa3 @ N4 ) @ ( X @ N4 ) ) ) ) ) ) ).
% times_real.abs_eq
thf(fact_9551_realrelI,axiom,
! [X9: nat > rat,Y7: nat > rat] :
( ( cauchy @ X9 )
=> ( ( cauchy @ Y7 )
=> ( ( vanishes
@ ^ [N4: nat] : ( minus_minus_rat @ ( X9 @ N4 ) @ ( Y7 @ N4 ) ) )
=> ( realrel @ X9 @ Y7 ) ) ) ) ).
% realrelI
thf(fact_9552_realrel__def,axiom,
( realrel
= ( ^ [X8: nat > rat,Y8: nat > rat] :
( ( cauchy @ X8 )
& ( cauchy @ Y8 )
& ( vanishes
@ ^ [N4: nat] : ( minus_minus_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) ) ) ) ) ) ).
% realrel_def
thf(fact_9553_Real_Opositive_Oabs__eq,axiom,
! [X: nat > rat] :
( ( realrel @ X @ X )
=> ( ( positive2 @ ( real2 @ X ) )
= ( ? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N4: nat] :
( ( ord_less_eq_nat @ K3 @ N4 )
=> ( ord_less_rat @ R5 @ ( X @ N4 ) ) ) ) ) ) ) ).
% Real.positive.abs_eq
thf(fact_9554_Real_Opositive__def,axiom,
( positive2
= ( map_fu1856342031159181835at_o_o @ rep_real @ id_o
@ ^ [X8: nat > rat] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N4: nat] :
( ( ord_less_eq_nat @ K3 @ N4 )
=> ( ord_less_rat @ R5 @ ( X8 @ N4 ) ) ) ) ) ) ).
% Real.positive_def
thf(fact_9555_inverse__real__def,axiom,
( inverse_inverse_real
= ( map_fu7146612038024189824t_real @ rep_real @ real2
@ ^ [X8: nat > rat] :
( if_nat_rat @ ( vanishes @ X8 )
@ ^ [N4: nat] : zero_zero_rat
@ ^ [N4: nat] : ( inverse_inverse_rat @ ( X8 @ N4 ) ) ) ) ) ).
% inverse_real_def
thf(fact_9556_uminus__real__def,axiom,
( uminus_uminus_real
= ( map_fu7146612038024189824t_real @ rep_real @ real2
@ ^ [X8: nat > rat,N4: nat] : ( uminus_uminus_rat @ ( X8 @ N4 ) ) ) ) ).
% uminus_real_def
thf(fact_9557_times__real__def,axiom,
( times_times_real
= ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
@ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) ) ) ) ).
% times_real_def
thf(fact_9558_plus__real__def,axiom,
( plus_plus_real
= ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
@ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) ) ) ) ).
% plus_real_def
thf(fact_9559_Real_Opositive_Orsp,axiom,
( bNF_re728719798268516973at_o_o @ realrel
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ ^ [X8: nat > rat] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N4: nat] :
( ( ord_less_eq_nat @ K3 @ N4 )
=> ( ord_less_rat @ R5 @ ( X8 @ N4 ) ) ) )
@ ^ [X8: nat > rat] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N4: nat] :
( ( ord_less_eq_nat @ K3 @ N4 )
=> ( ord_less_rat @ R5 @ ( X8 @ N4 ) ) ) ) ) ).
% Real.positive.rsp
thf(fact_9560_cr__real__def,axiom,
( cr_real
= ( ^ [X4: nat > rat,Y5: real] :
( ( realrel @ X4 @ X4 )
& ( ( real2 @ X4 )
= Y5 ) ) ) ) ).
% cr_real_def
thf(fact_9561_times__real_Orsp,axiom,
( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
@ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) )
@ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) ) ) ).
% times_real.rsp
thf(fact_9562_plus__real_Orsp,axiom,
( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
@ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) )
@ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) ) ) ).
% plus_real.rsp
thf(fact_9563_integer__of__natural_Orsp,axiom,
( bNF_re6650684261131312217nt_int
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ^ [Y4: int,Z: int] : Y4 = Z
@ semiri1314217659103216013at_int
@ semiri1314217659103216013at_int ) ).
% integer_of_natural.rsp
thf(fact_9564_uminus__integer_Orsp,axiom,
( bNF_re4712519889275205905nt_int
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ^ [Y4: int,Z: int] : Y4 = Z
@ uminus_uminus_int
@ uminus_uminus_int ) ).
% uminus_integer.rsp
thf(fact_9565_natural__of__integer_Orsp,axiom,
( bNF_re3715656647883201625at_nat
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ nat2
@ nat2 ) ).
% natural_of_integer.rsp
thf(fact_9566_Suc_Orsp,axiom,
( bNF_re5653821019739307937at_nat
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ suc
@ suc ) ).
% Suc.rsp
thf(fact_9567_less__integer_Orsp,axiom,
( bNF_re3403563459893282935_int_o
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ( bNF_re5089333283451836215nt_o_o
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ord_less_int
@ ord_less_int ) ).
% less_integer.rsp
thf(fact_9568_less__natural_Orsp,axiom,
( bNF_re578469030762574527_nat_o
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ( bNF_re4705727531993890431at_o_o
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ord_less_nat
@ ord_less_nat ) ).
% less_natural.rsp
thf(fact_9569_dup_Orsp,axiom,
( bNF_re4712519889275205905nt_int
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 )
@ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 ) ) ).
% dup.rsp
thf(fact_9570_sub_Orsp,axiom,
( bNF_re8402795839162346335um_int
@ ^ [Y4: num,Z: num] : Y4 = Z
@ ( bNF_re1822329894187522285nt_int
@ ^ [Y4: num,Z: num] : Y4 = Z
@ ^ [Y4: int,Z: int] : Y4 = Z )
@ ^ [M4: num,N4: num] : ( minus_minus_int @ ( numeral_numeral_int @ M4 ) @ ( numeral_numeral_int @ N4 ) )
@ ^ [M4: num,N4: num] : ( minus_minus_int @ ( numeral_numeral_int @ M4 ) @ ( numeral_numeral_int @ N4 ) ) ) ).
% sub.rsp
thf(fact_9571_minus__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ^ [Y4: int,Z: int] : Y4 = Z )
@ minus_minus_int
@ minus_minus_int ) ).
% minus_integer.rsp
thf(fact_9572_minus__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ^ [Y4: nat,Z: nat] : Y4 = Z )
@ minus_minus_nat
@ minus_minus_nat ) ).
% minus_natural.rsp
thf(fact_9573_plus__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ^ [Y4: int,Z: int] : Y4 = Z )
@ plus_plus_int
@ plus_plus_int ) ).
% plus_integer.rsp
thf(fact_9574_plus__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ^ [Y4: nat,Z: nat] : Y4 = Z )
@ plus_plus_nat
@ plus_plus_nat ) ).
% plus_natural.rsp
thf(fact_9575_num__of__integer_Orsp,axiom,
( bNF_re7626690874201225453um_num
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ^ [Y4: num,Z: num] : Y4 = Z
@ ( comp_nat_num_int @ num_of_nat @ nat2 )
@ ( comp_nat_num_int @ num_of_nat @ nat2 ) ) ).
% num_of_integer.rsp
thf(fact_9576_uminus__real_Orsp,axiom,
( bNF_re895249473297799549at_rat @ realrel @ realrel
@ ^ [X8: nat > rat,N4: nat] : ( uminus_uminus_rat @ ( X8 @ N4 ) )
@ ^ [X8: nat > rat,N4: nat] : ( uminus_uminus_rat @ ( X8 @ N4 ) ) ) ).
% uminus_real.rsp
thf(fact_9577_inverse__real_Orsp,axiom,
( bNF_re895249473297799549at_rat @ realrel @ realrel
@ ^ [X8: nat > rat] :
( if_nat_rat @ ( vanishes @ X8 )
@ ^ [N4: nat] : zero_zero_rat
@ ^ [N4: nat] : ( inverse_inverse_rat @ ( X8 @ N4 ) ) )
@ ^ [X8: nat > rat] :
( if_nat_rat @ ( vanishes @ X8 )
@ ^ [N4: nat] : zero_zero_rat
@ ^ [N4: nat] : ( inverse_inverse_rat @ ( X8 @ N4 ) ) ) ) ).
% inverse_real.rsp
thf(fact_9578_Real_Opositive_Otransfer,axiom,
( bNF_re4297313714947099218al_o_o @ pcr_real
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ ^ [X8: nat > rat] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N4: nat] :
( ( ord_less_eq_nat @ K3 @ N4 )
=> ( ord_less_rat @ R5 @ ( X8 @ N4 ) ) ) )
@ positive2 ) ).
% Real.positive.transfer
thf(fact_9579_times__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y5 @ U2 ) ) ) ) )
@ times_times_int ) ).
% times_int.transfer
thf(fact_9580_real_Orel__eq__transfer,axiom,
( bNF_re4521903465945308077real_o @ pcr_real
@ ( bNF_re4297313714947099218al_o_o @ pcr_real
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ realrel
@ ^ [Y4: real,Z: real] : Y4 = Z ) ).
% real.rel_eq_transfer
thf(fact_9581_real_Opcr__cr__eq,axiom,
pcr_real = cr_real ).
% real.pcr_cr_eq
thf(fact_9582_zero__real_Otransfer,axiom,
( pcr_real
@ ^ [N4: nat] : zero_zero_rat
@ zero_zero_real ) ).
% zero_real.transfer
thf(fact_9583_one__real_Otransfer,axiom,
( pcr_real
@ ^ [N4: nat] : one_one_rat
@ one_one_real ) ).
% one_real.transfer
thf(fact_9584_cr__real__eq,axiom,
( pcr_real
= ( ^ [X4: nat > rat,Y5: real] :
( ( cauchy @ X4 )
& ( ( real2 @ X4 )
= Y5 ) ) ) ) ).
% cr_real_eq
thf(fact_9585_uminus__real_Otransfer,axiom,
( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
@ ^ [X8: nat > rat,N4: nat] : ( uminus_uminus_rat @ ( X8 @ N4 ) )
@ uminus_uminus_real ) ).
% uminus_real.transfer
thf(fact_9586_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).
% zero_int.transfer
thf(fact_9587_plus__real_Otransfer,axiom,
( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
@ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) )
@ plus_plus_real ) ).
% plus_real.transfer
thf(fact_9588_times__real_Otransfer,axiom,
( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
@ ^ [X8: nat > rat,Y8: nat > rat,N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y8 @ N4 ) )
@ times_times_real ) ).
% times_real.transfer
thf(fact_9589_int__transfer,axiom,
( bNF_re6830278522597306478at_int
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ pcr_int
@ ^ [N4: nat] : ( product_Pair_nat_nat @ N4 @ zero_zero_nat )
@ semiri1314217659103216013at_int ) ).
% int_transfer
thf(fact_9590_uminus__int_Otransfer,axiom,
( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
@ ( produc2626176000494625587at_nat
@ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 ) )
@ uminus_uminus_int ) ).
% uminus_int.transfer
thf(fact_9591_nat_Otransfer,axiom,
( bNF_re4555766996558763186at_nat @ pcr_int
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ( produc6842872674320459806at_nat @ minus_minus_nat )
@ nat2 ) ).
% nat.transfer
thf(fact_9592_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% one_int.transfer
thf(fact_9593_inverse__real_Otransfer,axiom,
( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
@ ^ [X8: nat > rat] :
( if_nat_rat @ ( vanishes @ X8 )
@ ^ [N4: nat] : zero_zero_rat
@ ^ [N4: nat] : ( inverse_inverse_rat @ ( X8 @ N4 ) ) )
@ inverse_inverse_real ) ).
% inverse_real.transfer
thf(fact_9594_less__int_Otransfer,axiom,
( bNF_re717283939379294677_int_o @ pcr_int
@ ( bNF_re6644619430987730960nt_o_o @ pcr_int
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) )
@ ord_less_int ) ).
% less_int.transfer
thf(fact_9595_less__eq__int_Otransfer,axiom,
( bNF_re717283939379294677_int_o @ pcr_int
@ ( bNF_re6644619430987730960nt_o_o @ pcr_int
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) )
@ ord_less_eq_int ) ).
% less_eq_int.transfer
thf(fact_9596_plus__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) ) )
@ plus_plus_int ) ).
% plus_int.transfer
thf(fact_9597_minus__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y5 @ U2 ) ) ) )
@ minus_minus_int ) ).
% minus_int.transfer
thf(fact_9598_num__of__integer__def,axiom,
( code_num_of_integer
= ( map_fu1227494855608507351um_num @ code_int_of_integer @ id_num @ ( comp_nat_num_int @ num_of_nat @ nat2 ) ) ) ).
% num_of_integer_def
thf(fact_9599_times__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y5 @ U2 ) ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y5 @ U2 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_9600_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_9601_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_9602_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_9603_uminus__int_Orsp,axiom,
( bNF_re2241393799969408733at_nat @ intrel @ intrel
@ ( produc2626176000494625587at_nat
@ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 ) )
@ ( produc2626176000494625587at_nat
@ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 ) ) ) ).
% uminus_int.rsp
thf(fact_9604_nat_Orsp,axiom,
( bNF_re8246922863344978751at_nat @ intrel
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ( produc6842872674320459806at_nat @ minus_minus_nat )
@ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).
% nat.rsp
thf(fact_9605_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_9606_intrel__def,axiom,
( intrel
= ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] :
( ( plus_plus_nat @ X4 @ V4 )
= ( plus_plus_nat @ U2 @ Y5 ) ) ) ) ) ).
% intrel_def
thf(fact_9607_less__int_Orsp,axiom,
( bNF_re4202695980764964119_nat_o @ intrel
@ ( bNF_re3666534408544137501at_o_o @ intrel
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) ) ) ).
% less_int.rsp
thf(fact_9608_less__eq__int_Orsp,axiom,
( bNF_re4202695980764964119_nat_o @ intrel
@ ( bNF_re3666534408544137501at_o_o @ intrel
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_9609_int_Orel__eq__transfer,axiom,
( bNF_re717283939379294677_int_o @ pcr_int
@ ( bNF_re6644619430987730960nt_o_o @ pcr_int
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ intrel
@ ^ [Y4: int,Z: int] : Y4 = Z ) ).
% int.rel_eq_transfer
thf(fact_9610_minus__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y5 @ U2 ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y5 @ U2 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_9611_plus__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_9612_finite__le__enumerate,axiom,
! [S3: set_nat,N: nat] :
( ( finite_finite_nat @ S3 )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S3 ) )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S3 @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_9613_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= zero_zero_nat ) ) ).
% Least_eq_0
thf(fact_9614_Least__Suc2,axiom,
! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
( ( P @ N )
=> ( ( Q @ M )
=> ( ~ ( P @ zero_zero_nat )
=> ( ! [K2: nat] :
( ( P @ ( suc @ K2 ) )
= ( Q @ K2 ) )
=> ( ( ord_Least_nat @ P )
= ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_9615_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= ( suc
@ ( ord_Least_nat
@ ^ [M4: nat] : ( P @ ( suc @ M4 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_9616_Sup__real__def,axiom,
( comple1385675409528146559p_real
= ( ^ [X8: set_real] :
( ord_Least_real
@ ^ [Z3: real] :
! [X4: real] :
( ( member_real @ X4 @ X8 )
=> ( ord_less_eq_real @ X4 @ Z3 ) ) ) ) ) ).
% Sup_real_def
thf(fact_9617_rat__sgn__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( sgn_sgn_rat @ P5 ) )
= ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P5 ) ) ) @ one_one_int ) ) ).
% rat_sgn_code
thf(fact_9618_bezw_Oelims,axiom,
! [X: nat,Xa3: nat,Y: product_prod_int_int] :
( ( ( bezw @ X @ Xa3 )
= Y )
=> ( ( ( Xa3 = zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa3 != zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa3 @ ( modulo_modulo_nat @ X @ Xa3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa3 @ ( modulo_modulo_nat @ X @ Xa3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa3 @ ( modulo_modulo_nat @ X @ Xa3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa3 ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_9619_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_9620_quotient__of__denom__pos_H,axiom,
! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).
% quotient_of_denom_pos'
thf(fact_9621_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_9622_bezw_Osimps,axiom,
( bezw
= ( ^ [X4: nat,Y5: nat] : ( if_Pro3027730157355071871nt_int @ ( Y5 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X4 @ Y5 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X4 @ Y5 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X4 @ Y5 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Y5 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_9623_one__mod__minus__numeral,axiom,
! [N: num] :
( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% one_mod_minus_numeral
thf(fact_9624_minus__one__mod__numeral,axiom,
! [N: num] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% minus_one_mod_numeral
thf(fact_9625_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_9626_minus__numeral__mod__numeral,axiom,
! [M: num,N: num] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% minus_numeral_mod_numeral
thf(fact_9627_numeral__mod__minus__numeral,axiom,
! [M: num,N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).
% numeral_mod_minus_numeral
thf(fact_9628_Divides_Oadjust__mod__def,axiom,
( adjust_mod
= ( ^ [L3: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L3 @ R5 ) ) ) ) ).
% Divides.adjust_mod_def
thf(fact_9629_bezw_Opelims,axiom,
! [X: nat,Xa3: nat,Y: product_prod_int_int] :
( ( ( bezw @ X @ Xa3 )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa3 ) )
=> ~ ( ( ( ( Xa3 = zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa3 != zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa3 @ ( modulo_modulo_nat @ X @ Xa3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa3 @ ( modulo_modulo_nat @ X @ Xa3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa3 @ ( modulo_modulo_nat @ X @ Xa3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa3 ) ) ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa3 ) ) ) ) ) ).
