TPTP Problem File: ITP212_3.p
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%------------------------------------------------------------------------------
% File : ITP212_3 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Assertions 00821_023885
% 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 : 0024_Assertions_00821_023885 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 8903 (2939 unt;1727 typ; 0 def)
% Number of atoms : 16205 (6491 equ)
% Maximal formula atoms : 38 ( 2 avg)
% Number of connectives : 10511 (1831 ~; 287 |; 744 &)
% (1746 <=>;5903 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 4 avg)
% Maximal term depth : 38 ( 2 avg)
% Number of FOOLs : 1171 ( 380 fml; 791 var)
% Number of X terms : 307 ( 0 []; 288 ite; 19 let)
% Number of types : 330 ( 329 usr)
% Number of type conns : 1610 ( 976 >; 634 *; 0 +; 0 <<)
% Number of predicates : 163 ( 160 usr; 4 prp; 0-6 aty)
% Number of functors : 1243 (1243 usr; 425 con; 0-4 aty)
% Number of variables : 15774 (15543 !; 231 ?;15774 :)
% SPC : TX0_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 15:05:32.553
%------------------------------------------------------------------------------
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Code____Numeral__Onatural_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Code____Numeral__Onatural_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J,type,
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tff(ty_n_t__fun_It__Code____Numeral__Onatural_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Code____Numeral__Ointeger_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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tff(ty_n_t__fun_It__fun_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__fun_I_Eo_Mt__Rat__Orat_J_J,type,
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tff(ty_n_t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Product____Type__Ounit_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__fun_I_Eo_M_Eo_J_Mt__fun_I_Eo_M_Eo_J_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Assertions__Oassn_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Assertions__Oassn_J,type,
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tff(ty_n_t__fun_It__Assertions__Oassn_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Assertions__Oassn_Mt__Int__Oint_J,type,
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tff(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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tff(ty_n_t__Set__Oset_It__Code____Numeral__Onatural_J,type,
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tff(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Int__Oint_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Code____Numeral__Onatural_M_Eo_J,type,
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tff(ty_n_t__fun_It__Code____Numeral__Ointeger_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Code____Numeral__Ointeger_J,type,
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tff(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
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tff(ty_n_t__fun_It__Product____Type__Ounit_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Product____Type__Ounit_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Num__Onum_J,type,
fun_num_num: $tType ).
tff(ty_n_t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
fun_nat_rat: $tType ).
tff(ty_n_t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(ty_n_t__Set__Oset_It__Assertions__Oassn_J,type,
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tff(ty_n_t__fun_It__Assertions__Oassn_M_Eo_J,type,
fun_assn_o: $tType ).
tff(ty_n_t__fun_I_Eo_Mt__Assertions__Oassn_J,type,
fun_o_assn: $tType ).
tff(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
tff(ty_n_t__fun_It__Set__Oset_I_Eo_J_M_Eo_J,type,
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tff(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
option_num: $tType ).
tff(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
filter_nat: $tType ).
tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J,type,
fun_o_fun_o_o: $tType ).
tff(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
tff(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
tff(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
set_rat: $tType ).
tff(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
tff(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
tff(ty_n_t__Code____Numeral__Onatural,type,
code_natural: $tType ).
tff(ty_n_t__Code____Numeral__Ointeger,type,
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tff(ty_n_t__fun_It__Rat__Orat_M_Eo_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_M_Eo_J,type,
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tff(ty_n_t__fun_It__Int__Oint_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Int__Oint_J,type,
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tff(ty_n_t__Product____Type__Ounit,type,
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tff(ty_n_t__Typerep__Otyperep,type,
typerep: $tType ).
tff(ty_n_t__Assertions__Oassn,type,
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tff(ty_n_t__Set__Oset_I_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_M_Eo_J,type,
fun_o_o: $tType ).
tff(ty_n_t__Rat__Orat,type,
rat: $tType ).
tff(ty_n_t__Num__Onum,type,
num: $tType ).
tff(ty_n_t__Nat__Onat,type,
nat: $tType ).
tff(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (1398)
tff(sy_c_ATP_058Lamm__a____,type,
aTP_Lamm_a: set_Pr1261947904930325089at_nat > fun_Pr4796943543331071514_nat_o ).
tff(sy_c_ATP_058Lamm__aa____,type,
aTP_Lamm_aa: set_o > fun_o_o ).
tff(sy_c_ATP_058Lamm__ab____,type,
aTP_Lamm_ab: set_int > fun_int_o ).
tff(sy_c_ATP_058Lamm__ac____,type,
aTP_Lamm_ac: set_fu3555266933746557007_nat_o > fun_fu5135969500543783852at_o_o ).
tff(sy_c_ATP_058Lamm__ad____,type,
aTP_Lamm_ad: set_nat > fun_nat_o ).
tff(sy_c_ATP_058Lamm__ae____,type,
aTP_Lamm_ae: rat > fun_nat_fun_rat_rat ).
tff(sy_c_ATP_058Lamm__af____,type,
aTP_Lamm_af: fun_Pr6460944699146269935_nat_o ).
tff(sy_c_ATP_058Lamm__ag____,type,
aTP_Lamm_ag: fun_Pr6460944699146269935_nat_o > fun_fu8419763957888848259_nat_o ).
tff(sy_c_ATP_058Lamm__ah____,type,
aTP_Lamm_ah: fun_Pr6460944699146269935_nat_o > fun_fu8419763957888848259_nat_o ).
tff(sy_c_ATP_058Lamm__ai____,type,
aTP_Lamm_ai: produc3843707927480180839at_nat > fun_Pr3059359313727237556_nat_o ).
tff(sy_c_ATP_058Lamm__aj____,type,
aTP_Lamm_aj: fun_fu4164567611050918574at_o_o ).
tff(sy_c_ATP_058Lamm__ak____,type,
aTP_Lamm_ak: fun_Pr2028404973800837934_nat_o ).
tff(sy_c_ATP_058Lamm__al____,type,
aTP_Lamm_al: fun_o_fun_o_o ).
tff(sy_c_ATP_058Lamm__am____,type,
aTP_Lamm_am: fun_nat_fun_nat_o ).
tff(sy_c_ATP_058Lamm__an____,type,
aTP_Lamm_an: fun_int_fun_int_o ).
tff(sy_c_ATP_058Lamm__ao____,type,
aTP_Lamm_ao: $o > fun_fu5135969500543783852at_o_o ).
tff(sy_c_ATP_058Lamm__ap____,type,
aTP_Lamm_ap: $o > fun_Pr4796943543331071514_nat_o ).
tff(sy_c_ATP_058Lamm__aq____,type,
aTP_Lamm_aq: $o > fun_int_o ).
tff(sy_c_ATP_058Lamm__ar____,type,
aTP_Lamm_ar: fun_o_fun_o_o ).
tff(sy_c_ATP_058Lamm__as____,type,
aTP_Lamm_as: $o > fun_nat_o ).
tff(sy_c_ATP_058Lamm__at____,type,
aTP_Lamm_at: $o > fun_Product_unit_o ).
tff(sy_c_ATP_058Lamm__au____,type,
aTP_Lamm_au: fun_fu8419763957888848259_nat_o ).
tff(sy_c_ATP_058Lamm__av____,type,
aTP_Lamm_av: nat > fun_o_nat ).
tff(sy_c_ATP_058Lamm__aw____,type,
aTP_Lamm_aw: int > fun_o_int ).
tff(sy_c_ATP_058Lamm__ax____,type,
aTP_Lamm_ax: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__ay____,type,
aTP_Lamm_ay: int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__az____,type,
aTP_Lamm_az: nat > fun_Product_unit_nat ).
tff(sy_c_ATP_058Lamm__ba____,type,
aTP_Lamm_ba: int > fun_Product_unit_int ).
tff(sy_c_ATP_058Lamm__bb____,type,
aTP_Lamm_bb: set_nat > fun_nat_set_nat ).
tff(sy_c_ATP_058Lamm__bc____,type,
aTP_Lamm_bc: fun_Co8499238673658658103nteger ).
tff(sy_c_ATP_058Lamm__bd____,type,
aTP_Lamm_bd: fun_rat_rat ).
tff(sy_c_ATP_058Lamm__be____,type,
aTP_Lamm_be: fun_nat_nat ).
tff(sy_c_ATP_058Lamm__bf____,type,
aTP_Lamm_bf: fun_int_int ).
tff(sy_c_ATP_058Lamm__bg____,type,
aTP_Lamm_bg: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).
tff(sy_c_ATP_058Lamm__bh____,type,
aTP_Lamm_bh: set_Pr3486312109254157132nteger > fun_fu7695273699059480439eger_o ).
tff(sy_c_ATP_058Lamm__bi____,type,
aTP_Lamm_bi: set_Pr1312406793078187350et_nat > fun_fu8419763957888848259_nat_o ).
tff(sy_c_ATP_058Lamm__bj____,type,
aTP_Lamm_bj: set_Pr4040394263813967220et_nat > fun_fu2611741329380808165_nat_o ).
tff(sy_c_ATP_058Lamm__bk____,type,
aTP_Lamm_bk: set_Pr6700895561065273913nt_int > fun_fu6865642474282033866_int_o ).
tff(sy_c_ATP_058Lamm__bl____,type,
aTP_Lamm_bl: set_Pr4166680992137330533nt_int > fun_fu2978539155105838238_int_o ).
tff(sy_c_ATP_058Lamm__bm____,type,
aTP_Lamm_bm: set_Pr4329608150637261639at_nat > fun_se5639961202347202350_nat_o ).
tff(sy_c_ATP_058Lamm__bn____,type,
aTP_Lamm_bn: fun_fu5135969500543783852at_o_o ).
tff(sy_c_ATP_058Lamm__bo____,type,
aTP_Lamm_bo: fun_Pr4796943543331071514_nat_o ).
tff(sy_c_ATP_058Lamm__bp____,type,
aTP_Lamm_bp: fun_o_o ).
tff(sy_c_ATP_058Lamm__bq____,type,
aTP_Lamm_bq: fun_nat_o ).
tff(sy_c_ATP_058Lamm__br____,type,
aTP_Lamm_br: fun_int_o ).
tff(sy_c_ATP_058Lamm__bs____,type,
aTP_Lamm_bs: fun_Co8499238673658658103nteger ).
tff(sy_c_ATP_058Lamm__bt____,type,
aTP_Lamm_bt: fun_assn_assn ).
tff(sy_c_ATP_058Lamm__bu____,type,
aTP_Lamm_bu: fun_rat_rat ).
tff(sy_c_ATP_058Lamm__bv____,type,
aTP_Lamm_bv: fun_nat_nat ).
tff(sy_c_ATP_058Lamm__bw____,type,
aTP_Lamm_bw: fun_int_int ).
tff(sy_c_ATP_058Lamm__bx____,type,
aTP_Lamm_bx: produc3843707927480180839at_nat > fun_fu1322491050725084261_nat_o ).
tff(sy_c_ATP_058Lamm__by____,type,
aTP_Lamm_by: fun_Pr6460944699146269935_nat_o > fun_fu6295675835253126101at_o_o ).
tff(sy_c_ATP_058Lamm__bz____,type,
aTP_Lamm_bz: product_prod_nat_nat > fun_fu3275390434952296369_nat_o ).
tff(sy_c_ATP_058Lamm__ca____,type,
aTP_Lamm_ca: $o > fun_fun_o_o_fun_o_o ).
tff(sy_c_ATP_058Lamm__cb____,type,
aTP_Lamm_cb: nat > fun_fu4001317426178549763_nat_o ).
tff(sy_c_ATP_058Lamm__cc____,type,
aTP_Lamm_cc: int > fun_fu3883539188510166275_int_o ).
tff(sy_c_ATP_058Lamm__cd____,type,
aTP_Lamm_cd: produc3843707927480180839at_nat > fun_fu1322491050725084261_nat_o ).
tff(sy_c_ATP_058Lamm__ce____,type,
aTP_Lamm_ce: fun_Pr6460944699146269935_nat_o > fun_fu6295675835253126101at_o_o ).
tff(sy_c_ATP_058Lamm__cf____,type,
aTP_Lamm_cf: product_prod_nat_nat > fun_fu3275390434952296369_nat_o ).
tff(sy_c_ATP_058Lamm__cg____,type,
aTP_Lamm_cg: $o > fun_fun_o_o_fun_o_o ).
tff(sy_c_ATP_058Lamm__ch____,type,
aTP_Lamm_ch: nat > fun_fu4001317426178549763_nat_o ).
tff(sy_c_ATP_058Lamm__ci____,type,
aTP_Lamm_ci: int > fun_fu3883539188510166275_int_o ).
tff(sy_c_ATP_058Lamm__cj____,type,
aTP_Lamm_cj: fun_nat_Code_integer > fun_nat_Code_integer ).
tff(sy_c_ATP_058Lamm__ck____,type,
aTP_Lamm_ck: fun_nat_assn > fun_nat_assn ).
tff(sy_c_ATP_058Lamm__cl____,type,
aTP_Lamm_cl: fun_nat_rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__cm____,type,
aTP_Lamm_cm: fun_nat_nat > fun_nat_nat ).
tff(sy_c_ATP_058Lamm__cn____,type,
aTP_Lamm_cn: fun_nat_int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__co____,type,
aTP_Lamm_co: fun_o_Code_integer > fun_Co184521375657815108nteger ).
tff(sy_c_ATP_058Lamm__cp____,type,
aTP_Lamm_cp: fun_o_rat > fun_rat_fun_o_rat ).
tff(sy_c_ATP_058Lamm__cq____,type,
aTP_Lamm_cq: fun_o_nat > fun_nat_fun_o_nat ).
tff(sy_c_ATP_058Lamm__cr____,type,
aTP_Lamm_cr: fun_o_int > fun_int_fun_o_int ).
tff(sy_c_ATP_058Lamm__cs____,type,
aTP_Lamm_cs: fun_nat_Code_integer > fun_Co6226381647958292740nteger ).
tff(sy_c_ATP_058Lamm__ct____,type,
aTP_Lamm_ct: fun_nat_rat > fun_rat_fun_nat_rat ).
tff(sy_c_ATP_058Lamm__cu____,type,
aTP_Lamm_cu: fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__cv____,type,
aTP_Lamm_cv: fun_nat_int > fun_int_fun_nat_int ).
tff(sy_c_ATP_058Lamm__cw____,type,
aTP_Lamm_cw: fun_int_Code_integer > fun_Co3011819262099091808nteger ).
tff(sy_c_ATP_058Lamm__cx____,type,
aTP_Lamm_cx: fun_int_rat > fun_rat_fun_int_rat ).
tff(sy_c_ATP_058Lamm__cy____,type,
aTP_Lamm_cy: fun_nat_Code_integer > fun_na7078387940644931716nteger ).
tff(sy_c_ATP_058Lamm__cz____,type,
aTP_Lamm_cz: fun_nat_assn > fun_na2454419004871237530n_assn ).
tff(sy_c_ATP_058Lamm__da____,type,
aTP_Lamm_da: fun_nat_rat > fun_nat_fun_rat_rat ).
tff(sy_c_ATP_058Lamm__db____,type,
aTP_Lamm_db: fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__dc____,type,
aTP_Lamm_dc: fun_nat_int > fun_nat_fun_int_int ).
tff(sy_c_ATP_058Lamm__dd____,type,
aTP_Lamm_dd: nat > fun_int_nat ).
tff(sy_c_ATP_058Lamm__de____,type,
aTP_Lamm_de: fun_int_fun_int_int ).
tff(sy_c_ATP_058Lamm__df____,type,
aTP_Lamm_df: rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__dg____,type,
aTP_Lamm_dg: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__dh____,type,
aTP_Lamm_dh: assn > fun_as3747249873083764768_nat_o ).
tff(sy_c_ATP_058Lamm__di____,type,
aTP_Lamm_di: assn > fun_as3747249873083764768_nat_o ).
tff(sy_c_ATP_058Lamm__dj____,type,
aTP_Lamm_dj: fun_se5639961202347202350_nat_o ).
tff(sy_c_ATP_058Lamm__dk____,type,
aTP_Lamm_dk: fun_se5639961202347202350_nat_o ).
tff(sy_c_ATP_058Lamm__dl____,type,
aTP_Lamm_dl: fun_se5038631536132265326et_o_o ).
tff(sy_c_ATP_058Lamm__dm____,type,
aTP_Lamm_dm: fun_se5038631536132265326et_o_o ).
tff(sy_c_ATP_058Lamm__dn____,type,
aTP_Lamm_dn: fun_se2051630309975736070_int_o ).
tff(sy_c_ATP_058Lamm__do____,type,
aTP_Lamm_do: fun_se2051630309975736070_int_o ).
tff(sy_c_ATP_058Lamm__dp____,type,
aTP_Lamm_dp: fun_assn_fun_assn_o ).
tff(sy_c_ATP_058Lamm__dq____,type,
aTP_Lamm_dq: fun_assn_fun_assn_o ).
tff(sy_c_ATP_058Lamm__dr____,type,
aTP_Lamm_dr: fun_se2170385732985038782_nat_o ).
tff(sy_c_ATP_058Lamm__ds____,type,
aTP_Lamm_ds: fun_se2170385732985038782_nat_o ).
tff(sy_c_ATP_058Lamm__dt____,type,
aTP_Lamm_dt: fun_nat_fun_nat_o ).
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tff(sy_v_A,type,
a: assn ).
tff(sy_v_B,type,
b: assn ).
tff(sy_v_C,type,
c: assn ).
% Relevant facts (7163)
tff(fact_0_ent__iffI,axiom,
! [Aa: assn,Ba: assn] :
( aa_assn_o(entails(Aa),Ba)
=> ( aa_assn_o(entails(Ba),Aa)
=> ( Aa = Ba ) ) ) ).
% ent_iffI
tff(fact_1_ent__refl,axiom,
! [P: assn] : aa_assn_o(entails(P),P) ).
% ent_refl
tff(fact_2_ent__disjE,axiom,
! [Aa: assn,Ca: assn,Ba: assn] :
( aa_assn_o(entails(Aa),Ca)
=> ( aa_assn_o(entails(Ba),Ca)
=> aa_assn_o(entails(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Aa),Ba)),Ca) ) ) ).
% ent_disjE
tff(fact_3_ent__trans,axiom,
! [P: assn,Q: assn,R: assn] :
( aa_assn_o(entails(P),Q)
=> ( aa_assn_o(entails(Q),R)
=> aa_assn_o(entails(P),R) ) ) ).
% ent_trans
tff(fact_4_ent__disjI1,axiom,
! [P: assn,Q: assn,R: assn] :
( aa_assn_o(entails(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,P),Q)),R)
=> aa_assn_o(entails(P),R) ) ).
% ent_disjI1
tff(fact_5_ent__disjI2,axiom,
! [P: assn,Q: assn,R: assn] :
( aa_assn_o(entails(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,P),Q)),R)
=> aa_assn_o(entails(Q),R) ) ).
% ent_disjI2
tff(fact_6_ent__star__mono,axiom,
! [P: assn,P2: assn,Q: assn,Q2: assn] :
( aa_assn_o(entails(P),P2)
=> ( aa_assn_o(entails(Q),Q2)
=> aa_assn_o(entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),Q2)) ) ) ).
% ent_star_mono
tff(fact_7_assn__times__comm,axiom,
! [P: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),P) ).
% assn_times_comm
tff(fact_8_assn__times__assoc,axiom,
! [P: assn,Q: assn,R: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q)),R) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),R)) ).
% assn_times_assoc
tff(fact_9_ent__disjI1__direct,axiom,
! [Aa: assn,Ba: assn] : aa_assn_o(entails(Aa),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Aa),Ba)) ).
% ent_disjI1_direct
tff(fact_10_ent__disjI2__direct,axiom,
! [Ba: assn,Aa: assn] : aa_assn_o(entails(Ba),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Aa),Ba)) ).
% ent_disjI2_direct
tff(fact_11_sup_Oidem,axiom,
! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),A) = A ).
% sup.idem
tff(fact_12_sup_Oidem,axiom,
! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),A) = A ).
% sup.idem
tff(fact_13_sup_Oidem,axiom,
! [A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),A) = A ).
% sup.idem
tff(fact_14_sup__idem,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),X) = X ).
% sup_idem
tff(fact_15_sup__idem,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),X) = X ).
% sup_idem
tff(fact_16_sup__idem,axiom,
! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),X) = X ).
% sup_idem
tff(fact_17_sup_Oleft__idem,axiom,
! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ).
% sup.left_idem
tff(fact_18_sup_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ).
% sup.left_idem
tff(fact_19_sup_Oleft__idem,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ).
% sup.left_idem
tff(fact_20_sup__left__idem,axiom,
! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) ).
% sup_left_idem
tff(fact_21_sup__left__idem,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) ).
% sup_left_idem
tff(fact_22_sup__left__idem,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) ).
% sup_left_idem
tff(fact_23_sup_Oright__idem,axiom,
! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),B) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ).
% sup.right_idem
tff(fact_24_sup_Oright__idem,axiom,
! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ).
% sup.right_idem
tff(fact_25_sup_Oright__idem,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),B) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ).
% sup.right_idem
tff(fact_26_ent__mp,axiom,
! [P: assn,Q: assn] : aa_assn_o(entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,P),Q))),Q) ).
% ent_mp
tff(fact_27_ent__wandI,axiom,
! [Q: assn,P: assn,R: assn] :
( aa_assn_o(entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),P)),R)
=> aa_assn_o(entails(P),aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Q),R)) ) ).
% ent_wandI
tff(fact_28_ent__pure__pre__iff,axiom,
! [P: assn,B: $o,Q: assn] :
( aa_assn_o(entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B)))),Q)
<=> ( (B)
=> aa_assn_o(entails(P),Q) ) ) ).
% ent_pure_pre_iff
tff(fact_29_is__pure__assn__starI,axiom,
! [A: assn,B: assn] :
( aa_assn_o(is_pure_assn,A)
=> ( aa_assn_o(is_pure_assn,B)
=> aa_assn_o(is_pure_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)) ) ) ).
% is_pure_assn_starI
tff(fact_30_inf__sup__aci_I8_J,axiom,
! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) ).
% inf_sup_aci(8)
tff(fact_31_inf__sup__aci_I8_J,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) ).
% inf_sup_aci(8)
tff(fact_32_inf__sup__aci_I8_J,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) ).
% inf_sup_aci(8)
tff(fact_33_inf__sup__aci_I7_J,axiom,
! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).
% inf_sup_aci(7)
tff(fact_34_inf__sup__aci_I7_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ).
% inf_sup_aci(7)
tff(fact_35_inf__sup__aci_I7_J,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z)) ).
% inf_sup_aci(7)
tff(fact_36_pure__assn__eq__conv,axiom,
! [P: $o,Q: $o] :
( ( aa_o_assn(pure_assn,(P)) = aa_o_assn(pure_assn,(Q)) )
<=> ( (P)
<=> (Q) ) ) ).
% pure_assn_eq_conv
tff(fact_37_merge__pure__star,axiom,
! [A: $o,B: $o] :
aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,(A))),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
( (A)
& (B) )) ).
% merge_pure_star
tff(fact_38_merge__pure__or,axiom,
! [A: $o,B: $o] :
aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_o_assn(pure_assn,(A))),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
( (A)
| (B) )) ).
% merge_pure_or
tff(fact_39_is__pure__assn__pure,axiom,
! [P: $o] : aa_assn_o(is_pure_assn,aa_o_assn(pure_assn,(P))) ).
% is_pure_assn_pure
tff(fact_40_is__pure__assnE,axiom,
! [A: assn] :
( aa_assn_o(is_pure_assn,A)
=> ~ ! [P3: $o] : A != aa_o_assn(pure_assn,(P3)) ) ).
% is_pure_assnE
tff(fact_41_is__pure__assn__def,axiom,
! [A: assn] :
( aa_assn_o(is_pure_assn,A)
<=> ? [P4: $o] : A = aa_o_assn(pure_assn,(P4)) ) ).
% is_pure_assn_def
tff(fact_42_sup__left__commute,axiom,
! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).
% sup_left_commute
tff(fact_43_sup__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ).
% sup_left_commute
tff(fact_44_sup__left__commute,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z)) ).
% sup_left_commute
tff(fact_45_sup_Oleft__commute,axiom,
! [B: assn,A: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),C)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C)) ).
% sup.left_commute
tff(fact_46_sup_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),C)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)) ).
% sup.left_commute
tff(fact_47_sup_Oleft__commute,axiom,
! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),C)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C)) ).
% sup.left_commute
tff(fact_48_sup__commute,axiom,
! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X) ).
% sup_commute
tff(fact_49_sup__commute,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X) ).
% sup_commute
tff(fact_50_sup__commute,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),X) ).
% sup_commute
tff(fact_51_sup_Ocommute,axiom,
! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),A) ).
% sup.commute
tff(fact_52_sup_Ocommute,axiom,
! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),A) ).
% sup.commute
tff(fact_53_sup_Ocommute,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),A) ).
% sup.commute
tff(fact_54_sup__assoc,axiom,
! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ).
% sup_assoc
tff(fact_55_sup__assoc,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) ).
% sup_assoc
tff(fact_56_sup__assoc,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),Z) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) ).
% sup_assoc
tff(fact_57_sup_Oassoc,axiom,
! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C)) ).
% sup.assoc
tff(fact_58_sup_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),C) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)) ).
% sup.assoc
tff(fact_59_sup_Oassoc,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),C) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C)) ).
% sup.assoc
tff(fact_60_inf__sup__aci_I5_J,axiom,
! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X) ).
% inf_sup_aci(5)
tff(fact_61_inf__sup__aci_I5_J,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X) ).
% inf_sup_aci(5)
tff(fact_62_inf__sup__aci_I5_J,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),X) ).
% inf_sup_aci(5)
tff(fact_63_inf__sup__aci_I6_J,axiom,
! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ).
% inf_sup_aci(6)
tff(fact_64_inf__sup__aci_I6_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) ).
% inf_sup_aci(6)
tff(fact_65_inf__sup__aci_I6_J,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),Z) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) ).
% inf_sup_aci(6)
tff(fact_66_ent__pure__post__iff,axiom,
! [P: assn,Q: assn,B: $o] :
( aa_assn_o(entails(P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),aa_o_assn(pure_assn,(B))))
<=> ( ! [H: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),H)
=> (B) )
& aa_assn_o(entails(P),Q) ) ) ).
% ent_pure_post_iff
tff(fact_67_boolean__algebra__cancel_Osup2,axiom,
! [Ba: assn,K: assn,B: assn,A: assn] :
( ( Ba = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,K),B) )
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),Ba) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,K),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ) ).
% boolean_algebra_cancel.sup2
tff(fact_68_boolean__algebra__cancel_Osup2,axiom,
! [Ba: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( Ba = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),B) )
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),Ba) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).
% boolean_algebra_cancel.sup2
tff(fact_69_boolean__algebra__cancel_Osup2,axiom,
! [Ba: set_Pr4329608150637261639at_nat,K: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( ( Ba = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,K),B) )
=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),Ba) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,K),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ) ).
% boolean_algebra_cancel.sup2
tff(fact_70_boolean__algebra__cancel_Osup1,axiom,
! [Aa: assn,K: assn,A: assn,B: assn] :
( ( Aa = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,K),A) )
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Aa),B) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,K),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ) ).
% boolean_algebra_cancel.sup1
tff(fact_71_boolean__algebra__cancel_Osup1,axiom,
! [Aa: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( Aa = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),A) )
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Aa),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).
% boolean_algebra_cancel.sup1
tff(fact_72_boolean__algebra__cancel_Osup1,axiom,
! [Aa: set_Pr4329608150637261639at_nat,K: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( Aa = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,K),A) )
=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Aa),B) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,K),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ) ).
% boolean_algebra_cancel.sup1
tff(fact_73_ent__pure__pre__iff__sng,axiom,
! [B: $o,Q: assn] :
( aa_assn_o(entails(aa_o_assn(pure_assn,(B))),Q)
<=> ( (B)
=> aa_assn_o(entails(one_one_assn),Q) ) ) ).
% ent_pure_pre_iff_sng
tff(fact_74_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [B: code_integer,A: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).
% ab_semigroup_mult_class.mult.left_commute
tff(fact_75_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [B: assn,A: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ).
% ab_semigroup_mult_class.mult.left_commute
tff(fact_76_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).
% ab_semigroup_mult_class.mult.left_commute
tff(fact_77_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).
% ab_semigroup_mult_class.mult.left_commute
tff(fact_78_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).
% ab_semigroup_mult_class.mult.left_commute
tff(fact_79_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) ).
% ab_semigroup_mult_class.mult.commute
tff(fact_80_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A) ).
% ab_semigroup_mult_class.mult.commute
tff(fact_81_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A) ).
% ab_semigroup_mult_class.mult.commute
tff(fact_82_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) ).
% ab_semigroup_mult_class.mult.commute
tff(fact_83_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,B),A) ).
% ab_semigroup_mult_class.mult.commute
tff(fact_84_mult_Oassoc,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).
% mult.assoc
tff(fact_85_mult_Oassoc,axiom,
! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ).
% mult.assoc
tff(fact_86_mult_Oassoc,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).
% mult.assoc
tff(fact_87_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).
% mult.assoc
tff(fact_88_mult_Oassoc,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).
% mult.assoc
tff(fact_89_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),collec3392354462482085612at_nat(P))
<=> aa_Pro6150637604445964947_nat_o(P,A) ) ).
% mem_Collect_eq
tff(fact_90_mem__Collect__eq,axiom,
! [A: $o,P: fun_o_o] :
( aa_set_o_o(member_o((A)),collect_o(P))
<=> aa_o_o(P,(A)) ) ).
% mem_Collect_eq
tff(fact_91_mem__Collect__eq,axiom,
! [A: int,P: fun_int_o] :
( aa_set_int_o(member_int(A),collect_int(P))
<=> aa_int_o(P,A) ) ).
% mem_Collect_eq
tff(fact_92_mem__Collect__eq,axiom,
! [A: fun_Pr6460944699146269935_nat_o,P: fun_fu5135969500543783852at_o_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),collec6407654713642994138_nat_o(P))
<=> aa_fun1675921005893230629at_o_o(P,A) ) ).
% mem_Collect_eq
tff(fact_93_mem__Collect__eq,axiom,
! [A: nat,P: fun_nat_o] :
( aa_set_nat_o(member_nat(A),collect_nat(P))
<=> aa_nat_o(P,A) ) ).
% mem_Collect_eq
tff(fact_94_Collect__mem__eq,axiom,
! [Aa: set_Pr1261947904930325089at_nat] : collec3392354462482085612at_nat(aTP_Lamm_a(Aa)) = Aa ).
% Collect_mem_eq
tff(fact_95_Collect__mem__eq,axiom,
! [Aa: set_o] : collect_o(aTP_Lamm_aa(Aa)) = Aa ).
% Collect_mem_eq
tff(fact_96_Collect__mem__eq,axiom,
! [Aa: set_int] : collect_int(aTP_Lamm_ab(Aa)) = Aa ).
% Collect_mem_eq
tff(fact_97_Collect__mem__eq,axiom,
! [Aa: set_fu3555266933746557007_nat_o] : collec6407654713642994138_nat_o(aTP_Lamm_ac(Aa)) = Aa ).
% Collect_mem_eq
tff(fact_98_Collect__mem__eq,axiom,
! [Aa: set_nat] : collect_nat(aTP_Lamm_ad(Aa)) = Aa ).
% Collect_mem_eq
tff(fact_99_Collect__cong,axiom,
! [P: fun_fu5135969500543783852at_o_o,Q: fun_fu5135969500543783852at_o_o] :
( ! [X2: fun_Pr6460944699146269935_nat_o] :
( aa_fun1675921005893230629at_o_o(P,X2)
<=> aa_fun1675921005893230629at_o_o(Q,X2) )
=> ( collec6407654713642994138_nat_o(P) = collec6407654713642994138_nat_o(Q) ) ) ).