% bezw.pelims
thf(fact_9630_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa3: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa3 )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa3 ) )
=> ~ ( ( ( ( ord_less_eq_nat @ Xa3 @ X )
=> ( Y
= ( product_Pair_nat_nat @ Xa3 @ ( minus_minus_nat @ X @ Xa3 ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa3 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa3 @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa3 ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_9631_Rat_Opositive_Orep__eq,axiom,
( positive
= ( ^ [X4: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X4 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X4 ) ) ) ) ) ) ).
% Rat.positive.rep_eq
thf(fact_9632_Rat_Opositive__def,axiom,
( positive
= ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
@ ^ [X4: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ).
% Rat.positive_def
thf(fact_9633_normalize__def,axiom,
( normalize
= ( ^ [P6: product_prod_int_int] :
( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P6 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) )
@ ( if_Pro3027730157355071871nt_int
@ ( ( product_snd_int_int @ P6 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_9634_gcd__1__int,axiom,
! [M: int] :
( ( gcd_gcd_int @ M @ one_one_int )
= one_one_int ) ).
% gcd_1_int
thf(fact_9635_gcd__neg2__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) )
= ( gcd_gcd_int @ X @ Y ) ) ).
% gcd_neg2_int
thf(fact_9636_gcd__neg1__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y )
= ( gcd_gcd_int @ X @ Y ) ) ).
% gcd_neg1_int
thf(fact_9637_abs__gcd__int,axiom,
! [X: int,Y: int] :
( ( abs_abs_int @ ( gcd_gcd_int @ X @ Y ) )
= ( gcd_gcd_int @ X @ Y ) ) ).
% abs_gcd_int
thf(fact_9638_gcd__abs1__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ ( abs_abs_int @ X ) @ Y )
= ( gcd_gcd_int @ X @ Y ) ) ).
% gcd_abs1_int
thf(fact_9639_gcd__abs2__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ X @ ( abs_abs_int @ Y ) )
= ( gcd_gcd_int @ X @ Y ) ) ).
% gcd_abs2_int
thf(fact_9640_gcd__pos__int,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
= ( ( M != zero_zero_int )
| ( N != zero_zero_int ) ) ) ).
% gcd_pos_int
thf(fact_9641_gcd__neg__numeral__1__int,axiom,
! [N: num,X: int] :
( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X )
= ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X ) ) ).
% gcd_neg_numeral_1_int
thf(fact_9642_gcd__neg__numeral__2__int,axiom,
! [X: int,N: num] :
( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( gcd_gcd_int @ X @ ( numeral_numeral_int @ N ) ) ) ).
% gcd_neg_numeral_2_int
thf(fact_9643_gcd__0__int,axiom,
! [X: int] :
( ( gcd_gcd_int @ X @ zero_zero_int )
= ( abs_abs_int @ X ) ) ).
% gcd_0_int
thf(fact_9644_gcd__0__left__int,axiom,
! [X: int] :
( ( gcd_gcd_int @ zero_zero_int @ X )
= ( abs_abs_int @ X ) ) ).
% gcd_0_left_int
thf(fact_9645_gcd__proj1__if__dvd__int,axiom,
! [X: int,Y: int] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( gcd_gcd_int @ X @ Y )
= ( abs_abs_int @ X ) ) ) ).
% gcd_proj1_if_dvd_int
thf(fact_9646_gcd__proj2__if__dvd__int,axiom,
! [Y: int,X: int] :
( ( dvd_dvd_int @ Y @ X )
=> ( ( gcd_gcd_int @ X @ Y )
= ( abs_abs_int @ Y ) ) ) ).
% gcd_proj2_if_dvd_int
thf(fact_9647_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_9648_gcd__idem__int,axiom,
! [X: int] :
( ( gcd_gcd_int @ X @ X )
= ( abs_abs_int @ X ) ) ).
% gcd_idem_int
thf(fact_9649_gcd__red__int,axiom,
( gcd_gcd_int
= ( ^ [X4: int,Y5: int] : ( gcd_gcd_int @ Y5 @ ( modulo_modulo_int @ X4 @ Y5 ) ) ) ) ).
% gcd_red_int
thf(fact_9650_bezout__int,axiom,
! [X: int,Y: int] :
? [U3: int,V3: int] :
( ( plus_plus_int @ ( times_times_int @ U3 @ X ) @ ( times_times_int @ V3 @ Y ) )
= ( gcd_gcd_int @ X @ Y ) ) ).
% bezout_int
thf(fact_9651_gcd__mult__distrib__int,axiom,
! [K: int,M: int,N: int] :
( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
= ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).
% gcd_mult_distrib_int
thf(fact_9652_gcd__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ^ [Y4: int,Z: int] : Y4 = Z )
@ gcd_gcd_int
@ gcd_gcd_int ) ).
% gcd_integer.rsp
thf(fact_9653_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_9654_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_9655_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_9656_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_9657_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_9658_gcd__code__int,axiom,
( gcd_gcd_int
= ( ^ [K3: int,L3: int] : ( abs_abs_int @ ( if_int @ ( L3 = zero_zero_int ) @ K3 @ ( gcd_gcd_int @ L3 @ ( modulo_modulo_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L3 ) ) ) ) ) ) ) ).
% gcd_code_int
thf(fact_9659_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_9660_gcd__is__Max__divisors__int,axiom,
! [N: int,M: int] :
( ( N != zero_zero_int )
=> ( ( gcd_gcd_int @ M @ N )
= ( lattic8263393255366662781ax_int
@ ( collect_int
@ ^ [D3: int] :
( ( dvd_dvd_int @ D3 @ M )
& ( dvd_dvd_int @ D3 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_int
thf(fact_9661_plus__rat_Otransfer,axiom,
( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
@ ^ [X4: product_prod_int_int,Y5: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y5 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ X4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y5 ) ) )
@ plus_plus_rat ) ).
% plus_rat.transfer
thf(fact_9662_inverse__rat_Otransfer,axiom,
( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
@ ^ [X4: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X4 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X4 ) @ ( product_fst_int_int @ X4 ) ) )
@ inverse_inverse_rat ) ).
% inverse_rat.transfer
thf(fact_9663_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_9664_gcd__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( gcd_gcd_nat @ zero_zero_nat @ A )
= A ) ).
% gcd_nat.left_neutral
thf(fact_9665_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_9666_gcd__nat_Oright__neutral,axiom,
! [A: nat] :
( ( gcd_gcd_nat @ A @ zero_zero_nat )
= A ) ).
% gcd_nat.right_neutral
thf(fact_9667_gcd__0__nat,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ X @ zero_zero_nat )
= X ) ).
% gcd_0_nat
thf(fact_9668_gcd__0__left__nat,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ zero_zero_nat @ X )
= X ) ).
% gcd_0_left_nat
thf(fact_9669_gcd__1__nat,axiom,
! [M: nat] :
( ( gcd_gcd_nat @ M @ one_one_nat )
= one_one_nat ) ).
% gcd_1_nat
thf(fact_9670_gcd__proj2__if__dvd__nat,axiom,
! [Y: nat,X: nat] :
( ( dvd_dvd_nat @ Y @ X )
=> ( ( gcd_gcd_nat @ X @ Y )
= Y ) ) ).
% gcd_proj2_if_dvd_nat
thf(fact_9671_gcd__proj1__if__dvd__nat,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( gcd_gcd_nat @ X @ Y )
= X ) ) ).
% gcd_proj1_if_dvd_nat
thf(fact_9672_gcd__nat_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
= ( ( dvd_dvd_nat @ A @ B )
& ( dvd_dvd_nat @ A @ C ) ) ) ).
% gcd_nat.bounded_iff
thf(fact_9673_gcd__nat_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( gcd_gcd_nat @ A @ B )
= B ) ) ).
% gcd_nat.absorb2
thf(fact_9674_gcd__nat_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( gcd_gcd_nat @ A @ B )
= A ) ) ).
% gcd_nat.absorb1
thf(fact_9675_gcd__Suc__0,axiom,
! [M: nat] :
( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( suc @ zero_zero_nat ) ) ).
% gcd_Suc_0
thf(fact_9676_gcd__pos__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
= ( ( M != zero_zero_nat )
| ( N != zero_zero_nat ) ) ) ).
% gcd_pos_nat
thf(fact_9677_gcd__int__int__eq,axiom,
! [M: nat,N: nat] :
( ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ M @ N ) ) ) ).
% gcd_int_int_eq
thf(fact_9678_gcd__nat__abs__right__eq,axiom,
! [N: nat,K: int] :
( ( gcd_gcd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
= ( nat2 @ ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% gcd_nat_abs_right_eq
thf(fact_9679_gcd__nat__abs__left__eq,axiom,
! [K: int,N: nat] :
( ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
= ( nat2 @ ( gcd_gcd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% gcd_nat_abs_left_eq
thf(fact_9680_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_9681_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_9682_gcd__diff1__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
= ( gcd_gcd_nat @ M @ N ) ) ) ).
% gcd_diff1_nat
thf(fact_9683_gcd__diff2__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
= ( gcd_gcd_nat @ M @ N ) ) ) ).
% gcd_diff2_nat
thf(fact_9684_gcd__code__integer,axiom,
( gcd_gcd_Code_integer
= ( ^ [K3: code_integer,L3: code_integer] : ( abs_abs_Code_integer @ ( if_Code_integer @ ( L3 = zero_z3403309356797280102nteger ) @ K3 @ ( gcd_gcd_Code_integer @ L3 @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L3 ) ) ) ) ) ) ) ).
% gcd_code_integer
thf(fact_9685_gcd__integer_Orep__eq,axiom,
! [X: code_integer,Xa3: code_integer] :
( ( code_int_of_integer @ ( gcd_gcd_Code_integer @ X @ Xa3 ) )
= ( gcd_gcd_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% gcd_integer.rep_eq
thf(fact_9686_gcd__red__nat,axiom,
( gcd_gcd_nat
= ( ^ [X4: nat,Y5: nat] : ( gcd_gcd_nat @ Y5 @ ( modulo_modulo_nat @ X4 @ Y5 ) ) ) ) ).
% gcd_red_nat
thf(fact_9687_gcd__unique__nat,axiom,
! [D: nat,A: nat,B: nat] :
( ( ( dvd_dvd_nat @ D @ A )
& ( dvd_dvd_nat @ D @ B )
& ! [E3: nat] :
( ( ( dvd_dvd_nat @ E3 @ A )
& ( dvd_dvd_nat @ E3 @ B ) )
=> ( dvd_dvd_nat @ E3 @ D ) ) )
= ( D
= ( gcd_gcd_nat @ A @ B ) ) ) ).
% gcd_unique_nat
thf(fact_9688_gcd__nat_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
& ( ( gcd_gcd_nat @ A @ B )
!= C ) ) ) ).
% gcd_nat.strict_coboundedI2
thf(fact_9689_gcd__nat_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) )
=> ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
& ( ( gcd_gcd_nat @ A @ B )
!= C ) ) ) ).
% gcd_nat.strict_coboundedI1
thf(fact_9690_gcd__nat_Ostrict__order__iff,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
= ( ( A
= ( gcd_gcd_nat @ A @ B ) )
& ( A != B ) ) ) ).
% gcd_nat.strict_order_iff
thf(fact_9691_gcd__nat_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
& ( A
!= ( gcd_gcd_nat @ B @ C ) ) )
=> ~ ( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ~ ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_boundedE
thf(fact_9692_gcd__nat_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).
% gcd_nat.coboundedI2
thf(fact_9693_gcd__nat_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).
% gcd_nat.coboundedI1
thf(fact_9694_gcd__nat_Oabsorb__iff2,axiom,
( dvd_dvd_nat
= ( ^ [B3: nat,A3: nat] :
( ( gcd_gcd_nat @ A3 @ B3 )
= B3 ) ) ) ).
% gcd_nat.absorb_iff2
thf(fact_9695_gcd__nat_Oabsorb__iff1,axiom,
( dvd_dvd_nat
= ( ^ [A3: nat,B3: nat] :
( ( gcd_gcd_nat @ A3 @ B3 )
= A3 ) ) ) ).
% gcd_nat.absorb_iff1
thf(fact_9696_gcd__nat_Ocobounded2,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ).
% gcd_nat.cobounded2
thf(fact_9697_gcd__nat_Ocobounded1,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ).
% gcd_nat.cobounded1
thf(fact_9698_gcd__nat_Oorder__iff,axiom,
( dvd_dvd_nat
= ( ^ [A3: nat,B3: nat] :
( A3
= ( gcd_gcd_nat @ A3 @ B3 ) ) ) ) ).
% gcd_nat.order_iff
thf(fact_9699_gcd__nat_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) ) ) ) ).
% gcd_nat.boundedI
thf(fact_9700_gcd__nat_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
=> ~ ( ( dvd_dvd_nat @ A @ B )
=> ~ ( dvd_dvd_nat @ A @ C ) ) ) ).
% gcd_nat.boundedE
thf(fact_9701_gcd__nat_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ( dvd_dvd_nat @ B @ A )
& ( B != A ) )
=> ( ( gcd_gcd_nat @ A @ B )
= B ) ) ).
% gcd_nat.absorb4
thf(fact_9702_gcd__nat_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( ( gcd_gcd_nat @ A @ B )
= A ) ) ).
% gcd_nat.absorb3
thf(fact_9703_gcd__nat_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( gcd_gcd_nat @ A @ B ) )
=> ( dvd_dvd_nat @ A @ B ) ) ).
% gcd_nat.orderI
thf(fact_9704_gcd__nat_OorderE,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( A
= ( gcd_gcd_nat @ A @ B ) ) ) ).
% gcd_nat.orderE
thf(fact_9705_gcd__nat_Omono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ B @ D )
=> ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ ( gcd_gcd_nat @ C @ D ) ) ) ) ).
% gcd_nat.mono
thf(fact_9706_gcd__integer_Oabs__eq,axiom,
! [Xa3: int,X: int] :
( ( gcd_gcd_Code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( gcd_gcd_int @ Xa3 @ X ) ) ) ).
% gcd_integer.abs_eq
thf(fact_9707_gcd__mult__distrib__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
= ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% gcd_mult_distrib_nat
thf(fact_9708_gcd__nat_Oelims,axiom,
! [X: nat,Xa3: nat,Y: nat] :
( ( ( gcd_gcd_nat @ X @ Xa3 )
= Y )
=> ( ( ( Xa3 = zero_zero_nat )
=> ( Y = X ) )
& ( ( Xa3 != zero_zero_nat )
=> ( Y
= ( gcd_gcd_nat @ Xa3 @ ( modulo_modulo_nat @ X @ Xa3 ) ) ) ) ) ) ).
% gcd_nat.elims
thf(fact_9709_gcd__nat_Osimps,axiom,
( gcd_gcd_nat
= ( ^ [X4: nat,Y5: nat] : ( if_nat @ ( Y5 = zero_zero_nat ) @ X4 @ ( gcd_gcd_nat @ Y5 @ ( modulo_modulo_nat @ X4 @ Y5 ) ) ) ) ) ).
% gcd_nat.simps
thf(fact_9710_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_9711_Gcd__in,axiom,
! [A2: set_nat] :
( ! [A4: nat,B4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_nat @ B4 @ A2 )
=> ( member_nat @ ( gcd_gcd_nat @ A4 @ B4 ) @ A2 ) ) )
=> ( ( A2 != bot_bot_set_nat )
=> ( member_nat @ ( gcd_Gcd_nat @ A2 ) @ A2 ) ) ) ).
% Gcd_in
thf(fact_9712_bezout__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [X3: nat,Y3: nat] :
( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% bezout_nat
thf(fact_9713_bezout__gcd__nat_H,axiom,
! [B: nat,A: nat] :
? [X3: nat,Y3: nat] :
( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y3 ) @ ( times_times_nat @ A @ X3 ) )
& ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
= ( gcd_gcd_nat @ A @ B ) ) )
| ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y3 ) @ ( times_times_nat @ B @ X3 ) )
& ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
= ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_9714_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_9715_gcd__int__def,axiom,
( gcd_gcd_int
= ( ^ [X4: int,Y5: int] : ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ X4 ) ) @ ( nat2 @ ( abs_abs_int @ Y5 ) ) ) ) ) ) ).
% gcd_int_def
thf(fact_9716_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
@ ^ [M4: nat,N4: nat] :
( ( dvd_dvd_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_9717_one__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).
% one_rat.transfer
thf(fact_9718_gcd__is__Max__divisors__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( gcd_gcd_nat @ M @ N )
= ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [D3: nat] :
( ( dvd_dvd_nat @ D3 @ M )
& ( dvd_dvd_nat @ D3 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_9719_zero__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).
% zero_rat.transfer
thf(fact_9720_Fract_Otransfer,axiom,
( bNF_re3461391660133120880nt_rat
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ( bNF_re2214769303045360666nt_rat
@ ^ [Y4: int,Z: int] : Y4 = Z
@ pcr_rat )
@ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B3 ) )
@ fract ) ).
% Fract.transfer
thf(fact_9721_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_9722_uminus__rat_Otransfer,axiom,
( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
@ ^ [X4: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X4 ) ) @ ( product_snd_int_int @ X4 ) )
@ uminus_uminus_rat ) ).
% uminus_rat.transfer
thf(fact_9723_Rat_Opositive_Otransfer,axiom,
( bNF_re1494630372529172596at_o_o @ pcr_rat
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ ^ [X4: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) )
@ positive ) ).