% Collect_cong
tff(fact_100_Collect__cong,axiom,
! [P: fun_nat_o,Q: fun_nat_o] :
( ! [X2: nat] :
( aa_nat_o(P,X2)
<=> aa_nat_o(Q,X2) )
=> ( collect_nat(P) = collect_nat(Q) ) ) ).
% Collect_cong
tff(fact_101_mult_Oright__commute,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ).
% mult.right_commute
tff(fact_102_mult_Oright__commute,axiom,
! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C)),B) ).
% mult.right_commute
tff(fact_103_mult_Oright__commute,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ).
% mult.right_commute
tff(fact_104_mult_Oright__commute,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ).
% mult.right_commute
tff(fact_105_mult_Oright__commute,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ).
% mult.right_commute
tff(fact_106_mult_Oright__assoc,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).
% mult.right_assoc
tff(fact_107_mult_Oright__assoc,axiom,
! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ).
% mult.right_assoc
tff(fact_108_mult_Oright__assoc,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).
% mult.right_assoc
tff(fact_109_mult_Oright__assoc,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).
% mult.right_assoc
tff(fact_110_mult_Oright__assoc,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).
% mult.right_assoc
tff(fact_111_mod__pure__star__dist,axiom,
! [P: assn,B: $o,H2: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_o_assn(pure_assn,(B)))),H2)
<=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),H2)
& (B) ) ) ).
% mod_pure_star_dist
tff(fact_112_pure__true,axiom,
aa_o_assn(pure_assn,$true) = one_one_assn ).
% pure_true
tff(fact_113_pure__assn__eq__emp__iff,axiom,
! [P: $o] :
( ( aa_o_assn(pure_assn,(P)) = one_one_assn )
<=> (P) ) ).
% pure_assn_eq_emp_iff
tff(fact_114_Rep__assn__inject,axiom,
! [X: assn,Y: assn] :
( ( aa_ass4243947775541419017_nat_o(rep_assn,X) = aa_ass4243947775541419017_nat_o(rep_assn,Y) )
<=> ( X = Y ) ) ).
% Rep_assn_inject
tff(fact_115_mult_Oright__neutral,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),one_one_Code_integer) = A ).
% mult.right_neutral
tff(fact_116_mult_Oright__neutral,axiom,
! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),one_one_assn) = A ).
% mult.right_neutral
tff(fact_117_mult_Oright__neutral,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).
% mult.right_neutral
tff(fact_118_mult_Oright__neutral,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).
% mult.right_neutral
tff(fact_119_mult_Oright__neutral,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).
% mult.right_neutral
tff(fact_120_mult__1,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer),A) = A ).
% mult_1
tff(fact_121_mult__1,axiom,
! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ).
% mult_1
tff(fact_122_mult__1,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).
% mult_1
tff(fact_123_mult__1,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).
% mult_1
tff(fact_124_mult__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).
% mult_1
tff(fact_125_mod__or__dist,axiom,
! [P: assn,Q: assn,H2: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,P),Q)),H2)
<=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),H2)
| aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),H2) ) ) ).
% mod_or_dist
tff(fact_126_ent__pure__post__iff__sng,axiom,
! [P: assn,B: $o] :
( aa_assn_o(entails(P),aa_o_assn(pure_assn,(B)))
<=> ( ! [H: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),H)
=> (B) )
& aa_assn_o(entails(P),one_one_assn) ) ) ).
% ent_pure_post_iff_sng
tff(fact_127_one__reorient,axiom,
! [X: code_integer] :
( ( one_one_Code_integer = X )
<=> ( X = one_one_Code_integer ) ) ).
% one_reorient
tff(fact_128_one__reorient,axiom,
! [X: assn] :
( ( one_one_assn = X )
<=> ( X = one_one_assn ) ) ).
% one_reorient
tff(fact_129_one__reorient,axiom,
! [X: rat] :
( ( one_one_rat = X )
<=> ( X = one_one_rat ) ) ).
% one_reorient
tff(fact_130_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
<=> ( X = one_one_nat ) ) ).
% one_reorient
tff(fact_131_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
<=> ( X = one_one_int ) ) ).
% one_reorient
tff(fact_132_comm__monoid__mult__class_Omult__1,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer),A) = A ).
% comm_monoid_mult_class.mult_1
tff(fact_133_comm__monoid__mult__class_Omult__1,axiom,
! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ).
% comm_monoid_mult_class.mult_1
tff(fact_134_comm__monoid__mult__class_Omult__1,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).
% comm_monoid_mult_class.mult_1
tff(fact_135_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).
% comm_monoid_mult_class.mult_1
tff(fact_136_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).
% comm_monoid_mult_class.mult_1
tff(fact_137_mult_Ocomm__neutral,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),one_one_Code_integer) = A ).
% mult.comm_neutral
tff(fact_138_mult_Ocomm__neutral,axiom,
! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),one_one_assn) = A ).
% mult.comm_neutral
tff(fact_139_mult_Ocomm__neutral,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).
% mult.comm_neutral
tff(fact_140_mult_Ocomm__neutral,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).
% mult.comm_neutral
tff(fact_141_mult_Ocomm__neutral,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).
% mult.comm_neutral
tff(fact_142_ent__fwd,axiom,
! [P: assn,H2: produc3658429121746597890et_nat,Q: assn] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),H2)
=> ( aa_assn_o(entails(P),Q)
=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),H2) ) ) ).
% ent_fwd
tff(fact_143_entailsD,axiom,
! [P: assn,Q: assn,H2: produc3658429121746597890et_nat] :
( aa_assn_o(entails(P),Q)
=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),H2)
=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),H2) ) ) ).
% entailsD
tff(fact_144_entailsI,axiom,
! [P: assn,Q: assn] :
( ! [H3: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),H3)
=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),H3) )
=> aa_assn_o(entails(P),Q) ) ).
% entailsI
tff(fact_145_entails__def,axiom,
! [P: assn,Q: assn] :
( aa_assn_o(entails(P),Q)
<=> ! [H: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),H)
=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),H) ) ) ).
% entails_def
tff(fact_146_mod__starD,axiom,
! [Aa: assn,Ba: assn,H2: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Aa),Ba)),H2)
=> ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Aa),H1)
& aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Ba),H22) ) ) ).
% mod_starD
tff(fact_147_mod__starE,axiom,
! [A: assn,B: assn,H2: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),H2)
=> ~ ( ? [X_1: produc3658429121746597890et_nat] : aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,A),X_1)
=> ! [H_2: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,B),H_2) ) ) ).
% mod_starE
tff(fact_148_assn__one__left,axiom,
! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),P) = P ).
% assn_one_left
tff(fact_149_is__pure__assn__basic__simps_I2_J,axiom,
aa_assn_o(is_pure_assn,one_one_assn) ).
% is_pure_assn_basic_simps(2)
tff(fact_150_mult_Osafe__commute,axiom,
! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(syntax380677951125133566nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y)),A)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) ) ) ).
% mult.safe_commute
tff(fact_151_mult_Osafe__commute,axiom,
! [X: assn,Y: assn,A: assn,B: assn] :
( aa_assn_o(syntax7398250324933576852n_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),Y)),A)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A) ) ) ).
% mult.safe_commute
tff(fact_152_mult_Osafe__commute,axiom,
! [X: rat,Y: rat,A: rat,B: rat] :
( aa_rat_o(syntax3730441303064801268at_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)),A)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A) ) ) ).
% mult.safe_commute
tff(fact_153_mult_Osafe__commute,axiom,
! [X: nat,Y: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(syntax4682126007086162916at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) ) ) ).
% mult.safe_commute
tff(fact_154_mult_Osafe__commute,axiom,
! [X: int,Y: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(syntax5678989248478167196nt_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),A)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,B),A) ) ) ).
% mult.safe_commute
tff(fact_155_mult_Omonoid__axioms,axiom,
aa_Code_integer_o(monoid_Code_integer(times_3573771949741848930nteger),one_one_Code_integer) ).
% mult.monoid_axioms
tff(fact_156_mult_Omonoid__axioms,axiom,
aa_assn_o(monoid_assn(times_times_assn),one_one_assn) ).
% mult.monoid_axioms
tff(fact_157_mult_Omonoid__axioms,axiom,
aa_rat_o(monoid_rat(times_times_rat),one_one_rat) ).
% mult.monoid_axioms
tff(fact_158_mult_Omonoid__axioms,axiom,
aa_nat_o(monoid_nat(times_times_nat),one_one_nat) ).
% mult.monoid_axioms
tff(fact_159_mult_Omonoid__axioms,axiom,
aa_int_o(monoid_int(times_times_int),one_one_int) ).
% mult.monoid_axioms
tff(fact_160_ent__false__iff,axiom,
! [P: assn] :
( aa_assn_o(entails(P),bot_bot_assn)
<=> ! [H: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),H) ) ).
% ent_false_iff
tff(fact_161_mult_Ocomm__monoid__axioms,axiom,
aa_Code_integer_o(comm_m1654649897431166824nteger(times_3573771949741848930nteger),one_one_Code_integer) ).
% mult.comm_monoid_axioms
tff(fact_162_mult_Ocomm__monoid__axioms,axiom,
aa_assn_o(comm_monoid_assn(times_times_assn),one_one_assn) ).
% mult.comm_monoid_axioms
tff(fact_163_mult_Ocomm__monoid__axioms,axiom,
aa_rat_o(comm_monoid_rat(times_times_rat),one_one_rat) ).
% mult.comm_monoid_axioms
tff(fact_164_mult_Ocomm__monoid__axioms,axiom,
aa_nat_o(comm_monoid_nat(times_times_nat),one_one_nat) ).
% mult.comm_monoid_axioms
tff(fact_165_mult_Ocomm__monoid__axioms,axiom,
aa_int_o(comm_monoid_int(times_times_int),one_one_int) ).
% mult.comm_monoid_axioms
tff(fact_166_mod__star__trueE,axiom,
! [P: assn,H2: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),top_top_assn)),H2)
=> ~ ! [H4: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),H4) ) ).
% mod_star_trueE
tff(fact_167_mod__star__trueI,axiom,
! [P: assn,H2: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),H2)
=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),top_top_assn)),H2) ) ).
% mod_star_trueI
tff(fact_168_Rep__assn__inverse,axiom,
! [X: assn] : aa_fun5526454687433275355o_assn(abs_assn,aa_ass4243947775541419017_nat_o(rep_assn,X)) = X ).
% Rep_assn_inverse
tff(fact_169_wand__assn__def,axiom,
! [P: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,P),Q) = aa_fun5526454687433275355o_assn(abs_assn,aa_fun2638059832795814268_nat_o(wand_raw(aa_ass4243947775541419017_nat_o(rep_assn,P)),aa_ass4243947775541419017_nat_o(rep_assn,Q))) ).
% wand_assn_def
tff(fact_170_prod_Ocomm__monoid__list__set__axioms,axiom,
aa_Code_integer_o(groups7165323350619936122nteger(times_3573771949741848930nteger),one_one_Code_integer) ).
% prod.comm_monoid_list_set_axioms
tff(fact_171_prod_Ocomm__monoid__list__set__axioms,axiom,
aa_assn_o(groups5099259868496839749t_assn(times_times_assn),one_one_assn) ).
% prod.comm_monoid_list_set_axioms
tff(fact_172_prod_Ocomm__monoid__list__set__axioms,axiom,
aa_rat_o(groups2612948464029442805et_rat(times_times_rat),one_one_rat) ).
% prod.comm_monoid_list_set_axioms
tff(fact_173_prod_Ocomm__monoid__list__set__axioms,axiom,
aa_nat_o(groups3248078524115938541et_nat(times_times_nat),one_one_nat) ).
% prod.comm_monoid_list_set_axioms
tff(fact_174_prod_Ocomm__monoid__list__set__axioms,axiom,
aa_int_o(groups3245588053606888265et_int(times_times_int),one_one_int) ).
% prod.comm_monoid_list_set_axioms
tff(fact_175_mult_Oac__operator__axioms,axiom,
syntax414191934074104506nteger(times_3573771949741848930nteger) ).
% mult.ac_operator_axioms
tff(fact_176_mult_Oac__operator__axioms,axiom,
syntax3673046151410588037r_assn(times_times_assn) ).
% mult.ac_operator_axioms
tff(fact_177_mult_Oac__operator__axioms,axiom,
syntax8165409007564224565or_rat(times_times_rat) ).
% mult.ac_operator_axioms
tff(fact_178_mult_Oac__operator__axioms,axiom,
syntax8800539067650720301or_nat(times_times_nat) ).
% mult.ac_operator_axioms
tff(fact_179_mult_Oac__operator__axioms,axiom,
syntax8798048597141670025or_int(times_times_int) ).
% mult.ac_operator_axioms
tff(fact_180_prod__list_Ocomm__monoid__list__axioms,axiom,
aa_Code_integer_o(groups2567640417189887108nteger(times_3573771949741848930nteger),one_one_Code_integer) ).
% prod_list.comm_monoid_list_axioms
tff(fact_181_prod__list_Ocomm__monoid__list__axioms,axiom,
aa_assn_o(groups9150563453463953231t_assn(times_times_assn),one_one_assn) ).
% prod_list.comm_monoid_list_axioms
tff(fact_182_prod__list_Ocomm__monoid__list__axioms,axiom,
aa_rat_o(groups593308882113688063st_rat(times_times_rat),one_one_rat) ).
% prod_list.comm_monoid_list_axioms
tff(fact_183_prod__list_Ocomm__monoid__list__axioms,axiom,
aa_nat_o(groups1228438942200183799st_nat(times_times_nat),one_one_nat) ).
% prod_list.comm_monoid_list_axioms
tff(fact_184_prod__list_Ocomm__monoid__list__axioms,axiom,
aa_int_o(groups1225948471691133523st_int(times_times_int),one_one_int) ).
% prod_list.comm_monoid_list_axioms
tff(fact_185_sup__bot_Oright__neutral,axiom,
! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),bot_bot_assn) = A ).
% sup_bot.right_neutral
tff(fact_186_sup__bot_Oright__neutral,axiom,
! [A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),bot_bo228742789529271731at_nat) = A ).
% sup_bot.right_neutral
tff(fact_187_sup__bot_Oright__neutral,axiom,
! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),bot_bo2099793752762293965at_nat) = A ).
% sup_bot.right_neutral
tff(fact_188_sup__bot_Oright__neutral,axiom,
! [A: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),bot_bot_set_o) = A ).
% sup_bot.right_neutral
tff(fact_189_sup__bot_Oright__neutral,axiom,
! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),bot_bot_set_nat) = A ).
% sup_bot.right_neutral
tff(fact_190_sup__bot_Oright__neutral,axiom,
! [A: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),bot_bot_set_int) = A ).
% sup_bot.right_neutral
tff(fact_191_sup__bot_Oneutr__eq__iff,axiom,
! [A: assn,B: assn] :
( ( bot_bot_assn = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) )
<=> ( ( A = bot_bot_assn )
& ( B = bot_bot_assn ) ) ) ).
% sup_bot.neutr_eq_iff
tff(fact_192_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) )
<=> ( ( A = bot_bo228742789529271731at_nat )
& ( B = bot_bo228742789529271731at_nat ) ) ) ).
% sup_bot.neutr_eq_iff
tff(fact_193_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B) )
<=> ( ( A = bot_bo2099793752762293965at_nat )
& ( B = bot_bo2099793752762293965at_nat ) ) ) ).
% sup_bot.neutr_eq_iff
tff(fact_194_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_o,B: set_o] :
( ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B) )
<=> ( ( A = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% sup_bot.neutr_eq_iff
tff(fact_195_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
<=> ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
tff(fact_196_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_int,B: set_int] :
( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) )
<=> ( ( A = bot_bot_set_int )
& ( B = bot_bot_set_int ) ) ) ).
% sup_bot.neutr_eq_iff
tff(fact_197_sup__bot_Oleft__neutral,axiom,
! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,bot_bot_assn),A) = A ).
% sup_bot.left_neutral
tff(fact_198_sup__bot_Oleft__neutral,axiom,
! [A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),A) = A ).
% sup_bot.left_neutral
tff(fact_199_sup__bot_Oleft__neutral,axiom,
! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat),A) = A ).
% sup_bot.left_neutral
tff(fact_200_sup__bot_Oleft__neutral,axiom,
! [A: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,bot_bot_set_o),A) = A ).
% sup_bot.left_neutral
tff(fact_201_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),A) = A ).
% sup_bot.left_neutral
tff(fact_202_sup__bot_Oleft__neutral,axiom,
! [A: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),A) = A ).
% sup_bot.left_neutral
tff(fact_203_sup__bot_Oeq__neutr__iff,axiom,
! [A: assn,B: assn] :
( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = bot_bot_assn )
<=> ( ( A = bot_bot_assn )
& ( B = bot_bot_assn ) ) ) ).
% sup_bot.eq_neutr_iff
tff(fact_204_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = bot_bo228742789529271731at_nat )
<=> ( ( A = bot_bo228742789529271731at_nat )
& ( B = bot_bo228742789529271731at_nat ) ) ) ).
% sup_bot.eq_neutr_iff
tff(fact_205_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B) = bot_bo2099793752762293965at_nat )
<=> ( ( A = bot_bo2099793752762293965at_nat )
& ( B = bot_bo2099793752762293965at_nat ) ) ) ).
% sup_bot.eq_neutr_iff
tff(fact_206_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_o,B: set_o] :
( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B) = bot_bot_set_o )
<=> ( ( A = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% sup_bot.eq_neutr_iff
tff(fact_207_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = bot_bot_set_nat )
<=> ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
tff(fact_208_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_int,B: set_int] :
( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = bot_bot_set_int )
<=> ( ( A = bot_bot_set_int )
& ( B = bot_bot_set_int ) ) ) ).
% sup_bot.eq_neutr_iff
tff(fact_209_sup__eq__bot__iff,axiom,
! [X: assn,Y: assn] :
( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = bot_bot_assn )
<=> ( ( X = bot_bot_assn )
& ( Y = bot_bot_assn ) ) ) ).
% sup_eq_bot_iff
tff(fact_210_sup__eq__bot__iff,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = bot_bo228742789529271731at_nat )
<=> ( ( X = bot_bo228742789529271731at_nat )
& ( Y = bot_bo228742789529271731at_nat ) ) ) ).
% sup_eq_bot_iff
tff(fact_211_sup__eq__bot__iff,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y) = bot_bo2099793752762293965at_nat )
<=> ( ( X = bot_bo2099793752762293965at_nat )
& ( Y = bot_bo2099793752762293965at_nat ) ) ) ).
% sup_eq_bot_iff
tff(fact_212_sup__eq__bot__iff,axiom,
! [X: set_o,Y: set_o] :
( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),Y) = bot_bot_set_o )
<=> ( ( X = bot_bot_set_o )
& ( Y = bot_bot_set_o ) ) ) ).
% sup_eq_bot_iff
tff(fact_213_sup__eq__bot__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = bot_bot_set_nat )
<=> ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
tff(fact_214_sup__eq__bot__iff,axiom,
! [X: set_int,Y: set_int] :
( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = bot_bot_set_int )
<=> ( ( X = bot_bot_set_int )
& ( Y = bot_bot_set_int ) ) ) ).
% sup_eq_bot_iff
tff(fact_215_bot__eq__sup__iff,axiom,
! [X: assn,Y: assn] :
( ( bot_bot_assn = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) )
<=> ( ( X = bot_bot_assn )
& ( Y = bot_bot_assn ) ) ) ).
% bot_eq_sup_iff
tff(fact_216_bot__eq__sup__iff,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) )
<=> ( ( X = bot_bo228742789529271731at_nat )
& ( Y = bot_bo228742789529271731at_nat ) ) ) ).
% bot_eq_sup_iff
tff(fact_217_bot__eq__sup__iff,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y) )
<=> ( ( X = bot_bo2099793752762293965at_nat )
& ( Y = bot_bo2099793752762293965at_nat ) ) ) ).
% bot_eq_sup_iff
tff(fact_218_bot__eq__sup__iff,axiom,
! [X: set_o,Y: set_o] :
( ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),Y) )
<=> ( ( X = bot_bot_set_o )
& ( Y = bot_bot_set_o ) ) ) ).
% bot_eq_sup_iff
tff(fact_219_bot__eq__sup__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) )
<=> ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
tff(fact_220_bot__eq__sup__iff,axiom,
! [X: set_int,Y: set_int] :
( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) )
<=> ( ( X = bot_bot_set_int )
& ( Y = bot_bot_set_int ) ) ) ).
% bot_eq_sup_iff
tff(fact_221_sup__bot__right,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),bot_bot_assn) = X ).
% sup_bot_right
tff(fact_222_sup__bot__right,axiom,
! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),bot_bo228742789529271731at_nat) = X ).
% sup_bot_right
tff(fact_223_sup__bot__right,axiom,
! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),bot_bo2099793752762293965at_nat) = X ).
% sup_bot_right
tff(fact_224_sup__bot__right,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),bot_bot_set_o) = X ).
% sup_bot_right
tff(fact_225_sup__bot__right,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),bot_bot_set_nat) = X ).
% sup_bot_right
tff(fact_226_sup__bot__right,axiom,
! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),bot_bot_set_int) = X ).
% sup_bot_right
tff(fact_227_sup__bot__left,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,bot_bot_assn),X) = X ).
% sup_bot_left
tff(fact_228_sup__bot__left,axiom,
! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),X) = X ).
% sup_bot_left
tff(fact_229_sup__bot__left,axiom,
! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat),X) = X ).
% sup_bot_left
tff(fact_230_sup__bot__left,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,bot_bot_set_o),X) = X ).
% sup_bot_left
tff(fact_231_sup__bot__left,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),X) = X ).
% sup_bot_left
tff(fact_232_sup__bot__left,axiom,
! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),X) = X ).
% sup_bot_left
tff(fact_233_sup__top__right,axiom,
! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),top_to6833984726390702231at_nat) = top_to6833984726390702231at_nat ).
% sup_top_right
tff(fact_234_sup__top__right,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),top_top_assn) = top_top_assn ).
% sup_top_right
tff(fact_235_sup__top__right,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),top_top_set_o) = top_top_set_o ).
% sup_top_right
tff(fact_236_sup__top__right,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),top_top_set_nat) = top_top_set_nat ).
% sup_top_right
tff(fact_237_sup__top__right,axiom,
! [X: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,X),top_to1996260823553986621t_unit) = top_to1996260823553986621t_unit ).
% sup_top_right
tff(fact_238_sup__top__left,axiom,
! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,top_to6833984726390702231at_nat),X) = top_to6833984726390702231at_nat ).
% sup_top_left
tff(fact_239_sup__top__left,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,top_top_assn),X) = top_top_assn ).
% sup_top_left
tff(fact_240_sup__top__left,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,top_top_set_o),X) = top_top_set_o ).
% sup_top_left
tff(fact_241_sup__top__left,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,top_top_set_nat),X) = top_top_set_nat ).
% sup_top_left
tff(fact_242_sup__top__left,axiom,
! [X: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,top_to1996260823553986621t_unit),X) = top_to1996260823553986621t_unit ).
% sup_top_left
tff(fact_243_boolean__algebra_Odisj__one__right,axiom,
! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),top_to6833984726390702231at_nat) = top_to6833984726390702231at_nat ).
% boolean_algebra.disj_one_right
tff(fact_244_boolean__algebra_Odisj__one__right,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),top_top_assn) = top_top_assn ).
% boolean_algebra.disj_one_right
tff(fact_245_boolean__algebra_Odisj__one__right,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),top_top_set_o) = top_top_set_o ).
% boolean_algebra.disj_one_right
tff(fact_246_boolean__algebra_Odisj__one__right,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),top_top_set_nat) = top_top_set_nat ).
% boolean_algebra.disj_one_right
tff(fact_247_boolean__algebra_Odisj__one__right,axiom,
! [X: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,X),top_to1996260823553986621t_unit) = top_to1996260823553986621t_unit ).
% boolean_algebra.disj_one_right
tff(fact_248_boolean__algebra_Odisj__one__left,axiom,
! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,top_to6833984726390702231at_nat),X) = top_to6833984726390702231at_nat ).
% boolean_algebra.disj_one_left
tff(fact_249_boolean__algebra_Odisj__one__left,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,top_top_assn),X) = top_top_assn ).
% boolean_algebra.disj_one_left
tff(fact_250_boolean__algebra_Odisj__one__left,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,top_top_set_o),X) = top_top_set_o ).
% boolean_algebra.disj_one_left
tff(fact_251_boolean__algebra_Odisj__one__left,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,top_top_set_nat),X) = top_top_set_nat ).
% boolean_algebra.disj_one_left
tff(fact_252_boolean__algebra_Odisj__one__left,axiom,
! [X: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,top_to1996260823553986621t_unit),X) = top_to1996260823553986621t_unit ).
% boolean_algebra.disj_one_left
tff(fact_253_star__false__right,axiom,
! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),bot_bot_assn) = bot_bot_assn ).
% star_false_right
tff(fact_254_star__false__left,axiom,
! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,bot_bot_assn),P) = bot_bot_assn ).
% star_false_left
tff(fact_255_pure__false,axiom,
aa_o_assn(pure_assn,$false) = bot_bot_assn ).
% pure_false
tff(fact_256_pure__assn__eq__false__iff,axiom,
! [P: $o] :
( ( aa_o_assn(pure_assn,(P)) = bot_bot_assn )
<=> ~ (P) ) ).
% pure_assn_eq_false_iff
tff(fact_257_merge__true__star,axiom,
aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),top_top_assn) = top_top_assn ).
% merge_true_star
tff(fact_258_assn__basic__inequalities_I5_J,axiom,
top_top_assn != bot_bot_assn ).
% assn_basic_inequalities(5)
tff(fact_259_assn__basic__inequalities_I3_J,axiom,
bot_bot_assn != one_one_assn ).
% assn_basic_inequalities(3)
tff(fact_260_assn__basic__inequalities_I1_J,axiom,
top_top_assn != one_one_assn ).
% assn_basic_inequalities(1)
tff(fact_261_sup__bot_Ocomm__monoid__axioms,axiom,
aa_assn_o(comm_monoid_assn(sup_sup_assn),bot_bot_assn) ).
% sup_bot.comm_monoid_axioms
tff(fact_262_sup__bot_Ocomm__monoid__axioms,axiom,
aa_set5844364881195198285_nat_o(comm_m2457627138559323900at_nat(sup_su5525570899277871387at_nat),bot_bo228742789529271731at_nat) ).
% sup_bot.comm_monoid_axioms
tff(fact_263_sup__bot_Ocomm__monoid__axioms,axiom,
aa_set3702741264012733747_nat_o(comm_m5709050211238513046at_nat(sup_su6327502436637775413at_nat),bot_bo2099793752762293965at_nat) ).
% sup_bot.comm_monoid_axioms
tff(fact_264_sup__bot_Ocomm__monoid__axioms,axiom,
aa_set_o_o(comm_monoid_set_o(sup_sup_set_o),bot_bot_set_o) ).
% sup_bot.comm_monoid_axioms
tff(fact_265_sup__bot_Ocomm__monoid__axioms,axiom,
aa_set_nat_o(comm_monoid_set_nat(sup_sup_set_nat),bot_bot_set_nat) ).
% sup_bot.comm_monoid_axioms
tff(fact_266_sup__bot_Ocomm__monoid__axioms,axiom,
aa_set_int_o(comm_monoid_set_int(sup_sup_set_int),bot_bot_set_int) ).
% sup_bot.comm_monoid_axioms
tff(fact_267_sup__bot_Omonoid__axioms,axiom,
aa_assn_o(monoid_assn(sup_sup_assn),bot_bot_assn) ).
% sup_bot.monoid_axioms
tff(fact_268_sup__bot_Omonoid__axioms,axiom,
aa_set5844364881195198285_nat_o(monoid534952638934301128at_nat(sup_su5525570899277871387at_nat),bot_bo228742789529271731at_nat) ).
% sup_bot.monoid_axioms
tff(fact_269_sup__bot_Omonoid__axioms,axiom,
aa_set3702741264012733747_nat_o(monoid1519094961268053602at_nat(sup_su6327502436637775413at_nat),bot_bo2099793752762293965at_nat) ).
% sup_bot.monoid_axioms
tff(fact_270_sup__bot_Omonoid__axioms,axiom,
aa_set_o_o(monoid_set_o(sup_sup_set_o),bot_bot_set_o) ).
% sup_bot.monoid_axioms
tff(fact_271_sup__bot_Omonoid__axioms,axiom,
aa_set_nat_o(monoid_set_nat(sup_sup_set_nat),bot_bot_set_nat) ).
% sup_bot.monoid_axioms
tff(fact_272_sup__bot_Omonoid__axioms,axiom,
aa_set_int_o(monoid_set_int(sup_sup_set_int),bot_bot_set_int) ).
% sup_bot.monoid_axioms
tff(fact_273_boolean__algebra_Odisj__zero__right,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),bot_bot_assn) = X ).
% boolean_algebra.disj_zero_right
tff(fact_274_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),bot_bo228742789529271731at_nat) = X ).
% boolean_algebra.disj_zero_right
tff(fact_275_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),bot_bo2099793752762293965at_nat) = X ).
% boolean_algebra.disj_zero_right
tff(fact_276_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),bot_bot_set_o) = X ).
% boolean_algebra.disj_zero_right
tff(fact_277_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),bot_bot_set_nat) = X ).
% boolean_algebra.disj_zero_right
tff(fact_278_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),bot_bot_set_int) = X ).
% boolean_algebra.disj_zero_right
tff(fact_279_mod__false,axiom,
! [H2: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,bot_bot_assn),H2) ).
% mod_false
tff(fact_280_ent__true,axiom,
! [P: assn] : aa_assn_o(entails(P),top_top_assn) ).
% ent_true
tff(fact_281_ent__false,axiom,
! [P: assn] : aa_assn_o(entails(bot_bot_assn),P) ).
% ent_false
tff(fact_282_is__pure__assn__basic__simps_I1_J,axiom,
aa_assn_o(is_pure_assn,bot_bot_assn) ).
% is_pure_assn_basic_simps(1)
tff(fact_283_Abs__assn__eqI_I2_J,axiom,
! [P: fun_Pr6460944699146269935_nat_o,Pr: assn] :
( ! [H3: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(P,H3)
<=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pr),H3) )
=> ( Pr = aa_fun5526454687433275355o_assn(abs_assn,P) ) ) ).
% Abs_assn_eqI(2)
tff(fact_284_Abs__assn__eqI_I1_J,axiom,
! [P: fun_Pr6460944699146269935_nat_o,Pr: assn] :
( ! [H3: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(P,H3)
<=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pr),H3) )
=> ( aa_fun5526454687433275355o_assn(abs_assn,P) = Pr ) ) ).
% Abs_assn_eqI(1)
tff(fact_285_times__assn__def,axiom,
! [P: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q) = aa_fun5526454687433275355o_assn(abs_assn,aa_fun2638059832795814268_nat_o(times_assn_raw(aa_ass4243947775541419017_nat_o(rep_assn,P)),aa_ass4243947775541419017_nat_o(rep_assn,Q))) ).
% times_assn_def
tff(fact_286_one__assn__def,axiom,
one_one_assn = aa_fun5526454687433275355o_assn(abs_assn,one_assn_raw) ).
% one_assn_def
tff(fact_287_pure__assn__def,axiom,
! [B: $o] : aa_o_assn(pure_assn,(B)) = aa_fun5526454687433275355o_assn(abs_assn,pure_a825153325127701367it_nat((B))) ).
% pure_assn_def
tff(fact_288_mod__true,axiom,
! [H2: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,top_top_assn),H2)
<=> aa_Pro653082453696054616_nat_o(in_range,H2) ) ).
% mod_true
tff(fact_289_prod__list_Omonoid__list__axioms,axiom,
aa_Code_integer_o(groups5100778238730182072nteger(times_3573771949741848930nteger),one_one_Code_integer) ).
% prod_list.monoid_list_axioms
tff(fact_290_prod__list_Omonoid__list__axioms,axiom,
aa_assn_o(groups2340186759238748291t_assn(times_times_assn),one_one_assn) ).
% prod_list.monoid_list_axioms
tff(fact_291_prod__list_Omonoid__list__axioms,axiom,
aa_rat_o(groups7826448149055759155st_rat(times_times_rat),one_one_rat) ).
% prod_list.monoid_list_axioms
tff(fact_292_prod__list_Omonoid__list__axioms,axiom,
aa_nat_o(groups8461578209142254891st_nat(times_times_nat),one_one_nat) ).
% prod_list.monoid_list_axioms
tff(fact_293_prod__list_Omonoid__list__axioms,axiom,
aa_int_o(groups8459087738633204615st_int(times_times_int),one_one_int) ).
% prod_list.monoid_list_axioms
tff(fact_294_models__in__range,axiom,
! [P: assn,H2: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),H2)
=> aa_Pro653082453696054616_nat_o(in_range,H2) ) ).
% models_in_range
tff(fact_295_top__assn__def,axiom,
top_top_assn = aa_fun5526454687433275355o_assn(abs_assn,in_range) ).
% top_assn_def
tff(fact_296_pure__assn__proper,axiom,
! [B: $o] : aa_fun1675921005893230629at_o_o(proper,pure_a825153325127701367it_nat((B))) ).
% pure_assn_proper
tff(fact_297_sup__bot_Osemilattice__neutr__axioms,axiom,
aa_assn_o(semila5388846958306562758r_assn(sup_sup_assn),bot_bot_assn) ).
% sup_bot.semilattice_neutr_axioms
tff(fact_298_sup__bot_Osemilattice__neutr__axioms,axiom,
aa_set5844364881195198285_nat_o(semila3675781407871293737at_nat(sup_su5525570899277871387at_nat),bot_bo228742789529271731at_nat) ).
% sup_bot.semilattice_neutr_axioms
tff(fact_299_sup__bot_Osemilattice__neutr__axioms,axiom,
aa_set3702741264012733747_nat_o(semila858173770281612099at_nat(sup_su6327502436637775413at_nat),bot_bo2099793752762293965at_nat) ).
% sup_bot.semilattice_neutr_axioms
tff(fact_300_sup__bot_Osemilattice__neutr__axioms,axiom,
aa_set_o_o(semila1065760912024005978_set_o(sup_sup_set_o),bot_bot_set_o) ).
% sup_bot.semilattice_neutr_axioms
tff(fact_301_sup__bot_Osemilattice__neutr__axioms,axiom,
aa_set_nat_o(semila1241773964035338532et_nat(sup_sup_set_nat),bot_bot_set_nat) ).
% sup_bot.semilattice_neutr_axioms
tff(fact_302_sup__bot_Osemilattice__neutr__axioms,axiom,
aa_set_int_o(semila6287294981380917632et_int(sup_sup_set_int),bot_bot_set_int) ).
% sup_bot.semilattice_neutr_axioms
tff(fact_303_mod__h__bot__iff_I7_J,axiom,
! [P: assn,Q: assn,H2: heap_e7401611519738050253t_unit] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,P),Q)),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
<=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
| aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ) ) ).
% mod_h_bot_iff(7)
tff(fact_304_mod__h__bot__iff_I1_J,axiom,
! [B: $o,H2: heap_e7401611519738050253t_unit] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_o_assn(pure_assn,(B))),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
<=> (B) ) ).
% mod_h_bot_iff(1)
tff(fact_305_mod__h__bot__iff_I5_J,axiom,
! [P: assn,Q: assn,H2: heap_e7401611519738050253t_unit] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q)),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
<=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
& aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ) ) ).
% mod_h_bot_iff(5)
tff(fact_306_in__range__dist__union,axiom,
! [H2: heap_e7401611519738050253t_unit,As: set_nat,As2: set_nat] :
( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As2)))
<=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As))
& aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As2)) ) ) ).
% in_range_dist_union
tff(fact_307_bool__assn__proper_I1_J,axiom,
aa_fun1675921005893230629at_o_o(proper,in_range) ).
% bool_assn_proper(1)
tff(fact_308_wand__proper,axiom,
! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o] : aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(wand_raw(P),Q)) ).
% wand_proper
tff(fact_309_one__assn__proper,axiom,
aa_fun1675921005893230629at_o_o(proper,one_assn_raw) ).
% one_assn_proper
tff(fact_310_times__assn__proper,axiom,
! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o] :
( aa_fun1675921005893230629at_o_o(proper,P)
=> ( aa_fun1675921005893230629at_o_o(proper,Q)
=> aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(times_assn_raw(P),Q)) ) ) ).
% times_assn_proper
tff(fact_311_in__range__empty,axiom,
! [H2: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ).
% in_range_empty
tff(fact_312_one__assn__raw_Ocases,axiom,
! [X: produc3658429121746597890et_nat] :
~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] : X != produc7507926704131184380et_nat(H3,As3) ).
% one_assn_raw.cases
tff(fact_313_properD1,axiom,
! [P: fun_Pr6460944699146269935_nat_o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
( aa_fun1675921005893230629at_o_o(proper,P)
=> ( aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H2,As))
=> aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As)) ) ) ).
% properD1
tff(fact_314_mod__h__bot__indep,axiom,
! [P: assn,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
<=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),produc7507926704131184380et_nat(H5,bot_bot_set_nat)) ) ).
% mod_h_bot_indep
tff(fact_315_Rep__assn,axiom,
! [X: assn] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,X)),collec6407654713642994138_nat_o(proper)) ).
% Rep_assn
tff(fact_316_Rep__assn__cases,axiom,
! [Y: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),collec6407654713642994138_nat_o(proper))
=> ~ ! [X2: assn] : Y != aa_ass4243947775541419017_nat_o(rep_assn,X2) ) ).
% Rep_assn_cases
tff(fact_317_Rep__assn__induct,axiom,
! [Y: fun_Pr6460944699146269935_nat_o,P: fun_fu5135969500543783852at_o_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),collec6407654713642994138_nat_o(proper))
=> ( ! [X2: assn] : aa_fun1675921005893230629at_o_o(P,aa_ass4243947775541419017_nat_o(rep_assn,X2))
=> aa_fun1675921005893230629at_o_o(P,Y) ) ) ).
% Rep_assn_induct
tff(fact_318_mult_Osemigroup__axioms,axiom,
semigr7036097719740861967nteger(times_3573771949741848930nteger) ).
% mult.semigroup_axioms
tff(fact_319_mult_Osemigroup__axioms,axiom,
semigroup_assn(times_times_assn) ).
% mult.semigroup_axioms
tff(fact_320_mult_Osemigroup__axioms,axiom,
semigroup_rat(times_times_rat) ).
% mult.semigroup_axioms
tff(fact_321_mult_Osemigroup__axioms,axiom,
semigroup_nat(times_times_nat) ).
% mult.semigroup_axioms
tff(fact_322_mult_Osemigroup__axioms,axiom,
semigroup_int(times_times_int) ).
% mult.semigroup_axioms
tff(fact_323_sup_Osemigroup__axioms,axiom,
semigroup_assn(sup_sup_assn) ).
% sup.semigroup_axioms
tff(fact_324_sup_Osemigroup__axioms,axiom,
semigroup_set_nat(sup_sup_set_nat) ).
% sup.semigroup_axioms
tff(fact_325_sup_Osemigroup__axioms,axiom,
semigr6257743987242979861at_nat(sup_su5525570899277871387at_nat) ).
% sup.semigroup_axioms
tff(fact_326_pure__assn__raw_Osimps,axiom,
! [B: $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((B)),produc7507926704131184380et_nat(H2,As))
<=> ( ( As = bot_bot_set_nat )
& (B) ) ) ).
% pure_assn_raw.simps
tff(fact_327_pure__assn__raw_Oelims_I1_J,axiom,
! [X: $o,Xa: produc3658429121746597890et_nat,Y: $o] :
( ( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa)
<=> (Y) )
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xa = produc7507926704131184380et_nat(H3,As3) )
=> ( (Y)
<=> ~ ( ( As3 = bot_bot_set_nat )
& (X) ) ) ) ) ).
% pure_assn_raw.elims(1)
tff(fact_328_pure__assn__raw_Oelims_I2_J,axiom,
! [X: $o,Xa: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa)
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xa = produc7507926704131184380et_nat(H3,As3) )
=> ~ ( ( As3 = bot_bot_set_nat )
& (X) ) ) ) ).
% pure_assn_raw.elims(2)
tff(fact_329_pure__assn__raw_Oelims_I3_J,axiom,
! [X: $o,Xa: produc3658429121746597890et_nat] :
( ~ aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa)
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xa = produc7507926704131184380et_nat(H3,As3) )
=> ( ( As3 = bot_bot_set_nat )
& (X) ) ) ) ).
% pure_assn_raw.elims(3)
tff(fact_330_Abs__assn__cases,axiom,
! [X: assn] :
~ ! [Y2: fun_Pr6460944699146269935_nat_o] :
( ( X = aa_fun5526454687433275355o_assn(abs_assn,Y2) )
=> ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),collec6407654713642994138_nat_o(proper)) ) ).
% Abs_assn_cases
tff(fact_331_Abs__assn__induct,axiom,
! [P: fun_assn_o,X: assn] :
( ! [Y2: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),collec6407654713642994138_nat_o(proper))
=> aa_assn_o(P,aa_fun5526454687433275355o_assn(abs_assn,Y2)) )
=> aa_assn_o(P,X) ) ).
% Abs_assn_induct
tff(fact_332_Abs__assn__inject,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),collec6407654713642994138_nat_o(proper))
=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),collec6407654713642994138_nat_o(proper))
=> ( ( aa_fun5526454687433275355o_assn(abs_assn,X) = aa_fun5526454687433275355o_assn(abs_assn,Y) )
<=> ( X = Y ) ) ) ) ).
% Abs_assn_inject
tff(fact_333_one__assn__raw_Osimps,axiom,
! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
( aa_Pro653082453696054616_nat_o(one_assn_raw,produc7507926704131184380et_nat(H2,As))
<=> ( As = bot_bot_set_nat ) ) ).
% one_assn_raw.simps
tff(fact_334_one__assn__raw_Oelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat,Y: $o] :
( ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
<=> (Y) )
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( X = produc7507926704131184380et_nat(H3,As3) )
=> ( (Y)
<=> ( As3 != bot_bot_set_nat ) ) ) ) ).
% one_assn_raw.elims(1)
tff(fact_335_one__assn__raw_Oelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( X = produc7507926704131184380et_nat(H3,As3) )
=> ( As3 != bot_bot_set_nat ) ) ) ).
% one_assn_raw.elims(2)
tff(fact_336_one__assn__raw_Oelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat] :
( ~ aa_Pro653082453696054616_nat_o(one_assn_raw,X)
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( X = produc7507926704131184380et_nat(H3,As3) )
=> ( As3 = bot_bot_set_nat ) ) ) ).
% one_assn_raw.elims(3)
tff(fact_337_semilattice__neutr_Oaxioms_I2_J,axiom,
! [F: fun_nat_fun_nat_nat,Z: nat] :
( aa_nat_o(semila9081495762789891438tr_nat(F),Z)
=> aa_nat_o(comm_monoid_nat(F),Z) ) ).
% semilattice_neutr.axioms(2)
tff(fact_338_mod__h__bot__iff_I2_J,axiom,
! [H2: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,top_top_assn),produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ).
% mod_h_bot_iff(2)
tff(fact_339_mod__emp__simp,axiom,
! [H2: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,one_one_assn),produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ).
% mod_emp_simp
tff(fact_340_Abs__assn__inverse,axiom,
! [Y: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),collec6407654713642994138_nat_o(proper))
=> ( aa_ass4243947775541419017_nat_o(rep_assn,aa_fun5526454687433275355o_assn(abs_assn,Y)) = Y ) ) ).
% Abs_assn_inverse
tff(fact_341_empty__iff,axiom,
! [C: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),bot_bo8632255151191677115_nat_o) ).
% empty_iff
tff(fact_342_empty__iff,axiom,
! [C: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),bot_bo2099793752762293965at_nat) ).
% empty_iff
tff(fact_343_empty__iff,axiom,
! [C: $o] : ~ aa_set_o_o(member_o((C)),bot_bot_set_o) ).
% empty_iff
tff(fact_344_empty__iff,axiom,
! [C: nat] : ~ aa_set_nat_o(member_nat(C),bot_bot_set_nat) ).
% empty_iff
tff(fact_345_empty__iff,axiom,
! [C: int] : ~ aa_set_int_o(member_int(C),bot_bot_set_int) ).
% empty_iff
tff(fact_346_all__not__in__conv,axiom,
! [Aa: set_fu3555266933746557007_nat_o] :
( ! [X3: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),Aa)
<=> ( Aa = bot_bo8632255151191677115_nat_o ) ) ).
% all_not_in_conv
tff(fact_347_all__not__in__conv,axiom,
! [Aa: set_Pr1261947904930325089at_nat] :
( ! [X3: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),Aa)
<=> ( Aa = bot_bo2099793752762293965at_nat ) ) ).
% all_not_in_conv
tff(fact_348_all__not__in__conv,axiom,
! [Aa: set_o] :
( ! [X3: $o] : ~ aa_set_o_o(member_o((X3)),Aa)
<=> ( Aa = bot_bot_set_o ) ) ).
% all_not_in_conv
tff(fact_349_all__not__in__conv,axiom,
! [Aa: set_nat] :
( ! [X3: nat] : ~ aa_set_nat_o(member_nat(X3),Aa)
<=> ( Aa = bot_bot_set_nat ) ) ).
% all_not_in_conv
tff(fact_350_all__not__in__conv,axiom,
! [Aa: set_int] :
( ! [X3: int] : ~ aa_set_int_o(member_int(X3),Aa)
<=> ( Aa = bot_bot_set_int ) ) ).
% all_not_in_conv
tff(fact_351_Collect__empty__eq,axiom,
! [P: fun_fu5135969500543783852at_o_o] :
( ( collec6407654713642994138_nat_o(P) = bot_bo8632255151191677115_nat_o )
<=> ! [X3: fun_Pr6460944699146269935_nat_o] : ~ aa_fun1675921005893230629at_o_o(P,X3) ) ).
% Collect_empty_eq
tff(fact_352_Collect__empty__eq,axiom,
! [P: fun_Pr4796943543331071514_nat_o] :
( ( collec3392354462482085612at_nat(P) = bot_bo2099793752762293965at_nat )
<=> ! [X3: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(P,X3) ) ).
% Collect_empty_eq
tff(fact_353_Collect__empty__eq,axiom,
! [P: fun_o_o] :
( ( collect_o(P) = bot_bot_set_o )
<=> ! [X3: $o] : ~ aa_o_o(P,(X3)) ) ).
% Collect_empty_eq
tff(fact_354_Collect__empty__eq,axiom,
! [P: fun_nat_o] :
( ( collect_nat(P) = bot_bot_set_nat )
<=> ! [X3: nat] : ~ aa_nat_o(P,X3) ) ).
% Collect_empty_eq
tff(fact_355_Collect__empty__eq,axiom,
! [P: fun_int_o] :
( ( collect_int(P) = bot_bot_set_int )
<=> ! [X3: int] : ~ aa_int_o(P,X3) ) ).
% Collect_empty_eq
tff(fact_356_empty__Collect__eq,axiom,
! [P: fun_fu5135969500543783852at_o_o] :
( ( bot_bo8632255151191677115_nat_o = collec6407654713642994138_nat_o(P) )
<=> ! [X3: fun_Pr6460944699146269935_nat_o] : ~ aa_fun1675921005893230629at_o_o(P,X3) ) ).
% empty_Collect_eq
tff(fact_357_empty__Collect__eq,axiom,
! [P: fun_Pr4796943543331071514_nat_o] :
( ( bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(P) )
<=> ! [X3: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(P,X3) ) ).
% empty_Collect_eq
tff(fact_358_empty__Collect__eq,axiom,
! [P: fun_o_o] :
( ( bot_bot_set_o = collect_o(P) )
<=> ! [X3: $o] : ~ aa_o_o(P,(X3)) ) ).
% empty_Collect_eq
tff(fact_359_empty__Collect__eq,axiom,
! [P: fun_nat_o] :
( ( bot_bot_set_nat = collect_nat(P) )
<=> ! [X3: nat] : ~ aa_nat_o(P,X3) ) ).
% empty_Collect_eq
tff(fact_360_empty__Collect__eq,axiom,
! [P: fun_int_o] :
( ( bot_bot_set_int = collect_int(P) )
<=> ! [X3: int] : ~ aa_int_o(P,X3) ) ).
% empty_Collect_eq
tff(fact_361_Un__empty,axiom,
! [Aa: set_Pr4329608150637261639at_nat,Ba: set_Pr4329608150637261639at_nat] :
( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Aa),Ba) = bot_bo228742789529271731at_nat )
<=> ( ( Aa = bot_bo228742789529271731at_nat )
& ( Ba = bot_bo228742789529271731at_nat ) ) ) ).
% Un_empty
tff(fact_362_Un__empty,axiom,
! [Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Aa),Ba) = bot_bo2099793752762293965at_nat )
<=> ( ( Aa = bot_bo2099793752762293965at_nat )
& ( Ba = bot_bo2099793752762293965at_nat ) ) ) ).
% Un_empty
tff(fact_363_Un__empty,axiom,
! [Aa: set_o,Ba: set_o] :
( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Aa),Ba) = bot_bot_set_o )
<=> ( ( Aa = bot_bot_set_o )
& ( Ba = bot_bot_set_o ) ) ) ).
% Un_empty
tff(fact_364_Un__empty,axiom,
! [Aa: set_nat,Ba: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Aa),Ba) = bot_bot_set_nat )
<=> ( ( Aa = bot_bot_set_nat )
& ( Ba = bot_bot_set_nat ) ) ) ).
% Un_empty
tff(fact_365_Un__empty,axiom,
! [Aa: set_int,Ba: set_int] :
( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Aa),Ba) = bot_bot_set_int )
<=> ( ( Aa = bot_bot_set_int )
& ( Ba = bot_bot_set_int ) ) ) ).
% Un_empty
tff(fact_366_prod_Oinject,axiom,
! [X1: fun_Pr5098669352780829690e_term,X22: produc8923325533196201883nteger,Y1: fun_Pr5098669352780829690e_term,Y22: produc8923325533196201883nteger] :
( ( produc565448451787809096nteger(X1,X22) = produc565448451787809096nteger(Y1,Y22) )
<=> ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
tff(fact_367_prod_Oinject,axiom,
! [X1: fun_Pr6460944699146269935_nat_o,X22: produc3658429121746597890et_nat,Y1: fun_Pr6460944699146269935_nat_o,Y22: produc3658429121746597890et_nat] :
( ( produc6017135503889078682et_nat(X1,X22) = produc6017135503889078682et_nat(Y1,Y22) )
<=> ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
tff(fact_368_prod_Oinject,axiom,
! [X1: fun_Pr6460944699146269935_nat_o,X22: produc8799533983703829792et_nat,Y1: fun_Pr6460944699146269935_nat_o,Y22: produc8799533983703829792et_nat] :
( ( produc7291855603632349880et_nat(X1,X22) = produc7291855603632349880et_nat(Y1,Y22) )
<=> ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
tff(fact_369_prod_Oinject,axiom,
! [X1: fun_Pr7763680713840462443e_term,X22: product_prod_int_int,Y1: fun_Pr7763680713840462443e_term,Y22: product_prod_int_int] :
( ( produc7320969046608631221nt_int(X1,X22) = produc7320969046608631221nt_int(Y1,Y22) )
<=> ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
tff(fact_370_prod_Oinject,axiom,
! [X1: fun_in5730379987311621695e_term,X22: product_prod_int_int,Y1: fun_in5730379987311621695e_term,Y22: product_prod_int_int] :
( ( produc4729913460834754785nt_int(X1,X22) = produc4729913460834754785nt_int(Y1,Y22) )
<=> ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
tff(fact_371_old_Oprod_Oinject,axiom,
! [A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,A2: fun_Pr5098669352780829690e_term,B2: produc8923325533196201883nteger] :
( ( produc565448451787809096nteger(A,B) = produc565448451787809096nteger(A2,B2) )
<=> ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
tff(fact_372_old_Oprod_Oinject,axiom,
! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
( ( produc6017135503889078682et_nat(A,B) = produc6017135503889078682et_nat(A2,B2) )
<=> ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
tff(fact_373_old_Oprod_Oinject,axiom,
! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat] :
( ( produc7291855603632349880et_nat(A,B) = produc7291855603632349880et_nat(A2,B2) )
<=> ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
tff(fact_374_old_Oprod_Oinject,axiom,
! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,A2: fun_Pr7763680713840462443e_term,B2: product_prod_int_int] :
( ( produc7320969046608631221nt_int(A,B) = produc7320969046608631221nt_int(A2,B2) )
<=> ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
tff(fact_375_old_Oprod_Oinject,axiom,
! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,A2: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
( ( produc4729913460834754785nt_int(A,B) = produc4729913460834754785nt_int(A2,B2) )
<=> ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
tff(fact_376_mod__star__conv,axiom,
! [Aa: assn,Ba: assn,H2: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Aa),Ba)),H2)
<=> ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As22: set_nat] :
( ( H2 = produc7507926704131184380et_nat(Hr,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22)) )
& ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
& aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Aa),produc7507926704131184380et_nat(Hr,As1))
& aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Ba),produc7507926704131184380et_nat(Hr,As22)) ) ) ).
% mod_star_conv
tff(fact_377_star__assnI,axiom,
! [P: assn,H2: heap_e7401611519738050253t_unit,As: set_nat,Q: assn,As2: set_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),produc7507926704131184380et_nat(H2,As))
=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),produc7507926704131184380et_nat(H2,As2))
=> ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As2) = bot_bot_set_nat )
=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q)),produc7507926704131184380et_nat(H2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As2))) ) ) ) ).
% star_assnI
tff(fact_378_proper__iff,axiom,
! [P: fun_Pr6460944699146269935_nat_o,As: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
( aa_fun1675921005893230629at_o_o(proper,P)
=> ( aa_hea7994211679605476167unit_o(relH(As,H2),H5)
=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H5,As))
=> ( aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H2,As))
<=> aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H5,As)) ) ) ) ) ).
% proper_iff
tff(fact_379_proper__def,axiom,
! [P: fun_Pr6460944699146269935_nat_o] :
( aa_fun1675921005893230629at_o_o(proper,P)
<=> ! [H: heap_e7401611519738050253t_unit,H6: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H,As4))
=> aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As4)) )
& ( ( aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H,As4))
& aa_hea7994211679605476167unit_o(relH(As4,H),H6)
& aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H6,As4)) )
=> aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H6,As4)) ) ) ) ).
% proper_def
tff(fact_380_inf_Oidem,axiom,
! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),A) = A ).
% inf.idem
tff(fact_381_inf_Oidem,axiom,
! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),A) = A ).
% inf.idem
tff(fact_382_inf__idem,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),X) = X ).
% inf_idem
tff(fact_383_inf__idem,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),X) = X ).
% inf_idem
tff(fact_384_inf_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ).
% inf.left_idem
tff(fact_385_inf_Oleft__idem,axiom,
! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ).
% inf.left_idem
tff(fact_386_inf__left__idem,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) ).
% inf_left_idem
tff(fact_387_inf__left__idem,axiom,
! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).
% inf_left_idem
tff(fact_388_inf_Oright__idem,axiom,
! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ).
% inf.right_idem
tff(fact_389_inf_Oright__idem,axiom,
! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ).
% inf.right_idem
tff(fact_390_inf__right__idem,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) ).
% inf_right_idem
tff(fact_391_inf__right__idem,axiom,
! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).
% inf_right_idem
tff(fact_392_inf__bot__left,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,bot_bot_assn),X) = bot_bot_assn ).
% inf_bot_left
tff(fact_393_inf__bot__left,axiom,
! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),X) = bot_bo2099793752762293965at_nat ).
% inf_bot_left
tff(fact_394_inf__bot__left,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,bot_bot_set_o),X) = bot_bot_set_o ).
% inf_bot_left
tff(fact_395_inf__bot__left,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),X) = bot_bot_set_nat ).
% inf_bot_left
tff(fact_396_inf__bot__left,axiom,
! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,bot_bot_set_int),X) = bot_bot_set_int ).
% inf_bot_left
tff(fact_397_inf__bot__right,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),bot_bot_assn) = bot_bot_assn ).
% inf_bot_right
tff(fact_398_inf__bot__right,axiom,
! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).
% inf_bot_right
tff(fact_399_inf__bot__right,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),bot_bot_set_o) = bot_bot_set_o ).
% inf_bot_right
tff(fact_400_inf__bot__right,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),bot_bot_set_nat) = bot_bot_set_nat ).
% inf_bot_right
tff(fact_401_inf__bot__right,axiom,
! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),bot_bot_set_int) = bot_bot_set_int ).
% inf_bot_right
tff(fact_402_boolean__algebra_Oconj__zero__left,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,bot_bot_assn),X) = bot_bot_assn ).
% boolean_algebra.conj_zero_left
tff(fact_403_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),X) = bot_bo2099793752762293965at_nat ).
% boolean_algebra.conj_zero_left
tff(fact_404_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,bot_bot_set_o),X) = bot_bot_set_o ).
% boolean_algebra.conj_zero_left
tff(fact_405_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),X) = bot_bot_set_nat ).
% boolean_algebra.conj_zero_left
tff(fact_406_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,bot_bot_set_int),X) = bot_bot_set_int ).
% boolean_algebra.conj_zero_left
tff(fact_407_boolean__algebra_Oconj__zero__right,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),bot_bot_assn) = bot_bot_assn ).
% boolean_algebra.conj_zero_right
tff(fact_408_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).
% boolean_algebra.conj_zero_right
tff(fact_409_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),bot_bot_set_o) = bot_bot_set_o ).
% boolean_algebra.conj_zero_right
tff(fact_410_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),bot_bot_set_nat) = bot_bot_set_nat ).
% boolean_algebra.conj_zero_right
tff(fact_411_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),bot_bot_set_int) = bot_bot_set_int ).
% boolean_algebra.conj_zero_right
tff(fact_412_inf__top__left,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,top_top_assn),X) = X ).
% inf_top_left
tff(fact_413_inf__top__left,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,top_top_set_o),X) = X ).
% inf_top_left
tff(fact_414_inf__top__left,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,top_top_set_nat),X) = X ).
% inf_top_left
tff(fact_415_inf__top__left,axiom,
! [X: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,top_to1996260823553986621t_unit),X) = X ).
% inf_top_left
tff(fact_416_inf__top__right,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),top_top_assn) = X ).
% inf_top_right
tff(fact_417_inf__top__right,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),top_top_set_o) = X ).
% inf_top_right
tff(fact_418_inf__top__right,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),top_top_set_nat) = X ).
% inf_top_right
tff(fact_419_inf__top__right,axiom,
! [X: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,X),top_to1996260823553986621t_unit) = X ).
% inf_top_right
tff(fact_420_inf__eq__top__iff,axiom,
! [X: assn,Y: assn] :
( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = top_top_assn )
<=> ( ( X = top_top_assn )
& ( Y = top_top_assn ) ) ) ).
% inf_eq_top_iff
tff(fact_421_inf__eq__top__iff,axiom,
! [X: set_o,Y: set_o] :
( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),Y) = top_top_set_o )
<=> ( ( X = top_top_set_o )
& ( Y = top_top_set_o ) ) ) ).
% inf_eq_top_iff
tff(fact_422_inf__eq__top__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = top_top_set_nat )
<=> ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
tff(fact_423_inf__eq__top__iff,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,X),Y) = top_to1996260823553986621t_unit )
<=> ( ( X = top_to1996260823553986621t_unit )
& ( Y = top_to1996260823553986621t_unit ) ) ) ).
% inf_eq_top_iff
tff(fact_424_top__eq__inf__iff,axiom,
! [X: assn,Y: assn] :
( ( top_top_assn = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) )
<=> ( ( X = top_top_assn )
& ( Y = top_top_assn ) ) ) ).
% top_eq_inf_iff
tff(fact_425_top__eq__inf__iff,axiom,
! [X: set_o,Y: set_o] :
( ( top_top_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),Y) )
<=> ( ( X = top_top_set_o )
& ( Y = top_top_set_o ) ) ) ).
% top_eq_inf_iff
tff(fact_426_top__eq__inf__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( top_top_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) )
<=> ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
tff(fact_427_top__eq__inf__iff,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( top_to1996260823553986621t_unit = aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,X),Y) )
<=> ( ( X = top_to1996260823553986621t_unit )
& ( Y = top_to1996260823553986621t_unit ) ) ) ).
% top_eq_inf_iff
tff(fact_428_inf__top_Oeq__neutr__iff,axiom,
! [A: assn,B: assn] :
( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = top_top_assn )
<=> ( ( A = top_top_assn )
& ( B = top_top_assn ) ) ) ).
% inf_top.eq_neutr_iff
tff(fact_429_inf__top_Oeq__neutr__iff,axiom,
! [A: set_o,B: set_o] :
( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A),B) = top_top_set_o )
<=> ( ( A = top_top_set_o )
& ( B = top_top_set_o ) ) ) ).
% inf_top.eq_neutr_iff
tff(fact_430_inf__top_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = top_top_set_nat )
<=> ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
tff(fact_431_inf__top_Oeq__neutr__iff,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,A),B) = top_to1996260823553986621t_unit )
<=> ( ( A = top_to1996260823553986621t_unit )
& ( B = top_to1996260823553986621t_unit ) ) ) ).
% inf_top.eq_neutr_iff
tff(fact_432_inf__top_Oleft__neutral,axiom,
! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,top_top_assn),A) = A ).
% inf_top.left_neutral
tff(fact_433_inf__top_Oleft__neutral,axiom,
! [A: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,top_top_set_o),A) = A ).
% inf_top.left_neutral
tff(fact_434_inf__top_Oleft__neutral,axiom,
! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,top_top_set_nat),A) = A ).
% inf_top.left_neutral
tff(fact_435_inf__top_Oleft__neutral,axiom,
! [A: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,top_to1996260823553986621t_unit),A) = A ).
% inf_top.left_neutral
tff(fact_436_inf__top_Oneutr__eq__iff,axiom,
! [A: assn,B: assn] :
( ( top_top_assn = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) )
<=> ( ( A = top_top_assn )
& ( B = top_top_assn ) ) ) ).
% inf_top.neutr_eq_iff
tff(fact_437_inf__top_Oneutr__eq__iff,axiom,
! [A: set_o,B: set_o] :
( ( top_top_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A),B) )
<=> ( ( A = top_top_set_o )
& ( B = top_top_set_o ) ) ) ).
% inf_top.neutr_eq_iff
tff(fact_438_inf__top_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( top_top_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) )
<=> ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
tff(fact_439_inf__top_Oneutr__eq__iff,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( top_to1996260823553986621t_unit = aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,A),B) )
<=> ( ( A = top_to1996260823553986621t_unit )
& ( B = top_to1996260823553986621t_unit ) ) ) ).
% inf_top.neutr_eq_iff
tff(fact_440_inf__top_Oright__neutral,axiom,
! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),top_top_assn) = A ).
% inf_top.right_neutral
tff(fact_441_inf__top_Oright__neutral,axiom,
! [A: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A),top_top_set_o) = A ).
% inf_top.right_neutral
tff(fact_442_inf__top_Oright__neutral,axiom,
! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),top_top_set_nat) = A ).
% inf_top.right_neutral
tff(fact_443_inf__top_Oright__neutral,axiom,
! [A: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,A),top_to1996260823553986621t_unit) = A ).