% Rat.positive.transfer
thf(fact_9724_gcd__nat_Opelims,axiom,
! [X: nat,Xa3: nat,Y: nat] :
( ( ( gcd_gcd_nat @ X @ Xa3 )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa3 ) )
=> ~ ( ( ( ( Xa3 = zero_zero_nat )
=> ( Y = X ) )
& ( ( Xa3 != zero_zero_nat )
=> ( Y
= ( gcd_gcd_nat @ Xa3 @ ( modulo_modulo_nat @ X @ Xa3 ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa3 ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_9725_gcd__idem__nat,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ X @ X )
= X ) ).
% gcd_idem_nat
thf(fact_9726_gcd__nat_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( gcd_gcd_nat @ ( gcd_gcd_nat @ A @ B ) @ B )
= ( gcd_gcd_nat @ A @ B ) ) ).
% gcd_nat.right_idem
thf(fact_9727_gcd__nat_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( gcd_gcd_nat @ A @ ( gcd_gcd_nat @ A @ B ) )
= ( gcd_gcd_nat @ A @ B ) ) ).
% gcd_nat.left_idem
thf(fact_9728_gcd__nat_Oidem,axiom,
! [A: nat] :
( ( gcd_gcd_nat @ A @ A )
= A ) ).
% gcd_nat.idem
thf(fact_9729_of__rat__Real,axiom,
( field_7254667332652039916t_real
= ( ^ [X4: rat] :
( real2
@ ^ [N4: nat] : X4 ) ) ) ).
% of_rat_Real
thf(fact_9730_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_9731_less__RealD,axiom,
! [Y7: nat > rat,X: real] :
( ( cauchy @ Y7 )
=> ( ( ord_less_real @ X @ ( real2 @ Y7 ) )
=> ? [N2: nat] : ( ord_less_real @ X @ ( field_7254667332652039916t_real @ ( Y7 @ N2 ) ) ) ) ) ).
% less_RealD
thf(fact_9732_le__RealI,axiom,
! [Y7: nat > rat,X: real] :
( ( cauchy @ Y7 )
=> ( ! [N2: nat] : ( ord_less_eq_real @ X @ ( field_7254667332652039916t_real @ ( Y7 @ N2 ) ) )
=> ( ord_less_eq_real @ X @ ( real2 @ Y7 ) ) ) ) ).
% le_RealI
thf(fact_9733_Real__leI,axiom,
! [X9: nat > rat,Y: real] :
( ( cauchy @ X9 )
=> ( ! [N2: nat] : ( ord_less_eq_real @ ( field_7254667332652039916t_real @ ( X9 @ N2 ) ) @ Y )
=> ( ord_less_eq_real @ ( real2 @ X9 ) @ Y ) ) ) ).
% Real_leI
thf(fact_9734_plus__rat__def,axiom,
( plus_plus_rat
= ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
@ ^ [X4: product_prod_int_int,Y5: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y5 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ X4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y5 ) ) ) ) ) ).
% plus_rat_def
thf(fact_9735_inverse__rat__def,axiom,
( inverse_inverse_rat
= ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
@ ^ [X4: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X4 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X4 ) @ ( product_fst_int_int @ X4 ) ) ) ) ) ).
% inverse_rat_def
thf(fact_9736_one__rat__def,axiom,
( one_one_rat
= ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).
% one_rat_def
thf(fact_9737_Fract_Oabs__eq,axiom,
( fract
= ( ^ [Xa4: int,X4: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X4 ) ) ) ) ) ).
% Fract.abs_eq
thf(fact_9738_zero__rat__def,axiom,
( zero_zero_rat
= ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).
% zero_rat_def
thf(fact_9739_uminus__rat__def,axiom,
( uminus_uminus_rat
= ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
@ ^ [X4: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X4 ) ) @ ( product_snd_int_int @ X4 ) ) ) ) ).
% uminus_rat_def
thf(fact_9740_plus__rat_Oabs__eq,axiom,
! [Xa3: product_prod_int_int,X: product_prod_int_int] :
( ( ratrel @ Xa3 @ Xa3 )
=> ( ( ratrel @ X @ X )
=> ( ( plus_plus_rat @ ( abs_Rat @ Xa3 ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa3 ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa3 ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% plus_rat.abs_eq
thf(fact_9741_ratrel__iff,axiom,
( ratrel
= ( ^ [X4: product_prod_int_int,Y5: product_prod_int_int] :
( ( ( product_snd_int_int @ X4 )
!= zero_zero_int )
& ( ( product_snd_int_int @ Y5 )
!= zero_zero_int )
& ( ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y5 ) )
= ( times_times_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ) ).
% ratrel_iff
thf(fact_9742_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_9743_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_9744_Fract_Orsp,axiom,
( bNF_re157797125943740599nt_int
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ( bNF_re6250860962936578807nt_int
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ratrel )
@ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B3 ) )
@ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B3 ) ) ) ).
% Fract.rsp
thf(fact_9745_ratrel__def,axiom,
( ratrel
= ( ^ [X4: product_prod_int_int,Y5: product_prod_int_int] :
( ( ( product_snd_int_int @ X4 )
!= zero_zero_int )
& ( ( product_snd_int_int @ Y5 )
!= zero_zero_int )
& ( ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y5 ) )
= ( times_times_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ) ).
% ratrel_def
thf(fact_9746_uminus__rat_Orsp,axiom,
( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
@ ^ [X4: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X4 ) ) @ ( product_snd_int_int @ X4 ) )
@ ^ [X4: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X4 ) ) @ ( product_snd_int_int @ X4 ) ) ) ).
% uminus_rat.rsp
thf(fact_9747_Rat_Opositive_Orsp,axiom,
( bNF_re8699439704749558557nt_o_o @ ratrel
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ ^ [X4: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) )
@ ^ [X4: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) ) ) ).
% Rat.positive.rsp
thf(fact_9748_inverse__rat_Orsp,axiom,
( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
@ ^ [X4: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X4 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X4 ) @ ( product_fst_int_int @ X4 ) ) )
@ ^ [X4: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X4 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X4 ) @ ( product_fst_int_int @ X4 ) ) ) ) ).
% inverse_rat.rsp
thf(fact_9749_plus__rat_Orsp,axiom,
( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
@ ^ [X4: product_prod_int_int,Y5: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y5 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ X4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y5 ) ) )
@ ^ [X4: product_prod_int_int,Y5: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y5 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ X4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y5 ) ) ) ) ).
% plus_rat.rsp
thf(fact_9750_uminus__rat_Oabs__eq,axiom,
! [X: product_prod_int_int] :
( ( ratrel @ X @ X )
=> ( ( uminus_uminus_rat @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% uminus_rat.abs_eq
thf(fact_9751_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_9752_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_9753_quotient__of__def,axiom,
( quotient_of
= ( ^ [X4: rat] :
( the_Pr4378521158711661632nt_int
@ ^ [Pair: product_prod_int_int] :
( ( X4
= ( 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_9754_normalize__stable,axiom,
! [Q4: int,P5: int] :
( ( ord_less_int @ zero_zero_int @ Q4 )
=> ( ( algebr932160517623751201me_int @ P5 @ Q4 )
=> ( ( normalize @ ( product_Pair_int_int @ P5 @ Q4 ) )
= ( product_Pair_int_int @ P5 @ Q4 ) ) ) ) ).
% normalize_stable
thf(fact_9755_coprime__crossproduct__int,axiom,
! [A: int,D: int,B: int,C: int] :
( ( algebr932160517623751201me_int @ A @ D )
=> ( ( algebr932160517623751201me_int @ B @ C )
=> ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
= ( times_times_int @ ( abs_abs_int @ B ) @ ( abs_abs_int @ D ) ) )
= ( ( ( abs_abs_int @ A )
= ( abs_abs_int @ B ) )
& ( ( abs_abs_int @ C )
= ( abs_abs_int @ D ) ) ) ) ) ) ).
% coprime_crossproduct_int
thf(fact_9756_Rat__cases,axiom,
! [Q4: rat] :
~ ! [A4: int,B4: int] :
( ( Q4
= ( fract @ A4 @ B4 ) )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ~ ( algebr932160517623751201me_int @ A4 @ B4 ) ) ) ).
% Rat_cases
thf(fact_9757_Rat__induct,axiom,
! [P: rat > $o,Q4: rat] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( algebr932160517623751201me_int @ A4 @ B4 )
=> ( P @ ( fract @ A4 @ B4 ) ) ) )
=> ( P @ Q4 ) ) ).
% Rat_induct
thf(fact_9758_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_9759_Rat__cases__nonzero,axiom,
! [Q4: rat] :
( ! [A4: int,B4: int] :
( ( Q4
= ( fract @ A4 @ B4 ) )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( A4 != zero_zero_int )
=> ~ ( algebr932160517623751201me_int @ A4 @ B4 ) ) ) )
=> ( Q4 = zero_zero_rat ) ) ).
% Rat_cases_nonzero
thf(fact_9760_quotient__of__unique,axiom,
! [R2: rat] :
? [X3: product_prod_int_int] :
( ( R2
= ( fract @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
& ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X3 ) )
& ( algebr932160517623751201me_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) )
& ! [Y6: product_prod_int_int] :
( ( ( R2
= ( fract @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ Y6 ) ) )
& ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y6 ) )
& ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ Y6 ) ) )
=> ( Y6 = X3 ) ) ) ).
% quotient_of_unique
thf(fact_9761_coprime__int__iff,axiom,
! [M: nat,N: nat] :
( ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( algebr934650988132801477me_nat @ M @ N ) ) ).
% coprime_int_iff
thf(fact_9762_coprime__nat__abs__left__iff,axiom,
! [K: int,N: nat] :
( ( algebr934650988132801477me_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
= ( algebr932160517623751201me_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% coprime_nat_abs_left_iff
thf(fact_9763_coprime__nat__abs__right__iff,axiom,
! [N: nat,K: int] :
( ( algebr934650988132801477me_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
= ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% coprime_nat_abs_right_iff
thf(fact_9764_coprime__crossproduct__nat,axiom,
! [A: nat,D: nat,B: nat,C: nat] :
( ( algebr934650988132801477me_nat @ A @ D )
=> ( ( algebr934650988132801477me_nat @ B @ C )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ D ) )
= ( ( A = B )
& ( C = D ) ) ) ) ) ).
% coprime_crossproduct_nat
thf(fact_9765_coprime__Suc__left__nat,axiom,
! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ N ) @ N ) ).
% coprime_Suc_left_nat
thf(fact_9766_coprime__Suc__right__nat,axiom,
! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ N ) ) ).
% coprime_Suc_right_nat
thf(fact_9767_coprime__Suc__0__left,axiom,
! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).
% coprime_Suc_0_left
thf(fact_9768_coprime__Suc__0__right,axiom,
! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).
% coprime_Suc_0_right
thf(fact_9769_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_9770_coprime__diff__one__right__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% coprime_diff_one_right_nat
thf(fact_9771_coprime__diff__one__left__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% coprime_diff_one_left_nat
thf(fact_9772_Rats__abs__nat__div__natE,axiom,
! [X: real] :
( ( member_real @ X @ field_5140801741446780682s_real )
=> ~ ! [M2: nat,N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ( ( abs_abs_real @ X )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
=> ~ ( algebr934650988132801477me_nat @ M2 @ N2 ) ) ) ) ).
% Rats_abs_nat_div_natE
thf(fact_9773_product__atMost__eq__Un,axiom,
! [A2: set_nat,M: nat] :
( ( produc457027306803732586at_nat @ A2
@ ^ [Uu3: nat] : ( set_ord_atMost_nat @ M ) )
= ( sup_su6327502436637775413at_nat
@ ( produc457027306803732586at_nat @ A2
@ ^ [I3: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ I3 ) ) )
@ ( produc457027306803732586at_nat @ A2
@ ^ [I3: nat] : ( set_or6659071591806873216st_nat @ ( minus_minus_nat @ M @ I3 ) @ M ) ) ) ) ).
% product_atMost_eq_Un
thf(fact_9774_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_9775_Code__Target__Nat_ONat__def,axiom,
( code_Target_Nat
= ( map_fu6539832666145259331at_nat @ code_int_of_integer @ id_nat @ nat2 ) ) ).
% Code_Target_Nat.Nat_def
thf(fact_9776_nat__of__integer__def,axiom,
( code_nat_of_integer
= ( map_fu6539832666145259331at_nat @ code_int_of_integer @ id_nat @ nat2 ) ) ).
% nat_of_integer_def
thf(fact_9777_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_9778_last__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( last_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% last_upt
thf(fact_9779_atLeast__0,axiom,
( ( set_ord_atLeast_nat @ zero_zero_nat )
= top_top_set_nat ) ).
% atLeast_0
thf(fact_9780_atLeast__Suc__greaterThan,axiom,
! [K: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K ) )
= ( set_or1210151606488870762an_nat @ K ) ) ).
% atLeast_Suc_greaterThan
thf(fact_9781_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_9782_Bseq__monoseq__convergent_H__dec,axiom,
! [F: nat > real,M10: nat] :
( ( bfun_nat_real
@ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ M10 ) )
@ at_top_nat )
=> ( ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M10 @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ M2 ) ) ) )
=> ( topolo7531315842566124627t_real @ F ) ) ) ).
% Bseq_monoseq_convergent'_dec
thf(fact_9783_convergent__realpow,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( topolo7531315842566124627t_real @ ( power_power_real @ X ) ) ) ) ).
% convergent_realpow
thf(fact_9784_Bseq__monoseq__convergent_H__inc,axiom,
! [F: nat > real,M10: nat] :
( ( bfun_nat_real
@ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ M10 ) )
@ at_top_nat )
=> ( ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M10 @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_real @ ( F @ M2 ) @ ( F @ N2 ) ) ) )
=> ( topolo7531315842566124627t_real @ F ) ) ) ).
% Bseq_monoseq_convergent'_inc
thf(fact_9785_pair__lessI2,axiom,
! [A: nat,B: nat,S: nat,T: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ S @ T )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_9786_pair__less__iff1,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z2 ) ) @ fun_pair_less )
= ( ord_less_nat @ Y @ Z2 ) ) ).
% pair_less_iff1
thf(fact_9787_pair__lessI1,axiom,
! [A: nat,B: nat,S: nat,T: nat] :
( ( ord_less_nat @ A @ B )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_9788_pair__leqI1,axiom,
! [A: nat,B: nat,S: nat,T: nat] :
( ( ord_less_nat @ A @ B )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_9789_gcd__nat_Oordering__top__axioms,axiom,
( ordering_top_nat @ dvd_dvd_nat
@ ^ [M4: nat,N4: nat] :
( ( dvd_dvd_nat @ M4 @ N4 )
& ( M4 != N4 ) )
@ zero_zero_nat ) ).
% gcd_nat.ordering_top_axioms
thf(fact_9790_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X4: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X4 )
@ ^ [X4: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X4 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_9791_set__encode__vimage__Suc,axiom,
! [A2: set_nat] :
( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A2 ) )
= ( divide_divide_nat @ ( nat_set_encode @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% set_encode_vimage_Suc
thf(fact_9792_vimage__Suc__insert__Suc,axiom,
! [N: nat,A2: set_nat] :
( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N ) @ A2 ) )
= ( insert_nat @ N @ ( vimage_nat_nat @ suc @ A2 ) ) ) ).
% vimage_Suc_insert_Suc
thf(fact_9793_finite__vimage__Suc__iff,axiom,
! [F3: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F3 ) )
= ( finite_finite_nat @ F3 ) ) ).
% finite_vimage_Suc_iff
thf(fact_9794_vimage__Suc__insert__0,axiom,
! [A2: set_nat] :
( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A2 ) )
= ( vimage_nat_nat @ suc @ A2 ) ) ).
% vimage_Suc_insert_0
thf(fact_9795_euclidean__size__int__def,axiom,
( euclid4774559944035922753ze_int
= ( comp_int_nat_int @ nat2 @ abs_abs_int ) ) ).
% euclidean_size_int_def
thf(fact_9796_set__decode__div__2,axiom,
! [X: nat] :
( ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( vimage_nat_nat @ suc @ ( nat_set_decode @ X ) ) ) ).
% set_decode_div_2
thf(fact_9797_abs__division__segment,axiom,
! [K: int] :
( ( abs_abs_int @ ( euclid3395696857347342551nt_int @ K ) )
= one_one_int ) ).
% abs_division_segment
thf(fact_9798_division__segment__eq__sgn,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ( euclid3395696857347342551nt_int @ K )
= ( sgn_sgn_int @ K ) ) ) ).
% division_segment_eq_sgn
thf(fact_9799_division__segment__nat__def,axiom,
( euclid3398187327856392827nt_nat
= ( ^ [N4: nat] : one_one_nat ) ) ).
% division_segment_nat_def
thf(fact_9800_division__segment__int__def,axiom,
( euclid3395696857347342551nt_int
= ( ^ [K3: int] : ( if_int @ ( ord_less_eq_int @ zero_zero_int @ K3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% division_segment_int_def
thf(fact_9801_binomial__def,axiom,
( binomial
= ( ^ [N4: nat,K3: 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 )
= K3 ) ) ) ) ) ) ).
% binomial_def
thf(fact_9802_transp__realrel,axiom,
transp_nat_rat @ realrel ).
% transp_realrel
thf(fact_9803_pred__nat__def,axiom,
( pred_nat
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [M4: nat,N4: nat] :
( N4
= ( suc @ M4 ) ) ) ) ) ).