% inf_top.right_neutral
tff(fact_444_inf__sup__absorb,axiom,
! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = X ).
% inf_sup_absorb
tff(fact_445_inf__sup__absorb,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = X ).
% inf_sup_absorb
tff(fact_446_inf__sup__absorb,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) = X ).
% inf_sup_absorb
tff(fact_447_sup__inf__absorb,axiom,
! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = X ).
% sup_inf_absorb
tff(fact_448_sup__inf__absorb,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = X ).
% sup_inf_absorb
tff(fact_449_sup__inf__absorb,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)) = X ).
% sup_inf_absorb
tff(fact_450_relH__dist__union,axiom,
! [As: set_nat,As2: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
( aa_hea7994211679605476167unit_o(relH(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As2),H2),H5)
<=> ( aa_hea7994211679605476167unit_o(relH(As,H2),H5)
& aa_hea7994211679605476167unit_o(relH(As2,H2),H5) ) ) ).
% relH_dist_union
tff(fact_451_inf__sup__aci_I4_J,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) ).
% inf_sup_aci(4)
tff(fact_452_inf__sup__aci_I4_J,axiom,
! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).
% inf_sup_aci(4)
tff(fact_453_inf__sup__aci_I3_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ).
% inf_sup_aci(3)
tff(fact_454_inf__sup__aci_I3_J,axiom,
! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).
% inf_sup_aci(3)
tff(fact_455_inf__sup__aci_I2_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) ).
% inf_sup_aci(2)
tff(fact_456_inf__sup__aci_I2_J,axiom,
! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ).
% inf_sup_aci(2)
tff(fact_457_inf__sup__aci_I1_J,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X) ).
% inf_sup_aci(1)
tff(fact_458_inf__sup__aci_I1_J,axiom,
! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X) ).
% inf_sup_aci(1)
tff(fact_459_inf_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) ).
% inf.assoc
tff(fact_460_inf_Oassoc,axiom,
! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C)) ).
% inf.assoc
tff(fact_461_inf__assoc,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) ).
% inf_assoc
tff(fact_462_inf__assoc,axiom,
! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ).
% inf_assoc
tff(fact_463_inf_Ocommute,axiom,
! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),A) ).
% inf.commute
tff(fact_464_inf_Ocommute,axiom,
! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),A) ).
% inf.commute
tff(fact_465_inf__commute,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X) ).
% inf_commute
tff(fact_466_inf__commute,axiom,
! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X) ).
% inf_commute
tff(fact_467_inf_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),C)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) ).
% inf.left_commute
tff(fact_468_inf_Oleft__commute,axiom,
! [B: assn,A: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),C)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C)) ).
% inf.left_commute
tff(fact_469_inf__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ).
% inf_left_commute
tff(fact_470_inf__left__commute,axiom,
! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).
% inf_left_commute
tff(fact_471_relH__trans,axiom,
! [As: set_nat,H12: heap_e7401611519738050253t_unit,H23: heap_e7401611519738050253t_unit,H32: heap_e7401611519738050253t_unit] :
( aa_hea7994211679605476167unit_o(relH(As,H12),H23)
=> ( aa_hea7994211679605476167unit_o(relH(As,H23),H32)
=> aa_hea7994211679605476167unit_o(relH(As,H12),H32) ) ) ).
% relH_trans
tff(fact_472_relH__sym,axiom,
! [As: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
( aa_hea7994211679605476167unit_o(relH(As,H2),H5)
=> aa_hea7994211679605476167unit_o(relH(As,H5),H2) ) ).
% relH_sym
tff(fact_473_boolean__algebra__cancel_Oinf1,axiom,
! [Aa: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( Aa = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),A) )
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).
% boolean_algebra_cancel.inf1
tff(fact_474_boolean__algebra__cancel_Oinf1,axiom,
! [Aa: assn,K: assn,A: assn,B: assn] :
( ( Aa = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),A) )
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Aa),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) ) ) ).
% boolean_algebra_cancel.inf1
tff(fact_475_boolean__algebra__cancel_Oinf2,axiom,
! [Ba: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( Ba = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),B) )
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),Ba) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).
% boolean_algebra_cancel.inf2
tff(fact_476_boolean__algebra__cancel_Oinf2,axiom,
! [Ba: assn,K: assn,B: assn,A: assn] :
( ( Ba = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),B) )
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),Ba) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) ) ) ).
% boolean_algebra_cancel.inf2
tff(fact_477_times__assn__raw_Ocases,axiom,
! [X: produc3112576180687729086et_nat] :
~ ! [P3: fun_Pr6460944699146269935_nat_o,Q3: fun_Pr6460944699146269935_nat_o,H3: heap_e7401611519738050253t_unit,As3: set_nat] : X != produc7291855603632349880et_nat(P3,produc6017135503889078682et_nat(Q3,produc7507926704131184380et_nat(H3,As3))) ).
% times_assn_raw.cases
tff(fact_478_disjoint__iff__not__equal,axiom,
! [Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Aa),Ba) = bot_bo2099793752762293965at_nat )
<=> ! [X3: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),Aa)
=> ! [Xa2: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa2),Ba)
=> ( X3 != Xa2 ) ) ) ) ).
% disjoint_iff_not_equal
tff(fact_479_disjoint__iff__not__equal,axiom,
! [Aa: set_o,Ba: set_o] :
( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Aa),Ba) = bot_bot_set_o )
<=> ! [X3: $o] :
( aa_set_o_o(member_o((X3)),Aa)
=> ! [Xa2: $o] :
( aa_set_o_o(member_o((Xa2)),Ba)
=> ( (X3)
<=> ~ (Xa2) ) ) ) ) ).
% disjoint_iff_not_equal
tff(fact_480_disjoint__iff__not__equal,axiom,
! [Aa: set_nat,Ba: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),Ba) = bot_bot_set_nat )
<=> ! [X3: nat] :
( aa_set_nat_o(member_nat(X3),Aa)
=> ! [Xa2: nat] :
( aa_set_nat_o(member_nat(Xa2),Ba)
=> ( X3 != Xa2 ) ) ) ) ).
% disjoint_iff_not_equal
tff(fact_481_disjoint__iff__not__equal,axiom,
! [Aa: set_int,Ba: set_int] :
( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Aa),Ba) = bot_bot_set_int )
<=> ! [X3: int] :
( aa_set_int_o(member_int(X3),Aa)
=> ! [Xa2: int] :
( aa_set_int_o(member_int(Xa2),Ba)
=> ( X3 != Xa2 ) ) ) ) ).
% disjoint_iff_not_equal
tff(fact_482_Int__empty__right,axiom,
! [Aa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Aa),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).
% Int_empty_right
tff(fact_483_Int__empty__right,axiom,
! [Aa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Aa),bot_bot_set_o) = bot_bot_set_o ).
% Int_empty_right
tff(fact_484_Int__empty__right,axiom,
! [Aa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),bot_bot_set_nat) = bot_bot_set_nat ).
% Int_empty_right
tff(fact_485_Int__empty__right,axiom,
! [Aa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Aa),bot_bot_set_int) = bot_bot_set_int ).
% Int_empty_right
tff(fact_486_Int__empty__left,axiom,
! [Ba: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),Ba) = bot_bo2099793752762293965at_nat ).
% Int_empty_left
tff(fact_487_Int__empty__left,axiom,
! [Ba: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,bot_bot_set_o),Ba) = bot_bot_set_o ).
% Int_empty_left
tff(fact_488_Int__empty__left,axiom,
! [Ba: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),Ba) = bot_bot_set_nat ).
% Int_empty_left
tff(fact_489_Int__empty__left,axiom,
! [Ba: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,bot_bot_set_int),Ba) = bot_bot_set_int ).
% Int_empty_left
tff(fact_490_disjoint__iff,axiom,
! [Aa: set_fu3555266933746557007_nat_o,Ba: set_fu3555266933746557007_nat_o] :
( ( inf_in3376754634429691197_nat_o(Aa,Ba) = bot_bo8632255151191677115_nat_o )
<=> ! [X3: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),Aa)
=> ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),Ba) ) ) ).
% disjoint_iff
tff(fact_491_disjoint__iff,axiom,
! [Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Aa),Ba) = bot_bo2099793752762293965at_nat )
<=> ! [X3: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),Aa)
=> ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),Ba) ) ) ).
% disjoint_iff
tff(fact_492_disjoint__iff,axiom,
! [Aa: set_o,Ba: set_o] :
( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Aa),Ba) = bot_bot_set_o )
<=> ! [X3: $o] :
( aa_set_o_o(member_o((X3)),Aa)
=> ~ aa_set_o_o(member_o((X3)),Ba) ) ) ).
% disjoint_iff
tff(fact_493_disjoint__iff,axiom,
! [Aa: set_nat,Ba: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),Ba) = bot_bot_set_nat )
<=> ! [X3: nat] :
( aa_set_nat_o(member_nat(X3),Aa)
=> ~ aa_set_nat_o(member_nat(X3),Ba) ) ) ).
% disjoint_iff
tff(fact_494_disjoint__iff,axiom,
! [Aa: set_int,Ba: set_int] :
( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Aa),Ba) = bot_bot_set_int )
<=> ! [X3: int] :
( aa_set_int_o(member_int(X3),Aa)
=> ~ aa_set_int_o(member_int(X3),Ba) ) ) ).
% disjoint_iff
tff(fact_495_Int__emptyI,axiom,
! [Aa: set_fu3555266933746557007_nat_o,Ba: set_fu3555266933746557007_nat_o] :
( ! [X2: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),Aa)
=> ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),Ba) )
=> ( inf_in3376754634429691197_nat_o(Aa,Ba) = bot_bo8632255151191677115_nat_o ) ) ).
% Int_emptyI
tff(fact_496_Int__emptyI,axiom,
! [Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
( ! [X2: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),Aa)
=> ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),Ba) )
=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Aa),Ba) = bot_bo2099793752762293965at_nat ) ) ).
% Int_emptyI
tff(fact_497_Int__emptyI,axiom,
! [Aa: set_o,Ba: set_o] :
( ! [X2: $o] :
( aa_set_o_o(member_o((X2)),Aa)
=> ~ aa_set_o_o(member_o((X2)),Ba) )
=> ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Aa),Ba) = bot_bot_set_o ) ) ).
% Int_emptyI
tff(fact_498_Int__emptyI,axiom,
! [Aa: set_nat,Ba: set_nat] :
( ! [X2: nat] :
( aa_set_nat_o(member_nat(X2),Aa)
=> ~ aa_set_nat_o(member_nat(X2),Ba) )
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),Ba) = bot_bot_set_nat ) ) ).
% Int_emptyI
tff(fact_499_Int__emptyI,axiom,
! [Aa: set_int,Ba: set_int] :
( ! [X2: int] :
( aa_set_int_o(member_int(X2),Aa)
=> ~ aa_set_int_o(member_int(X2),Ba) )
=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Aa),Ba) = bot_bot_set_int ) ) ).
% Int_emptyI
tff(fact_500_boolean__algebra_Oconj__one__right,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),top_top_assn) = X ).
% boolean_algebra.conj_one_right
tff(fact_501_boolean__algebra_Oconj__one__right,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),top_top_set_o) = X ).
% boolean_algebra.conj_one_right
tff(fact_502_boolean__algebra_Oconj__one__right,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),top_top_set_nat) = X ).
% boolean_algebra.conj_one_right
tff(fact_503_boolean__algebra_Oconj__one__right,axiom,
! [X: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,X),top_to1996260823553986621t_unit) = X ).
% boolean_algebra.conj_one_right
tff(fact_504_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).
% boolean_algebra.conj_disj_distrib
tff(fact_505_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ).
% boolean_algebra.conj_disj_distrib
tff(fact_506_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Z)) ).
% boolean_algebra.conj_disj_distrib
tff(fact_507_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).
% boolean_algebra.disj_conj_distrib
tff(fact_508_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ).
% boolean_algebra.disj_conj_distrib
tff(fact_509_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z)) ).
% boolean_algebra.disj_conj_distrib
tff(fact_510_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Z),X)) ).
% boolean_algebra.conj_disj_distrib2
tff(fact_511_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Z),X)) ).
% boolean_algebra.conj_disj_distrib2
tff(fact_512_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)),X) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),X)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Z),X)) ).
% boolean_algebra.conj_disj_distrib2
tff(fact_513_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Z),X)) ).
% boolean_algebra.disj_conj_distrib2
tff(fact_514_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Z),X)) ).
% boolean_algebra.disj_conj_distrib2
tff(fact_515_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z)),X) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),X)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Z),X)) ).
% boolean_algebra.disj_conj_distrib2
tff(fact_516_distrib__imp1,axiom,
! [X: assn,Y: assn,Z: assn] :
( ! [X2: assn,Y2: assn,Z2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y2),Z2)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X2),Y2)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X2),Z2))
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ) ) ).
% distrib_imp1
tff(fact_517_distrib__imp1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ! [X2: set_nat,Y2: set_nat,Z2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y2),Z2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),Y2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),Z2))
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ) ) ).
% distrib_imp1
tff(fact_518_distrib__imp1,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y2),Z2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X2),Y2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X2),Z2))
=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z)) ) ) ).
% distrib_imp1
tff(fact_519_distrib__imp2,axiom,
! [X: assn,Y: assn,Z: assn] :
( ! [X2: assn,Y2: assn,Z2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X2),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y2),Z2)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X2),Y2)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X2),Z2))
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ) ) ).
% distrib_imp2
tff(fact_520_distrib__imp2,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ! [X2: set_nat,Y2: set_nat,Z2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y2),Z2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),Y2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),Z2))
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ) ) ).
% distrib_imp2
tff(fact_521_distrib__imp2,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y2),Z2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X2),Y2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X2),Z2))
=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Z)) ) ) ).
% distrib_imp2
tff(fact_522_inf__sup__distrib1,axiom,
! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).
% inf_sup_distrib1
tff(fact_523_inf__sup__distrib1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ).
% inf_sup_distrib1
tff(fact_524_inf__sup__distrib1,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Z)) ).
% inf_sup_distrib1
tff(fact_525_inf__sup__distrib2,axiom,
! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Z),X)) ).
% inf_sup_distrib2
tff(fact_526_inf__sup__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Z),X)) ).
% inf_sup_distrib2
tff(fact_527_inf__sup__distrib2,axiom,
! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)),X) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),X)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Z),X)) ).
% inf_sup_distrib2
tff(fact_528_sup__inf__distrib1,axiom,
! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).
% sup_inf_distrib1
tff(fact_529_sup__inf__distrib1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ).
% sup_inf_distrib1
tff(fact_530_sup__inf__distrib1,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z)) ).
% sup_inf_distrib1
tff(fact_531_sup__inf__distrib2,axiom,
! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Z),X)) ).
% sup_inf_distrib2
tff(fact_532_sup__inf__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Z),X)) ).
% sup_inf_distrib2
tff(fact_533_sup__inf__distrib2,axiom,
! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z)),X) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),X)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Z),X)) ).
% sup_inf_distrib2
tff(fact_534_inf_Osemigroup__axioms,axiom,
semigroup_set_nat(inf_inf_set_nat) ).
% inf.semigroup_axioms
tff(fact_535_inf_Osemigroup__axioms,axiom,
semigroup_assn(inf_inf_assn) ).
% inf.semigroup_axioms
tff(fact_536_inf__top_Ocomm__monoid__axioms,axiom,
aa_assn_o(comm_monoid_assn(inf_inf_assn),top_top_assn) ).
% inf_top.comm_monoid_axioms
tff(fact_537_inf__top_Ocomm__monoid__axioms,axiom,
aa_set_o_o(comm_monoid_set_o(inf_inf_set_o),top_top_set_o) ).
% inf_top.comm_monoid_axioms
tff(fact_538_inf__top_Ocomm__monoid__axioms,axiom,
aa_set_nat_o(comm_monoid_set_nat(inf_inf_set_nat),top_top_set_nat) ).
% inf_top.comm_monoid_axioms
tff(fact_539_inf__top_Ocomm__monoid__axioms,axiom,
comm_m7591727228438362018t_unit(inf_in4660618365625256667t_unit,top_to1996260823553986621t_unit) ).
% inf_top.comm_monoid_axioms
tff(fact_540_inf__top_Omonoid__axioms,axiom,
aa_assn_o(monoid_assn(inf_inf_assn),top_top_assn) ).
% inf_top.monoid_axioms
tff(fact_541_inf__top_Omonoid__axioms,axiom,
aa_set_o_o(monoid_set_o(inf_inf_set_o),top_top_set_o) ).
% inf_top.monoid_axioms
tff(fact_542_inf__top_Omonoid__axioms,axiom,
aa_set_nat_o(monoid_set_nat(inf_inf_set_nat),top_top_set_nat) ).
% inf_top.monoid_axioms
tff(fact_543_inf__top_Omonoid__axioms,axiom,
monoid202360481972076142t_unit(inf_in4660618365625256667t_unit,top_to1996260823553986621t_unit) ).
% inf_top.monoid_axioms
tff(fact_544_inf__top_Osemilattice__neutr__axioms,axiom,
aa_assn_o(semila5388846958306562758r_assn(inf_inf_assn),top_top_assn) ).
% inf_top.semilattice_neutr_axioms
tff(fact_545_inf__top_Osemilattice__neutr__axioms,axiom,
aa_set_o_o(semila1065760912024005978_set_o(inf_inf_set_o),top_top_set_o) ).
% inf_top.semilattice_neutr_axioms
tff(fact_546_inf__top_Osemilattice__neutr__axioms,axiom,
aa_set_nat_o(semila1241773964035338532et_nat(inf_inf_set_nat),top_top_set_nat) ).
% inf_top.semilattice_neutr_axioms
tff(fact_547_inf__top_Osemilattice__neutr__axioms,axiom,
semila7638931295954862799t_unit(inf_in4660618365625256667t_unit,top_to1996260823553986621t_unit) ).
% inf_top.semilattice_neutr_axioms
tff(fact_548_wand__raw_Osimps,axiom,
! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(P),Q),produc7507926704131184380et_nat(H2,As))
<=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As))
& ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As5) = bot_bot_set_nat )
& aa_hea7994211679605476167unit_o(relH(As,H2),H6)
& aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H6,As))
& aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H6,As5)) )
=> aa_Pro653082453696054616_nat_o(Q,produc7507926704131184380et_nat(H6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As5))) ) ) ) ).
% wand_raw.simps
tff(fact_549_wand__raw_Oelims_I1_J,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
<=> (Y) )
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xb = produc7507926704131184380et_nat(H3,As3) )
=> ( (Y)
<=> ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As3))
& ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As3),As5) = bot_bot_set_nat )
& aa_hea7994211679605476167unit_o(relH(As3,H3),H6)
& aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H6,As3))
& aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H6,As5)) )
=> aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As3),As5))) ) ) ) ) ) ).
% wand_raw.elims(1)
tff(fact_550_wand__raw_Oelims_I2_J,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xb = produc7507926704131184380et_nat(H3,As3) )
=> ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As3))
& ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As3),As6) = bot_bot_set_nat )
& aa_hea7994211679605476167unit_o(relH(As3,H3),H7)
& aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H7,As3))
& aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H7,As6)) )
=> aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H7,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As3),As6))) ) ) ) ) ).
% wand_raw.elims(2)
tff(fact_551_wand__raw_Oelims_I3_J,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xb = produc7507926704131184380et_nat(H3,As3) )
=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As3))
& ! [H4: heap_e7401611519738050253t_unit,As7: set_nat] :
( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As3),As7) = bot_bot_set_nat )
& aa_hea7994211679605476167unit_o(relH(As3,H3),H4)
& aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As3))
& aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H4,As7)) )
=> aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H4,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As3),As7))) ) ) ) ) ).
% wand_raw.elims(3)
tff(fact_552_mod__relH,axiom,
! [As: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,P: assn] :
( aa_hea7994211679605476167unit_o(relH(As,H2),H5)
=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),produc7507926704131184380et_nat(H2,As))
<=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),produc7507926704131184380et_nat(H5,As)) ) ) ).
% mod_relH
tff(fact_553_relH__refl,axiom,
! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As))
=> aa_hea7994211679605476167unit_o(relH(As,H2),H2) ) ).
% relH_refl
tff(fact_554_relH__in__rangeI_I1_J,axiom,
! [As: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
( aa_hea7994211679605476167unit_o(relH(As,H2),H5)
=> aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As)) ) ).
% relH_in_rangeI(1)
tff(fact_555_relH__in__rangeI_I2_J,axiom,
! [As: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
( aa_hea7994211679605476167unit_o(relH(As,H2),H5)
=> aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H5,As)) ) ).
% relH_in_rangeI(2)
tff(fact_556_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),X) = bot_bo228742789529271731at_nat )
=> ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),X) = top_to6833984726390702231at_nat )
=> ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),Y) = bot_bo228742789529271731at_nat )
=> ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),Y) = top_to6833984726390702231at_nat )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
tff(fact_557_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),X) = bot_bo2099793752762293965at_nat )
=> ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),X) = top_to4669805908274784177at_nat )
=> ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),Y) = bot_bo2099793752762293965at_nat )
=> ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),Y) = top_to4669805908274784177at_nat )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
tff(fact_558_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_int,X: set_int,Y: set_int] :
( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),X) = bot_bot_set_int )
=> ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),X) = top_top_set_int )
=> ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),Y) = bot_bot_set_int )
=> ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),Y) = top_top_set_int )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
tff(fact_559_boolean__algebra_Ocomplement__unique,axiom,
! [A: assn,X: assn,Y: assn] :
( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),X) = bot_bot_assn )
=> ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),X) = top_top_assn )
=> ( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),Y) = bot_bot_assn )
=> ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),Y) = top_top_assn )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
tff(fact_560_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_o,X: set_o,Y: set_o] :
( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A),X) = bot_bot_set_o )
=> ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),X) = top_top_set_o )
=> ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A),Y) = bot_bot_set_o )
=> ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),Y) = top_top_set_o )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
tff(fact_561_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_nat,X: set_nat,Y: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),X) = bot_bot_set_nat )
=> ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),X) = top_top_set_nat )
=> ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),Y) = bot_bot_set_nat )
=> ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),Y) = top_top_set_nat )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
tff(fact_562_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_Product_unit,X: set_Product_unit,Y: set_Product_unit] :
( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,A),X) = bot_bo3957492148770167129t_unit )
=> ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),X) = top_to1996260823553986621t_unit )
=> ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,A),Y) = bot_bo3957492148770167129t_unit )
=> ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),Y) = top_to1996260823553986621t_unit )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
tff(fact_563_prod__induct4,axiom,
! [P: fun_Pr1227177680593602867_nat_o,X: produc3112576180687729086et_nat] :
( ! [A3: fun_Pr6460944699146269935_nat_o,B3: fun_Pr6460944699146269935_nat_o,C2: heap_e7401611519738050253t_unit,D: set_nat] : aa_Pro2671238023309943196_nat_o(P,produc7291855603632349880et_nat(A3,produc6017135503889078682et_nat(B3,produc7507926704131184380et_nat(C2,D))))
=> aa_Pro2671238023309943196_nat_o(P,X) ) ).
% prod_induct4
tff(fact_564_prod__induct3,axiom,
! [P: fun_Pr7015759342434529115eger_o,X: produc2453742332425399446nteger] :
( ! [A3: fun_Pr5098669352780829690e_term,B3: code_integer,C2: code_integer] : aa_Pro4148064003102580676eger_o(P,produc565448451787809096nteger(A3,produc1086072967326762835nteger(B3,C2)))
=> aa_Pro4148064003102580676eger_o(P,X) ) ).
% prod_induct3
tff(fact_565_prod__induct3,axiom,
! [P: fun_Pr2620521676128474833_nat_o,X: produc8799533983703829792et_nat] :
( ! [A3: fun_Pr6460944699146269935_nat_o,B3: heap_e7401611519738050253t_unit,C2: set_nat] : aa_Pro5643009357358181178_nat_o(P,produc6017135503889078682et_nat(A3,produc7507926704131184380et_nat(B3,C2)))
=> aa_Pro5643009357358181178_nat_o(P,X) ) ).
% prod_induct3
tff(fact_566_prod__induct3,axiom,
! [P: fun_Pr1227177680593602867_nat_o,X: produc3112576180687729086et_nat] :
( ! [A3: fun_Pr6460944699146269935_nat_o,B3: fun_Pr6460944699146269935_nat_o,C2: produc3658429121746597890et_nat] : aa_Pro2671238023309943196_nat_o(P,produc7291855603632349880et_nat(A3,produc6017135503889078682et_nat(B3,C2)))
=> aa_Pro2671238023309943196_nat_o(P,X) ) ).
% prod_induct3
tff(fact_567_prod__induct3,axiom,
! [P: fun_Pr3540825087398332590_int_o,X: produc3043004577601930755nt_int] :
( ! [A3: fun_Pr7763680713840462443e_term,B3: int,C2: int] : aa_Pro2601676399929874711_int_o(P,produc7320969046608631221nt_int(A3,aa_int7462565939943104719nt_int(product_Pair_int_int(B3),C2)))
=> aa_Pro2601676399929874711_int_o(P,X) ) ).
% prod_induct3
tff(fact_568_prod__induct3,axiom,
! [P: fun_Pr1196100989263358594_int_o,X: produc4788131091073775919nt_int] :
( ! [A3: fun_in5730379987311621695e_term,B3: int,C2: int] : aa_Pro5451197003417746155_int_o(P,produc4729913460834754785nt_int(A3,aa_int7462565939943104719nt_int(product_Pair_int_int(B3),C2)))
=> aa_Pro5451197003417746155_int_o(P,X) ) ).
% prod_induct3
tff(fact_569_prod__cases4,axiom,
! [Y: produc3112576180687729086et_nat] :
~ ! [A3: fun_Pr6460944699146269935_nat_o,B3: fun_Pr6460944699146269935_nat_o,C2: heap_e7401611519738050253t_unit,D: set_nat] : Y != produc7291855603632349880et_nat(A3,produc6017135503889078682et_nat(B3,produc7507926704131184380et_nat(C2,D))) ).
% prod_cases4
tff(fact_570_prod__cases3,axiom,
! [Y: produc2453742332425399446nteger] :
~ ! [A3: fun_Pr5098669352780829690e_term,B3: code_integer,C2: code_integer] : Y != produc565448451787809096nteger(A3,produc1086072967326762835nteger(B3,C2)) ).
% prod_cases3
tff(fact_571_prod__cases3,axiom,
! [Y: produc8799533983703829792et_nat] :
~ ! [A3: fun_Pr6460944699146269935_nat_o,B3: heap_e7401611519738050253t_unit,C2: set_nat] : Y != produc6017135503889078682et_nat(A3,produc7507926704131184380et_nat(B3,C2)) ).
% prod_cases3
tff(fact_572_prod__cases3,axiom,
! [Y: produc3112576180687729086et_nat] :
~ ! [A3: fun_Pr6460944699146269935_nat_o,B3: fun_Pr6460944699146269935_nat_o,C2: produc3658429121746597890et_nat] : Y != produc7291855603632349880et_nat(A3,produc6017135503889078682et_nat(B3,C2)) ).
% prod_cases3
tff(fact_573_prod__cases3,axiom,
! [Y: produc3043004577601930755nt_int] :
~ ! [A3: fun_Pr7763680713840462443e_term,B3: int,C2: int] : Y != produc7320969046608631221nt_int(A3,aa_int7462565939943104719nt_int(product_Pair_int_int(B3),C2)) ).
% prod_cases3
tff(fact_574_prod__cases3,axiom,
! [Y: produc4788131091073775919nt_int] :
~ ! [A3: fun_in5730379987311621695e_term,B3: int,C2: int] : Y != produc4729913460834754785nt_int(A3,aa_int7462565939943104719nt_int(product_Pair_int_int(B3),C2)) ).
% prod_cases3
tff(fact_575_Pair__inject,axiom,
! [A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,A2: fun_Pr5098669352780829690e_term,B2: produc8923325533196201883nteger] :
( ( produc565448451787809096nteger(A,B) = produc565448451787809096nteger(A2,B2) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
tff(fact_576_Pair__inject,axiom,
! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
( ( produc6017135503889078682et_nat(A,B) = produc6017135503889078682et_nat(A2,B2) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
tff(fact_577_Pair__inject,axiom,
! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat] :
( ( produc7291855603632349880et_nat(A,B) = produc7291855603632349880et_nat(A2,B2) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
tff(fact_578_Pair__inject,axiom,
! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,A2: fun_Pr7763680713840462443e_term,B2: product_prod_int_int] :
( ( produc7320969046608631221nt_int(A,B) = produc7320969046608631221nt_int(A2,B2) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
tff(fact_579_Pair__inject,axiom,
! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,A2: fun_in5730379987311621695e_term,B2: product_prod_int_int] :
( ( produc4729913460834754785nt_int(A,B) = produc4729913460834754785nt_int(A2,B2) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
tff(fact_580_prod__cases,axiom,
! [P: fun_Pr7015759342434529115eger_o,P5: produc2453742332425399446nteger] :
( ! [A3: fun_Pr5098669352780829690e_term,B3: produc8923325533196201883nteger] : aa_Pro4148064003102580676eger_o(P,produc565448451787809096nteger(A3,B3))
=> aa_Pro4148064003102580676eger_o(P,P5) ) ).
% prod_cases
tff(fact_581_prod__cases,axiom,
! [P: fun_Pr2620521676128474833_nat_o,P5: produc8799533983703829792et_nat] :
( ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] : aa_Pro5643009357358181178_nat_o(P,produc6017135503889078682et_nat(A3,B3))
=> aa_Pro5643009357358181178_nat_o(P,P5) ) ).
% prod_cases
tff(fact_582_prod__cases,axiom,
! [P: fun_Pr1227177680593602867_nat_o,P5: produc3112576180687729086et_nat] :
( ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] : aa_Pro2671238023309943196_nat_o(P,produc7291855603632349880et_nat(A3,B3))
=> aa_Pro2671238023309943196_nat_o(P,P5) ) ).
% prod_cases
tff(fact_583_prod__cases,axiom,
! [P: fun_Pr3540825087398332590_int_o,P5: produc3043004577601930755nt_int] :
( ! [A3: fun_Pr7763680713840462443e_term,B3: product_prod_int_int] : aa_Pro2601676399929874711_int_o(P,produc7320969046608631221nt_int(A3,B3))
=> aa_Pro2601676399929874711_int_o(P,P5) ) ).
% prod_cases
tff(fact_584_prod__cases,axiom,
! [P: fun_Pr1196100989263358594_int_o,P5: produc4788131091073775919nt_int] :
( ! [A3: fun_in5730379987311621695e_term,B3: product_prod_int_int] : aa_Pro5451197003417746155_int_o(P,produc4729913460834754785nt_int(A3,B3))
=> aa_Pro5451197003417746155_int_o(P,P5) ) ).
% prod_cases
tff(fact_585_surj__pair,axiom,
! [P5: produc2453742332425399446nteger] :
? [X2: fun_Pr5098669352780829690e_term,Y2: produc8923325533196201883nteger] : P5 = produc565448451787809096nteger(X2,Y2) ).
% surj_pair
tff(fact_586_surj__pair,axiom,
! [P5: produc8799533983703829792et_nat] :
? [X2: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat] : P5 = produc6017135503889078682et_nat(X2,Y2) ).
% surj_pair
tff(fact_587_surj__pair,axiom,
! [P5: produc3112576180687729086et_nat] :
? [X2: fun_Pr6460944699146269935_nat_o,Y2: produc8799533983703829792et_nat] : P5 = produc7291855603632349880et_nat(X2,Y2) ).
% surj_pair
tff(fact_588_surj__pair,axiom,
! [P5: produc3043004577601930755nt_int] :
? [X2: fun_Pr7763680713840462443e_term,Y2: product_prod_int_int] : P5 = produc7320969046608631221nt_int(X2,Y2) ).
% surj_pair
tff(fact_589_surj__pair,axiom,
! [P5: produc4788131091073775919nt_int] :
? [X2: fun_in5730379987311621695e_term,Y2: product_prod_int_int] : P5 = produc4729913460834754785nt_int(X2,Y2) ).
% surj_pair
tff(fact_590_old_Oprod_Oexhaust,axiom,
! [Y: produc2453742332425399446nteger] :
~ ! [A3: fun_Pr5098669352780829690e_term,B3: produc8923325533196201883nteger] : Y != produc565448451787809096nteger(A3,B3) ).
% old.prod.exhaust
tff(fact_591_old_Oprod_Oexhaust,axiom,
! [Y: produc8799533983703829792et_nat] :
~ ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] : Y != produc6017135503889078682et_nat(A3,B3) ).
% old.prod.exhaust
tff(fact_592_old_Oprod_Oexhaust,axiom,
! [Y: produc3112576180687729086et_nat] :
~ ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] : Y != produc7291855603632349880et_nat(A3,B3) ).
% old.prod.exhaust
tff(fact_593_old_Oprod_Oexhaust,axiom,
! [Y: produc3043004577601930755nt_int] :
~ ! [A3: fun_Pr7763680713840462443e_term,B3: product_prod_int_int] : Y != produc7320969046608631221nt_int(A3,B3) ).
% old.prod.exhaust
tff(fact_594_old_Oprod_Oexhaust,axiom,
! [Y: produc4788131091073775919nt_int] :
~ ! [A3: fun_in5730379987311621695e_term,B3: product_prod_int_int] : Y != produc4729913460834754785nt_int(A3,B3) ).
% old.prod.exhaust
tff(fact_595_Un__empty__left,axiom,
! [Ba: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),Ba) = Ba ).
% Un_empty_left
tff(fact_596_Un__empty__left,axiom,
! [Ba: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat),Ba) = Ba ).
% Un_empty_left
tff(fact_597_Un__empty__left,axiom,
! [Ba: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,bot_bot_set_o),Ba) = Ba ).
% Un_empty_left
tff(fact_598_Un__empty__left,axiom,
! [Ba: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),Ba) = Ba ).
% Un_empty_left
tff(fact_599_Un__empty__left,axiom,
! [Ba: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),Ba) = Ba ).
% Un_empty_left
tff(fact_600_Un__empty__right,axiom,
! [Aa: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Aa),bot_bo228742789529271731at_nat) = Aa ).
% Un_empty_right
tff(fact_601_Un__empty__right,axiom,
! [Aa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Aa),bot_bo2099793752762293965at_nat) = Aa ).
% Un_empty_right
tff(fact_602_Un__empty__right,axiom,
! [Aa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Aa),bot_bot_set_o) = Aa ).
% Un_empty_right
tff(fact_603_Un__empty__right,axiom,
! [Aa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Aa),bot_bot_set_nat) = Aa ).
% Un_empty_right
tff(fact_604_Un__empty__right,axiom,
! [Aa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Aa),bot_bot_set_int) = Aa ).
% Un_empty_right
tff(fact_605_empty__not__UNIV,axiom,
bot_bo2099793752762293965at_nat != top_to4669805908274784177at_nat ).
% empty_not_UNIV
tff(fact_606_empty__not__UNIV,axiom,
bot_bot_set_int != top_top_set_int ).
% empty_not_UNIV
tff(fact_607_empty__not__UNIV,axiom,
bot_bot_set_o != top_top_set_o ).
% empty_not_UNIV
tff(fact_608_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
tff(fact_609_empty__not__UNIV,axiom,
bot_bo3957492148770167129t_unit != top_to1996260823553986621t_unit ).
% empty_not_UNIV
tff(fact_610_bot__set__def,axiom,
bot_bo8632255151191677115_nat_o = collec6407654713642994138_nat_o(bot_bo8793957029302321216at_o_o) ).
% bot_set_def
tff(fact_611_bot__set__def,axiom,
bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(bot_bo5074786513356115630_nat_o) ).
% bot_set_def
tff(fact_612_bot__set__def,axiom,
bot_bot_set_o = collect_o(bot_bot_fun_o_o) ).
% bot_set_def
tff(fact_613_bot__set__def,axiom,
bot_bot_set_nat = collect_nat(bot_bot_fun_nat_o) ).
% bot_set_def
tff(fact_614_bot__set__def,axiom,
bot_bot_set_int = collect_int(bot_bot_fun_int_o) ).
% bot_set_def
tff(fact_615_ex__in__conv,axiom,
! [Aa: set_fu3555266933746557007_nat_o] :
( ? [X3: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),Aa)
<=> ( Aa != bot_bo8632255151191677115_nat_o ) ) ).
% ex_in_conv
tff(fact_616_ex__in__conv,axiom,
! [Aa: set_Pr1261947904930325089at_nat] :
( ? [X3: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),Aa)
<=> ( Aa != bot_bo2099793752762293965at_nat ) ) ).
% ex_in_conv
tff(fact_617_ex__in__conv,axiom,
! [Aa: set_o] :
( ? [X3: $o] : aa_set_o_o(member_o((X3)),Aa)
<=> ( Aa != bot_bot_set_o ) ) ).
% ex_in_conv
tff(fact_618_ex__in__conv,axiom,
! [Aa: set_nat] :
( ? [X3: nat] : aa_set_nat_o(member_nat(X3),Aa)
<=> ( Aa != bot_bot_set_nat ) ) ).
% ex_in_conv
tff(fact_619_ex__in__conv,axiom,
! [Aa: set_int] :
( ? [X3: int] : aa_set_int_o(member_int(X3),Aa)
<=> ( Aa != bot_bot_set_int ) ) ).
% ex_in_conv
tff(fact_620_equals0I,axiom,
! [Aa: set_fu3555266933746557007_nat_o] :
( ! [Y2: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),Aa)
=> ( Aa = bot_bo8632255151191677115_nat_o ) ) ).
% equals0I
tff(fact_621_equals0I,axiom,
! [Aa: set_Pr1261947904930325089at_nat] :
( ! [Y2: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y2),Aa)
=> ( Aa = bot_bo2099793752762293965at_nat ) ) ).
% equals0I
tff(fact_622_equals0I,axiom,
! [Aa: set_o] :
( ! [Y2: $o] : ~ aa_set_o_o(member_o((Y2)),Aa)
=> ( Aa = bot_bot_set_o ) ) ).
% equals0I
tff(fact_623_equals0I,axiom,
! [Aa: set_nat] :
( ! [Y2: nat] : ~ aa_set_nat_o(member_nat(Y2),Aa)
=> ( Aa = bot_bot_set_nat ) ) ).
% equals0I
tff(fact_624_equals0I,axiom,
! [Aa: set_int] :
( ! [Y2: int] : ~ aa_set_int_o(member_int(Y2),Aa)
=> ( Aa = bot_bot_set_int ) ) ).
% equals0I
tff(fact_625_equals0D,axiom,
! [Aa: set_fu3555266933746557007_nat_o,A: fun_Pr6460944699146269935_nat_o] :
( ( Aa = bot_bo8632255151191677115_nat_o )
=> ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),Aa) ) ).
% equals0D
tff(fact_626_equals0D,axiom,
! [Aa: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ( Aa = bot_bo2099793752762293965at_nat )
=> ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),Aa) ) ).
% equals0D
tff(fact_627_equals0D,axiom,
! [Aa: set_o,A: $o] :
( ( Aa = bot_bot_set_o )
=> ~ aa_set_o_o(member_o((A)),Aa) ) ).
% equals0D
tff(fact_628_equals0D,axiom,
! [Aa: set_nat,A: nat] :
( ( Aa = bot_bot_set_nat )
=> ~ aa_set_nat_o(member_nat(A),Aa) ) ).
% equals0D
tff(fact_629_equals0D,axiom,
! [Aa: set_int,A: int] :
( ( Aa = bot_bot_set_int )
=> ~ aa_set_int_o(member_int(A),Aa) ) ).
% equals0D
tff(fact_630_emptyE,axiom,
! [A: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),bot_bo8632255151191677115_nat_o) ).
% emptyE
tff(fact_631_emptyE,axiom,
! [A: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),bot_bo2099793752762293965at_nat) ).
% emptyE
tff(fact_632_emptyE,axiom,
! [A: $o] : ~ aa_set_o_o(member_o((A)),bot_bot_set_o) ).
% emptyE
tff(fact_633_emptyE,axiom,
! [A: nat] : ~ aa_set_nat_o(member_nat(A),bot_bot_set_nat) ).
% emptyE
tff(fact_634_emptyE,axiom,
! [A: int] : ~ aa_set_int_o(member_int(A),bot_bot_set_int) ).
% emptyE
tff(fact_635_wand__assnI,axiom,
! [H2: heap_e7401611519738050253t_unit,As: set_nat,Q: assn,R: assn] :
( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As))
=> ( ! [H4: heap_e7401611519738050253t_unit,As7: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As7) = bot_bot_set_nat )
=> ( aa_hea7994211679605476167unit_o(relH(As,H2),H4)
=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As))
=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),produc7507926704131184380et_nat(H4,As7))
=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,R),produc7507926704131184380et_nat(H4,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As7))) ) ) ) )
=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Q),R)),produc7507926704131184380et_nat(H2,As)) ) ) ).
% wand_assnI
tff(fact_636_times__assn__raw_Osimps,axiom,
! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(P),Q),produc7507926704131184380et_nat(H2,As))
<=> ? [As1: set_nat,As22: set_nat] :
( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22) )
& ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
& aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H2,As1))
& aa_Pro653082453696054616_nat_o(Q,produc7507926704131184380et_nat(H2,As22)) ) ) ).
% times_assn_raw.simps
tff(fact_637_times__assn__raw_Oelims_I1_J,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
<=> (Y) )
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xb = produc7507926704131184380et_nat(H3,As3) )
=> ( (Y)
<=> ~ ? [As1: set_nat,As22: set_nat] :
( ( As3 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22) )
& ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
& aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H3,As1))
& aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H3,As22)) ) ) ) ) ).
% times_assn_raw.elims(1)
tff(fact_638_times__assn__raw_Oelims_I2_J,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xb = produc7507926704131184380et_nat(H3,As3) )
=> ~ ? [As12: set_nat,As23: set_nat] :
( ( As3 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As12),As23) )
& ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As12),As23) = bot_bot_set_nat )
& aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H3,As12))
& aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H3,As23)) ) ) ) ).
% times_assn_raw.elims(2)
tff(fact_639_times__assn__raw_Oelims_I3_J,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xb = produc7507926704131184380et_nat(H3,As3) )
=> ? [As13: set_nat,As24: set_nat] :
( ( As3 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As13),As24) )
& ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As13),As24) = bot_bot_set_nat )
& aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H3,As13))
& aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H3,As24)) ) ) ) ).
% times_assn_raw.elims(3)
tff(fact_640_properI,axiom,
! [P: fun_Pr6460944699146269935_nat_o] :
( ! [As3: set_nat,H3: heap_e7401611519738050253t_unit] :
( aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H3,As3))
=> aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As3)) )
=> ( ! [As3: set_nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
( aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H3,As3))
=> ( aa_hea7994211679605476167unit_o(relH(As3,H3),H4)
=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As3))
=> aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H4,As3)) ) ) )
=> aa_fun1675921005893230629at_o_o(proper,P) ) ) ).
% properI
tff(fact_641_properD2,axiom,
! [P: fun_Pr6460944699146269935_nat_o,H2: heap_e7401611519738050253t_unit,As: set_nat,H5: heap_e7401611519738050253t_unit] :
( aa_fun1675921005893230629at_o_o(proper,P)
=> ( aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H2,As))
=> ( aa_hea7994211679605476167unit_o(relH(As,H2),H5)
=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H5,As))
=> aa_Pro653082453696054616_nat_o(P,produc7507926704131184380et_nat(H5,As)) ) ) ) ) ).
% properD2
tff(fact_642_disjointI,axiom,
! [A: set_fu3555266933746557007_nat_o,B: set_fu3555266933746557007_nat_o] :
( ! [X2: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A)
=> ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),B) )
=> ( inf_in3376754634429691197_nat_o(A,B) = bot_bo8632255151191677115_nat_o ) ) ).
% disjointI
tff(fact_643_disjointI,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ! [X2: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),A)
=> ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),B) )
=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat ) ) ).
% disjointI
tff(fact_644_disjointI,axiom,
! [A: set_o,B: set_o] :
( ! [X2: $o] :
( aa_set_o_o(member_o((X2)),A)
=> ~ aa_set_o_o(member_o((X2)),B) )
=> ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A),B) = bot_bot_set_o ) ) ).
% disjointI
tff(fact_645_disjointI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( aa_set_nat_o(member_nat(X2),A)
=> ~ aa_set_nat_o(member_nat(X2),B) )
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat ) ) ).
% disjointI
tff(fact_646_disjointI,axiom,
! [A: set_int,B: set_int] :
( ! [X2: int] :
( aa_set_int_o(member_int(X2),A)
=> ~ aa_set_int_o(member_int(X2),B) )
=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = bot_bot_set_int ) ) ).
% disjointI
tff(fact_647_wand__raw_Opelims_I3_J,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
=> ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xb = produc7507926704131184380et_nat(H3,As3) )
=> ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H3,As3))))
=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As3))
& ! [H4: heap_e7401611519738050253t_unit,As7: set_nat] :
( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As3),As7) = bot_bot_set_nat )
& aa_hea7994211679605476167unit_o(relH(As3,H3),H4)
& aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As3))
& aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H4,As7)) )
=> aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H4,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As3),As7))) ) ) ) ) ) ) ).
% wand_raw.pelims(3)
tff(fact_648_wand__raw_Opelims_I2_J,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
=> ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xb = produc7507926704131184380et_nat(H3,As3) )
=> ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H3,As3))))
=> ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As3))
& ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As3),As6) = bot_bot_set_nat )
& aa_hea7994211679605476167unit_o(relH(As3,H3),H7)
& aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H7,As3))
& aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H7,As6)) )
=> aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H7,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As3),As6))) ) ) ) ) ) ) ).
% wand_raw.pelims(2)
tff(fact_649_wand__raw_Opelims_I1_J,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
<=> (Y) )
=> ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xb = produc7507926704131184380et_nat(H3,As3) )
=> ( ( (Y)
<=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As3))
& ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As3),As5) = bot_bot_set_nat )
& aa_hea7994211679605476167unit_o(relH(As3,H3),H6)
& aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H6,As3))
& aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H6,As5)) )
=> aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As3),As5))) ) ) )
=> ~ aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H3,As3)))) ) ) ) ) ).
% wand_raw.pelims(1)
tff(fact_650_Set_Ois__empty__def,axiom,
! [Aa: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(is_emp1662574758705540307at_nat,Aa)
<=> ( Aa = bot_bo2099793752762293965at_nat ) ) ).
% Set.is_empty_def
tff(fact_651_Set_Ois__empty__def,axiom,
! [Aa: set_o] :
( aa_set_o_o(is_empty_o,Aa)
<=> ( Aa = bot_bot_set_o ) ) ).
% Set.is_empty_def
tff(fact_652_Set_Ois__empty__def,axiom,
! [Aa: set_nat] :
( aa_set_nat_o(is_empty_nat,Aa)
<=> ( Aa = bot_bot_set_nat ) ) ).
% Set.is_empty_def
tff(fact_653_Set_Ois__empty__def,axiom,
! [Aa: set_int] :
( aa_set_int_o(is_empty_int,Aa)
<=> ( Aa = bot_bot_set_int ) ) ).
% Set.is_empty_def
tff(fact_654_times__assn__raw_Opelims_I3_J,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
=> ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xb = produc7507926704131184380et_nat(H3,As3) )
=> ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H3,As3))))
=> ? [As13: set_nat,As24: set_nat] :
( ( As3 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As13),As24) )
& ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As13),As24) = bot_bot_set_nat )
& aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H3,As13))
& aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H3,As24)) ) ) ) ) ) ).
% times_assn_raw.pelims(3)
tff(fact_655_times__assn__raw_Opelims_I2_J,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
=> ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xb = produc7507926704131184380et_nat(H3,As3) )
=> ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H3,As3))))
=> ~ ? [As12: set_nat,As23: set_nat] :
( ( As3 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As12),As23) )
& ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As12),As23) = bot_bot_set_nat )
& aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H3,As12))
& aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H3,As23)) ) ) ) ) ) ).
% times_assn_raw.pelims(2)
tff(fact_656_times__assn__raw_Opelims_I1_J,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
<=> (Y) )
=> ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xb = produc7507926704131184380et_nat(H3,As3) )
=> ( ( (Y)
<=> ? [As1: set_nat,As22: set_nat] :
( ( As3 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22) )
& ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
& aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H3,As1))
& aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H3,As22)) ) )
=> ~ aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H3,As3)))) ) ) ) ) ).
% times_assn_raw.pelims(1)
tff(fact_657_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y) = bot_bo228742789529271731at_nat )
=> ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = top_to6833984726390702231at_nat )
=> ( aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,X) = Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_658_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = bot_bo2099793752762293965at_nat )
=> ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y) = top_to4669805908274784177at_nat )
=> ( aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X) = Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_659_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_int,Y: set_int] :
( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = bot_bot_set_int )
=> ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = top_top_set_int )
=> ( aa_set_int_set_int(uminus1532241313380277803et_int,X) = Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_660_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_o,Y: set_o] :
( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),Y) = bot_bot_set_o )
=> ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),Y) = top_top_set_o )
=> ( aa_set_o_set_o(uminus_uminus_set_o,X) = Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_661_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_nat,Y: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = bot_bot_set_nat )
=> ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = top_top_set_nat )
=> ( aa_set_nat_set_nat(uminus5710092332889474511et_nat,X) = Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_662_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,X),Y) = bot_bo3957492148770167129t_unit )
=> ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,X),Y) = top_to1996260823553986621t_unit )
=> ( aa_set8324764978855220220t_unit(uminus5944136376168626660t_unit,X) = Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_663_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: assn,Y: assn] :
( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = bot_bot_assn )
=> ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = top_top_assn )
=> ( aa_assn_assn(uminus_uminus_assn,X) = Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_664_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X: assn,Y: assn] :
( ( aa_assn_assn(uminus_uminus_assn,X) = aa_assn_assn(uminus_uminus_assn,Y) )
<=> ( X = Y ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
tff(fact_665_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X: assn] : aa_assn_assn(uminus_uminus_assn,aa_assn_assn(uminus_uminus_assn,X)) = X ).
% boolean_algebra_class.boolean_algebra.double_compl
tff(fact_666_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) )
<=> ( A = B ) ) ).
% neg_equal_iff_equal
tff(fact_667_neg__equal__iff__equal,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) )
<=> ( A = B ) ) ).
% neg_equal_iff_equal
tff(fact_668_neg__equal__iff__equal,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
<=> ( A = B ) ) ).
% neg_equal_iff_equal
tff(fact_669_add_Oinverse__inverse,axiom,
! [A: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,A)) = A ).
% add.inverse_inverse
tff(fact_670_add_Oinverse__inverse,axiom,
! [A: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,A)) = A ).
% add.inverse_inverse
tff(fact_671_add_Oinverse__inverse,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).
% add.inverse_inverse
tff(fact_672_merge__pure__and,axiom,
! [A: $o,B: $o] :
aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_o_assn(pure_assn,(A))),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
( (A)
& (B) )) ).
% merge_pure_and
tff(fact_673_Compl__disjoint2,axiom,
! [Aa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Aa)),Aa) = bot_bo2099793752762293965at_nat ).
% Compl_disjoint2
tff(fact_674_Compl__disjoint2,axiom,
! [Aa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(uminus_uminus_set_o,Aa)),Aa) = bot_bot_set_o ).
% Compl_disjoint2
tff(fact_675_Compl__disjoint2,axiom,
! [Aa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,Aa)),Aa) = bot_bot_set_nat ).
% Compl_disjoint2
tff(fact_676_Compl__disjoint2,axiom,
! [Aa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,Aa)),Aa) = bot_bot_set_int ).
% Compl_disjoint2
tff(fact_677_Compl__disjoint,axiom,
! [Aa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Aa),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Aa)) = bot_bo2099793752762293965at_nat ).
% Compl_disjoint
tff(fact_678_Compl__disjoint,axiom,
! [Aa: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Aa),aa_set_o_set_o(uminus_uminus_set_o,Aa)) = bot_bot_set_o ).
% Compl_disjoint
tff(fact_679_Compl__disjoint,axiom,
! [Aa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Aa)) = bot_bot_set_nat ).
% Compl_disjoint
tff(fact_680_Compl__disjoint,axiom,
! [Aa: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Aa),aa_set_int_set_int(uminus1532241313380277803et_int,Aa)) = bot_bot_set_int ).
% Compl_disjoint
tff(fact_681_mod__not__dist,axiom,
! [P: assn,H2: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(uminus_uminus_assn,P)),H2)
<=> ( aa_Pro653082453696054616_nat_o(in_range,H2)
& ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),H2) ) ) ).
% mod_not_dist
tff(fact_682_boolean__algebra_Ocompl__zero,axiom,
aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,bot_bo2099793752762293965at_nat) = top_to4669805908274784177at_nat ).
% boolean_algebra.compl_zero
tff(fact_683_boolean__algebra_Ocompl__zero,axiom,
aa_set_int_set_int(uminus1532241313380277803et_int,bot_bot_set_int) = top_top_set_int ).
% boolean_algebra.compl_zero
tff(fact_684_boolean__algebra_Ocompl__zero,axiom,
aa_set_o_set_o(uminus_uminus_set_o,bot_bot_set_o) = top_top_set_o ).
% boolean_algebra.compl_zero
tff(fact_685_boolean__algebra_Ocompl__zero,axiom,
aa_set_nat_set_nat(uminus5710092332889474511et_nat,bot_bot_set_nat) = top_top_set_nat ).
% boolean_algebra.compl_zero
tff(fact_686_boolean__algebra_Ocompl__zero,axiom,
aa_set8324764978855220220t_unit(uminus5944136376168626660t_unit,bot_bo3957492148770167129t_unit) = top_to1996260823553986621t_unit ).
% boolean_algebra.compl_zero
tff(fact_687_boolean__algebra_Ocompl__zero,axiom,
aa_assn_assn(uminus_uminus_assn,bot_bot_assn) = top_top_assn ).
% boolean_algebra.compl_zero
tff(fact_688_boolean__algebra_Ocompl__one,axiom,
aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,top_to4669805908274784177at_nat) = bot_bo2099793752762293965at_nat ).
% boolean_algebra.compl_one
tff(fact_689_boolean__algebra_Ocompl__one,axiom,
aa_set_int_set_int(uminus1532241313380277803et_int,top_top_set_int) = bot_bot_set_int ).
% boolean_algebra.compl_one
tff(fact_690_boolean__algebra_Ocompl__one,axiom,
aa_set_o_set_o(uminus_uminus_set_o,top_top_set_o) = bot_bot_set_o ).
% boolean_algebra.compl_one
tff(fact_691_boolean__algebra_Ocompl__one,axiom,
aa_set_nat_set_nat(uminus5710092332889474511et_nat,top_top_set_nat) = bot_bot_set_nat ).
% boolean_algebra.compl_one
tff(fact_692_boolean__algebra_Ocompl__one,axiom,
aa_set8324764978855220220t_unit(uminus5944136376168626660t_unit,top_to1996260823553986621t_unit) = bot_bo3957492148770167129t_unit ).
% boolean_algebra.compl_one
tff(fact_693_boolean__algebra_Ocompl__one,axiom,
aa_assn_assn(uminus_uminus_assn,top_top_assn) = bot_bot_assn ).
% boolean_algebra.compl_one
tff(fact_694_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)) = bot_bo2099793752762293965at_nat ).
% boolean_algebra.conj_cancel_right
tff(fact_695_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),aa_set_o_set_o(uminus_uminus_set_o,X)) = bot_bot_set_o ).
% boolean_algebra.conj_cancel_right
tff(fact_696_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)) = bot_bot_set_nat ).
% boolean_algebra.conj_cancel_right
tff(fact_697_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),aa_set_int_set_int(uminus1532241313380277803et_int,X)) = bot_bot_set_int ).
% boolean_algebra.conj_cancel_right
tff(fact_698_boolean__algebra_Oconj__cancel__right,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(uminus_uminus_assn,X)) = bot_bot_assn ).
% boolean_algebra.conj_cancel_right
tff(fact_699_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),X) = bot_bo2099793752762293965at_nat ).
% boolean_algebra.conj_cancel_left
tff(fact_700_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),X) = bot_bot_set_o ).
% boolean_algebra.conj_cancel_left
tff(fact_701_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),X) = bot_bot_set_nat ).
% boolean_algebra.conj_cancel_left
tff(fact_702_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),X) = bot_bot_set_int ).
% boolean_algebra.conj_cancel_left
tff(fact_703_boolean__algebra_Oconj__cancel__left,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),X) = bot_bot_assn ).
% boolean_algebra.conj_cancel_left
tff(fact_704_inf__compl__bot__right,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X))) = bot_bo2099793752762293965at_nat ).
% inf_compl_bot_right
tff(fact_705_inf__compl__bot__right,axiom,
! [X: set_o,Y: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Y),aa_set_o_set_o(uminus_uminus_set_o,X))) = bot_bot_set_o ).
% inf_compl_bot_right
tff(fact_706_inf__compl__bot__right,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),aa_set_nat_set_nat(uminus5710092332889474511et_nat,X))) = bot_bot_set_nat ).
% inf_compl_bot_right
tff(fact_707_inf__compl__bot__right,axiom,
! [X: set_int,Y: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Y),aa_set_int_set_int(uminus1532241313380277803et_int,X))) = bot_bot_set_int ).
% inf_compl_bot_right
tff(fact_708_inf__compl__bot__right,axiom,
! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(uminus_uminus_assn,X))) = bot_bot_assn ).
% inf_compl_bot_right
tff(fact_709_inf__compl__bot__left2,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),Y)) = bot_bo2099793752762293965at_nat ).
% inf_compl_bot_left2
tff(fact_710_inf__compl__bot__left2,axiom,
! [X: set_o,Y: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),Y)) = bot_bot_set_o ).
% inf_compl_bot_left2
tff(fact_711_inf__compl__bot__left2,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),Y)) = bot_bot_set_nat ).
% inf_compl_bot_left2
tff(fact_712_inf__compl__bot__left2,axiom,
! [X: set_int,Y: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),Y)) = bot_bot_set_int ).
% inf_compl_bot_left2
tff(fact_713_inf__compl__bot__left2,axiom,
! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),Y)) = bot_bot_assn ).
% inf_compl_bot_left2
tff(fact_714_inf__compl__bot__left1,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = bot_bo2099793752762293965at_nat ).
% inf_compl_bot_left1
tff(fact_715_inf__compl__bot__left1,axiom,
! [X: set_o,Y: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),Y)) = bot_bot_set_o ).
% inf_compl_bot_left1
tff(fact_716_inf__compl__bot__left1,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = bot_bot_set_nat ).
% inf_compl_bot_left1
tff(fact_717_inf__compl__bot__left1,axiom,
! [X: set_int,Y: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)) = bot_bot_set_int ).
% inf_compl_bot_left1
tff(fact_718_inf__compl__bot__left1,axiom,
! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = bot_bot_assn ).
% inf_compl_bot_left1
tff(fact_719_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,X)) = top_to6833984726390702231at_nat ).
% boolean_algebra.disj_cancel_right
tff(fact_720_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),aa_set_o_set_o(uminus_uminus_set_o,X)) = top_top_set_o ).
% boolean_algebra.disj_cancel_right
tff(fact_721_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)) = top_top_set_nat ).
% boolean_algebra.disj_cancel_right
tff(fact_722_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,X),aa_set8324764978855220220t_unit(uminus5944136376168626660t_unit,X)) = top_to1996260823553986621t_unit ).
% boolean_algebra.disj_cancel_right
tff(fact_723_boolean__algebra_Odisj__cancel__right,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(uminus_uminus_assn,X)) = top_top_assn ).
% boolean_algebra.disj_cancel_right
tff(fact_724_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,X)),X) = top_to6833984726390702231at_nat ).
% boolean_algebra.disj_cancel_left
tff(fact_725_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),X) = top_top_set_o ).
% boolean_algebra.disj_cancel_left
tff(fact_726_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),X) = top_top_set_nat ).
% boolean_algebra.disj_cancel_left
tff(fact_727_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,aa_set8324764978855220220t_unit(uminus5944136376168626660t_unit,X)),X) = top_to1996260823553986621t_unit ).
% boolean_algebra.disj_cancel_left
tff(fact_728_boolean__algebra_Odisj__cancel__left,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(uminus_uminus_assn,X)),X) = top_top_assn ).
% boolean_algebra.disj_cancel_left
tff(fact_729_sup__compl__top__left2,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,X)),Y)) = top_to6833984726390702231at_nat ).
% sup_compl_top_left2
tff(fact_730_sup__compl__top__left2,axiom,
! [X: set_o,Y: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),Y)) = top_top_set_o ).
% sup_compl_top_left2
tff(fact_731_sup__compl__top__left2,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),Y)) = top_top_set_nat ).
% sup_compl_top_left2
tff(fact_732_sup__compl__top__left2,axiom,
! [X: set_Product_unit,Y: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,X),aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,aa_set8324764978855220220t_unit(uminus5944136376168626660t_unit,X)),Y)) = top_to1996260823553986621t_unit ).
% sup_compl_top_left2
tff(fact_733_sup__compl__top__left2,axiom,
! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(uminus_uminus_assn,X)),Y)) = top_top_assn ).
% sup_compl_top_left2
tff(fact_734_sup__compl__top__left1,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,X)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) = top_to6833984726390702231at_nat ).
% sup_compl_top_left1
tff(fact_735_sup__compl__top__left1,axiom,
! [X: set_o,Y: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),Y)) = top_top_set_o ).
% sup_compl_top_left1
tff(fact_736_sup__compl__top__left1,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = top_top_set_nat ).
% sup_compl_top_left1
tff(fact_737_sup__compl__top__left1,axiom,
! [X: set_Product_unit,Y: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,aa_set8324764978855220220t_unit(uminus5944136376168626660t_unit,X)),aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,X),Y)) = top_to1996260823553986621t_unit ).
% sup_compl_top_left1
tff(fact_738_sup__compl__top__left1,axiom,
! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = top_top_assn ).
% sup_compl_top_left1
tff(fact_739_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)) ).
% boolean_algebra.de_Morgan_disj
tff(fact_740_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,X)),aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,Y)) ).
% boolean_algebra.de_Morgan_disj
tff(fact_741_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: assn,Y: assn] : aa_assn_assn(uminus_uminus_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(uminus_uminus_assn,Y)) ).
% boolean_algebra.de_Morgan_disj
tff(fact_742_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)) ).
% boolean_algebra.de_Morgan_conj
tff(fact_743_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,X)),aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,Y)) ).
% boolean_algebra.de_Morgan_conj
tff(fact_744_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: assn,Y: assn] : aa_assn_assn(uminus_uminus_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(uminus_uminus_assn,Y)) ).
% boolean_algebra.de_Morgan_conj
tff(fact_745_mod__h__bot__iff_I6_J,axiom,
! [P: assn,Q: assn,H2: heap_e7401611519738050253t_unit] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P),Q)),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
<=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
& aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ) ) ).
% mod_h_bot_iff(6)
tff(fact_746_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( aa_int_int(uminus_uminus_int,A) = B )
<=> ( aa_int_int(uminus_uminus_int,B) = A ) ) ).
% minus_equation_iff
tff(fact_747_minus__equation__iff,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(uminus_uminus_rat,A) = B )
<=> ( aa_rat_rat(uminus_uminus_rat,B) = A ) ) ).
% minus_equation_iff
tff(fact_748_minus__equation__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
<=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) = A ) ) ).