% pred_nat_def
thf(fact_9804_less__enat__def,axiom,
( ord_le72135733267957522d_enat
= ( ^ [M4: extended_enat,N4: extended_enat] :
( extended_case_enat_o
@ ^ [M1: nat] : ( extended_case_enat_o @ ( ord_less_nat @ M1 ) @ $true @ N4 )
@ $false
@ M4 ) ) ) ).
% less_enat_def
thf(fact_9805_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( produc8211389475949308722nt_int
@ ^ [Q6: int,R5: int] : ( plus_plus_int @ Q6 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_9806_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N: nat] :
( ( order_5726023648592871131at_nat @ F )
=> ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_9807_prod__decode__triangle__add,axiom,
! [K: nat,M: nat] :
( ( nat_prod_decode @ ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) )
= ( nat_prod_decode_aux @ K @ M ) ) ).
% prod_decode_triangle_add
thf(fact_9808_prod__decode__eq,axiom,
! [X: nat,Y: nat] :
( ( ( nat_prod_decode @ X )
= ( nat_prod_decode @ Y ) )
= ( X = Y ) ) ).
% prod_decode_eq
thf(fact_9809_prod__decode__inverse,axiom,
! [N: nat] :
( ( nat_prod_encode @ ( nat_prod_decode @ N ) )
= N ) ).
% prod_decode_inverse
thf(fact_9810_prod__encode__inverse,axiom,
! [X: product_prod_nat_nat] :
( ( nat_prod_decode @ ( nat_prod_encode @ X ) )
= X ) ).
% prod_encode_inverse
thf(fact_9811_inj__prod__decode,axiom,
! [A2: set_nat] : ( inj_on5538052773655684606at_nat @ nat_prod_decode @ A2 ) ).
% inj_prod_decode
thf(fact_9812_prod__decode__def,axiom,
( nat_prod_decode
= ( nat_prod_decode_aux @ zero_zero_nat ) ) ).
% prod_decode_def
thf(fact_9813_surj__prod__decode,axiom,
( ( image_5846123807819985514at_nat @ nat_prod_decode @ top_top_set_nat )
= top_to4669805908274784177at_nat ) ).
% surj_prod_decode
thf(fact_9814_bij__prod__decode,axiom,
bij_be8693218025023041337at_nat @ nat_prod_decode @ top_top_set_nat @ top_to4669805908274784177at_nat ).
% bij_prod_decode
thf(fact_9815_list__decode_Opinduct,axiom,
! [A0: nat,P: nat > $o] :
( ( accp_nat @ nat_list_decode_rel @ A0 )
=> ( ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
=> ( P @ zero_zero_nat ) )
=> ( ! [N2: nat] :
( ( accp_nat @ nat_list_decode_rel @ ( suc @ N2 ) )
=> ( ! [X5: nat,Y6: nat] :
( ( ( product_Pair_nat_nat @ X5 @ Y6 )
= ( nat_prod_decode @ N2 ) )
=> ( P @ Y6 ) )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% list_decode.pinduct
thf(fact_9816_list__decode_Oelims,axiom,
! [X: nat,Y: list_nat] :
( ( ( nat_list_decode @ X )
= Y )
=> ( ( ( X = zero_zero_nat )
=> ( Y != nil_nat ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ N2 ) )
=> ( Y
!= ( produc2761476792215241774st_nat
@ ^ [X4: nat,Y5: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y5 ) )
@ ( nat_prod_decode @ N2 ) ) ) ) ) ) ).
% list_decode.elims
thf(fact_9817_list__decode__inverse,axiom,
! [N: nat] :
( ( nat_list_encode @ ( nat_list_decode @ N ) )
= N ) ).
% list_decode_inverse
thf(fact_9818_list__encode__inverse,axiom,
! [X: list_nat] :
( ( nat_list_decode @ ( nat_list_encode @ X ) )
= X ) ).
% list_encode_inverse
thf(fact_9819_list__decode__eq,axiom,
! [X: nat,Y: nat] :
( ( ( nat_list_decode @ X )
= ( nat_list_decode @ Y ) )
= ( X = Y ) ) ).
% list_decode_eq
thf(fact_9820_inj__list__decode,axiom,
! [A2: set_nat] : ( inj_on_nat_list_nat @ nat_list_decode @ A2 ) ).
% inj_list_decode
thf(fact_9821_list__decode_Opsimps_I1_J,axiom,
( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
=> ( ( nat_list_decode @ zero_zero_nat )
= nil_nat ) ) ).
% list_decode.psimps(1)
thf(fact_9822_list__decode_Osimps_I1_J,axiom,
( ( nat_list_decode @ zero_zero_nat )
= nil_nat ) ).
% list_decode.simps(1)
thf(fact_9823_list__decode_Opsimps_I2_J,axiom,
! [N: nat] :
( ( accp_nat @ nat_list_decode_rel @ ( suc @ N ) )
=> ( ( nat_list_decode @ ( suc @ N ) )
= ( produc2761476792215241774st_nat
@ ^ [X4: nat,Y5: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y5 ) )
@ ( nat_prod_decode @ N ) ) ) ) ).
% list_decode.psimps(2)
thf(fact_9824_bij__list__decode,axiom,
bij_be6293887246118711976st_nat @ nat_list_decode @ top_top_set_nat @ top_top_set_list_nat ).
% bij_list_decode
thf(fact_9825_surj__list__decode,axiom,
( ( image_nat_list_nat @ nat_list_decode @ top_top_set_nat )
= top_top_set_list_nat ) ).
% surj_list_decode
thf(fact_9826_list__decode_Osimps_I2_J,axiom,
! [N: nat] :
( ( nat_list_decode @ ( suc @ N ) )
= ( produc2761476792215241774st_nat
@ ^ [X4: nat,Y5: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y5 ) )
@ ( nat_prod_decode @ N ) ) ) ).
% list_decode.simps(2)
thf(fact_9827_list__decode_Opelims,axiom,
! [X: nat,Y: list_nat] :
( ( ( nat_list_decode @ X )
= Y )
=> ( ( accp_nat @ nat_list_decode_rel @ X )
=> ( ( ( X = zero_zero_nat )
=> ( ( Y = nil_nat )
=> ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ N2 ) )
=> ( ( Y
= ( produc2761476792215241774st_nat
@ ^ [X4: nat,Y5: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y5 ) )
@ ( nat_prod_decode @ N2 ) ) )
=> ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N2 ) ) ) ) ) ) ) ).
% list_decode.pelims
thf(fact_9828_num__of__integer_Otransfer,axiom,
( bNF_re6718328864250387230um_num @ code_pcr_integer
@ ^ [Y4: num,Z: num] : Y4 = Z
@ ( comp_nat_num_int @ num_of_nat @ nat2 )
@ code_num_of_integer ) ).
% num_of_integer.transfer
thf(fact_9829_compute__powr__real,axiom,
( powr_real2
= ( ^ [B3: real,I3: real] :
( if_real @ ( ord_less_eq_real @ B3 @ zero_zero_real )
@ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ^ [Uu3: product_unit] : ( powr_real2 @ B3 @ I3 ) )
@ ( if_real
@ ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ I3 ) )
= I3 )
@ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ I3 ) @ ( power_power_real @ B3 @ ( nat2 @ ( archim6058952711729229775r_real @ I3 ) ) ) @ ( divide_divide_real @ one_one_real @ ( power_power_real @ B3 @ ( nat2 @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ I3 ) ) ) ) ) )
@ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ^ [Uu3: product_unit] : ( powr_real2 @ B3 @ I3 ) ) ) ) ) ) ).
% compute_powr_real
thf(fact_9830_Code__Target__Nat_ONat_Otransfer,axiom,
( bNF_re2807294637932363402at_nat @ code_pcr_integer
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ nat2
@ code_Target_Nat ) ).
% Code_Target_Nat.Nat.transfer
thf(fact_9831_gcd__integer_Otransfer,axiom,
bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ gcd_gcd_int @ gcd_gcd_Code_integer ).
% gcd_integer.transfer
thf(fact_9832_nat__of__integer_Otransfer,axiom,
( bNF_re2807294637932363402at_nat @ code_pcr_integer
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ nat2
@ code_nat_of_integer ) ).
% nat_of_integer.transfer
thf(fact_9833_plus__integer_Otransfer,axiom,
bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ plus_plus_int @ plus_p5714425477246183910nteger ).
% plus_integer.transfer
thf(fact_9834_minus__integer_Otransfer,axiom,
bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ minus_minus_int @ minus_8373710615458151222nteger ).
% minus_integer.transfer
thf(fact_9835_less__integer_Otransfer,axiom,
( bNF_re6321650412969554871eger_o @ code_pcr_integer
@ ( bNF_re6574881592172037608er_o_o @ code_pcr_integer
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ord_less_int
@ ord_le6747313008572928689nteger ) ).
% less_integer.transfer
thf(fact_9836_uminus__integer_Otransfer,axiom,
bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer @ uminus_uminus_int @ uminus1351360451143612070nteger ).
% uminus_integer.transfer
thf(fact_9837_one__integer_Otransfer,axiom,
code_pcr_integer @ one_one_int @ one_one_Code_integer ).
% one_integer.transfer
thf(fact_9838_zero__integer_Otransfer,axiom,
code_pcr_integer @ zero_zero_int @ zero_z3403309356797280102nteger ).
% zero_integer.transfer
thf(fact_9839_int__of__integer__integer__of__nat,axiom,
! [N: nat] :
( ( code_int_of_integer @ ( code_integer_of_nat @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% int_of_integer_integer_of_nat
thf(fact_9840_integer__of__nat_Orep__eq,axiom,
! [X: nat] :
( ( code_int_of_integer @ ( code_integer_of_nat @ X ) )
= ( semiri1314217659103216013at_int @ X ) ) ).
% integer_of_nat.rep_eq
thf(fact_9841_integer__of__nat__0,axiom,
( ( code_integer_of_nat @ zero_zero_nat )
= zero_z3403309356797280102nteger ) ).
% integer_of_nat_0
thf(fact_9842_integer__of__nat_Oabs__eq,axiom,
( code_integer_of_nat
= ( ^ [X4: nat] : ( code_integer_of_int @ ( semiri1314217659103216013at_int @ X4 ) ) ) ) ).
% integer_of_nat.abs_eq
thf(fact_9843_integer__of__nat_Otransfer,axiom,
( bNF_re4153400068438556298nteger
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ code_pcr_integer
@ semiri1314217659103216013at_int
@ code_integer_of_nat ) ).
% integer_of_nat.transfer
thf(fact_9844_integer__of__nat__1,axiom,
( ( code_integer_of_nat @ one_one_nat )
= one_one_Code_integer ) ).
% integer_of_nat_1
thf(fact_9845_dup_Otransfer,axiom,
( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer
@ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 )
@ code_dup ) ).
% dup.transfer
thf(fact_9846_integer__of__nat__def,axiom,
( code_integer_of_nat
= ( map_fu6290471996055670595nteger @ id_nat @ code_integer_of_int @ semiri1314217659103216013at_int ) ) ).
% integer_of_nat_def
thf(fact_9847_dup_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( code_dup @ X ) )
= ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ X ) ) ) ).
% dup.rep_eq
thf(fact_9848_dup_Oabs__eq,axiom,
! [X: int] :
( ( code_dup @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( plus_plus_int @ X @ X ) ) ) ).
% dup.abs_eq
thf(fact_9849_Code__Numeral_Osub__code_I9_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( minus_8373710615458151222nteger @ ( code_dup @ ( code_sub @ M @ N ) ) @ one_one_Code_integer ) ) ).
% Code_Numeral.sub_code(9)
thf(fact_9850_Code__Numeral_Osub__code_I8_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( plus_p5714425477246183910nteger @ ( code_dup @ ( code_sub @ M @ N ) ) @ one_one_Code_integer ) ) ).
% Code_Numeral.sub_code(8)
thf(fact_9851_sub_Orep__eq,axiom,
! [X: num,Xa3: num] :
( ( code_int_of_integer @ ( code_sub @ X @ Xa3 ) )
= ( minus_minus_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Xa3 ) ) ) ).
% sub.rep_eq
thf(fact_9852_sub_Oabs__eq,axiom,
( code_sub
= ( ^ [Xa4: num,X4: num] : ( code_integer_of_int @ ( minus_minus_int @ ( numeral_numeral_int @ Xa4 ) @ ( numeral_numeral_int @ X4 ) ) ) ) ) ).
% sub.abs_eq
thf(fact_9853_sub_Otransfer,axiom,
( bNF_re7876454716742015248nteger
@ ^ [Y4: num,Z: num] : Y4 = Z
@ ( bNF_re6501075790457514782nteger
@ ^ [Y4: num,Z: num] : Y4 = Z
@ code_pcr_integer )
@ ^ [M4: num,N4: num] : ( minus_minus_int @ ( numeral_numeral_int @ M4 ) @ ( numeral_numeral_int @ N4 ) )
@ code_sub ) ).
% sub.transfer
thf(fact_9854_less__than__iff,axiom,
! [X: nat,Y: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ less_than )
= ( ord_less_nat @ X @ Y ) ) ).
% less_than_iff
thf(fact_9855_elimnum,axiom,
! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) ) ) ).
% elimnum
thf(fact_9856_at__top__to__right,axiom,
( at_top_real
= ( filtermap_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% at_top_to_right
thf(fact_9857_idiff__enat__0,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ zero_zero_nat ) @ N )
= ( extended_enat2 @ zero_zero_nat ) ) ).
% idiff_enat_0
thf(fact_9858_idiff__enat__0__right,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ N @ ( extended_enat2 @ zero_zero_nat ) )
= N ) ).
% idiff_enat_0_right
thf(fact_9859_plus__enat__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( plus_p3455044024723400733d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( plus_plus_nat @ M @ N ) ) ) ).
% plus_enat_simps(1)
thf(fact_9860_idiff__enat__enat,axiom,
! [A: nat,B: nat] :
( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ A ) @ ( extended_enat2 @ B ) )
= ( extended_enat2 @ ( minus_minus_nat @ A @ B ) ) ) ).
% idiff_enat_enat
thf(fact_9861_enat__ord__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% enat_ord_simps(2)
thf(fact_9862_numeral__less__enat__iff,axiom,
! [M: num,N: nat] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% numeral_less_enat_iff
thf(fact_9863_zero__enat__def,axiom,
( zero_z5237406670263579293d_enat
= ( extended_enat2 @ zero_zero_nat ) ) ).
% zero_enat_def
thf(fact_9864_enat__0__iff_I1_J,axiom,
! [X: nat] :
( ( ( extended_enat2 @ X )
= zero_z5237406670263579293d_enat )
= ( X = zero_zero_nat ) ) ).
% enat_0_iff(1)
thf(fact_9865_enat__0__iff_I2_J,axiom,
! [X: nat] :
( ( zero_z5237406670263579293d_enat
= ( extended_enat2 @ X ) )
= ( X = zero_zero_nat ) ) ).
% enat_0_iff(2)
thf(fact_9866_one__enat__def,axiom,
( one_on7984719198319812577d_enat
= ( extended_enat2 @ one_one_nat ) ) ).
% one_enat_def
thf(fact_9867_enat__1__iff_I1_J,axiom,
! [X: nat] :
( ( ( extended_enat2 @ X )
= one_on7984719198319812577d_enat )
= ( X = one_one_nat ) ) ).
% enat_1_iff(1)
thf(fact_9868_enat__1__iff_I2_J,axiom,
! [X: nat] :
( ( one_on7984719198319812577d_enat
= ( extended_enat2 @ X ) )
= ( X = one_one_nat ) ) ).
% enat_1_iff(2)
thf(fact_9869_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
! [A: $o,B: $o,Uu: extended_enat] :
( ( vEBT_VEBT_elim_dead @ ( vEBT_Leaf @ A @ B ) @ Uu )
= ( vEBT_Leaf @ A @ B ) ) ).
% VEBT_internal.elim_dead.simps(1)
thf(fact_9870_less__enatE,axiom,
! [N: extended_enat,M: nat] :
( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M ) )
=> ~ ! [K2: nat] :
( ( N
= ( extended_enat2 @ K2 ) )
=> ~ ( ord_less_nat @ K2 @ M ) ) ) ).
% less_enatE
thf(fact_9871_iadd__le__enat__iff,axiom,
! [X: extended_enat,Y: extended_enat,N: nat] :
( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ ( extended_enat2 @ N ) )
= ( ? [Y9: nat,X10: nat] :
( ( X
= ( extended_enat2 @ X10 ) )
& ( Y
= ( extended_enat2 @ Y9 ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ X10 @ Y9 ) @ N ) ) ) ) ).
% iadd_le_enat_iff
thf(fact_9872_Suc__ile__eq,axiom,
! [M: nat,N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N )
= ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% Suc_ile_eq
thf(fact_9873_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,L: nat] :
( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ L ) )
= ( vEBT_Node @ Info2 @ Deg
@ ( take_VEBT_VEBT @ ( divide_divide_nat @ L @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ ( map_VE8901447254227204932T_VEBT
@ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ TreeList2 ) )
@ ( vEBT_VEBT_elim_dead @ Summary @ ( extended_enat2 @ ( divide_divide_nat @ L @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.elim_dead.simps(3)
thf(fact_9874_at__left__minus,axiom,
! [A: real] :
( ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) )
= ( filtermap_real_real @ uminus_uminus_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ A ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ A ) ) ) ) ) ).
% at_left_minus
thf(fact_9875_at__right__minus,axiom,
! [A: real] :
( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
= ( filtermap_real_real @ uminus_uminus_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ A ) @ ( set_or5984915006950818249n_real @ ( uminus_uminus_real @ A ) ) ) ) ) ).