% minus_equation_iff
tff(fact_749_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A = aa_int_int(uminus_uminus_int,B) )
<=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).
% equation_minus_iff
tff(fact_750_equation__minus__iff,axiom,
! [A: rat,B: rat] :
( ( A = aa_rat_rat(uminus_uminus_rat,B) )
<=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).
% equation_minus_iff
tff(fact_751_equation__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
<=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).
% equation_minus_iff
tff(fact_752_Compl__UNIV__eq,axiom,
aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,top_to4669805908274784177at_nat) = bot_bo2099793752762293965at_nat ).
% Compl_UNIV_eq
tff(fact_753_Compl__UNIV__eq,axiom,
aa_set_int_set_int(uminus1532241313380277803et_int,top_top_set_int) = bot_bot_set_int ).
% Compl_UNIV_eq
tff(fact_754_Compl__UNIV__eq,axiom,
aa_set_o_set_o(uminus_uminus_set_o,top_top_set_o) = bot_bot_set_o ).
% Compl_UNIV_eq
tff(fact_755_Compl__UNIV__eq,axiom,
aa_set_nat_set_nat(uminus5710092332889474511et_nat,top_top_set_nat) = bot_bot_set_nat ).
% Compl_UNIV_eq
tff(fact_756_Compl__UNIV__eq,axiom,
aa_set8324764978855220220t_unit(uminus5944136376168626660t_unit,top_to1996260823553986621t_unit) = bot_bo3957492148770167129t_unit ).
% Compl_UNIV_eq
tff(fact_757_Compl__empty__eq,axiom,
aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,bot_bo2099793752762293965at_nat) = top_to4669805908274784177at_nat ).
% Compl_empty_eq
tff(fact_758_Compl__empty__eq,axiom,
aa_set_int_set_int(uminus1532241313380277803et_int,bot_bot_set_int) = top_top_set_int ).
% Compl_empty_eq
tff(fact_759_Compl__empty__eq,axiom,
aa_set_o_set_o(uminus_uminus_set_o,bot_bot_set_o) = top_top_set_o ).
% Compl_empty_eq
tff(fact_760_Compl__empty__eq,axiom,
aa_set_nat_set_nat(uminus5710092332889474511et_nat,bot_bot_set_nat) = top_top_set_nat ).
% Compl_empty_eq
tff(fact_761_Compl__empty__eq,axiom,
aa_set8324764978855220220t_unit(uminus5944136376168626660t_unit,bot_bo3957492148770167129t_unit) = top_to1996260823553986621t_unit ).
% Compl_empty_eq
tff(fact_762_mod__and__dist,axiom,
! [P: assn,Q: assn,H2: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P),Q)),H2)
<=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),H2)
& aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),H2) ) ) ).
% mod_and_dist
tff(fact_763_ent__conjE2,axiom,
! [Ba: assn,Ca: assn,Aa: assn] :
( aa_assn_o(entails(Ba),Ca)
=> aa_assn_o(entails(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Aa),Ba)),Ca) ) ).
% ent_conjE2
tff(fact_764_ent__conjE1,axiom,
! [Aa: assn,Ca: assn,Ba: assn] :
( aa_assn_o(entails(Aa),Ca)
=> aa_assn_o(entails(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Aa),Ba)),Ca) ) ).
% ent_conjE1
tff(fact_765_ent__conjI,axiom,
! [Aa: assn,Ba: assn,Ca: assn] :
( aa_assn_o(entails(Aa),Ba)
=> ( aa_assn_o(entails(Aa),Ca)
=> aa_assn_o(entails(Aa),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Ba),Ca)) ) ) ).
% ent_conjI
tff(fact_766_inf__cancel__left2,axiom,
! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),A)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),B)) = bot_bo2099793752762293965at_nat ).
% inf_cancel_left2
tff(fact_767_inf__cancel__left2,axiom,
! [X: set_o,A: set_o,B: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),A)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),B)) = bot_bot_set_o ).
% inf_cancel_left2
tff(fact_768_inf__cancel__left2,axiom,
! [X: set_nat,A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),B)) = bot_bot_set_nat ).
% inf_cancel_left2
tff(fact_769_inf__cancel__left2,axiom,
! [X: set_int,A: set_int,B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),A)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),B)) = bot_bot_set_int ).
% inf_cancel_left2
tff(fact_770_inf__cancel__left2,axiom,
! [X: assn,A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),A)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),B)) = bot_bot_assn ).
% inf_cancel_left2
tff(fact_771_inf__cancel__left1,axiom,
! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),A)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),B)) = bot_bo2099793752762293965at_nat ).
% inf_cancel_left1
tff(fact_772_inf__cancel__left1,axiom,
! [X: set_o,A: set_o,B: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),A)),aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),B)) = bot_bot_set_o ).
% inf_cancel_left1
tff(fact_773_inf__cancel__left1,axiom,
! [X: set_nat,A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),B)) = bot_bot_set_nat ).
% inf_cancel_left1
tff(fact_774_inf__cancel__left1,axiom,
! [X: set_int,A: set_int,B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),A)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),B)) = bot_bot_set_int ).
% inf_cancel_left1
tff(fact_775_inf__cancel__left1,axiom,
! [X: assn,A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),A)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),B)) = bot_bot_assn ).
% inf_cancel_left1
tff(fact_776_sup__cancel__left2,axiom,
! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,X)),A)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),B)) = top_to6833984726390702231at_nat ).
% sup_cancel_left2
tff(fact_777_sup__cancel__left2,axiom,
! [X: set_o,A: set_o,B: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),A)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),B)) = top_top_set_o ).
% sup_cancel_left2
tff(fact_778_sup__cancel__left2,axiom,
! [X: set_nat,A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),B)) = top_top_set_nat ).
% sup_cancel_left2
tff(fact_779_sup__cancel__left2,axiom,
! [X: set_Product_unit,A: set_Product_unit,B: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,aa_set8324764978855220220t_unit(uminus5944136376168626660t_unit,X)),A)),aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,X),B)) = top_to1996260823553986621t_unit ).
% sup_cancel_left2
tff(fact_780_sup__cancel__left2,axiom,
! [X: assn,A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(uminus_uminus_assn,X)),A)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),B)) = top_top_assn ).
% sup_cancel_left2
tff(fact_781_sup__cancel__left1,axiom,
! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),A)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,X)),B)) = top_to6833984726390702231at_nat ).
% sup_cancel_left1
tff(fact_782_sup__cancel__left1,axiom,
! [X: set_o,A: set_o,B: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),A)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),B)) = top_top_set_o ).
% sup_cancel_left1
tff(fact_783_sup__cancel__left1,axiom,
! [X: set_nat,A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),B)) = top_top_set_nat ).
% sup_cancel_left1
tff(fact_784_sup__cancel__left1,axiom,
! [X: set_Product_unit,A: set_Product_unit,B: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,X),A)),aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,aa_set8324764978855220220t_unit(uminus5944136376168626660t_unit,X)),B)) = top_to1996260823553986621t_unit ).
% sup_cancel_left1
tff(fact_785_sup__cancel__left1,axiom,
! [X: assn,A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),A)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(uminus_uminus_assn,X)),B)) = top_top_assn ).
% sup_cancel_left1
tff(fact_786_pairself_Ocases,axiom,
! [X: produc4788131091073775919nt_int] :
~ ! [F2: fun_in5730379987311621695e_term,A3: int,B3: int] : X != produc4729913460834754785nt_int(F2,aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B3)) ).
% pairself.cases
tff(fact_787_bex2I,axiom,
! [A: nat,B: nat,S: set_Pr1261947904930325089at_nat,P: fun_nat_fun_nat_o] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),S)
=> ( ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),S)
=> aa_nat_o(aa_nat_fun_nat_o(P,A),B) )
=> ? [A3: nat,B3: nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3)),S)
& aa_nat_o(aa_nat_fun_nat_o(P,A3),B3) ) ) ) ).
% bex2I
tff(fact_788_bex2I,axiom,
! [A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,S: set_Pr3486312109254157132nteger,P: fun_fu7695273699059480439eger_o] :
( member8978266922497061677nteger(produc565448451787809096nteger(A,B),S)
=> ( ( member8978266922497061677nteger(produc565448451787809096nteger(A,B),S)
=> aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(P,A),B) )
=> ? [A3: fun_Pr5098669352780829690e_term,B3: produc8923325533196201883nteger] :
( member8978266922497061677nteger(produc565448451787809096nteger(A3,B3),S)
& aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(P,A3),B3) ) ) ) ).
% bex2I
tff(fact_789_bex2I,axiom,
! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,S: set_Pr1312406793078187350et_nat,P: fun_fu8419763957888848259_nat_o] :
( member4513763875858039863et_nat(produc6017135503889078682et_nat(A,B),S)
=> ( ( member4513763875858039863et_nat(produc6017135503889078682et_nat(A,B),S)
=> aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,A),B) )
=> ? [A3: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] :
( member4513763875858039863et_nat(produc6017135503889078682et_nat(A3,B3),S)
& aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,A3),B3) ) ) ) ).
% bex2I
tff(fact_790_bex2I,axiom,
! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,S: set_Pr4040394263813967220et_nat,P: fun_fu2611741329380808165_nat_o] :
( member9174161442226468181et_nat(produc7291855603632349880et_nat(A,B),S)
=> ( ( member9174161442226468181et_nat(produc7291855603632349880et_nat(A,B),S)
=> aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,A),B) )
=> ? [A3: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] :
( member9174161442226468181et_nat(produc7291855603632349880et_nat(A3,B3),S)
& aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,A3),B3) ) ) ) ).
% bex2I
tff(fact_791_bex2I,axiom,
! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,S: set_Pr6700895561065273913nt_int,P: fun_fu6865642474282033866_int_o] :
( member753276645541486746nt_int(produc7320969046608631221nt_int(A,B),S)
=> ( ( member753276645541486746nt_int(produc7320969046608631221nt_int(A,B),S)
=> aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(P,A),B) )
=> ? [A3: fun_Pr7763680713840462443e_term,B3: product_prod_int_int] :
( member753276645541486746nt_int(produc7320969046608631221nt_int(A3,B3),S)
& aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(P,A3),B3) ) ) ) ).
% bex2I
tff(fact_792_bex2I,axiom,
! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,S: set_Pr4166680992137330533nt_int,P: fun_fu2978539155105838238_int_o] :
( member5890090881338784710nt_int(produc4729913460834754785nt_int(A,B),S)
=> ( ( member5890090881338784710nt_int(produc4729913460834754785nt_int(A,B),S)
=> aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(P,A),B) )
=> ? [A3: fun_in5730379987311621695e_term,B3: product_prod_int_int] :
( member5890090881338784710nt_int(produc4729913460834754785nt_int(A3,B3),S)
& aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(P,A3),B3) ) ) ) ).
% bex2I
tff(fact_793_set__notEmptyE,axiom,
! [S: set_fu3555266933746557007_nat_o] :
( ( S != bot_bo8632255151191677115_nat_o )
=> ~ ! [X2: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),S) ) ).
% set_notEmptyE
tff(fact_794_set__notEmptyE,axiom,
! [S: set_Pr1261947904930325089at_nat] :
( ( S != bot_bo2099793752762293965at_nat )
=> ~ ! [X2: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),S) ) ).
% set_notEmptyE
tff(fact_795_set__notEmptyE,axiom,
! [S: set_o] :
( ( S != bot_bot_set_o )
=> ~ ! [X2: $o] : ~ aa_set_o_o(member_o((X2)),S) ) ).
% set_notEmptyE
tff(fact_796_set__notEmptyE,axiom,
! [S: set_nat] :
( ( S != bot_bot_set_nat )
=> ~ ! [X2: nat] : ~ aa_set_nat_o(member_nat(X2),S) ) ).
% set_notEmptyE
tff(fact_797_set__notEmptyE,axiom,
! [S: set_int] :
( ( S != bot_bot_set_int )
=> ~ ! [X2: int] : ~ aa_set_int_o(member_int(X2),S) ) ).
% set_notEmptyE
tff(fact_798_memb__imp__not__empty,axiom,
! [X: fun_Pr6460944699146269935_nat_o,S: set_fu3555266933746557007_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),S)
=> ( S != bot_bo8632255151191677115_nat_o ) ) ).
% memb_imp_not_empty
tff(fact_799_memb__imp__not__empty,axiom,
! [X: product_prod_nat_nat,S: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),S)
=> ( S != bot_bo2099793752762293965at_nat ) ) ).
% memb_imp_not_empty
tff(fact_800_memb__imp__not__empty,axiom,
! [X: $o,S: set_o] :
( aa_set_o_o(member_o((X)),S)
=> ( S != bot_bot_set_o ) ) ).
% memb_imp_not_empty
tff(fact_801_memb__imp__not__empty,axiom,
! [X: nat,S: set_nat] :
( aa_set_nat_o(member_nat(X),S)
=> ( S != bot_bot_set_nat ) ) ).
% memb_imp_not_empty
tff(fact_802_memb__imp__not__empty,axiom,
! [X: int,S: set_int] :
( aa_set_int_o(member_int(X),S)
=> ( S != bot_bot_set_int ) ) ).
% memb_imp_not_empty
tff(fact_803_mult__minus1,axiom,
! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,one_one_int)),Z) = aa_int_int(uminus_uminus_int,Z) ).
% mult_minus1
tff(fact_804_mult__minus1,axiom,
! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Z) = aa_rat_rat(uminus_uminus_rat,Z) ).
% mult_minus1
tff(fact_805_mult__minus1,axiom,
! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Z) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).
% mult_minus1
tff(fact_806_mult__minus1__right,axiom,
! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Z) ).
% mult_minus1_right
tff(fact_807_mult__minus1__right,axiom,
! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Z) ).
% mult_minus1_right
tff(fact_808_mult__minus1__right,axiom,
! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).
% mult_minus1_right
tff(fact_809_mult__minus__right,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).
% mult_minus_right
tff(fact_810_mult__minus__right,axiom,
! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).
% mult_minus_right
tff(fact_811_mult__minus__right,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).
% mult_minus_right
tff(fact_812_minus__mult__minus,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),B) ).
% minus_mult_minus
tff(fact_813_minus__mult__minus,axiom,
! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) ).
% minus_mult_minus
tff(fact_814_minus__mult__minus,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) ).
% minus_mult_minus
tff(fact_815_mult__minus__left,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).
% mult_minus_left
tff(fact_816_mult__minus__left,axiom,
! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).
% mult_minus_left
tff(fact_817_mult__minus__left,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).
% mult_minus_left
tff(fact_818_square__eq__1__iff,axiom,
! [X: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),X) = one_one_int )
<=> ( ( X = one_one_int )
| ( X = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).
% square_eq_1_iff
tff(fact_819_square__eq__1__iff,axiom,
! [X: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X) = one_one_rat )
<=> ( ( X = one_one_rat )
| ( X = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).
% square_eq_1_iff
tff(fact_820_square__eq__1__iff,axiom,
! [X: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X) = one_one_Code_integer )
<=> ( ( X = one_one_Code_integer )
| ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).
% square_eq_1_iff
tff(fact_821_bot__empty__eq,axiom,
! [X4: fun_Pr6460944699146269935_nat_o] :
( aa_fun1675921005893230629at_o_o(bot_bo8793957029302321216at_o_o,X4)
<=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),bot_bo8632255151191677115_nat_o) ) ).
% bot_empty_eq
tff(fact_822_bot__empty__eq,axiom,
! [X4: product_prod_nat_nat] :
( aa_Pro6150637604445964947_nat_o(bot_bo5074786513356115630_nat_o,X4)
<=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),bot_bo2099793752762293965at_nat) ) ).
% bot_empty_eq
tff(fact_823_bot__empty__eq,axiom,
! [X4: $o] :
( aa_o_o(bot_bot_fun_o_o,(X4))
<=> aa_set_o_o(member_o((X4)),bot_bot_set_o) ) ).
% bot_empty_eq
tff(fact_824_bot__empty__eq,axiom,
! [X4: nat] :
( aa_nat_o(bot_bot_fun_nat_o,X4)
<=> aa_set_nat_o(member_nat(X4),bot_bot_set_nat) ) ).
% bot_empty_eq
tff(fact_825_bot__empty__eq,axiom,
! [X4: int] :
( aa_int_o(bot_bot_fun_int_o,X4)
<=> aa_set_int_o(member_int(X4),bot_bot_set_int) ) ).
% bot_empty_eq
tff(fact_826_Collect__empty__eq__bot,axiom,
! [P: fun_fu5135969500543783852at_o_o] :
( ( collec6407654713642994138_nat_o(P) = bot_bo8632255151191677115_nat_o )
<=> ( P = bot_bo8793957029302321216at_o_o ) ) ).
% Collect_empty_eq_bot
tff(fact_827_Collect__empty__eq__bot,axiom,
! [P: fun_Pr4796943543331071514_nat_o] :
( ( collec3392354462482085612at_nat(P) = bot_bo2099793752762293965at_nat )
<=> ( P = bot_bo5074786513356115630_nat_o ) ) ).
% Collect_empty_eq_bot
tff(fact_828_Collect__empty__eq__bot,axiom,
! [P: fun_o_o] :
( ( collect_o(P) = bot_bot_set_o )
<=> ( P = bot_bot_fun_o_o ) ) ).
% Collect_empty_eq_bot
tff(fact_829_Collect__empty__eq__bot,axiom,
! [P: fun_nat_o] :
( ( collect_nat(P) = bot_bot_set_nat )
<=> ( P = bot_bot_fun_nat_o ) ) ).
% Collect_empty_eq_bot
tff(fact_830_Collect__empty__eq__bot,axiom,
! [P: fun_int_o] :
( ( collect_int(P) = bot_bot_set_int )
<=> ( P = bot_bot_fun_int_o ) ) ).
% Collect_empty_eq_bot
tff(fact_831_one__neq__neg__one,axiom,
one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).
% one_neq_neg_one
tff(fact_832_one__neq__neg__one,axiom,
one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% one_neq_neg_one
tff(fact_833_one__neq__neg__one,axiom,
one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% one_neq_neg_one
tff(fact_834_minus__mult__commute,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) ).
% minus_mult_commute
tff(fact_835_minus__mult__commute,axiom,
! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).
% minus_mult_commute
tff(fact_836_minus__mult__commute,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).
% minus_mult_commute
tff(fact_837_square__eq__iff,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),A) = aa_int_int(aa_int_fun_int_int(times_times_int,B),B) )
<=> ( ( A = B )
| ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ).
% square_eq_iff
tff(fact_838_square__eq__iff,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),B) )
<=> ( ( A = B )
| ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).
% square_eq_iff
tff(fact_839_square__eq__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),B) )
<=> ( ( A = B )
| ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ).
% square_eq_iff
tff(fact_840_dbl__inc__simps_I4_J,axiom,
aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).
% dbl_inc_simps(4)
tff(fact_841_dbl__inc__simps_I4_J,axiom,
neg_nu5219082963157363817nc_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% dbl_inc_simps(4)
tff(fact_842_dbl__inc__simps_I4_J,axiom,
aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% dbl_inc_simps(4)
tff(fact_843_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
aa_set5844364881195198285_nat_o(boolea9150928655215452349at_nat(inf_in7913087082777306421at_nat,sup_su5525570899277871387at_nat,uminus935396558254630718at_nat,bot_bo228742789529271731at_nat),top_to6833984726390702231at_nat) ).
% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_844_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
aa_set3702741264012733747_nat_o(boolea949959532490082519at_nat(inf_in2572325071724192079at_nat,sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat,bot_bo2099793752762293965at_nat),top_to4669805908274784177at_nat) ).
% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_845_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
aa_set_int_o(boolea5824373010784320748et_int(inf_inf_set_int,sup_sup_set_int,uminus1532241313380277803et_int,bot_bot_set_int),top_top_set_int) ).
% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_846_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
aa_set_o_o(boolea379910186789422830_set_o(inf_inf_set_o,sup_sup_set_o,uminus_uminus_set_o,bot_bot_set_o),top_top_set_o) ).
% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_847_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
aa_set_nat_o(boolea778851993438741648et_nat(inf_inf_set_nat,sup_sup_set_nat,uminus5710092332889474511et_nat,bot_bot_set_nat),top_top_set_nat) ).
% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_848_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea3191171876098928227t_unit(inf_in4660618365625256667t_unit,sup_su793286257634532545t_unit,uminus5944136376168626660t_unit,bot_bo3957492148770167129t_unit,top_to1996260823553986621t_unit) ).
% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_849_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
aa_assn_o(boolea5875918072483028530a_assn(inf_inf_assn,sup_sup_assn,uminus_uminus_assn,bot_bot_assn),top_top_assn) ).
% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_850_pairself_Opelims,axiom,
! [X: fun_in5730379987311621695e_term,Xa: product_prod_int_int,Y: produc6576344331059438605e_term] :
( ( pairse3534876335208825186e_term(X,Xa) = Y )
=> ( aa_Pro5451197003417746155_int_o(accp_P3253464921323968230nt_int(pairse6848479906795794847e_term),produc4729913460834754785nt_int(X,Xa))
=> ~ ! [A3: int,B3: int] :
( ( Xa = aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B3) )
=> ( ( Y = produc5950683997804057413e_term(aa_int3262262799336221824e_term(X,A3),aa_int3262262799336221824e_term(X,B3)) )
=> ~ aa_Pro5451197003417746155_int_o(accp_P3253464921323968230nt_int(pairse6848479906795794847e_term),produc4729913460834754785nt_int(X,aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B3))) ) ) ) ) ).
% pairself.pelims
tff(fact_851_mod__pure,axiom,
! [B: $o,H2: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_o_assn(pure_assn,(B))),H2)
<=> ( ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,H2) = bot_bot_set_nat )
& (B) ) ) ).
% mod_pure
tff(fact_852_type__definition__assn,axiom,
aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(rep_assn,abs_assn),collec6407654713642994138_nat_o(proper)) ).
% type_definition_assn
tff(fact_853_pure__assn__raw_Opelims_I1_J,axiom,
! [X: $o,Xa: produc3658429121746597890et_nat,Y: $o] :
( ( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa)
<=> (Y) )
=> ( accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),Xa))
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xa = produc7507926704131184380et_nat(H3,As3) )
=> ( ( (Y)
<=> ( ( As3 = bot_bot_set_nat )
& (X) ) )
=> ~ accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),produc7507926704131184380et_nat(H3,As3))) ) ) ) ) ).
% pure_assn_raw.pelims(1)
tff(fact_854_pure__assn__raw_Opelims_I2_J,axiom,
! [X: $o,Xa: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa)
=> ( accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),Xa))
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xa = produc7507926704131184380et_nat(H3,As3) )
=> ( accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),produc7507926704131184380et_nat(H3,As3)))
=> ~ ( ( As3 = bot_bot_set_nat )
& (X) ) ) ) ) ) ).
% pure_assn_raw.pelims(2)
tff(fact_855_pure__assn__raw_Opelims_I3_J,axiom,
! [X: $o,Xa: produc3658429121746597890et_nat] :
( ~ aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa)
=> ( accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),Xa))
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( Xa = produc7507926704131184380et_nat(H3,As3) )
=> ( accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),produc7507926704131184380et_nat(H3,As3)))
=> ( ( As3 = bot_bot_set_nat )
& (X) ) ) ) ) ) ).
% pure_assn_raw.pelims(3)
tff(fact_856_dbl__dec__simps_I3_J,axiom,
aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,one_one_Code_integer) = one_one_Code_integer ).
% dbl_dec_simps(3)
tff(fact_857_dbl__dec__simps_I3_J,axiom,
neg_nu3179335615603231917ec_rat(one_one_rat) = one_one_rat ).
% dbl_dec_simps(3)
tff(fact_858_dbl__dec__simps_I3_J,axiom,
aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).
% dbl_dec_simps(3)
tff(fact_859_mod__emp,axiom,
! [H2: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,one_one_assn),H2)
<=> ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,H2) = bot_bot_set_nat ) ) ).
% mod_emp
tff(fact_860_sndE,axiom,
! [X: produc2453742332425399446nteger,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,P: fun_Pr7211993375558241536eger_o] :
( ( X = produc565448451787809096nteger(A,B) )
=> ( aa_Pro4926521453222654329eger_o(P,aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,X))
=> aa_Pro4926521453222654329eger_o(P,B) ) ) ).
% sndE
tff(fact_861_sndE,axiom,
! [X: produc8799533983703829792et_nat,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,P: fun_Pr6460944699146269935_nat_o] :
( ( X = produc6017135503889078682et_nat(A,B) )
=> ( aa_Pro653082453696054616_nat_o(P,aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,X))
=> aa_Pro653082453696054616_nat_o(P,B) ) ) ).
% sndE
tff(fact_862_sndE,axiom,
! [X: produc3112576180687729086et_nat,A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,P: fun_Pr2620521676128474833_nat_o] :
( ( X = produc7291855603632349880et_nat(A,B) )
=> ( aa_Pro5643009357358181178_nat_o(P,aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,X))
=> aa_Pro5643009357358181178_nat_o(P,B) ) ) ).
% sndE
tff(fact_863_sndE,axiom,
! [X: produc3043004577601930755nt_int,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
( ( X = produc7320969046608631221nt_int(A,B) )
=> ( aa_Pro4592068669816069595_int_o(P,aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,X))
=> aa_Pro4592068669816069595_int_o(P,B) ) ) ).
% sndE
tff(fact_864_sndE,axiom,
! [X: produc4788131091073775919nt_int,A: fun_in5730379987311621695e_term,B: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
( ( X = produc4729913460834754785nt_int(A,B) )
=> ( aa_Pro4592068669816069595_int_o(P,aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,X))
=> aa_Pro4592068669816069595_int_o(P,B) ) ) ).
% sndE
tff(fact_865_sndE,axiom,
! [X: produc3658429121746597890et_nat,A: heap_e7401611519738050253t_unit,B: set_nat,P: fun_set_nat_o] :
( ( X = produc7507926704131184380et_nat(A,B) )
=> ( aa_set_nat_o(P,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X))
=> aa_set_nat_o(P,B) ) ) ).
% sndE
tff(fact_866_sndE,axiom,
! [X: product_prod_int_int,A: int,B: int,P: fun_int_o] :
( ( X = aa_int7462565939943104719nt_int(product_Pair_int_int(A),B) )
=> ( aa_int_o(P,aa_Pro6799258434465214121nt_int(product_snd_int_int,X))
=> aa_int_o(P,B) ) ) ).
% sndE
tff(fact_867_sndE,axiom,
! [X: produc5835291356934675326atural,A: code_natural,B: produc7822875418678951345atural,P: fun_Pr8359263388657976682ural_o] :
( ( X = aa_Pro7165575656943818355atural(produc6639722614265839536atural(A),B) )
=> ( aa_Pro6073791466322389475ural_o(P,aa_Pro3673693039631929973atural(produc6591912806276919810atural,X))
=> aa_Pro6073791466322389475ural_o(P,B) ) ) ).
% sndE
tff(fact_868_snd__eqD,axiom,
! [X: fun_Pr5098669352780829690e_term,Y: produc8923325533196201883nteger,A: produc8923325533196201883nteger] :
( ( aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,produc565448451787809096nteger(X,Y)) = A )
=> ( Y = A ) ) ).
% snd_eqD
tff(fact_869_snd__eqD,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat,A: produc3658429121746597890et_nat] :
( ( aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,produc6017135503889078682et_nat(X,Y)) = A )
=> ( Y = A ) ) ).
% snd_eqD
tff(fact_870_snd__eqD,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat,A: produc8799533983703829792et_nat] :
( ( aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,produc7291855603632349880et_nat(X,Y)) = A )
=> ( Y = A ) ) ).
% snd_eqD
tff(fact_871_snd__eqD,axiom,
! [X: fun_Pr7763680713840462443e_term,Y: product_prod_int_int,A: product_prod_int_int] :
( ( aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,produc7320969046608631221nt_int(X,Y)) = A )
=> ( Y = A ) ) ).
% snd_eqD
tff(fact_872_snd__eqD,axiom,
! [X: fun_in5730379987311621695e_term,Y: product_prod_int_int,A: product_prod_int_int] :
( ( aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,produc4729913460834754785nt_int(X,Y)) = A )
=> ( Y = A ) ) ).
% snd_eqD
tff(fact_873_snd__eqD,axiom,
! [X: heap_e7401611519738050253t_unit,Y: set_nat,A: set_nat] :
( ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,produc7507926704131184380et_nat(X,Y)) = A )
=> ( Y = A ) ) ).
% snd_eqD
tff(fact_874_snd__eqD,axiom,
! [X: int,Y: int,A: int] :
( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y)) = A )
=> ( Y = A ) ) ).
% snd_eqD
tff(fact_875_snd__eqD,axiom,
! [X: code_natural,Y: produc7822875418678951345atural,A: produc7822875418678951345atural] :
( ( aa_Pro3673693039631929973atural(produc6591912806276919810atural,aa_Pro7165575656943818355atural(produc6639722614265839536atural(X),Y)) = A )
=> ( Y = A ) ) ).
% snd_eqD
tff(fact_876_snd__conv,axiom,
! [X1: fun_Pr5098669352780829690e_term,X22: produc8923325533196201883nteger] : aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,produc565448451787809096nteger(X1,X22)) = X22 ).
% snd_conv
tff(fact_877_snd__conv,axiom,
! [X1: fun_Pr6460944699146269935_nat_o,X22: produc3658429121746597890et_nat] : aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,produc6017135503889078682et_nat(X1,X22)) = X22 ).
% snd_conv
tff(fact_878_snd__conv,axiom,
! [X1: fun_Pr6460944699146269935_nat_o,X22: produc8799533983703829792et_nat] : aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,produc7291855603632349880et_nat(X1,X22)) = X22 ).
% snd_conv
tff(fact_879_snd__conv,axiom,
! [X1: fun_Pr7763680713840462443e_term,X22: product_prod_int_int] : aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,produc7320969046608631221nt_int(X1,X22)) = X22 ).
% snd_conv
tff(fact_880_snd__conv,axiom,
! [X1: fun_in5730379987311621695e_term,X22: product_prod_int_int] : aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,produc4729913460834754785nt_int(X1,X22)) = X22 ).
% snd_conv
tff(fact_881_snd__conv,axiom,
! [X1: heap_e7401611519738050253t_unit,X22: set_nat] : aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,produc7507926704131184380et_nat(X1,X22)) = X22 ).
% snd_conv
tff(fact_882_snd__conv,axiom,
! [X1: int,X22: int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(product_Pair_int_int(X1),X22)) = X22 ).
% snd_conv
tff(fact_883_snd__conv,axiom,
! [X1: code_natural,X22: produc7822875418678951345atural] : aa_Pro3673693039631929973atural(produc6591912806276919810atural,aa_Pro7165575656943818355atural(produc6639722614265839536atural(X1),X22)) = X22 ).
% snd_conv
tff(fact_884_sndI,axiom,
! [X: produc2453742332425399446nteger,Y: fun_Pr5098669352780829690e_term,Z: produc8923325533196201883nteger] :
( ( X = produc565448451787809096nteger(Y,Z) )
=> ( aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,X) = Z ) ) ).
% sndI
tff(fact_885_sndI,axiom,
! [X: produc8799533983703829792et_nat,Y: fun_Pr6460944699146269935_nat_o,Z: produc3658429121746597890et_nat] :
( ( X = produc6017135503889078682et_nat(Y,Z) )
=> ( aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,X) = Z ) ) ).
% sndI
tff(fact_886_sndI,axiom,
! [X: produc3112576180687729086et_nat,Y: fun_Pr6460944699146269935_nat_o,Z: produc8799533983703829792et_nat] :
( ( X = produc7291855603632349880et_nat(Y,Z) )
=> ( aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,X) = Z ) ) ).