% at_right_minus
thf(fact_9876_at__right__to__top,axiom,
( ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) )
= ( filtermap_real_real @ inverse_inverse_real @ at_top_real ) ) ).
% at_right_to_top
thf(fact_9877_VEBT__internal_Oelim__dead_Oelims,axiom,
! [X: vEBT_VEBT,Xa3: extended_enat,Y: vEBT_VEBT] :
( ( ( vEBT_VEBT_elim_dead @ X @ Xa3 )
= Y )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
!= ( vEBT_Leaf @ A4 @ B4 ) ) )
=> ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( Xa3 = extend5688581933313929465d_enat )
=> ( Y
!= ( vEBT_Node @ Info @ Deg2
@ ( map_VE8901447254227204932T_VEBT
@ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ TreeList )
@ ( vEBT_VEBT_elim_dead @ Summary2 @ extend5688581933313929465d_enat ) ) ) ) )
=> ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
=> ! [L2: nat] :
( ( Xa3
= ( extended_enat2 @ L2 ) )
=> ( Y
!= ( vEBT_Node @ Info @ Deg2
@ ( take_VEBT_VEBT @ ( divide_divide_nat @ L2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ ( map_VE8901447254227204932T_VEBT
@ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ TreeList ) )
@ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide_nat @ L2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.elim_dead.elims
thf(fact_9878_elimcomplete,axiom,
! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ extend5688581933313929465d_enat )
= ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) ) ) ).
% elimcomplete
thf(fact_9879_times__enat__simps_I3_J,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
= zero_z5237406670263579293d_enat ) )
& ( ( N != zero_zero_nat )
=> ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
= extend5688581933313929465d_enat ) ) ) ).
% times_enat_simps(3)
thf(fact_9880_times__enat__simps_I4_J,axiom,
! [M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
= zero_z5237406670263579293d_enat ) )
& ( ( M != zero_zero_nat )
=> ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
= extend5688581933313929465d_enat ) ) ) ).
% times_enat_simps(4)
thf(fact_9881_infinity__ne__i1,axiom,
extend5688581933313929465d_enat != one_on7984719198319812577d_enat ).
% infinity_ne_i1
thf(fact_9882_zero__one__enat__neq_I1_J,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_one_enat_neq(1)
thf(fact_9883_VEBT__internal_Oelim__dead_Ocases,axiom,
! [X: produc7272778201969148633d_enat] :
( ! [A4: $o,B4: $o,Uu2: extended_enat] :
( X
!= ( produc581526299967858633d_enat @ ( vEBT_Leaf @ A4 @ B4 ) @ Uu2 ) )
=> ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( X
!= ( produc581526299967858633d_enat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ extend5688581933313929465d_enat ) )
=> ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,L2: nat] :
( X
!= ( produc581526299967858633d_enat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ ( extended_enat2 @ L2 ) ) ) ) ) ).
% VEBT_internal.elim_dead.cases
thf(fact_9884_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ extend5688581933313929465d_enat )
= ( vEBT_Node @ Info2 @ Deg
@ ( map_VE8901447254227204932T_VEBT
@ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ TreeList2 )
@ ( vEBT_VEBT_elim_dead @ Summary @ extend5688581933313929465d_enat ) ) ) ).
% VEBT_internal.elim_dead.simps(2)
thf(fact_9885_VEBT__internal_Oelim__dead_Opelims,axiom,
! [X: vEBT_VEBT,Xa3: extended_enat,Y: vEBT_VEBT] :
( ( ( vEBT_VEBT_elim_dead @ X @ Xa3 )
= Y )
=> ( ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ X @ Xa3 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa3 ) ) ) )
=> ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( Xa3 = extend5688581933313929465d_enat )
=> ( ( Y
= ( vEBT_Node @ Info @ Deg2
@ ( map_VE8901447254227204932T_VEBT
@ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ TreeList )
@ ( vEBT_VEBT_elim_dead @ Summary2 @ extend5688581933313929465d_enat ) ) )
=> ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ extend5688581933313929465d_enat ) ) ) ) )
=> ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
=> ! [L2: nat] :
( ( Xa3
= ( extended_enat2 @ L2 ) )
=> ( ( Y
= ( vEBT_Node @ Info @ Deg2
@ ( take_VEBT_VEBT @ ( divide_divide_nat @ L2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ ( map_VE8901447254227204932T_VEBT
@ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ TreeList ) )
@ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide_nat @ L2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
=> ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ ( extended_enat2 @ L2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.elim_dead.pelims
thf(fact_9886_diff__enat__def,axiom,
( minus_3235023915231533773d_enat
= ( ^ [A3: extended_enat,B3: extended_enat] :
( extend3600170679010898289d_enat
@ ^ [X4: nat] :
( extend3600170679010898289d_enat
@ ^ [Y5: nat] : ( extended_enat2 @ ( minus_minus_nat @ X4 @ Y5 ) )
@ zero_z5237406670263579293d_enat
@ B3 )
@ extend5688581933313929465d_enat
@ A3 ) ) ) ).
% diff_enat_def
thf(fact_9887_plus__enat__def,axiom,
( plus_p3455044024723400733d_enat
= ( ^ [M4: extended_enat,N4: extended_enat] :
( extend3600170679010898289d_enat
@ ^ [O: nat] :
( extend3600170679010898289d_enat
@ ^ [P6: nat] : ( extended_enat2 @ ( plus_plus_nat @ O @ P6 ) )
@ extend5688581933313929465d_enat
@ N4 )
@ extend5688581933313929465d_enat
@ M4 ) ) ) ).
% plus_enat_def
thf(fact_9888_times__enat__def,axiom,
( times_7803423173614009249d_enat
= ( ^ [M4: extended_enat,N4: extended_enat] :
( extend3600170679010898289d_enat
@ ^ [O: nat] :
( extend3600170679010898289d_enat
@ ^ [P6: nat] : ( extended_enat2 @ ( times_times_nat @ O @ P6 ) )
@ ( if_Extended_enat @ ( O = zero_zero_nat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
@ N4 )
@ ( if_Extended_enat @ ( N4 = zero_z5237406670263579293d_enat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
@ M4 ) ) ) ).
% times_enat_def
thf(fact_9889_eSuc__def,axiom,
( extended_eSuc
= ( extend3600170679010898289d_enat
@ ^ [N4: nat] : ( extended_enat2 @ ( suc @ N4 ) )
@ extend5688581933313929465d_enat ) ) ).
% eSuc_def
thf(fact_9890_Code__Numeral_Odup__def,axiom,
( code_dup
= ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int
@ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 ) ) ) ).
% Code_Numeral.dup_def
thf(fact_9891_eSuc__minus__1,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ ( extended_eSuc @ N ) @ one_on7984719198319812577d_enat )
= N ) ).
% eSuc_minus_1
thf(fact_9892_enat__eSuc__iff,axiom,
! [Y: nat,X: extended_enat] :
( ( ( extended_enat2 @ Y )
= ( extended_eSuc @ X ) )
= ( ? [N4: nat] :
( ( Y
= ( suc @ N4 ) )
& ( ( extended_enat2 @ N4 )
= X ) ) ) ) ).
% enat_eSuc_iff
thf(fact_9893_eSuc__enat__iff,axiom,
! [X: extended_enat,Y: nat] :
( ( ( extended_eSuc @ X )
= ( extended_enat2 @ Y ) )
= ( ? [N4: nat] :
( ( Y
= ( suc @ N4 ) )
& ( X
= ( extended_enat2 @ N4 ) ) ) ) ) ).
% eSuc_enat_iff
thf(fact_9894_eSuc__enat,axiom,
! [N: nat] :
( ( extended_eSuc @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( suc @ N ) ) ) ).
% eSuc_enat
thf(fact_9895_one__eSuc,axiom,
( one_on7984719198319812577d_enat
= ( extended_eSuc @ zero_z5237406670263579293d_enat ) ) ).
% one_eSuc
thf(fact_9896_eSuc__plus__1,axiom,
( extended_eSuc
= ( ^ [N4: extended_enat] : ( plus_p3455044024723400733d_enat @ N4 @ one_on7984719198319812577d_enat ) ) ) ).
% eSuc_plus_1
thf(fact_9897_plus__1__eSuc_I1_J,axiom,
! [Q4: extended_enat] :
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ Q4 )
= ( extended_eSuc @ Q4 ) ) ).
% plus_1_eSuc(1)
thf(fact_9898_plus__1__eSuc_I2_J,axiom,
! [Q4: extended_enat] :
( ( plus_p3455044024723400733d_enat @ Q4 @ one_on7984719198319812577d_enat )
= ( extended_eSuc @ Q4 ) ) ).
% plus_1_eSuc(2)
thf(fact_9899_uminus__integer__def,axiom,
( uminus1351360451143612070nteger
= ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int @ uminus_uminus_int ) ) ).
% uminus_integer_def
thf(fact_9900_gcd__integer__def,axiom,
( gcd_gcd_Code_integer
= ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ gcd_gcd_int ) ) ).
% gcd_integer_def
thf(fact_9901_minus__integer__def,axiom,
( minus_8373710615458151222nteger
= ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ minus_minus_int ) ) ).
% minus_integer_def
thf(fact_9902_plus__integer__def,axiom,
( plus_p5714425477246183910nteger
= ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ plus_plus_int ) ) ).
% plus_integer_def
thf(fact_9903_has__vector__derivative__id,axiom,
! [Net: filter_real] :
( has_ve631408500373753343e_real
@ ^ [X4: real] : X4
@ one_one_real
@ Net ) ).
% has_vector_derivative_id
thf(fact_9904_Quotient__real,axiom,
quotie3684837364556693515t_real @ realrel @ real2 @ rep_real @ cr_real ).
% Quotient_real
thf(fact_9905_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_9906_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_9907_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_9908_char_Osize_I2_J,axiom,
! [X1: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_size_char @ ( char2 @ X1 @ X2 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= zero_zero_nat ) ).
% char.size(2)
thf(fact_9909_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_9910_char_Osize__gen,axiom,
! [X1: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_char @ ( char2 @ X1 @ X2 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= zero_zero_nat ) ).
% char.size_gen
thf(fact_9911_cr__int__def,axiom,
( cr_int
= ( ^ [X4: product_prod_nat_nat] :
( ^ [Y4: int,Z: int] : Y4 = Z
@ ( abs_Integ @ X4 ) ) ) ) ).
% cr_int_def
thf(fact_9912_int_Opcr__cr__eq,axiom,
pcr_int = cr_int ).
% int.pcr_cr_eq
thf(fact_9913_less__eq,axiom,
! [M: nat,N: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
= ( ord_less_nat @ M @ N ) ) ).
% less_eq
thf(fact_9914_Quotient__int,axiom,
quotie1194848508323700631at_int @ intrel @ abs_Integ @ rep_Integ @ cr_int ).
% Quotient_int
thf(fact_9915_gcd__nat_Osemilattice__neutr__axioms,axiom,
semila9081495762789891438tr_nat @ gcd_gcd_nat @ zero_zero_nat ).
% gcd_nat.semilattice_neutr_axioms
thf(fact_9916_max__nat_Osemilattice__neutr__axioms,axiom,
semila9081495762789891438tr_nat @ ord_max_nat @ zero_zero_nat ).
% max_nat.semilattice_neutr_axioms
thf(fact_9917_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).
% natLess_def
thf(fact_9918_Restr__natLeq,axiom,
! [N: nat] :
( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
@ ( produc457027306803732586at_nat
@ ( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ X4 @ N ) )
@ ^ [Uu3: nat] :
( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ X4 @ N ) ) ) )
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ N )
& ( ord_less_nat @ Y5 @ N )
& ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_9919_Restr__natLeq2,axiom,
! [N: nat] :
( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
@ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
@ ^ [Uu3: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ N )
& ( ord_less_nat @ Y5 @ N )
& ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_9920_rcis__cnj,axiom,
( cnj
= ( ^ [A3: complex] : ( rcis @ ( real_V1022390504157884413omplex @ A3 ) @ ( uminus_uminus_real @ ( arg @ A3 ) ) ) ) ) ).
% rcis_cnj
thf(fact_9921_complex__cnj__one__iff,axiom,
! [Z2: complex] :
( ( ( cnj @ Z2 )
= one_one_complex )
= ( Z2 = one_one_complex ) ) ).
% complex_cnj_one_iff
thf(fact_9922_complex__cnj__one,axiom,
( ( cnj @ one_one_complex )
= one_one_complex ) ).
% complex_cnj_one
thf(fact_9923_complex__cnj__minus,axiom,
! [X: complex] :
( ( cnj @ ( uminus1482373934393186551omplex @ X ) )
= ( uminus1482373934393186551omplex @ ( cnj @ X ) ) ) ).
% complex_cnj_minus
thf(fact_9924_complex__cnj__inverse,axiom,
! [X: complex] :
( ( cnj @ ( invers8013647133539491842omplex @ X ) )
= ( invers8013647133539491842omplex @ ( cnj @ X ) ) ) ).
% complex_cnj_inverse
thf(fact_9925_complex__cnj__of__int,axiom,
! [Z2: int] :
( ( cnj @ ( ring_17405671764205052669omplex @ Z2 ) )
= ( ring_17405671764205052669omplex @ Z2 ) ) ).
% complex_cnj_of_int
thf(fact_9926_complex__cnj__i,axiom,
( ( cnj @ imaginary_unit )
= ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% complex_cnj_i
thf(fact_9927_complex__cnj__neg__numeral,axiom,
! [W2: num] :
( ( cnj @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ).
% complex_cnj_neg_numeral
thf(fact_9928_natLeq__underS__less,axiom,
! [N: nat] :
( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
= ( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ X4 @ N ) ) ) ).
% natLeq_underS_less
thf(fact_9929_cnj_Osimps_I2_J,axiom,
! [Z2: complex] :
( ( im @ ( cnj @ Z2 ) )
= ( uminus_uminus_real @ ( im @ Z2 ) ) ) ).
% cnj.simps(2)
thf(fact_9930_complex__cnj,axiom,
! [A: real,B: real] :
( ( cnj @ ( complex2 @ A @ B ) )
= ( complex2 @ A @ ( uminus_uminus_real @ B ) ) ) ).
% complex_cnj
thf(fact_9931_cis__cnj,axiom,
! [T: real] :
( ( cnj @ ( cis @ T ) )
= ( cis @ ( uminus_uminus_real @ T ) ) ) ).
% cis_cnj
thf(fact_9932_cnj_Ocode,axiom,
( cnj
= ( ^ [Z3: complex] : ( complex2 @ ( re @ Z3 ) @ ( uminus_uminus_real @ ( im @ Z3 ) ) ) ) ) ).
% cnj.code
thf(fact_9933_gcd__nat_Omonoid__axioms,axiom,
monoid_nat @ gcd_gcd_nat @ zero_zero_nat ).
% gcd_nat.monoid_axioms
thf(fact_9934_max__nat_Omonoid__axioms,axiom,
monoid_nat @ ord_max_nat @ zero_zero_nat ).
% max_nat.monoid_axioms
thf(fact_9935_Lcm__eq__0__I__nat,axiom,
! [A2: set_nat] :
( ( member_nat @ zero_zero_nat @ A2 )
=> ( ( gcd_Lcm_nat @ A2 )
= zero_zero_nat ) ) ).
% Lcm_eq_0_I_nat
thf(fact_9936_abs__Lcm__eq,axiom,
! [K5: set_int] :
( ( abs_abs_int @ ( gcd_Lcm_int @ K5 ) )
= ( gcd_Lcm_int @ K5 ) ) ).
% abs_Lcm_eq
thf(fact_9937_Lcm__0__iff__nat,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( gcd_Lcm_nat @ A2 )
= zero_zero_nat )
= ( member_nat @ zero_zero_nat @ A2 ) ) ) ).
% Lcm_0_iff_nat
thf(fact_9938_Lcm__abs__eq,axiom,
! [K5: set_int] :
( ( gcd_Lcm_int @ ( image_int_int @ abs_abs_int @ K5 ) )
= ( gcd_Lcm_int @ K5 ) ) ).
% Lcm_abs_eq
thf(fact_9939_Lcm__int__eq,axiom,
! [N5: set_nat] :
( ( gcd_Lcm_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N5 ) )
= ( semiri1314217659103216013at_int @ ( gcd_Lcm_nat @ N5 ) ) ) ).
% Lcm_int_eq
thf(fact_9940_Lcm__nat__abs__eq,axiom,
! [K5: set_int] :
( ( gcd_Lcm_nat
@ ( image_int_nat
@ ^ [K3: int] : ( nat2 @ ( abs_abs_int @ K3 ) )
@ K5 ) )
= ( nat2 @ ( gcd_Lcm_int @ K5 ) ) ) ).
% Lcm_nat_abs_eq
thf(fact_9941_Lcm__nat__infinite,axiom,
! [M10: set_nat] :
( ~ ( finite_finite_nat @ M10 )
=> ( ( gcd_Lcm_nat @ M10 )
= zero_zero_nat ) ) ).
% Lcm_nat_infinite
thf(fact_9942_Lcm__least__int,axiom,
! [A2: set_int,A: int] :
( ! [B4: int] :
( ( member_int @ B4 @ A2 )
=> ( dvd_dvd_int @ B4 @ A ) )
=> ( dvd_dvd_int @ ( gcd_Lcm_int @ A2 ) @ A ) ) ).