% sndI
tff(fact_887_sndI,axiom,
! [X: produc3043004577601930755nt_int,Y: fun_Pr7763680713840462443e_term,Z: product_prod_int_int] :
( ( X = produc7320969046608631221nt_int(Y,Z) )
=> ( aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,X) = Z ) ) ).
% sndI
tff(fact_888_sndI,axiom,
! [X: produc4788131091073775919nt_int,Y: fun_in5730379987311621695e_term,Z: product_prod_int_int] :
( ( X = produc4729913460834754785nt_int(Y,Z) )
=> ( aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,X) = Z ) ) ).
% sndI
tff(fact_889_sndI,axiom,
! [X: produc3658429121746597890et_nat,Y: heap_e7401611519738050253t_unit,Z: set_nat] :
( ( X = produc7507926704131184380et_nat(Y,Z) )
=> ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X) = Z ) ) ).
% sndI
tff(fact_890_sndI,axiom,
! [X: product_prod_int_int,Y: int,Z: int] :
( ( X = aa_int7462565939943104719nt_int(product_Pair_int_int(Y),Z) )
=> ( aa_Pro6799258434465214121nt_int(product_snd_int_int,X) = Z ) ) ).
% sndI
tff(fact_891_sndI,axiom,
! [X: produc5835291356934675326atural,Y: code_natural,Z: produc7822875418678951345atural] :
( ( X = aa_Pro7165575656943818355atural(produc6639722614265839536atural(Y),Z) )
=> ( aa_Pro3673693039631929973atural(produc6591912806276919810atural,X) = Z ) ) ).
% sndI
tff(fact_892_eq__snd__iff,axiom,
! [B: produc8923325533196201883nteger,P5: produc2453742332425399446nteger] :
( ( B = aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,P5) )
<=> ? [A4: fun_Pr5098669352780829690e_term] : P5 = produc565448451787809096nteger(A4,B) ) ).
% eq_snd_iff
tff(fact_893_eq__snd__iff,axiom,
! [B: produc3658429121746597890et_nat,P5: produc8799533983703829792et_nat] :
( ( B = aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,P5) )
<=> ? [A4: fun_Pr6460944699146269935_nat_o] : P5 = produc6017135503889078682et_nat(A4,B) ) ).
% eq_snd_iff
tff(fact_894_eq__snd__iff,axiom,
! [B: produc8799533983703829792et_nat,P5: produc3112576180687729086et_nat] :
( ( B = aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,P5) )
<=> ? [A4: fun_Pr6460944699146269935_nat_o] : P5 = produc7291855603632349880et_nat(A4,B) ) ).
% eq_snd_iff
tff(fact_895_eq__snd__iff,axiom,
! [B: product_prod_int_int,P5: produc3043004577601930755nt_int] :
( ( B = aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,P5) )
<=> ? [A4: fun_Pr7763680713840462443e_term] : P5 = produc7320969046608631221nt_int(A4,B) ) ).
% eq_snd_iff
tff(fact_896_eq__snd__iff,axiom,
! [B: product_prod_int_int,P5: produc4788131091073775919nt_int] :
( ( B = aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,P5) )
<=> ? [A4: fun_in5730379987311621695e_term] : P5 = produc4729913460834754785nt_int(A4,B) ) ).
% eq_snd_iff
tff(fact_897_eq__snd__iff,axiom,
! [B: set_nat,P5: produc3658429121746597890et_nat] :
( ( B = aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,P5) )
<=> ? [A4: heap_e7401611519738050253t_unit] : P5 = produc7507926704131184380et_nat(A4,B) ) ).
% eq_snd_iff
tff(fact_898_eq__snd__iff,axiom,
! [B: int,P5: product_prod_int_int] :
( ( B = aa_Pro6799258434465214121nt_int(product_snd_int_int,P5) )
<=> ? [A4: int] : P5 = aa_int7462565939943104719nt_int(product_Pair_int_int(A4),B) ) ).
% eq_snd_iff
tff(fact_899_eq__snd__iff,axiom,
! [B: produc7822875418678951345atural,P5: produc5835291356934675326atural] :
( ( B = aa_Pro3673693039631929973atural(produc6591912806276919810atural,P5) )
<=> ? [A4: code_natural] : P5 = aa_Pro7165575656943818355atural(produc6639722614265839536atural(A4),B) ) ).
% eq_snd_iff
tff(fact_900_dbl__dec__simps_I2_J,axiom,
aa_int_int(neg_nu3811975205180677377ec_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).
% dbl_dec_simps(2)
tff(fact_901_dbl__dec__simps_I2_J,axiom,
neg_nu3179335615603231917ec_rat(zero_zero_rat) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% dbl_dec_simps(2)
tff(fact_902_dbl__dec__simps_I2_J,axiom,
aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% dbl_dec_simps(2)
tff(fact_903_dbl__inc__simps_I2_J,axiom,
aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,zero_z3403309356797280102nteger) = one_one_Code_integer ).
% dbl_inc_simps(2)
tff(fact_904_dbl__inc__simps_I2_J,axiom,
neg_nu5219082963157363817nc_rat(zero_zero_rat) = one_one_rat ).
% dbl_inc_simps(2)
tff(fact_905_dbl__inc__simps_I2_J,axiom,
aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).
% dbl_inc_simps(2)
tff(fact_906_disjoint__eq__subset__Compl,axiom,
! [Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Aa),Ba) = bot_bo2099793752762293965at_nat )
<=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Aa),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Ba)) ) ).
% disjoint_eq_subset_Compl
tff(fact_907_disjoint__eq__subset__Compl,axiom,
! [Aa: set_o,Ba: set_o] :
( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Aa),Ba) = bot_bot_set_o )
<=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Aa),aa_set_o_set_o(uminus_uminus_set_o,Ba)) ) ).
% disjoint_eq_subset_Compl
tff(fact_908_disjoint__eq__subset__Compl,axiom,
! [Aa: set_int,Ba: set_int] :
( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Aa),Ba) = bot_bot_set_int )
<=> aa_set_int_o(ord_less_eq_set_int(Aa),aa_set_int_set_int(uminus1532241313380277803et_int,Ba)) ) ).
% disjoint_eq_subset_Compl
tff(fact_909_disjoint__eq__subset__Compl,axiom,
! [Aa: set_nat,Ba: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),Ba) = bot_bot_set_nat )
<=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Ba)) ) ).
% disjoint_eq_subset_Compl
tff(fact_910_semilattice__neutr__def,axiom,
! [F: fun_nat_fun_nat_nat,Z: nat] :
( aa_nat_o(semila9081495762789891438tr_nat(F),Z)
<=> ( semilattice_nat(F)
& aa_nat_o(comm_monoid_nat(F),Z) ) ) ).
% semilattice_neutr_def
tff(fact_911_semilattice__neutr_Ointro,axiom,
! [F: fun_nat_fun_nat_nat,Z: nat] :
( semilattice_nat(F)
=> ( aa_nat_o(comm_monoid_nat(F),Z)
=> aa_nat_o(semila9081495762789891438tr_nat(F),Z) ) ) ).
% semilattice_neutr.intro
tff(fact_912_minus__assn__def,axiom,
! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(minus_minus_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(uminus_uminus_assn,B)) ).
% minus_assn_def
tff(fact_913_order__refl,axiom,
! [X: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),X) ).
% order_refl
tff(fact_914_order__refl,axiom,
! [X: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),X) ).
% order_refl
tff(fact_915_order__refl,axiom,
! [X: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),X) ).
% order_refl
tff(fact_916_order__refl,axiom,
! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),X) ).
% order_refl
tff(fact_917_order__refl,axiom,
! [X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),X) ).
% order_refl
tff(fact_918_dual__order_Orefl,axiom,
! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),A) ).
% dual_order.refl
tff(fact_919_dual__order_Orefl,axiom,
! [A: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),A) ).
% dual_order.refl
tff(fact_920_dual__order_Orefl,axiom,
! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),A) ).
% dual_order.refl
tff(fact_921_dual__order_Orefl,axiom,
! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).
% dual_order.refl
tff(fact_922_dual__order_Orefl,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).
% dual_order.refl
tff(fact_923_le__zero__eq,axiom,
! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),zero_zero_nat)
<=> ( N = zero_zero_nat ) ) ).
% le_zero_eq
tff(fact_924_mult__zero__left,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ).
% mult_zero_left
tff(fact_925_mult__zero__left,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A) = zero_zero_rat ).
% mult_zero_left
tff(fact_926_mult__zero__left,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A) = zero_zero_nat ).
% mult_zero_left
tff(fact_927_mult__zero__left,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),A) = zero_zero_int ).
% mult_zero_left
tff(fact_928_mult__zero__right,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% mult_zero_right
tff(fact_929_mult__zero__right,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),zero_zero_rat) = zero_zero_rat ).
% mult_zero_right
tff(fact_930_mult__zero__right,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),zero_zero_nat) = zero_zero_nat ).
% mult_zero_right
tff(fact_931_mult__zero__right,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),zero_zero_int) = zero_zero_int ).
% mult_zero_right
tff(fact_932_mult__eq__0__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = zero_z3403309356797280102nteger )
<=> ( ( A = zero_z3403309356797280102nteger )
| ( B = zero_z3403309356797280102nteger ) ) ) ).
% mult_eq_0_iff
tff(fact_933_mult__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
<=> ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% mult_eq_0_iff
tff(fact_934_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
<=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
tff(fact_935_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
<=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
tff(fact_936_mult__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
<=> ( ( C = zero_z3403309356797280102nteger )
| ( A = B ) ) ) ).
% mult_cancel_left
tff(fact_937_mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
<=> ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_left
tff(fact_938_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
<=> ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
tff(fact_939_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
<=> ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
tff(fact_940_mult__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
<=> ( ( C = zero_z3403309356797280102nteger )
| ( A = B ) ) ) ).
% mult_cancel_right
tff(fact_941_mult__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
<=> ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_right
tff(fact_942_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
<=> ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
tff(fact_943_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
<=> ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
tff(fact_944_diff__self,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),A) = zero_z3403309356797280102nteger ).
% diff_self
tff(fact_945_diff__self,axiom,
! [A: rat] : minus_minus_rat(A,A) = zero_zero_rat ).
% diff_self
tff(fact_946_diff__self,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).
% diff_self
tff(fact_947_diff__0__right,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),zero_z3403309356797280102nteger) = A ).
% diff_0_right
tff(fact_948_diff__0__right,axiom,
! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).
% diff_0_right
tff(fact_949_diff__0__right,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).
% diff_0_right
tff(fact_950_zero__diff,axiom,
! [A: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(zero_z1048942125864253310at_nat,A) = zero_z1048942125864253310at_nat ).
% zero_diff
tff(fact_951_zero__diff,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),A) = zero_zero_nat ).
% zero_diff
tff(fact_952_diff__zero,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),zero_z3403309356797280102nteger) = A ).
% diff_zero
tff(fact_953_diff__zero,axiom,
! [A: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(A,zero_z1048942125864253310at_nat) = A ).
% diff_zero
tff(fact_954_diff__zero,axiom,
! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).
% diff_zero
tff(fact_955_diff__zero,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),zero_zero_nat) = A ).
% diff_zero
tff(fact_956_diff__zero,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).
% diff_zero
tff(fact_957_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),A) = zero_z3403309356797280102nteger ).
% cancel_comm_monoid_add_class.diff_cancel
tff(fact_958_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(A,A) = zero_z1048942125864253310at_nat ).
% cancel_comm_monoid_add_class.diff_cancel
tff(fact_959_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
tff(fact_960_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),A) = zero_zero_nat ).
% cancel_comm_monoid_add_class.diff_cancel
tff(fact_961_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).
% cancel_comm_monoid_add_class.diff_cancel
tff(fact_962_neg__le__iff__le,axiom,
! [B: rat,A: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
<=> aa_rat_o(ord_less_eq_rat(A),B) ) ).
% neg_le_iff_le
tff(fact_963_neg__le__iff__le,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).
% neg_le_iff_le
tff(fact_964_neg__le__iff__le,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% neg_le_iff_le
tff(fact_965_compl__le__compl__iff,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(uminus_uminus_assn,Y))
<=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X) ) ).
% compl_le_compl_iff
tff(fact_966_compl__le__compl__iff,axiom,
! [X: set_nat,Y: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y))
<=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X) ) ).
% compl_le_compl_iff
tff(fact_967_add_Oinverse__neutral,axiom,
aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).
% add.inverse_neutral
tff(fact_968_add_Oinverse__neutral,axiom,
aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).
% add.inverse_neutral
tff(fact_969_add_Oinverse__neutral,axiom,
aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% add.inverse_neutral
tff(fact_970_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int = aa_int_int(uminus_uminus_int,A) )
<=> ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
tff(fact_971_neg__0__equal__iff__equal,axiom,
! [A: rat] :
( ( zero_zero_rat = aa_rat_rat(uminus_uminus_rat,A) )
<=> ( zero_zero_rat = A ) ) ).
% neg_0_equal_iff_equal
tff(fact_972_neg__0__equal__iff__equal,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
<=> ( zero_z3403309356797280102nteger = A ) ) ).
% neg_0_equal_iff_equal
tff(fact_973_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( aa_int_int(uminus_uminus_int,A) = zero_zero_int )
<=> ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
tff(fact_974_neg__equal__0__iff__equal,axiom,
! [A: rat] :
( ( aa_rat_rat(uminus_uminus_rat,A) = zero_zero_rat )
<=> ( A = zero_zero_rat ) ) ).
% neg_equal_0_iff_equal
tff(fact_975_neg__equal__0__iff__equal,axiom,
! [A: code_integer] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = zero_z3403309356797280102nteger )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% neg_equal_0_iff_equal
tff(fact_976_equal__neg__zero,axiom,
! [A: int] :
( ( A = aa_int_int(uminus_uminus_int,A) )
<=> ( A = zero_zero_int ) ) ).
% equal_neg_zero
tff(fact_977_equal__neg__zero,axiom,
! [A: rat] :
( ( A = aa_rat_rat(uminus_uminus_rat,A) )
<=> ( A = zero_zero_rat ) ) ).
% equal_neg_zero
tff(fact_978_equal__neg__zero,axiom,
! [A: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% equal_neg_zero
tff(fact_979_neg__equal__zero,axiom,
! [A: int] :
( ( aa_int_int(uminus_uminus_int,A) = A )
<=> ( A = zero_zero_int ) ) ).
% neg_equal_zero
tff(fact_980_neg__equal__zero,axiom,
! [A: rat] :
( ( aa_rat_rat(uminus_uminus_rat,A) = A )
<=> ( A = zero_zero_rat ) ) ).
% neg_equal_zero
tff(fact_981_neg__equal__zero,axiom,
! [A: code_integer] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% neg_equal_zero
tff(fact_982_le__inf__iff,axiom,
! [X: assn,Y: assn,Z: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z))
<=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
& aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Z) ) ) ).
% le_inf_iff
tff(fact_983_le__inf__iff,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z))
<=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
& aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z) ) ) ).
% le_inf_iff
tff(fact_984_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z) ) ) ).
% le_inf_iff
tff(fact_985_le__inf__iff,axiom,
! [X: int,Y: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z))
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z) ) ) ).
% le_inf_iff
tff(fact_986_inf_Obounded__iff,axiom,
! [A: assn,B: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C))
<=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
& aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C) ) ) ).
% inf.bounded_iff
tff(fact_987_inf_Obounded__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C))
<=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
& aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C) ) ) ).
% inf.bounded_iff
tff(fact_988_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).
% inf.bounded_iff
tff(fact_989_inf_Obounded__iff,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C))
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).
% inf.bounded_iff
tff(fact_990_le__sup__iff,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),Z)
<=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),Z)
& aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),Z) ) ) ).
% le_sup_iff
tff(fact_991_le__sup__iff,axiom,
! [X: assn,Y: assn,Z: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),Z)
<=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Z)
& aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),Z) ) ) ).
% le_sup_iff
tff(fact_992_le__sup__iff,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z)
<=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z)
& aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Z) ) ) ).
% le_sup_iff
tff(fact_993_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),Z)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z) ) ) ).
% le_sup_iff
tff(fact_994_le__sup__iff,axiom,
! [X: int,Y: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),Z)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z) ) ) ).
% le_sup_iff
tff(fact_995_sup_Obounded__iff,axiom,
! [B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C)),A)
<=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
& aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),A) ) ) ).
% sup.bounded_iff
tff(fact_996_sup_Obounded__iff,axiom,
! [B: assn,C: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C)),A)
<=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
& aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),A) ) ) ).
% sup.bounded_iff
tff(fact_997_sup_Obounded__iff,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A)
<=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
& aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A) ) ) ).
% sup.bounded_iff
tff(fact_998_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).
% sup.bounded_iff
tff(fact_999_sup_Obounded__iff,axiom,
! [B: int,C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).
% sup.bounded_iff
tff(fact_1000_minus__diff__eq,axiom,
! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).
% minus_diff_eq
tff(fact_1001_minus__diff__eq,axiom,
! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,minus_minus_rat(A,B)) = minus_minus_rat(B,A) ).
% minus_diff_eq
tff(fact_1002_minus__diff__eq,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A) ).
% minus_diff_eq
tff(fact_1003_empty__subsetI,axiom,
! [Aa: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(bot_bo2099793752762293965at_nat),Aa) ).
% empty_subsetI
tff(fact_1004_empty__subsetI,axiom,
! [Aa: set_o] : aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,bot_bot_set_o),Aa) ).
% empty_subsetI
tff(fact_1005_empty__subsetI,axiom,
! [Aa: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),Aa) ).
% empty_subsetI
tff(fact_1006_empty__subsetI,axiom,
! [Aa: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),Aa) ).
% empty_subsetI
tff(fact_1007_subset__empty,axiom,
! [Aa: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Aa),bot_bo2099793752762293965at_nat)
<=> ( Aa = bot_bo2099793752762293965at_nat ) ) ).
% subset_empty
tff(fact_1008_subset__empty,axiom,
! [Aa: set_o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Aa),bot_bot_set_o)
<=> ( Aa = bot_bot_set_o ) ) ).
% subset_empty
tff(fact_1009_subset__empty,axiom,
! [Aa: set_int] :
( aa_set_int_o(ord_less_eq_set_int(Aa),bot_bot_set_int)
<=> ( Aa = bot_bot_set_int ) ) ).
% subset_empty
tff(fact_1010_subset__empty,axiom,
! [Aa: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),bot_bot_set_nat)
<=> ( Aa = bot_bot_set_nat ) ) ).
% subset_empty
tff(fact_1011_diff__ge__0__iff__ge,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),minus_minus_rat(A,B))
<=> aa_rat_o(ord_less_eq_rat(B),A) ) ).
% diff_ge_0_iff_ge
tff(fact_1012_diff__ge__0__iff__ge,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ).
% diff_ge_0_iff_ge
tff(fact_1013_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).
% diff_ge_0_iff_ge
tff(fact_1014_neg__0__le__iff__le,axiom,
! [A: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
<=> aa_rat_o(ord_less_eq_rat(A),zero_zero_rat) ) ).
% neg_0_le_iff_le
tff(fact_1015_neg__0__le__iff__le,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).
% neg_0_le_iff_le
tff(fact_1016_neg__0__le__iff__le,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).
% neg_0_le_iff_le
tff(fact_1017_neg__le__0__iff__le,axiom,
! [A: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
<=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),A) ) ).
% neg_le_0_iff_le
tff(fact_1018_neg__le__0__iff__le,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).
% neg_le_0_iff_le
tff(fact_1019_neg__le__0__iff__le,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).
% neg_le_0_iff_le
tff(fact_1020_less__eq__neg__nonpos,axiom,
! [A: rat] :
( aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(uminus_uminus_rat,A))
<=> aa_rat_o(ord_less_eq_rat(A),zero_zero_rat) ) ).
% less_eq_neg_nonpos
tff(fact_1021_less__eq__neg__nonpos,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).
% less_eq_neg_nonpos
tff(fact_1022_less__eq__neg__nonpos,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).
% less_eq_neg_nonpos
tff(fact_1023_neg__less__eq__nonneg,axiom,
! [A: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,A)),A)
<=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),A) ) ).
% neg_less_eq_nonneg
tff(fact_1024_neg__less__eq__nonneg,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).
% neg_less_eq_nonneg
tff(fact_1025_neg__less__eq__nonneg,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).
% neg_less_eq_nonneg
tff(fact_1026_mult__cancel__right2,axiom,
! [A: code_integer,C: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) = C )
<=> ( ( C = zero_z3403309356797280102nteger )
| ( A = one_one_Code_integer ) ) ) ).
% mult_cancel_right2
tff(fact_1027_mult__cancel__right2,axiom,
! [A: rat,C: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = C )
<=> ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_right2
tff(fact_1028_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = C )
<=> ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
tff(fact_1029_mult__cancel__right1,axiom,
! [C: code_integer,B: code_integer] :
( ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
<=> ( ( C = zero_z3403309356797280102nteger )
| ( B = one_one_Code_integer ) ) ) ).
% mult_cancel_right1
tff(fact_1030_mult__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
<=> ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_right1
tff(fact_1031_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
<=> ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
tff(fact_1032_mult__cancel__left2,axiom,
! [C: code_integer,A: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) = C )
<=> ( ( C = zero_z3403309356797280102nteger )
| ( A = one_one_Code_integer ) ) ) ).
% mult_cancel_left2
tff(fact_1033_mult__cancel__left2,axiom,
! [C: rat,A: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = C )
<=> ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_left2
tff(fact_1034_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = C )
<=> ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
tff(fact_1035_mult__cancel__left1,axiom,
! [C: code_integer,B: code_integer] :
( ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
<=> ( ( C = zero_z3403309356797280102nteger )
| ( B = one_one_Code_integer ) ) ) ).
% mult_cancel_left1
tff(fact_1036_mult__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
<=> ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_left1
tff(fact_1037_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
<=> ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
tff(fact_1038_diff__numeral__special_I9_J,axiom,
aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).
% diff_numeral_special(9)
tff(fact_1039_diff__numeral__special_I9_J,axiom,
minus_minus_rat(one_one_rat,one_one_rat) = zero_zero_rat ).
% diff_numeral_special(9)
tff(fact_1040_diff__numeral__special_I9_J,axiom,
aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),one_one_int) = zero_zero_int ).
% diff_numeral_special(9)
tff(fact_1041_diff__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),A) = aa_int_int(uminus_uminus_int,A) ).
% diff_0
tff(fact_1042_diff__0,axiom,
! [A: rat] : minus_minus_rat(zero_zero_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ).
% diff_0
tff(fact_1043_diff__0,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(zero_z3403309356797280102nteger),A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).
% diff_0
tff(fact_1044_diff__numeral__special_I12_J,axiom,
aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).
% diff_numeral_special(12)
tff(fact_1045_diff__numeral__special_I12_J,axiom,
minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).
% diff_numeral_special(12)
tff(fact_1046_diff__numeral__special_I12_J,axiom,
aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).
% diff_numeral_special(12)
tff(fact_1047_diff__eq__diff__eq,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( minus_minus_rat(A,B) = minus_minus_rat(C,D2) )
=> ( ( A = B )
<=> ( C = D2 ) ) ) ).
% diff_eq_diff_eq
tff(fact_1048_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D2) )
=> ( ( A = B )
<=> ( C = D2 ) ) ) ).
% diff_eq_diff_eq
tff(fact_1049_eq__iff__diff__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( A = B )
<=> ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B) = zero_z3403309356797280102nteger ) ) ).
% eq_iff_diff_eq_0
tff(fact_1050_eq__iff__diff__eq__0,axiom,
! [A: rat,B: rat] :
( ( A = B )
<=> ( minus_minus_rat(A,B) = zero_zero_rat ) ) ).
% eq_iff_diff_eq_0
tff(fact_1051_eq__iff__diff__eq__0,axiom,
! [A: int,B: int] :
( ( A = B )
<=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int ) ) ).
% eq_iff_diff_eq_0
tff(fact_1052_diff__mono,axiom,
! [A: rat,B: rat,D2: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(A),B)
=> ( aa_rat_o(ord_less_eq_rat(D2),C)
=> aa_rat_o(ord_less_eq_rat(minus_minus_rat(A,C)),minus_minus_rat(B,D2)) ) ) ).
% diff_mono
tff(fact_1053_diff__mono,axiom,
! [A: code_integer,B: code_integer,D2: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,D2),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),C)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),D2)) ) ) ).
% diff_mono
tff(fact_1054_diff__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D2),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D2)) ) ) ).
% diff_mono
tff(fact_1055_diff__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(B),A)
=> aa_rat_o(ord_less_eq_rat(minus_minus_rat(C,A)),minus_minus_rat(C,B)) ) ).
% diff_left_mono
tff(fact_1056_diff__left__mono,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(C),A)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(C),B)) ) ).
% diff_left_mono
tff(fact_1057_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).
% diff_left_mono
tff(fact_1058_diff__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(A),B)
=> aa_rat_o(ord_less_eq_rat(minus_minus_rat(A,C)),minus_minus_rat(B,C)) ) ).
% diff_right_mono
tff(fact_1059_diff__right__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),C)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) ) ).
% diff_right_mono
tff(fact_1060_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).
% diff_right_mono
tff(fact_1061_le__iff__diff__le__0,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(A),B)
<=> aa_rat_o(ord_less_eq_rat(minus_minus_rat(A,B)),zero_zero_rat) ) ).
% le_iff_diff_le_0
tff(fact_1062_le__iff__diff__le__0,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),zero_z3403309356797280102nteger) ) ).
% le_iff_diff_le_0
tff(fact_1063_le__iff__diff__le__0,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).
% le_iff_diff_le_0
tff(fact_1064_zero__le,axiom,
! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X) ).
% zero_le
tff(fact_1065_diff__eq__diff__less__eq,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( minus_minus_rat(A,B) = minus_minus_rat(C,D2) )
=> ( aa_rat_o(ord_less_eq_rat(A),B)
<=> aa_rat_o(ord_less_eq_rat(C),D2) ) ) ).
% diff_eq_diff_less_eq
tff(fact_1066_diff__eq__diff__less__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(C),D2) )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D2) ) ) ).
% diff_eq_diff_less_eq
tff(fact_1067_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D2) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2) ) ) ).
% diff_eq_diff_less_eq
tff(fact_1068_cancel__ab__semigroup__add__class_Odiff__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) ).
% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_1069_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) ).
% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_1070_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).
% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_1071_zero__reorient,axiom,
! [X: code_integer] :
( ( zero_z3403309356797280102nteger = X )
<=> ( X = zero_z3403309356797280102nteger ) ) ).
% zero_reorient
tff(fact_1072_zero__reorient,axiom,
! [X: multis2468970476368604999at_nat] :
( ( zero_z1048942125864253310at_nat = X )
<=> ( X = zero_z1048942125864253310at_nat ) ) ).
% zero_reorient
tff(fact_1073_zero__reorient,axiom,
! [X: rat] :
( ( zero_zero_rat = X )
<=> ( X = zero_zero_rat ) ) ).
% zero_reorient
tff(fact_1074_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
<=> ( X = zero_zero_nat ) ) ).
% zero_reorient
tff(fact_1075_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
<=> ( X = zero_zero_int ) ) ).
% zero_reorient
tff(fact_1076_nle__le,axiom,
! [A: code_integer,B: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
& ( B != A ) ) ) ).
% nle_le
tff(fact_1077_nle__le,axiom,
! [A: nat,B: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
& ( B != A ) ) ) ).
% nle_le
tff(fact_1078_nle__le,axiom,
! [A: int,B: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
& ( B != A ) ) ) ).
% nle_le
tff(fact_1079_le__cases3,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),Z) )
=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X)
=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Z) )
=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Z)
=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),Y) )
=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),Y)
=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X) )
=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),Z)
=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),X) )
=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),X)
=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y) ) ) ) ) ) ) ).
% le_cases3
tff(fact_1080_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z) )
=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z) )
=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) )
=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) )
=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X) )
=> ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ) ) ) ) ).
% le_cases3
tff(fact_1081_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z) )
=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z) )
=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) )
=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) )
=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X) )
=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ) ) ) ) ).
% le_cases3
tff(fact_1082_order__class_Oorder__eq__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( X = Y )
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X) ) ) ).
% order_class.order_eq_iff
tff(fact_1083_order__class_Oorder__eq__iff,axiom,
! [X: assn,Y: assn] :
( ( X = Y )
<=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
& aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X) ) ) ).
% order_class.order_eq_iff
tff(fact_1084_order__class_Oorder__eq__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
<=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
& aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X) ) ) ).
% order_class.order_eq_iff
tff(fact_1085_order__class_Oorder__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ) ).
% order_class.order_eq_iff
tff(fact_1086_order__class_Oorder__eq__iff,axiom,
! [X: int,Y: int] :
( ( X = Y )
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ) ).
% order_class.order_eq_iff
tff(fact_1087_ord__eq__le__trans,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A = B )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),C) ) ) ).
% ord_eq_le_trans
tff(fact_1088_ord__eq__le__trans,axiom,
! [A: assn,B: assn,C: assn] :
( ( A = B )
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C) ) ) ).
% ord_eq_le_trans
tff(fact_1089_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C) ) ) ).
% ord_eq_le_trans
tff(fact_1090_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).
% ord_eq_le_trans
tff(fact_1091_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).
% ord_eq_le_trans
tff(fact_1092_ord__le__eq__trans,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( ( B = C )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),C) ) ) ).
% ord_le_eq_trans
tff(fact_1093_ord__le__eq__trans,axiom,
! [A: assn,B: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( ( B = C )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C) ) ) ).
% ord_le_eq_trans
tff(fact_1094_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
=> ( ( B = C )
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C) ) ) ).
% ord_le_eq_trans
tff(fact_1095_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( ( B = C )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).
% ord_le_eq_trans
tff(fact_1096_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( ( B = C )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).
% ord_le_eq_trans
tff(fact_1097_order__antisym,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X)
=> ( X = Y ) ) ) ).
% order_antisym
tff(fact_1098_order__antisym,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
=> ( X = Y ) ) ) ).
% order_antisym
tff(fact_1099_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
=> ( X = Y ) ) ) ).
% order_antisym
tff(fact_1100_order__antisym,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
=> ( X = Y ) ) ) ).
% order_antisym
tff(fact_1101_order__antisym,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
=> ( X = Y ) ) ) ).
% order_antisym
tff(fact_1102_order_Otrans,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),C) ) ) ).
% order.trans
tff(fact_1103_order_Otrans,axiom,
! [A: assn,B: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C) ) ) ).
% order.trans
tff(fact_1104_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C) ) ) ).
% order.trans
tff(fact_1105_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).
% order.trans
tff(fact_1106_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).
% order.trans
tff(fact_1107_order__trans,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),Z)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Z) ) ) ).
% order_trans
tff(fact_1108_order__trans,axiom,
! [X: assn,Y: assn,Z: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),Z)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Z) ) ) ).
% order_trans
tff(fact_1109_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Z)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z) ) ) ).
% order_trans
tff(fact_1110_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z) ) ) ).
% order_trans
tff(fact_1111_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z) ) ) ).
% order_trans
tff(fact_1112_linorder__wlog,axiom,
! [P: fun_Co2081214437782444048eger_o,A: code_integer,B: code_integer] :
( ! [A3: code_integer,B3: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A3),B3)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A3),B3) )
=> ( ! [A3: code_integer,B3: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,B3),A3)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A3),B3) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A),B) ) ) ).
% linorder_wlog
tff(fact_1113_linorder__wlog,axiom,
! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),B3)
=> aa_nat_o(aa_nat_fun_nat_o(P,A3),B3) )
=> ( ! [A3: nat,B3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(P,B3),A3)
=> aa_nat_o(aa_nat_fun_nat_o(P,A3),B3) )
=> aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ).
% linorder_wlog
tff(fact_1114_linorder__wlog,axiom,
! [P: fun_int_fun_int_o,A: int,B: int] :
( ! [A3: int,B3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),B3)
=> aa_int_o(aa_int_fun_int_o(P,A3),B3) )
=> ( ! [A3: int,B3: int] :
( aa_int_o(aa_int_fun_int_o(P,B3),A3)
=> aa_int_o(aa_int_fun_int_o(P,A3),B3) )
=> aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ).