% Lcm_least_int
thf(fact_9943_dvd__Lcm__int,axiom,
! [M: int,M10: set_int] :
( ( member_int @ M @ M10 )
=> ( dvd_dvd_int @ M @ ( gcd_Lcm_int @ M10 ) ) ) ).
% dvd_Lcm_int
thf(fact_9944_dvd__Lcm__nat,axiom,
! [M: nat,M10: set_nat] :
( ( member_nat @ M @ M10 )
=> ( dvd_dvd_nat @ M @ ( gcd_Lcm_nat @ M10 ) ) ) ).
% dvd_Lcm_nat
thf(fact_9945_Lcm__dvd__nat,axiom,
! [M10: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ M10 )
=> ( dvd_dvd_nat @ X3 @ N ) )
=> ( dvd_dvd_nat @ ( gcd_Lcm_nat @ M10 ) @ N ) ) ).
% Lcm_dvd_nat
thf(fact_9946_Lcm__int__greater__eq__0,axiom,
! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Lcm_int @ K5 ) ) ).
% Lcm_int_greater_eq_0
thf(fact_9947_Lcm__nat__empty,axiom,
( ( gcd_Lcm_nat @ bot_bot_set_nat )
= one_one_nat ) ).
% Lcm_nat_empty
thf(fact_9948_Gcd__nat__def,axiom,
( gcd_Gcd_nat
= ( ^ [M7: set_nat] :
( gcd_Lcm_nat
@ ( collect_nat
@ ^ [D3: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ M7 )
=> ( dvd_dvd_nat @ D3 @ X4 ) ) ) ) ) ) ).
% Gcd_nat_def
thf(fact_9949_Lcm__int__def,axiom,
( gcd_Lcm_int
= ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Lcm_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).
% Lcm_int_def
thf(fact_9950_Lcm__eq__Max__nat,axiom,
! [M10: set_nat] :
( ( finite_finite_nat @ M10 )
=> ( ( M10 != bot_bot_set_nat )
=> ( ~ ( member_nat @ zero_zero_nat @ M10 )
=> ( ! [M2: nat,N2: nat] :
( ( member_nat @ M2 @ M10 )
=> ( ( member_nat @ N2 @ M10 )
=> ( member_nat @ ( gcd_lcm_nat @ M2 @ N2 ) @ M10 ) ) )
=> ( ( gcd_Lcm_nat @ M10 )
= ( lattic8265883725875713057ax_nat @ M10 ) ) ) ) ) ) ).
% Lcm_eq_Max_nat
thf(fact_9951_Lcm__int__set__eq__fold,axiom,
! [Xs: list_int] :
( ( gcd_Lcm_int @ ( set_int2 @ Xs ) )
= ( fold_int_int @ gcd_lcm_int @ Xs @ one_one_int ) ) ).
% Lcm_int_set_eq_fold
thf(fact_9952_lcm__0__iff__int,axiom,
! [M: int,N: int] :
( ( ( gcd_lcm_int @ M @ N )
= zero_zero_int )
= ( ( M = zero_zero_int )
| ( N = zero_zero_int ) ) ) ).
% lcm_0_iff_int
thf(fact_9953_lcm__0__iff__nat,axiom,
! [M: nat,N: nat] :
( ( ( gcd_lcm_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% lcm_0_iff_nat
thf(fact_9954_abs__lcm__int,axiom,
! [I: int,J: int] :
( ( abs_abs_int @ ( gcd_lcm_int @ I @ J ) )
= ( gcd_lcm_int @ I @ J ) ) ).
% abs_lcm_int
thf(fact_9955_lcm__abs1__int,axiom,
! [X: int,Y: int] :
( ( gcd_lcm_int @ ( abs_abs_int @ X ) @ Y )
= ( gcd_lcm_int @ X @ Y ) ) ).
% lcm_abs1_int
thf(fact_9956_lcm__abs2__int,axiom,
! [X: int,Y: int] :
( ( gcd_lcm_int @ X @ ( abs_abs_int @ Y ) )
= ( gcd_lcm_int @ X @ Y ) ) ).
% lcm_abs2_int
thf(fact_9957_lcm__proj2__if__dvd__nat,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( gcd_lcm_nat @ X @ Y )
= Y ) ) ).
% lcm_proj2_if_dvd_nat
thf(fact_9958_lcm__proj1__if__dvd__nat,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( gcd_lcm_nat @ Y @ X )
= Y ) ) ).
% lcm_proj1_if_dvd_nat
thf(fact_9959_lcm__proj2__iff__nat,axiom,
! [M: nat,N: nat] :
( ( ( gcd_lcm_nat @ M @ N )
= N )
= ( dvd_dvd_nat @ M @ N ) ) ).
% lcm_proj2_iff_nat
thf(fact_9960_lcm__proj1__iff__nat,axiom,
! [M: nat,N: nat] :
( ( ( gcd_lcm_nat @ M @ N )
= M )
= ( dvd_dvd_nat @ N @ M ) ) ).
% lcm_proj1_iff_nat
thf(fact_9961_lcm__1__iff__nat,axiom,
! [M: nat,N: nat] :
( ( ( gcd_lcm_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% lcm_1_iff_nat
thf(fact_9962_lcm__1__iff__int,axiom,
! [M: int,N: int] :
( ( ( gcd_lcm_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) )
& ( ( N = one_one_int )
| ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% lcm_1_iff_int
thf(fact_9963_lcm__proj1__iff__int,axiom,
! [M: int,N: int] :
( ( ( gcd_lcm_int @ M @ N )
= ( abs_abs_int @ M ) )
= ( dvd_dvd_int @ N @ M ) ) ).
% lcm_proj1_iff_int
thf(fact_9964_lcm__proj2__iff__int,axiom,
! [M: int,N: int] :
( ( ( gcd_lcm_int @ M @ N )
= ( abs_abs_int @ N ) )
= ( dvd_dvd_int @ M @ N ) ) ).
% lcm_proj2_iff_int
thf(fact_9965_lcm__proj1__if__dvd__int,axiom,
! [X: int,Y: int] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( gcd_lcm_int @ Y @ X )
= ( abs_abs_int @ Y ) ) ) ).
% lcm_proj1_if_dvd_int
thf(fact_9966_lcm__proj2__if__dvd__int,axiom,
! [X: int,Y: int] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( gcd_lcm_int @ X @ Y )
= ( abs_abs_int @ Y ) ) ) ).
% lcm_proj2_if_dvd_int
thf(fact_9967_lcm__int__int__eq,axiom,
! [M: nat,N: nat] :
( ( gcd_lcm_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ ( gcd_lcm_nat @ M @ N ) ) ) ).
% lcm_int_int_eq
thf(fact_9968_lcm__nat__abs__right__eq,axiom,
! [N: nat,K: int] :
( ( gcd_lcm_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
= ( nat2 @ ( gcd_lcm_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% lcm_nat_abs_right_eq
thf(fact_9969_lcm__nat__abs__left__eq,axiom,
! [K: int,N: nat] :
( ( gcd_lcm_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
= ( nat2 @ ( gcd_lcm_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% lcm_nat_abs_left_eq
thf(fact_9970_Lcm__nat__insert,axiom,
! [N: nat,M10: set_nat] :
( ( gcd_Lcm_nat @ ( insert_nat @ N @ M10 ) )
= ( gcd_lcm_nat @ N @ ( gcd_Lcm_nat @ M10 ) ) ) ).
% Lcm_nat_insert
thf(fact_9971_lcm__nat__def,axiom,
( gcd_lcm_nat
= ( ^ [X4: nat,Y5: nat] : ( divide_divide_nat @ ( times_times_nat @ X4 @ Y5 ) @ ( gcd_gcd_nat @ X4 @ Y5 ) ) ) ) ).
% lcm_nat_def
thf(fact_9972_lcm__unique__int,axiom,
! [D: int,A: int,B: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ D )
& ( dvd_dvd_int @ A @ D )
& ( dvd_dvd_int @ B @ D )
& ! [E3: int] :
( ( ( dvd_dvd_int @ A @ E3 )
& ( dvd_dvd_int @ B @ E3 ) )
=> ( dvd_dvd_int @ D @ E3 ) ) )
= ( D
= ( gcd_lcm_int @ A @ B ) ) ) ).
% lcm_unique_int
thf(fact_9973_prod__gcd__lcm__nat,axiom,
( times_times_nat
= ( ^ [M4: nat,N4: nat] : ( times_times_nat @ ( gcd_gcd_nat @ M4 @ N4 ) @ ( gcd_lcm_nat @ M4 @ N4 ) ) ) ) ).
% prod_gcd_lcm_nat
thf(fact_9974_lcm__neg1__int,axiom,
! [X: int,Y: int] :
( ( gcd_lcm_int @ ( uminus_uminus_int @ X ) @ Y )
= ( gcd_lcm_int @ X @ Y ) ) ).
% lcm_neg1_int
thf(fact_9975_lcm__neg2__int,axiom,
! [X: int,Y: int] :
( ( gcd_lcm_int @ X @ ( uminus_uminus_int @ Y ) )
= ( gcd_lcm_int @ X @ Y ) ) ).
% lcm_neg2_int
thf(fact_9976_lcm__int__def,axiom,
( gcd_lcm_int
= ( ^ [X4: int,Y5: int] : ( semiri1314217659103216013at_int @ ( gcd_lcm_nat @ ( nat2 @ ( abs_abs_int @ X4 ) ) @ ( nat2 @ ( abs_abs_int @ Y5 ) ) ) ) ) ) ).
% lcm_int_def
thf(fact_9977_lcm__unique__nat,axiom,
! [A: nat,D: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ D )
& ( dvd_dvd_nat @ B @ D )
& ! [E3: nat] :
( ( ( dvd_dvd_nat @ A @ E3 )
& ( dvd_dvd_nat @ B @ E3 ) )
=> ( dvd_dvd_nat @ D @ E3 ) ) )
= ( D
= ( gcd_lcm_nat @ A @ B ) ) ) ).
% lcm_unique_nat
thf(fact_9978_dvd__lcm__I2__nat,axiom,
! [K: nat,N: nat,M: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( gcd_lcm_nat @ M @ N ) ) ) ).
% dvd_lcm_I2_nat
thf(fact_9979_dvd__lcm__I1__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( dvd_dvd_nat @ K @ ( gcd_lcm_nat @ M @ N ) ) ) ).
% dvd_lcm_I1_nat
thf(fact_9980_dvd__lcm__I2__int,axiom,
! [I: int,N: int,M: int] :
( ( dvd_dvd_int @ I @ N )
=> ( dvd_dvd_int @ I @ ( gcd_lcm_int @ M @ N ) ) ) ).
% dvd_lcm_I2_int
thf(fact_9981_dvd__lcm__I1__int,axiom,
! [I: int,M: int,N: int] :
( ( dvd_dvd_int @ I @ M )
=> ( dvd_dvd_int @ I @ ( gcd_lcm_int @ M @ N ) ) ) ).
% dvd_lcm_I1_int
thf(fact_9982_lcm__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ^ [Y4: int,Z: int] : Y4 = Z )
@ gcd_lcm_int
@ gcd_lcm_int ) ).
% lcm_integer.rsp
thf(fact_9983_lcm__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_lcm_int @ X @ Y ) ) ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( P @ ( gcd_lcm_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
=> ( ( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( P @ ( gcd_lcm_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
=> ( ( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( P @ ( gcd_lcm_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
=> ( P @ ( gcd_lcm_int @ X @ Y ) ) ) ) ) ) ).
% lcm_cases_int
thf(fact_9984_lcm__ge__0__int,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_lcm_int @ X @ Y ) ) ).
% lcm_ge_0_int
thf(fact_9985_lcm__pos__int,axiom,
! [M: int,N: int] :
( ( M != zero_zero_int )
=> ( ( N != zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( gcd_lcm_int @ M @ N ) ) ) ) ).
% lcm_pos_int
thf(fact_9986_lcm__pos__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( gcd_lcm_nat @ M @ N ) ) ) ) ).
% lcm_pos_nat
thf(fact_9987_prod__gcd__lcm__int,axiom,
! [M: int,N: int] :
( ( times_times_int @ ( abs_abs_int @ M ) @ ( abs_abs_int @ N ) )
= ( times_times_int @ ( gcd_gcd_int @ M @ N ) @ ( gcd_lcm_int @ M @ N ) ) ) ).
% prod_gcd_lcm_int
thf(fact_9988_Lcm__in__lcm__closed__set__nat,axiom,
! [M10: set_nat] :
( ( finite_finite_nat @ M10 )
=> ( ( M10 != bot_bot_set_nat )
=> ( ! [M2: nat,N2: nat] :
( ( member_nat @ M2 @ M10 )
=> ( ( member_nat @ N2 @ M10 )
=> ( member_nat @ ( gcd_lcm_nat @ M2 @ N2 ) @ M10 ) ) )
=> ( member_nat @ ( gcd_Lcm_nat @ M10 ) @ M10 ) ) ) ) ).
% Lcm_in_lcm_closed_set_nat
thf(fact_9989_lcm__altdef__int,axiom,
( gcd_lcm_int
= ( ^ [A3: int,B3: int] : ( divide_divide_int @ ( times_times_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) @ ( gcd_gcd_int @ A3 @ B3 ) ) ) ) ).
% lcm_altdef_int
thf(fact_9990_Lcm__nat__set__eq__fold,axiom,
! [Xs: list_nat] :
( ( gcd_Lcm_nat @ ( set_nat2 @ Xs ) )
= ( fold_nat_nat @ gcd_lcm_nat @ Xs @ one_one_nat ) ) ).
% Lcm_nat_set_eq_fold
thf(fact_9991_Lcm__nat__def,axiom,
( gcd_Lcm_nat
= ( ^ [M7: set_nat] : ( if_nat @ ( finite_finite_nat @ M7 ) @ ( lattic7826324295020591184_F_nat @ gcd_lcm_nat @ one_one_nat @ M7 ) @ zero_zero_nat ) ) ) ).
% Lcm_nat_def
thf(fact_9992_lcm__integer_Orep__eq,axiom,
! [X: code_integer,Xa3: code_integer] :
( ( code_int_of_integer @ ( gcd_lcm_Code_integer @ X @ Xa3 ) )
= ( gcd_lcm_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% lcm_integer.rep_eq
thf(fact_9993_lcm__integer_Oabs__eq,axiom,
! [Xa3: int,X: int] :
( ( gcd_lcm_Code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( gcd_lcm_int @ Xa3 @ X ) ) ) ).
% lcm_integer.abs_eq
thf(fact_9994_lcm__code__integer,axiom,
( gcd_lcm_Code_integer
= ( ^ [A3: code_integer,B3: code_integer] : ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( gcd_gcd_Code_integer @ A3 @ B3 ) ) ) ) ).
% lcm_code_integer
thf(fact_9995_lcm__integer_Otransfer,axiom,
bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ gcd_lcm_int @ gcd_lcm_Code_integer ).
% lcm_integer.transfer
thf(fact_9996_lcm__integer__def,axiom,
( gcd_lcm_Code_integer
= ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ gcd_lcm_int ) ) ).
% lcm_integer_def
thf(fact_9997_unit__factor__simps_I1_J,axiom,
( ( unit_f2748546683901255202or_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% unit_factor_simps(1)
thf(fact_9998_unit__factor__simps_I2_J,axiom,
! [N: nat] :
( ( unit_f2748546683901255202or_nat @ ( suc @ N ) )
= one_one_nat ) ).
% unit_factor_simps(2)
thf(fact_9999_unit__factor__nat__def,axiom,
( unit_f2748546683901255202or_nat
= ( ^ [N4: nat] : ( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% unit_factor_nat_def
thf(fact_10000_gcd__nat_Ocomm__monoid__axioms,axiom,
comm_monoid_nat @ gcd_gcd_nat @ zero_zero_nat ).
% gcd_nat.comm_monoid_axioms
thf(fact_10001_max__nat_Ocomm__monoid__axioms,axiom,
comm_monoid_nat @ ord_max_nat @ zero_zero_nat ).
% max_nat.comm_monoid_axioms
thf(fact_10002_times__num__def,axiom,
( times_times_num
= ( ^ [M4: num,N4: num] : ( num_of_nat @ ( times_times_nat @ ( nat_of_num @ M4 ) @ ( nat_of_num @ N4 ) ) ) ) ) ).
% times_num_def
thf(fact_10003_nat__of__num__mult,axiom,
! [X: num,Y: num] :
( ( nat_of_num @ ( times_times_num @ X @ Y ) )
= ( times_times_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).
% nat_of_num_mult
thf(fact_10004_nat__of__num__sqr,axiom,
! [X: num] :
( ( nat_of_num @ ( sqr @ X ) )
= ( times_times_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% nat_of_num_sqr
thf(fact_10005_nat__of__num__inc,axiom,
! [X: num] :
( ( nat_of_num @ ( inc @ X ) )
= ( suc @ ( nat_of_num @ X ) ) ) ).
% nat_of_num_inc
thf(fact_10006_nat__of__num__inverse,axiom,
! [X: num] :
( ( num_of_nat @ ( nat_of_num @ X ) )
= X ) ).
% nat_of_num_inverse
thf(fact_10007_num__eq__iff,axiom,
( ( ^ [Y4: num,Z: num] : Y4 = Z )
= ( ^ [X4: num,Y5: num] :
( ( nat_of_num @ X4 )
= ( nat_of_num @ Y5 ) ) ) ) ).
% num_eq_iff
thf(fact_10008_nat__of__num__numeral,axiom,
nat_of_num = numeral_numeral_nat ).
% nat_of_num_numeral
thf(fact_10009_less__num__def,axiom,
( ord_less_num
= ( ^ [M4: num,N4: num] : ( ord_less_nat @ ( nat_of_num @ M4 ) @ ( nat_of_num @ N4 ) ) ) ) ).
% less_num_def
thf(fact_10010_nat__of__num__pos,axiom,
! [X: num] : ( ord_less_nat @ zero_zero_nat @ ( nat_of_num @ X ) ) ).
% nat_of_num_pos
thf(fact_10011_nat__of__num__neq__0,axiom,
! [X: num] :
( ( nat_of_num @ X )
!= zero_zero_nat ) ).
% nat_of_num_neq_0
thf(fact_10012_nat__of__num__code_I2_J,axiom,
! [N: num] :
( ( nat_of_num @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ).
% nat_of_num_code(2)
thf(fact_10013_nat__of__num__code_I1_J,axiom,
( ( nat_of_num @ one )
= one_one_nat ) ).
% nat_of_num_code(1)
thf(fact_10014_nat__of__num_Osimps_I2_J,axiom,
! [X: num] :
( ( nat_of_num @ ( bit0 @ X ) )
= ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% nat_of_num.simps(2)
thf(fact_10015_nat__of__num__add,axiom,
! [X: num,Y: num] :
( ( nat_of_num @ ( plus_plus_num @ X @ Y ) )
= ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).
% nat_of_num_add
thf(fact_10016_less__eq__num__def,axiom,
( ord_less_eq_num
= ( ^ [M4: num,N4: num] : ( ord_less_eq_nat @ ( nat_of_num @ M4 ) @ ( nat_of_num @ N4 ) ) ) ) ).
% less_eq_num_def
thf(fact_10017_nat__of__num_Osimps_I1_J,axiom,
( ( nat_of_num @ one )
= ( suc @ zero_zero_nat ) ) ).
% nat_of_num.simps(1)
thf(fact_10018_nat__of__num_Osimps_I3_J,axiom,
! [X: num] :
( ( nat_of_num @ ( bit1 @ X ) )
= ( suc @ ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ) ).
% nat_of_num.simps(3)
thf(fact_10019_num__of__nat__inverse,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nat_of_num @ ( num_of_nat @ N ) )
= N ) ) ).
% num_of_nat_inverse
thf(fact_10020_nat__of__num__code_I3_J,axiom,
! [N: num] :
( ( nat_of_num @ ( bit1 @ N ) )
= ( suc @ ( plus_plus_nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ) ).
% nat_of_num_code(3)
thf(fact_10021_plus__num__def,axiom,
( plus_plus_num
= ( ^ [M4: num,N4: num] : ( num_of_nat @ ( plus_plus_nat @ ( nat_of_num @ M4 ) @ ( nat_of_num @ N4 ) ) ) ) ) ).
% plus_num_def
thf(fact_10022_real__floor__code,axiom,
! [X: rat] :
( ( archim6058952711729229775r_real @ ( ratreal @ X ) )
= ( archim3151403230148437115or_rat @ X ) ) ).
% real_floor_code
thf(fact_10023_real__minus__code,axiom,
! [X: rat,Y: rat] :
( ( minus_minus_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ratreal @ ( minus_minus_rat @ X @ Y ) ) ) ).
% real_minus_code
thf(fact_10024_real__less__eq__code,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ord_less_eq_rat @ X @ Y ) ) ).
% real_less_eq_code
thf(fact_10025_real__less__code,axiom,
! [X: rat,Y: rat] :
( ( ord_less_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ord_less_rat @ X @ Y ) ) ).
% real_less_code
thf(fact_10026_zero__real__code,axiom,
( zero_zero_real
= ( ratreal @ zero_zero_rat ) ) ).
% zero_real_code
thf(fact_10027_Ratreal__def,axiom,
ratreal = field_7254667332652039916t_real ).
% Ratreal_def
thf(fact_10028_real__uminus__code,axiom,
! [X: rat] :
( ( uminus_uminus_real @ ( ratreal @ X ) )
= ( ratreal @ ( uminus_uminus_rat @ X ) ) ) ).
% real_uminus_code
thf(fact_10029_one__real__code,axiom,
( one_one_real
= ( ratreal @ one_one_rat ) ) ).
% one_real_code
thf(fact_10030_real__times__code,axiom,
! [X: rat,Y: rat] :
( ( times_times_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ratreal @ ( times_times_rat @ X @ Y ) ) ) ).
% real_times_code
thf(fact_10031_real__plus__code,axiom,
! [X: rat,Y: rat] :
( ( plus_plus_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ratreal @ ( plus_plus_rat @ X @ Y ) ) ) ).
% real_plus_code
thf(fact_10032_real__inverse__code,axiom,
! [X: rat] :
( ( inverse_inverse_real @ ( ratreal @ X ) )
= ( ratreal @ ( inverse_inverse_rat @ X ) ) ) ).
% real_inverse_code
thf(fact_10033_real__divide__code,axiom,
! [X: rat,Y: rat] :
( ( divide_divide_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ratreal @ ( divide_divide_rat @ X @ Y ) ) ) ).
% real_divide_code
thf(fact_10034_natLeq__on__wo__rel,axiom,
! [N: nat] :
( bNF_We3818239936649020644el_nat
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ N )
& ( ord_less_nat @ Y5 @ N )
& ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) ) ).
% natLeq_on_wo_rel
thf(fact_10035_gcd__nat_Osemilattice__order__axioms,axiom,
( semila1248733672344298208er_nat @ gcd_gcd_nat @ dvd_dvd_nat
@ ^ [M4: nat,N4: nat] :
( ( dvd_dvd_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ).
% gcd_nat.semilattice_order_axioms
thf(fact_10036_natural__decr,axiom,
! [N: code_natural] :
( ( N != zero_z2226904508553997617atural )
=> ( ord_less_nat @ ( minus_minus_nat @ ( code_nat_of_natural @ N ) @ ( suc @ zero_zero_nat ) ) @ ( code_nat_of_natural @ N ) ) ) ).
% natural_decr
thf(fact_10037_plus__natural_Orep__eq,axiom,
! [X: code_natural,Xa3: code_natural] :
( ( code_nat_of_natural @ ( plus_p4538020629002901425atural @ X @ Xa3 ) )
= ( plus_plus_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% plus_natural.rep_eq
thf(fact_10038_minus__natural_Orep__eq,axiom,
! [X: code_natural,Xa3: code_natural] :
( ( code_nat_of_natural @ ( minus_7197305767214868737atural @ X @ Xa3 ) )
= ( minus_minus_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% minus_natural.rep_eq
thf(fact_10039_one__natural_Orep__eq,axiom,
( ( code_nat_of_natural @ one_one_Code_natural )
= one_one_nat ) ).
% one_natural.rep_eq
thf(fact_10040_zero__natural_Orep__eq,axiom,
( ( code_nat_of_natural @ zero_z2226904508553997617atural )
= zero_zero_nat ) ).
% zero_natural.rep_eq
thf(fact_10041_less__natural_Orep__eq,axiom,
( ord_le5570908160329646204atural
= ( ^ [X4: code_natural,Xa4: code_natural] : ( ord_less_nat @ ( code_nat_of_natural @ X4 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).
% less_natural.rep_eq
thf(fact_10042_natural__zero__minus__one,axiom,
( ( minus_7197305767214868737atural @ zero_z2226904508553997617atural @ one_one_Code_natural )
= zero_z2226904508553997617atural ) ).
% natural_zero_minus_one
thf(fact_10043_integer__of__natural_Orep__eq,axiom,
! [X: code_natural] :
( ( code_int_of_integer @ ( code_i5400310926305786745atural @ X ) )
= ( semiri1314217659103216013at_int @ ( code_nat_of_natural @ X ) ) ) ).
% integer_of_natural.rep_eq
thf(fact_10044_int__of__integer__of__natural,axiom,
! [N: code_natural] :
( ( code_int_of_integer @ ( code_i5400310926305786745atural @ N ) )
= ( semiri1314217659103216013at_int @ ( code_nat_of_natural @ N ) ) ) ).
% int_of_integer_of_natural
thf(fact_10045_log_Oelims,axiom,
! [X: code_natural,Xa3: code_natural,Y: code_natural] :
( ( ( log @ X @ Xa3 )
= Y )
=> ( ( ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
| ( ord_le5570908160329646204atural @ Xa3 @ X ) )
=> ( Y = one_one_Code_natural ) )
& ( ~ ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
| ( ord_le5570908160329646204atural @ Xa3 @ X ) )
=> ( Y
= ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ X @ ( divide5121882707175180666atural @ Xa3 @ X ) ) ) ) ) ) ) ).
% log.elims
thf(fact_10046_log_Osimps,axiom,
( log
= ( ^ [B3: code_natural,I3: code_natural] :
( if_Code_natural
@ ( ( ord_le1926595141338095240atural @ B3 @ one_one_Code_natural )
| ( ord_le5570908160329646204atural @ I3 @ B3 ) )
@ one_one_Code_natural
@ ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ B3 @ ( divide5121882707175180666atural @ I3 @ B3 ) ) ) ) ) ) ).
% log.simps
thf(fact_10047_next_Osimps,axiom,
! [V: code_natural,W2: code_natural] :
( ( next @ ( produc3574140220909816553atural @ V @ W2 ) )
= ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ one_one_Code_natural ) ) @ one_one_Code_natural ) @ ( produc3574140220909816553atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% next.simps
thf(fact_10048_Code__Numeral_ONat_Orep__eq,axiom,
! [X: code_integer] :
( ( code_nat_of_natural @ ( code_Nat @ X ) )
= ( nat2 @ ( code_int_of_integer @ X ) ) ) ).
% Code_Numeral.Nat.rep_eq
thf(fact_10049_Random_Orange__def,axiom,
( range
= ( ^ [K3: code_natural] :
( produc5538323210962509403atural
@ ( iterat8892046348760725948atural @ ( log @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K3 )
@ ^ [L3: code_natural] :
( produc5538323210962509403atural @ next
@ ^ [V4: code_natural] : ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ V4 @ ( times_2397367101498566445atural @ L3 @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ one_one_Code_natural )
@ ^ [V4: code_natural] : ( produc6639722614265839536atural @ ( modulo8411746178871703098atural @ V4 @ K3 ) ) ) ) ) ).
% Random.range_def
thf(fact_10050_inc__shift__def,axiom,
( inc_shift
= ( ^ [V4: code_natural,K3: code_natural] : ( if_Code_natural @ ( V4 = K3 ) @ one_one_Code_natural @ ( plus_p4538020629002901425atural @ K3 @ one_one_Code_natural ) ) ) ) ).
% inc_shift_def
thf(fact_10051_log_Opelims,axiom,
! [X: code_natural,Xa3: code_natural,Y: code_natural] :
( ( ( log @ X @ Xa3 )
= Y )
=> ( ( accp_P8126237942716283194atural @ log_rel @ ( produc3574140220909816553atural @ X @ Xa3 ) )
=> ~ ( ( ( ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
| ( ord_le5570908160329646204atural @ Xa3 @ X ) )
=> ( Y = one_one_Code_natural ) )
& ( ~ ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
| ( ord_le5570908160329646204atural @ Xa3 @ X ) )
=> ( Y
= ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ X @ ( divide5121882707175180666atural @ Xa3 @ X ) ) ) ) ) )
=> ~ ( accp_P8126237942716283194atural @ log_rel @ ( produc3574140220909816553atural @ X @ Xa3 ) ) ) ) ) ).
% log.pelims
thf(fact_10052_natural__of__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_nat_of_natural @ ( code_n4118661773612635043nteger @ X ) )
= ( nat2 @ ( code_int_of_integer @ X ) ) ) ).
% natural_of_integer.rep_eq
thf(fact_10053_integer__of__natural__def,axiom,
( code_i5400310926305786745atural
= ( map_fu2787874002554666395nteger @ code_nat_of_natural @ code_integer_of_int @ semiri1314217659103216013at_int ) ) ).
% integer_of_natural_def
thf(fact_10054_zero__natural__def,axiom,
( zero_z2226904508553997617atural
= ( code_natural_of_nat @ zero_zero_nat ) ) ).
% zero_natural_def
thf(fact_10055_less__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( ord_le5570908160329646204atural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
= ( ord_less_nat @ Xa3 @ X ) ) ).
% less_natural.abs_eq
thf(fact_10056_plus__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( plus_p4538020629002901425atural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( plus_plus_nat @ Xa3 @ X ) ) ) ).
% plus_natural.abs_eq
thf(fact_10057_minus__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( minus_7197305767214868737atural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( minus_minus_nat @ Xa3 @ X ) ) ) ).
% minus_natural.abs_eq
thf(fact_10058_one__natural__def,axiom,
( one_one_Code_natural
= ( code_natural_of_nat @ one_one_nat ) ) ).
% one_natural_def
thf(fact_10059_integer__of__natural_Oabs__eq,axiom,
! [X: nat] :
( ( code_i5400310926305786745atural @ ( code_natural_of_nat @ X ) )
= ( code_integer_of_int @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% integer_of_natural.abs_eq
thf(fact_10060_natural__of__integer_Oabs__eq,axiom,
! [X: int] :
( ( code_n4118661773612635043nteger @ ( code_integer_of_int @ X ) )
= ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).
% natural_of_integer.abs_eq
thf(fact_10061_Code__Numeral_ONat_Oabs__eq,axiom,
! [X: int] :
( ( code_Nat @ ( code_integer_of_int @ X ) )
= ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).
% Code_Numeral.Nat.abs_eq
thf(fact_10062_Code__Numeral_ONat__def,axiom,
( code_Nat
= ( map_fu1051355602067684763atural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).
% Code_Numeral.Nat_def
thf(fact_10063_natural__of__integer__def,axiom,
( code_n4118661773612635043nteger
= ( map_fu1051355602067684763atural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).
% natural_of_integer_def
thf(fact_10064_Suc_Orep__eq,axiom,
! [X: code_natural] :
( ( code_nat_of_natural @ ( code_Suc @ X ) )
= ( suc @ ( code_nat_of_natural @ X ) ) ) ).
% Suc.rep_eq
thf(fact_10065_Code__Numeral_ONat_Otransfer,axiom,
bNF_re1831474436612530402atural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_Nat ).
% Code_Numeral.Nat.transfer
thf(fact_10066_zero__natural_Otransfer,axiom,
code_pcr_natural @ zero_zero_nat @ zero_z2226904508553997617atural ).
% zero_natural.transfer
thf(fact_10067_Suc_Otransfer,axiom,
bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural @ suc @ code_Suc ).
% Suc.transfer
thf(fact_10068_less__natural_Otransfer,axiom,
( bNF_re1639080489988575423ural_o @ code_pcr_natural
@ ( bNF_re2785088596696291543al_o_o @ code_pcr_natural
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ord_less_nat
@ ord_le5570908160329646204atural ) ).
% less_natural.transfer
thf(fact_10069_minus__natural_Otransfer,axiom,
bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ minus_minus_nat @ minus_7197305767214868737atural ).
% minus_natural.transfer
thf(fact_10070_plus__natural_Otransfer,axiom,
bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ plus_plus_nat @ plus_p4538020629002901425atural ).
% plus_natural.transfer
thf(fact_10071_one__natural_Otransfer,axiom,
code_pcr_natural @ one_one_nat @ one_one_Code_natural ).
% one_natural.transfer
thf(fact_10072_Suc_Oabs__eq,axiom,
! [X: nat] :
( ( code_Suc @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( suc @ X ) ) ) ).
% Suc.abs_eq
thf(fact_10073_integer__of__natural_Otransfer,axiom,
bNF_re5252274238750452962nteger @ code_pcr_natural @ code_pcr_integer @ semiri1314217659103216013at_int @ code_i5400310926305786745atural ).
% integer_of_natural.transfer
thf(fact_10074_natural__of__integer_Otransfer,axiom,
bNF_re1831474436612530402atural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_n4118661773612635043nteger ).
% natural_of_integer.transfer
thf(fact_10075_Suc__natural__minus__one,axiom,
! [N: code_natural] :
( ( minus_7197305767214868737atural @ ( code_Suc @ N ) @ one_one_Code_natural )
= N ) ).
% Suc_natural_minus_one
thf(fact_10076_Code__Numeral_OSuc__def,axiom,
( code_Suc
= ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat @ suc ) ) ).
% Code_Numeral.Suc_def
thf(fact_10077_Code__Numeral_Osub__def,axiom,
( code_sub
= ( map_fu6891787308814931657nteger @ id_num @ ( map_fu8638147718074629079nteger @ id_num @ code_integer_of_int )
@ ^ [M4: num,N4: num] : ( minus_minus_int @ ( numeral_numeral_int @ M4 ) @ ( numeral_numeral_int @ N4 ) ) ) ) ).
% Code_Numeral.sub_def
thf(fact_10078_minus__natural__def,axiom,
( minus_7197305767214868737atural
= ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ minus_minus_nat ) ) ).
% minus_natural_def
thf(fact_10079_plus__natural__def,axiom,
( plus_p4538020629002901425atural
= ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ plus_plus_nat ) ) ).
% plus_natural_def
thf(fact_10080_Quotient3__int,axiom,
quotie6776551016481293500at_int @ intrel @ abs_Integ @ rep_Integ ).
% Quotient3_int
thf(fact_10081_Field__natLeq__on,axiom,
! [N: nat] :
( ( field_nat
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ N )
& ( ord_less_nat @ Y5 @ N )
& ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) )
= ( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ X4 @ N ) ) ) ).
% Field_natLeq_on
thf(fact_10082_natLeq__on__Well__order,axiom,
! [N: nat] :
( order_2888998067076097458on_nat
@ ( field_nat
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ N )
& ( ord_less_nat @ Y5 @ N )
& ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ N )
& ( ord_less_nat @ Y5 @ N )
& ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) ) ).
% natLeq_on_Well_order
thf(fact_10083_VEBT__internal_OminNull_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y )
=> ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv2: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ~ Y
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
=> ( ! [Uu2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ~ Y
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
=> ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ( Y
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> ( ~ Y
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(1)
thf(fact_10084_VEBT__internal_OminNull_Opelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ! [Uv2: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
=> ( ! [Uu2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(3)
thf(fact_10085_VEBT__internal_OminNull_Opelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(2)
thf(fact_10086_natLeq__on__well__order__on,axiom,
! [N: nat] :
( order_2888998067076097458on_nat
@ ( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ X4 @ N ) )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ N )
& ( ord_less_nat @ Y5 @ N )
& ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) ) ).
% natLeq_on_well_order_on
thf(fact_10087_less__eq__int__code_I9_J,axiom,
! [K: num,L: num] :
( ( ord_less_eq_int @ ( neg @ K ) @ ( neg @ L ) )
= ( ord_less_eq_num @ L @ K ) ) ).
% less_eq_int_code(9)
thf(fact_10088_less__eq__int__code_I7_J,axiom,
! [K: num] : ( ord_less_eq_int @ ( neg @ K ) @ zero_zero_int ) ).
% less_eq_int_code(7)
thf(fact_10089_less__eq__int__code_I3_J,axiom,
! [L: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( neg @ L ) ) ).
% less_eq_int_code(3)
thf(fact_10090_nat__code_I1_J,axiom,
! [K: num] :
( ( nat2 @ ( neg @ K ) )
= zero_zero_nat ) ).
% nat_code(1)
thf(fact_10091_less__int__code_I9_J,axiom,
! [K: num,L: num] :
( ( ord_less_int @ ( neg @ K ) @ ( neg @ L ) )
= ( ord_less_num @ L @ K ) ) ).
% less_int_code(9)
thf(fact_10092_plus__int__code_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( neg @ M ) @ ( neg @ N ) )
= ( neg @ ( plus_plus_num @ M @ N ) ) ) ).
% plus_int_code(6)
thf(fact_10093_less__int__code_I3_J,axiom,
! [L: num] :
~ ( ord_less_int @ zero_zero_int @ ( neg @ L ) ) ).
% less_int_code(3)
thf(fact_10094_less__int__code_I7_J,axiom,
! [K: num] : ( ord_less_int @ ( neg @ K ) @ zero_zero_int ) ).
% less_int_code(7)
thf(fact_10095_Int_Osub__code_I4_J,axiom,
! [N: num] :
( ( sub @ one @ ( bit0 @ N ) )
= ( neg @ ( bitM @ N ) ) ) ).
% Int.sub_code(4)
thf(fact_10096_Int_Osub__code_I5_J,axiom,
! [N: num] :
( ( sub @ one @ ( bit1 @ N ) )
= ( neg @ ( bit0 @ N ) ) ) ).
% Int.sub_code(5)
thf(fact_10097_Int_Osub__code_I1_J,axiom,
( ( sub @ one @ one )
= zero_zero_int ) ).
% Int.sub_code(1)
thf(fact_10098_Int_Osub__def,axiom,
( sub
= ( ^ [M4: num,N4: num] : ( minus_minus_int @ ( numeral_numeral_int @ M4 ) @ ( numeral_numeral_int @ N4 ) ) ) ) ).
% Int.sub_def
thf(fact_10099_minus__int__code_I6_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( neg @ M ) @ ( neg @ N ) )
= ( sub @ N @ M ) ) ).
% minus_int_code(6)
thf(fact_10100_Int_Osub__code_I9_J,axiom,
! [M: num,N: num] :
( ( sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( minus_minus_int @ ( dup @ ( sub @ M @ N ) ) @ one_one_int ) ) ).
% Int.sub_code(9)
thf(fact_10101_Int_Osub__code_I8_J,axiom,
! [M: num,N: num] :
( ( sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( plus_plus_int @ ( dup @ ( sub @ M @ N ) ) @ one_one_int ) ) ).
% Int.sub_code(8)
thf(fact_10102_Int_Odup__code_I1_J,axiom,
( ( dup @ zero_zero_int )
= zero_zero_int ) ).
% Int.dup_code(1)
thf(fact_10103_Int_Odup__def,axiom,
( dup
= ( ^ [K3: int] : ( plus_plus_int @ K3 @ K3 ) ) ) ).
% Int.dup_def
thf(fact_10104_Int_Odup__code_I3_J,axiom,
! [N: num] :
( ( dup @ ( neg @ N ) )
= ( neg @ ( bit0 @ N ) ) ) ).
% Int.dup_code(3)
thf(fact_10105_Int_Osub__code_I6_J,axiom,
! [M: num,N: num] :
( ( sub @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( dup @ ( sub @ M @ N ) ) ) ).
% Int.sub_code(6)
thf(fact_10106_Int_Osub__code_I7_J,axiom,
! [M: num,N: num] :
( ( sub @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( dup @ ( sub @ M @ N ) ) ) ).
% Int.sub_code(7)
thf(fact_10107_Int_Osub__code_I2_J,axiom,
! [M: num] :
( ( sub @ ( bit0 @ M ) @ one )
= ( pos @ ( bitM @ M ) ) ) ).
% Int.sub_code(2)
thf(fact_10108_Int_Osub__code_I3_J,axiom,
! [M: num] :
( ( sub @ ( bit1 @ M ) @ one )
= ( pos @ ( bit0 @ M ) ) ) ).
% Int.sub_code(3)
thf(fact_10109_Int_Odup__code_I2_J,axiom,
! [N: num] :
( ( dup @ ( pos @ N ) )
= ( pos @ ( bit0 @ N ) ) ) ).
% Int.dup_code(2)
thf(fact_10110_less__eq__int__code_I8_J,axiom,
! [K: num,L: num] : ( ord_less_eq_int @ ( neg @ K ) @ ( pos @ L ) ) ).
% less_eq_int_code(8)
thf(fact_10111_less__eq__int__code_I6_J,axiom,
! [K: num,L: num] :
~ ( ord_less_eq_int @ ( pos @ K ) @ ( neg @ L ) ) ).
% less_eq_int_code(6)
thf(fact_10112_uminus__int__code_I3_J,axiom,
! [M: num] :
( ( uminus_uminus_int @ ( neg @ M ) )
= ( pos @ M ) ) ).
% uminus_int_code(3)
thf(fact_10113_uminus__int__code_I2_J,axiom,
! [M: num] :
( ( uminus_uminus_int @ ( pos @ M ) )
= ( neg @ M ) ) ).
% uminus_int_code(2)
thf(fact_10114_Int_ONeg__def,axiom,
( neg
= ( ^ [N4: num] : ( uminus_uminus_int @ ( pos @ N4 ) ) ) ) ).
% Int.Neg_def
thf(fact_10115_less__int__code_I6_J,axiom,
! [K: num,L: num] :
~ ( ord_less_int @ ( pos @ K ) @ ( neg @ L ) ) ).
% less_int_code(6)
thf(fact_10116_less__int__code_I8_J,axiom,
! [K: num,L: num] : ( ord_less_int @ ( neg @ K ) @ ( pos @ L ) ) ).
% less_int_code(8)
thf(fact_10117_less__int__code_I4_J,axiom,
! [K: num] :
~ ( ord_less_int @ ( pos @ K ) @ zero_zero_int ) ).
% less_int_code(4)
thf(fact_10118_less__int__code_I2_J,axiom,
! [L: num] : ( ord_less_int @ zero_zero_int @ ( pos @ L ) ) ).
% less_int_code(2)
thf(fact_10119_one__int__code,axiom,
( one_one_int
= ( pos @ one ) ) ).
% one_int_code
thf(fact_10120_plus__int__code_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( pos @ M ) @ ( pos @ N ) )
= ( pos @ ( plus_plus_num @ M @ N ) ) ) ).
% plus_int_code(3)
thf(fact_10121_less__int__code_I5_J,axiom,
! [K: num,L: num] :
( ( ord_less_int @ ( pos @ K ) @ ( pos @ L ) )
= ( ord_less_num @ K @ L ) ) ).
% less_int_code(5)
thf(fact_10122_nat__code_I3_J,axiom,
! [K: num] :
( ( nat2 @ ( pos @ K ) )
= ( nat_of_num @ K ) ) ).
% nat_code(3)
thf(fact_10123_less__eq__int__code_I2_J,axiom,
! [L: num] : ( ord_less_eq_int @ zero_zero_int @ ( pos @ L ) ) ).
% less_eq_int_code(2)
thf(fact_10124_less__eq__int__code_I4_J,axiom,
! [K: num] :
~ ( ord_less_eq_int @ ( pos @ K ) @ zero_zero_int ) ).
% less_eq_int_code(4)
thf(fact_10125_less__eq__int__code_I5_J,axiom,
! [K: num,L: num] :
( ( ord_less_eq_int @ ( pos @ K ) @ ( pos @ L ) )
= ( ord_less_eq_num @ K @ L ) ) ).
% less_eq_int_code(5)
thf(fact_10126_Int_OPos__def,axiom,
pos = numeral_numeral_int ).
% Int.Pos_def
thf(fact_10127_times__int__code_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( pos @ M ) @ ( pos @ N ) )
= ( pos @ ( times_times_num @ M @ N ) ) ) ).
% times_int_code(3)
thf(fact_10128_minus__int__code_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( pos @ M ) @ ( pos @ N ) )
= ( sub @ M @ N ) ) ).
% minus_int_code(3)
thf(fact_10129_minus__int__code_I5_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( neg @ M ) @ ( pos @ N ) )
= ( neg @ ( plus_plus_num @ M @ N ) ) ) ).
% minus_int_code(5)
thf(fact_10130_minus__int__code_I4_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( pos @ M ) @ ( neg @ N ) )
= ( pos @ ( plus_plus_num @ M @ N ) ) ) ).
% minus_int_code(4)
thf(fact_10131_times__int__code_I4_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( pos @ M ) @ ( neg @ N ) )
= ( neg @ ( times_times_num @ M @ N ) ) ) ).
% times_int_code(4)
thf(fact_10132_times__int__code_I5_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( neg @ M ) @ ( pos @ N ) )
= ( neg @ ( times_times_num @ M @ N ) ) ) ).
% times_int_code(5)
thf(fact_10133_times__int__code_I6_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( neg @ M ) @ ( neg @ N ) )
= ( pos @ ( times_times_num @ M @ N ) ) ) ).
% times_int_code(6)
thf(fact_10134_plus__int__code_I4_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( pos @ M ) @ ( neg @ N ) )
= ( sub @ M @ N ) ) ).
% plus_int_code(4)
thf(fact_10135_plus__int__code_I5_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( neg @ M ) @ ( pos @ N ) )
= ( sub @ N @ M ) ) ).
% plus_int_code(5)
thf(fact_10136_Quotient3__real,axiom,
quotie8700032322157300518t_real @ realrel @ real2 @ rep_real ).
% Quotient3_real
thf(fact_10137_sum__encode__def,axiom,
( nat_sum_encode
= ( sum_ca6763686470577984908at_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ^ [B3: nat] : ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ) ).
% sum_encode_def
thf(fact_10138_inj__sum__encode,axiom,
! [A2: set_Sum_sum_nat_nat] : ( inj_on6343450744447823682at_nat @ nat_sum_encode @ A2 ) ).
% inj_sum_encode
thf(fact_10139_bij__sum__encode,axiom,
bij_be5432664580149595207at_nat @ nat_sum_encode @ top_to6661820994512907621at_nat @ top_top_set_nat ).
% bij_sum_encode
thf(fact_10140_sum__encode__eq,axiom,
! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] :
( ( ( nat_sum_encode @ X )
= ( nat_sum_encode @ Y ) )
= ( X = Y ) ) ).
% sum_encode_eq
thf(fact_10141_surj__sum__encode,axiom,
( ( image_1320371278474632150at_nat @ nat_sum_encode @ top_to6661820994512907621at_nat )
= top_top_set_nat ) ).
% surj_sum_encode
thf(fact_10142_int__encode__def,axiom,
( nat_int_encode
= ( ^ [I3: int] : ( nat_sum_encode @ ( if_Sum_sum_nat_nat @ ( ord_less_eq_int @ zero_zero_int @ I3 ) @ ( sum_Inl_nat_nat @ ( nat2 @ I3 ) ) @ ( sum_Inr_nat_nat @ ( nat2 @ ( minus_minus_int @ ( uminus_uminus_int @ I3 ) @ one_one_int ) ) ) ) ) ) ) ).
% int_encode_def
thf(fact_10143_sum__decode__def,axiom,
( nat_sum_decode
= ( ^ [N4: nat] : ( if_Sum_sum_nat_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( sum_Inl_nat_nat @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( sum_Inr_nat_nat @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% sum_decode_def
thf(fact_10144_sum__encode__inverse,axiom,
! [X: sum_sum_nat_nat] :
( ( nat_sum_decode @ ( nat_sum_encode @ X ) )
= X ) ).
% sum_encode_inverse
thf(fact_10145_sum__decode__inverse,axiom,
! [N: nat] :
( ( nat_sum_encode @ ( nat_sum_decode @ N ) )
= N ) ).
% sum_decode_inverse
thf(fact_10146_surj__sum__decode,axiom,
( ( image_678696785212003926at_nat @ nat_sum_decode @ top_top_set_nat )
= top_to6661820994512907621at_nat ) ).
% surj_sum_decode
thf(fact_10147_bij__sum__decode,axiom,
bij_be4790990086886966983at_nat @ nat_sum_decode @ top_top_set_nat @ top_to6661820994512907621at_nat ).
% bij_sum_decode
thf(fact_10148_inj__sum__decode,axiom,
! [A2: set_nat] : ( inj_on5701776251185195458at_nat @ nat_sum_decode @ A2 ) ).
% inj_sum_decode
thf(fact_10149_sum__decode__eq,axiom,
! [X: nat,Y: nat] :
( ( ( nat_sum_decode @ X )
= ( nat_sum_decode @ Y ) )
= ( X = Y ) ) ).
% sum_decode_eq
thf(fact_10150_nth__item_Opinduct,axiom,
! [A0: nat,P: nat > $o] :
( ( accp_nat @ nth_item_rel @ A0 )
=> ( ( ( accp_nat @ nth_item_rel @ zero_zero_nat )
=> ( P @ zero_zero_nat ) )
=> ( ! [N2: nat] :
( ( accp_nat @ nth_item_rel @ ( suc @ N2 ) )
=> ( ! [A7: nat,Aa: nat] :
( ( ( nat_sum_decode @ N2 )
= ( sum_Inl_nat_nat @ A7 ) )
=> ( ( ( nat_sum_decode @ A7 )
= ( sum_Inl_nat_nat @ Aa ) )
=> ( P @ Aa ) ) )
=> ( ! [A7: nat,B9: nat] :
( ( ( nat_sum_decode @ N2 )
= ( sum_Inl_nat_nat @ A7 ) )
=> ( ( ( nat_sum_decode @ A7 )
= ( sum_Inr_nat_nat @ B9 ) )
=> ( P @ B9 ) ) )
=> ( ! [B9: nat,Ba: nat,X5: nat,Y6: nat] :
( ( ( nat_sum_decode @ N2 )
= ( sum_Inr_nat_nat @ B9 ) )
=> ( ( ( nat_sum_decode @ B9 )
= ( sum_Inr_nat_nat @ Ba ) )
=> ( ( ( product_Pair_nat_nat @ X5 @ Y6 )
= ( nat_prod_decode @ Ba ) )
=> ( P @ X5 ) ) ) )
=> ( ! [B9: nat,Ba: nat,X5: nat,Y6: nat] :
( ( ( nat_sum_decode @ N2 )
= ( sum_Inr_nat_nat @ B9 ) )
=> ( ( ( nat_sum_decode @ B9 )
= ( sum_Inr_nat_nat @ Ba ) )
=> ( ( ( product_Pair_nat_nat @ X5 @ Y6 )
= ( nat_prod_decode @ Ba ) )
=> ( P @ Y6 ) ) ) )
=> ( P @ ( suc @ N2 ) ) ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% nth_item.pinduct
thf(fact_10151_int__decode__def,axiom,
( nat_int_decode
= ( ^ [N4: nat] :
( sum_ca7763040182479039464nt_nat @ semiri1314217659103216013at_int
@ ^ [B3: nat] : ( minus_minus_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ B3 ) ) @ one_one_int )
@ ( nat_sum_decode @ N4 ) ) ) ) ).
% int_decode_def
% 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 ) )
= ( ? [X8: real] : ( P @ X8 ) ) ) ).
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__Extended____Nat__Oenat_T,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( if_Extended_enat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( if_Extended_enat @ $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__Code____Numeral__Onatural_T,axiom,
! [X: code_natural,Y: code_natural] :
( ( if_Code_natural @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Code____Numeral__Onatural_T,axiom,
! [X: code_natural,Y: code_natural] :
( ( if_Code_natural @ $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__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__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] :
( ( if_Sum_sum_nat_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] :
( ( if_Sum_sum_nat_nat @ $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_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 (1)
thf(conj_0,conjecture,
ord_less_nat @ zero_zero_nat @ m ).
%------------------------------------------------------------------------------