% linorder_wlog
tff(fact_1115_dual__order_Oeq__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( A = B )
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).
% dual_order.eq_iff
tff(fact_1116_dual__order_Oeq__iff,axiom,
! [A: assn,B: assn] :
( ( A = B )
<=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
& aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B) ) ) ).
% dual_order.eq_iff
tff(fact_1117_dual__order_Oeq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
<=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
& aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ) ).
% dual_order.eq_iff
tff(fact_1118_dual__order_Oeq__iff,axiom,
! [A: nat,B: nat] :
( ( A = B )
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).
% dual_order.eq_iff
tff(fact_1119_dual__order_Oeq__iff,axiom,
! [A: int,B: int] :
( ( A = B )
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).
% dual_order.eq_iff
tff(fact_1120_dual__order_Oantisym,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( A = B ) ) ) ).
% dual_order.antisym
tff(fact_1121_dual__order_Oantisym,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( A = B ) ) ) ).
% dual_order.antisym
tff(fact_1122_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
=> ( A = B ) ) ) ).
% dual_order.antisym
tff(fact_1123_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( A = B ) ) ) ).
% dual_order.antisym
tff(fact_1124_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( A = B ) ) ) ).
% dual_order.antisym
tff(fact_1125_dual__order_Otrans,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),A) ) ) ).
% dual_order.trans
tff(fact_1126_dual__order_Otrans,axiom,
! [B: assn,A: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),B)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),A) ) ) ).
% dual_order.trans
tff(fact_1127_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),B)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A) ) ) ).
% dual_order.trans
tff(fact_1128_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).
% dual_order.trans
tff(fact_1129_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).
% dual_order.trans
tff(fact_1130_antisym,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
=> ( A = B ) ) ) ).
% antisym
tff(fact_1131_antisym,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
=> ( A = B ) ) ) ).
% antisym
tff(fact_1132_antisym,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
=> ( A = B ) ) ) ).
% antisym
tff(fact_1133_antisym,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ( A = B ) ) ) ).
% antisym
tff(fact_1134_antisym,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( A = B ) ) ) ).
% antisym
tff(fact_1135_Orderings_Oorder__eq__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( A = B )
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ).
% Orderings.order_eq_iff
tff(fact_1136_Orderings_Oorder__eq__iff,axiom,
! [A: assn,B: assn] :
( ( A = B )
<=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
& aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A) ) ) ).
% Orderings.order_eq_iff
tff(fact_1137_Orderings_Oorder__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
<=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
& aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A) ) ) ).
% Orderings.order_eq_iff
tff(fact_1138_Orderings_Oorder__eq__iff,axiom,
! [A: nat,B: nat] :
( ( A = B )
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).
% Orderings.order_eq_iff
tff(fact_1139_Orderings_Oorder__eq__iff,axiom,
! [A: int,B: int] :
( ( A = B )
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).
% Orderings.order_eq_iff
tff(fact_1140_order__subst1,axiom,
! [A: code_integer,F: fun_Co8499238673658658103nteger,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(F,B))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F,X2)),aa_Cod1783974652533396728nteger(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(F,C)) ) ) ) ).
% order_subst1
tff(fact_1141_order__subst1,axiom,
! [A: code_integer,F: fun_as9138348466139350018nteger,B: assn,C: assn] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_assn_Code_integer(F,B))
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_assn_Code_integer(F,X2)),aa_assn_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_assn_Code_integer(F,C)) ) ) ) ).
% order_subst1
tff(fact_1142_order__subst1,axiom,
! [A: code_integer,F: fun_nat_Code_integer,B: nat,C: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_nat_Code_integer(F,B))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X2)),aa_nat_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_nat_Code_integer(F,C)) ) ) ) ).
% order_subst1
tff(fact_1143_order__subst1,axiom,
! [A: code_integer,F: fun_int_Code_integer,B: int,C: int] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_int_Code_integer(F,B))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
=> ( ! [X2: int,Y2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,X2)),aa_int_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_int_Code_integer(F,C)) ) ) ) ).
% order_subst1
tff(fact_1144_order__subst1,axiom,
! [A: assn,F: fun_Co4490034186337852034r_assn,B: code_integer,C: code_integer] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_Code_integer_assn(F,B))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_Code_integer_assn(F,X2)),aa_Code_integer_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_Code_integer_assn(F,C)) ) ) ) ).
% order_subst1
tff(fact_1145_order__subst1,axiom,
! [A: assn,F: fun_assn_assn,B: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(F,B))
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(F,C)) ) ) ) ).
% order_subst1
tff(fact_1146_order__subst1,axiom,
! [A: assn,F: fun_nat_assn,B: nat,C: nat] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_nat_assn(F,B))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_nat_assn(F,C)) ) ) ) ).
% order_subst1
tff(fact_1147_order__subst1,axiom,
! [A: assn,F: fun_int_assn,B: int,C: int] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_int_assn(F,B))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
=> ( ! [X2: int,Y2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_int_assn(F,X2)),aa_int_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_int_assn(F,C)) ) ) ) ).
% order_subst1
tff(fact_1148_order__subst1,axiom,
! [A: nat,F: fun_Code_integer_nat,B: code_integer,C: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_Code_integer_nat(F,B))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(F,X2)),aa_Code_integer_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_Code_integer_nat(F,C)) ) ) ) ).
% order_subst1
tff(fact_1149_order__subst1,axiom,
! [A: nat,F: fun_assn_nat,B: assn,C: assn] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_assn_nat(F,B))
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_assn_nat(F,C)) ) ) ) ).
% order_subst1
tff(fact_1150_order__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Co8499238673658658103nteger,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F,B)),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F,X2)),aa_Cod1783974652533396728nteger(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_1151_order__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Co4490034186337852034r_assn,C: assn] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_Code_integer_assn(F,B)),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_Code_integer_assn(F,X2)),aa_Code_integer_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_Code_integer_assn(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_1152_order__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Code_integer_nat,C: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(F,B)),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(F,X2)),aa_Code_integer_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_1153_order__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Code_integer_int,C: int] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(F,B)),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(F,X2)),aa_Code_integer_int(F,Y2)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_1154_order__subst2,axiom,
! [A: assn,B: assn,F: fun_as9138348466139350018nteger,C: code_integer] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_assn_Code_integer(F,B)),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_assn_Code_integer(F,X2)),aa_assn_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_assn_Code_integer(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_1155_order__subst2,axiom,
! [A: assn,B: assn,F: fun_assn_assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,B)),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_1156_order__subst2,axiom,
! [A: assn,B: assn,F: fun_assn_nat,C: nat] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,B)),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_1157_order__subst2,axiom,
! [A: assn,B: assn,F: fun_assn_int,C: int] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,B)),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_1158_order__subst2,axiom,
! [A: nat,B: nat,F: fun_nat_Code_integer,C: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,B)),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X2)),aa_nat_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_1159_order__subst2,axiom,
! [A: nat,B: nat,F: fun_nat_assn,C: assn] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,B)),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,A)),C) ) ) ) ).
% order_subst2
tff(fact_1160_order__eq__refl,axiom,
! [X: code_integer,Y: code_integer] :
( ( X = Y )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y) ) ).
% order_eq_refl
tff(fact_1161_order__eq__refl,axiom,
! [X: assn,Y: assn] :
( ( X = Y )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y) ) ).
% order_eq_refl
tff(fact_1162_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y) ) ).
% order_eq_refl
tff(fact_1163_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).
% order_eq_refl
tff(fact_1164_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).
% order_eq_refl
tff(fact_1165_linorder__linear,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X) ) ).
% linorder_linear
tff(fact_1166_linorder__linear,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).
% linorder_linear
tff(fact_1167_linorder__linear,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
| aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).
% linorder_linear
tff(fact_1168_ord__eq__le__subst,axiom,
! [A: code_integer,F: fun_Co8499238673658658103nteger,B: code_integer,C: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(F,B) )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F,X2)),aa_Cod1783974652533396728nteger(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_1169_ord__eq__le__subst,axiom,
! [A: assn,F: fun_Co4490034186337852034r_assn,B: code_integer,C: code_integer] :
( ( A = aa_Code_integer_assn(F,B) )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_Code_integer_assn(F,X2)),aa_Code_integer_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_Code_integer_assn(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_1170_ord__eq__le__subst,axiom,
! [A: nat,F: fun_Code_integer_nat,B: code_integer,C: code_integer] :
( ( A = aa_Code_integer_nat(F,B) )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(F,X2)),aa_Code_integer_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_Code_integer_nat(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_1171_ord__eq__le__subst,axiom,
! [A: int,F: fun_Code_integer_int,B: code_integer,C: code_integer] :
( ( A = aa_Code_integer_int(F,B) )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(F,X2)),aa_Code_integer_int(F,Y2)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_Code_integer_int(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_1172_ord__eq__le__subst,axiom,
! [A: code_integer,F: fun_as9138348466139350018nteger,B: assn,C: assn] :
( ( A = aa_assn_Code_integer(F,B) )
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_assn_Code_integer(F,X2)),aa_assn_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_assn_Code_integer(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_1173_ord__eq__le__subst,axiom,
! [A: assn,F: fun_assn_assn,B: assn,C: assn] :
( ( A = aa_assn_assn(F,B) )
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_1174_ord__eq__le__subst,axiom,
! [A: nat,F: fun_assn_nat,B: assn,C: assn] :
( ( A = aa_assn_nat(F,B) )
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_assn_nat(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_1175_ord__eq__le__subst,axiom,
! [A: int,F: fun_assn_int,B: assn,C: assn] :
( ( A = aa_assn_int(F,B) )
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_assn_int(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_1176_ord__eq__le__subst,axiom,
! [A: code_integer,F: fun_nat_Code_integer,B: nat,C: nat] :
( ( A = aa_nat_Code_integer(F,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X2)),aa_nat_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_nat_Code_integer(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_1177_ord__eq__le__subst,axiom,
! [A: assn,F: fun_nat_assn,B: nat,C: nat] :
( ( A = aa_nat_assn(F,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_nat_assn(F,C)) ) ) ) ).
% ord_eq_le_subst
tff(fact_1178_ord__le__eq__subst,axiom,
! [A: code_integer,B: code_integer,F: fun_Co8499238673658658103nteger,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( ( aa_Cod1783974652533396728nteger(F,B) = C )
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F,X2)),aa_Cod1783974652533396728nteger(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_1179_ord__le__eq__subst,axiom,
! [A: code_integer,B: code_integer,F: fun_Co4490034186337852034r_assn,C: assn] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( ( aa_Code_integer_assn(F,B) = C )
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_Code_integer_assn(F,X2)),aa_Code_integer_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_Code_integer_assn(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_1180_ord__le__eq__subst,axiom,
! [A: code_integer,B: code_integer,F: fun_Code_integer_nat,C: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( ( aa_Code_integer_nat(F,B) = C )
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(F,X2)),aa_Code_integer_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_1181_ord__le__eq__subst,axiom,
! [A: code_integer,B: code_integer,F: fun_Code_integer_int,C: int] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( ( aa_Code_integer_int(F,B) = C )
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(F,X2)),aa_Code_integer_int(F,Y2)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_1182_ord__le__eq__subst,axiom,
! [A: assn,B: assn,F: fun_as9138348466139350018nteger,C: code_integer] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( ( aa_assn_Code_integer(F,B) = C )
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_assn_Code_integer(F,X2)),aa_assn_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_assn_Code_integer(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_1183_ord__le__eq__subst,axiom,
! [A: assn,B: assn,F: fun_assn_assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( ( aa_assn_assn(F,B) = C )
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_1184_ord__le__eq__subst,axiom,
! [A: assn,B: assn,F: fun_assn_nat,C: nat] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( ( aa_assn_nat(F,B) = C )
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_1185_ord__le__eq__subst,axiom,
! [A: assn,B: assn,F: fun_assn_int,C: int] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( ( aa_assn_int(F,B) = C )
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_1186_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: fun_nat_Code_integer,C: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( ( aa_nat_Code_integer(F,B) = C )
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X2)),aa_nat_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_1187_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: fun_nat_assn,C: assn] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( ( aa_nat_assn(F,B) = C )
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,A)),C) ) ) ) ).
% ord_le_eq_subst
tff(fact_1188_linorder__le__cases,axiom,
! [X: code_integer,Y: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X) ) ).
% linorder_le_cases
tff(fact_1189_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).
% linorder_le_cases
tff(fact_1190_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).
% linorder_le_cases
tff(fact_1191_order__antisym__conv,axiom,
! [Y: code_integer,X: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
<=> ( X = Y ) ) ) ).
% order_antisym_conv
tff(fact_1192_order__antisym__conv,axiom,
! [Y: assn,X: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
<=> ( X = Y ) ) ) ).
% order_antisym_conv
tff(fact_1193_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
<=> ( X = Y ) ) ) ).
% order_antisym_conv
tff(fact_1194_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
<=> ( X = Y ) ) ) ).
% order_antisym_conv
tff(fact_1195_order__antisym__conv,axiom,
! [Y: int,X: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
<=> ( X = Y ) ) ) ).
% order_antisym_conv
tff(fact_1196_diff__shunt__var,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat(X,Y) = bot_bo2099793752762293965at_nat )
<=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(X),Y) ) ).
% diff_shunt_var
tff(fact_1197_diff__shunt__var,axiom,
! [X: set_o,Y: set_o] :
( ( minus_minus_set_o(X,Y) = bot_bot_set_o )
<=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,X),Y) ) ).
% diff_shunt_var
tff(fact_1198_diff__shunt__var,axiom,
! [X: set_int,Y: set_int] :
( ( minus_minus_set_int(X,Y) = bot_bot_set_int )
<=> aa_set_int_o(ord_less_eq_set_int(X),Y) ) ).
% diff_shunt_var
tff(fact_1199_diff__shunt__var,axiom,
! [X: assn,Y: assn] :
( ( aa_assn_assn(aa_ass954072293579886259n_assn(minus_minus_assn,X),Y) = bot_bot_assn )
<=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y) ) ).
% diff_shunt_var
tff(fact_1200_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X),Y) = bot_bot_set_nat )
<=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y) ) ).
% diff_shunt_var
tff(fact_1201_mult__mono,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( aa_rat_o(ord_less_eq_rat(A),B)
=> ( aa_rat_o(ord_less_eq_rat(C),D2)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),B)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).
% mult_mono
tff(fact_1202_mult__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D2)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).
% mult_mono
tff(fact_1203_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).
% mult_mono
tff(fact_1204_mult__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).
% mult_mono
tff(fact_1205_mult__mono_H,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( aa_rat_o(ord_less_eq_rat(A),B)
=> ( aa_rat_o(ord_less_eq_rat(C),D2)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).
% mult_mono'
tff(fact_1206_mult__mono_H,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D2)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).
% mult_mono'
tff(fact_1207_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).
% mult_mono'
tff(fact_1208_mult__mono_H,axiom,
! [A: int,B: int,C: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).
% mult_mono'
tff(fact_1209_zero__le__square,axiom,
! [A: rat] : aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)) ).
% zero_le_square
tff(fact_1210_zero__le__square,axiom,
! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A)) ).
% zero_le_square
tff(fact_1211_zero__le__square,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),A)) ).
% zero_le_square
tff(fact_1212_split__mult__pos__le,axiom,
! [A: rat,B: rat] :
( ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
& aa_rat_o(ord_less_eq_rat(zero_zero_rat),B) )
| ( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
& aa_rat_o(ord_less_eq_rat(B),zero_zero_rat) ) )
=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ).
% split_mult_pos_le
tff(fact_1213_split__mult__pos__le,axiom,
! [A: code_integer,B: code_integer] :
( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) )
| ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ).
% split_mult_pos_le
tff(fact_1214_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).
% split_mult_pos_le
tff(fact_1215_mult__left__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(B),A)
=> ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).
% mult_left_mono_neg
tff(fact_1216_mult__left__mono__neg,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).
% mult_left_mono_neg
tff(fact_1217_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).
% mult_left_mono_neg
tff(fact_1218_mult__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
=> ( aa_rat_o(ord_less_eq_rat(B),zero_zero_rat)
=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).
% mult_nonpos_nonpos
tff(fact_1219_mult__nonpos__nonpos,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ) ).
% mult_nonpos_nonpos
tff(fact_1220_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).
% mult_nonpos_nonpos
tff(fact_1221_mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(A),B)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).
% mult_left_mono
tff(fact_1222_mult__left__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).
% mult_left_mono
tff(fact_1223_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).
% mult_left_mono
tff(fact_1224_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).
% mult_left_mono
tff(fact_1225_mult__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(B),A)
=> ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).
% mult_right_mono_neg
tff(fact_1226_mult__right__mono__neg,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).
% mult_right_mono_neg
tff(fact_1227_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).
% mult_right_mono_neg
tff(fact_1228_mult__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(A),B)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).
% mult_right_mono
tff(fact_1229_mult__right__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).
% mult_right_mono
tff(fact_1230_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).
% mult_right_mono
tff(fact_1231_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).
% mult_right_mono
tff(fact_1232_mult__le__0__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
<=> ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
& aa_rat_o(ord_less_eq_rat(B),zero_zero_rat) )
| ( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
& aa_rat_o(ord_less_eq_rat(zero_zero_rat),B) ) ) ) ).
% mult_le_0_iff
tff(fact_1233_mult__le__0__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger)
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) )
| ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) ) ) ).
% mult_le_0_iff
tff(fact_1234_mult__le__0__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ) ) ).
% mult_le_0_iff
tff(fact_1235_split__mult__neg__le,axiom,
! [A: rat,B: rat] :
( ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
& aa_rat_o(ord_less_eq_rat(B),zero_zero_rat) )
| ( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
& aa_rat_o(ord_less_eq_rat(zero_zero_rat),B) ) )
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ).
% split_mult_neg_le
tff(fact_1236_split__mult__neg__le,axiom,
! [A: code_integer,B: code_integer] :
( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) )
| ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ).
% split_mult_neg_le
tff(fact_1237_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat) )
| ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ).
% split_mult_neg_le
tff(fact_1238_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ).
% split_mult_neg_le
tff(fact_1239_mult__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),B)
=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).
% mult_nonneg_nonneg
tff(fact_1240_mult__nonneg__nonneg,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ) ).
% mult_nonneg_nonneg
tff(fact_1241_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).
% mult_nonneg_nonneg
tff(fact_1242_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).
% mult_nonneg_nonneg
tff(fact_1243_mult__nonneg__nonpos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(B),zero_zero_rat)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).
% mult_nonneg_nonpos
tff(fact_1244_mult__nonneg__nonpos,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).
% mult_nonneg_nonpos
tff(fact_1245_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).
% mult_nonneg_nonpos
tff(fact_1246_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).
% mult_nonneg_nonpos
tff(fact_1247_mult__nonpos__nonneg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),B)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).
% mult_nonpos_nonneg
tff(fact_1248_mult__nonpos__nonneg,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).
% mult_nonpos_nonneg
tff(fact_1249_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).
% mult_nonpos_nonneg
tff(fact_1250_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).
% mult_nonpos_nonneg
tff(fact_1251_mult__nonneg__nonpos2,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(B),zero_zero_rat)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat) ) ) ).
% mult_nonneg_nonpos2
tff(fact_1252_mult__nonneg__nonpos2,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),zero_z3403309356797280102nteger) ) ) ).
% mult_nonneg_nonpos2
tff(fact_1253_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).
% mult_nonneg_nonpos2
tff(fact_1254_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).
% mult_nonneg_nonpos2
tff(fact_1255_zero__le__mult__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
<=> ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
& aa_rat_o(ord_less_eq_rat(zero_zero_rat),B) )
| ( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
& aa_rat_o(ord_less_eq_rat(B),zero_zero_rat) ) ) ) ).
% zero_le_mult_iff
tff(fact_1256_zero__le__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) )
| ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) ) ) ).
% zero_le_mult_iff
tff(fact_1257_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) ) ) ).
% zero_le_mult_iff
tff(fact_1258_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(A),B)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1259_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1260_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1261_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1262_not__one__le__zero,axiom,
~ aa_rat_o(ord_less_eq_rat(one_one_rat),zero_zero_rat) ).
% not_one_le_zero
tff(fact_1263_not__one__le__zero,axiom,
~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),zero_z3403309356797280102nteger) ).
% not_one_le_zero
tff(fact_1264_not__one__le__zero,axiom,
~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),zero_zero_nat) ).
% not_one_le_zero
tff(fact_1265_not__one__le__zero,axiom,
~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),zero_zero_int) ).
% not_one_le_zero
tff(fact_1266_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
aa_rat_o(ord_less_eq_rat(zero_zero_rat),one_one_rat) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1267_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),one_one_Code_integer) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1268_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1269_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1270_zero__less__one__class_Ozero__le__one,axiom,
aa_rat_o(ord_less_eq_rat(zero_zero_rat),one_one_rat) ).
% zero_less_one_class.zero_le_one
tff(fact_1271_zero__less__one__class_Ozero__le__one,axiom,
aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),one_one_Code_integer) ).
% zero_less_one_class.zero_le_one
tff(fact_1272_zero__less__one__class_Ozero__le__one,axiom,
aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).
% zero_less_one_class.zero_le_one
tff(fact_1273_zero__less__one__class_Ozero__le__one,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).
% zero_less_one_class.zero_le_one
tff(fact_1274_mult__left__le,axiom,
! [C: rat,A: rat] :
( aa_rat_o(ord_less_eq_rat(C),one_one_rat)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),A) ) ) ).
% mult_left_le
tff(fact_1275_mult__left__le,axiom,
! [C: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),A) ) ) ).
% mult_left_le
tff(fact_1276_mult__left__le,axiom,
! [C: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A) ) ) ).
% mult_left_le
tff(fact_1277_mult__left__le,axiom,
! [C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A) ) ) ).
% mult_left_le
tff(fact_1278_mult__le__one,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(A),one_one_rat)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),B)
=> ( aa_rat_o(ord_less_eq_rat(B),one_one_rat)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),one_one_rat) ) ) ) ).
% mult_le_one
tff(fact_1279_mult__le__one,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer) ) ) ) ).
% mult_le_one
tff(fact_1280_mult__le__one,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ) ).
% mult_le_one
tff(fact_1281_mult__le__one,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ) ).
% mult_le_one
tff(fact_1282_mult__right__le__one__le,axiom,
! [X: rat,Y: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),X)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),Y)
=> ( aa_rat_o(ord_less_eq_rat(Y),one_one_rat)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)),X) ) ) ) ).
% mult_right_le_one_le
tff(fact_1283_mult__right__le__one__le,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y)),X) ) ) ) ).
% mult_right_le_one_le
tff(fact_1284_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),X) ) ) ) ).
% mult_right_le_one_le
tff(fact_1285_mult__left__le__one__le,axiom,
! [X: rat,Y: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),X)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),Y)
=> ( aa_rat_o(ord_less_eq_rat(Y),one_one_rat)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X)),X) ) ) ) ).
% mult_left_le_one_le
tff(fact_1286_mult__left__le__one__le,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)),X) ) ) ) ).
% mult_left_le_one_le
tff(fact_1287_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)),X) ) ) ) ).
% mult_left_le_one_le
tff(fact_1288_le__minus__one__simps_I1_J,axiom,
aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).
% le_minus_one_simps(1)
tff(fact_1289_le__minus__one__simps_I1_J,axiom,
aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).
% le_minus_one_simps(1)
tff(fact_1290_le__minus__one__simps_I1_J,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).
% le_minus_one_simps(1)
tff(fact_1291_le__minus__one__simps_I3_J,axiom,
~ aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).
% le_minus_one_simps(3)
tff(fact_1292_le__minus__one__simps_I3_J,axiom,
~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% le_minus_one_simps(3)
tff(fact_1293_le__minus__one__simps_I3_J,axiom,
~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).
% le_minus_one_simps(3)
tff(fact_1294_left__diff__distrib,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).
% left_diff_distrib
tff(fact_1295_left__diff__distrib,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).
% left_diff_distrib
tff(fact_1296_left__diff__distrib,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).
% left_diff_distrib
tff(fact_1297_right__diff__distrib,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ).
% right_diff_distrib
tff(fact_1298_right__diff__distrib,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).
% right_diff_distrib
tff(fact_1299_right__diff__distrib,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).
% right_diff_distrib
tff(fact_1300_left__diff__distrib_H,axiom,
! [B: code_integer,C: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)),A) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)) ).
% left_diff_distrib'
tff(fact_1301_left__diff__distrib_H,axiom,
! [B: rat,C: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(B,C)),A) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)) ).
% left_diff_distrib'
tff(fact_1302_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ).
% left_diff_distrib'
tff(fact_1303_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)) ).
% left_diff_distrib'
tff(fact_1304_right__diff__distrib_H,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ).
% right_diff_distrib'
tff(fact_1305_right__diff__distrib_H,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).
% right_diff_distrib'
tff(fact_1306_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).
% right_diff_distrib'
tff(fact_1307_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).
% right_diff_distrib'
tff(fact_1308_minus__diff__commute,axiom,
! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,B)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),B) ).
% minus_diff_commute
tff(fact_1309_minus__diff__commute,axiom,
! [B: rat,A: rat] : minus_minus_rat(aa_rat_rat(uminus_uminus_rat,B),A) = minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A),B) ).
% minus_diff_commute
tff(fact_1310_minus__diff__commute,axiom,
! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ).
% minus_diff_commute
tff(fact_1311_le__numeral__extra_I4_J,axiom,
aa_rat_o(ord_less_eq_rat(one_one_rat),one_one_rat) ).
% le_numeral_extra(4)
tff(fact_1312_le__numeral__extra_I4_J,axiom,
aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),one_one_Code_integer) ).
% le_numeral_extra(4)
tff(fact_1313_le__numeral__extra_I4_J,axiom,
aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),one_one_nat) ).
% le_numeral_extra(4)
tff(fact_1314_le__numeral__extra_I4_J,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),one_one_int) ).
% le_numeral_extra(4)
tff(fact_1315_bot_Oextremum__uniqueI,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A),bot_bo2099793752762293965at_nat)
=> ( A = bot_bo2099793752762293965at_nat ) ) ).
% bot.extremum_uniqueI
tff(fact_1316_bot_Oextremum__uniqueI,axiom,
! [A: set_o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),bot_bot_set_o)
=> ( A = bot_bot_set_o ) ) ).
% bot.extremum_uniqueI
tff(fact_1317_bot_Oextremum__uniqueI,axiom,
! [A: set_int] :
( aa_set_int_o(ord_less_eq_set_int(A),bot_bot_set_int)
=> ( A = bot_bot_set_int ) ) ).
% bot.extremum_uniqueI
tff(fact_1318_bot_Oextremum__uniqueI,axiom,
! [A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),bot_bot_assn)
=> ( A = bot_bot_assn ) ) ).
% bot.extremum_uniqueI
tff(fact_1319_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),bot_bot_set_nat)
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
tff(fact_1320_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
tff(fact_1321_bot_Oextremum__unique,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A),bot_bo2099793752762293965at_nat)
<=> ( A = bot_bo2099793752762293965at_nat ) ) ).
% bot.extremum_unique
tff(fact_1322_bot_Oextremum__unique,axiom,
! [A: set_o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),bot_bot_set_o)
<=> ( A = bot_bot_set_o ) ) ).
% bot.extremum_unique
tff(fact_1323_bot_Oextremum__unique,axiom,
! [A: set_int] :
( aa_set_int_o(ord_less_eq_set_int(A),bot_bot_set_int)
<=> ( A = bot_bot_set_int ) ) ).
% bot.extremum_unique
tff(fact_1324_bot_Oextremum__unique,axiom,
! [A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),bot_bot_assn)
<=> ( A = bot_bot_assn ) ) ).
% bot.extremum_unique
tff(fact_1325_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),bot_bot_set_nat)
<=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
tff(fact_1326_bot_Oextremum__unique,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
<=> ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
tff(fact_1327_bot_Oextremum,axiom,
! [A: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(bot_bo2099793752762293965at_nat),A) ).
% bot.extremum
tff(fact_1328_bot_Oextremum,axiom,
! [A: set_o] : aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,bot_bot_set_o),A) ).
% bot.extremum
tff(fact_1329_bot_Oextremum,axiom,
! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A) ).
% bot.extremum
tff(fact_1330_bot_Oextremum,axiom,
! [A: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,bot_bot_assn),A) ).
% bot.extremum
tff(fact_1331_bot_Oextremum,axiom,
! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A) ).
% bot.extremum
tff(fact_1332_bot_Oextremum,axiom,
! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,bot_bot_nat),A) ).
% bot.extremum
tff(fact_1333_subrelI,axiom,
! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ! [X2: nat,Y2: nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2)),R2)
=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2)),S2) )
=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(R2),S2) ) ).
% subrelI
tff(fact_1334_subrelI,axiom,
! [R2: set_Pr3486312109254157132nteger,S2: set_Pr3486312109254157132nteger] :
( ! [X2: fun_Pr5098669352780829690e_term,Y2: produc8923325533196201883nteger] :
( member8978266922497061677nteger(produc565448451787809096nteger(X2,Y2),R2)
=> member8978266922497061677nteger(produc565448451787809096nteger(X2,Y2),S2) )
=> ord_le2932375828722672556nteger(R2,S2) ) ).
% subrelI
tff(fact_1335_subrelI,axiom,
! [R2: set_Pr1312406793078187350et_nat,S2: set_Pr1312406793078187350et_nat] :
( ! [X2: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat] :
( member4513763875858039863et_nat(produc6017135503889078682et_nat(X2,Y2),R2)
=> member4513763875858039863et_nat(produc6017135503889078682et_nat(X2,Y2),S2) )
=> ord_le3919382008627836342et_nat(R2,S2) ) ).
% subrelI
tff(fact_1336_subrelI,axiom,
! [R2: set_Pr4040394263813967220et_nat,S2: set_Pr4040394263813967220et_nat] :
( ! [X2: fun_Pr6460944699146269935_nat_o,Y2: produc8799533983703829792et_nat] :
( member9174161442226468181et_nat(produc7291855603632349880et_nat(X2,Y2),R2)
=> member9174161442226468181et_nat(produc7291855603632349880et_nat(X2,Y2),S2) )
=> ord_le5189643068899110356et_nat(R2,S2) ) ).
% subrelI
tff(fact_1337_subrelI,axiom,
! [R2: set_Pr6700895561065273913nt_int,S2: set_Pr6700895561065273913nt_int] :
( ! [X2: fun_Pr7763680713840462443e_term,Y2: product_prod_int_int] :
( member753276645541486746nt_int(produc7320969046608631221nt_int(X2,Y2),R2)
=> member753276645541486746nt_int(produc7320969046608631221nt_int(X2,Y2),S2) )
=> ord_le3990897393281686169nt_int(R2,S2) ) ).
% subrelI
tff(fact_1338_subrelI,axiom,
! [R2: set_Pr4166680992137330533nt_int,S2: set_Pr4166680992137330533nt_int] :
( ! [X2: fun_in5730379987311621695e_term,Y2: product_prod_int_int] :
( member5890090881338784710nt_int(produc4729913460834754785nt_int(X2,Y2),R2)
=> member5890090881338784710nt_int(produc4729913460834754785nt_int(X2,Y2),S2) )
=> ord_le3197784583289274309nt_int(R2,S2) ) ).
% subrelI
tff(fact_1339_le__minus__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(uminus_uminus_rat,B))
<=> aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(uminus_uminus_rat,A)) ) ).
% le_minus_iff
tff(fact_1340_le__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).
% le_minus_iff
tff(fact_1341_le__minus__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A)) ) ).
% le_minus_iff
tff(fact_1342_minus__le__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,A)),B)
<=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,B)),A) ) ).
% minus_le_iff
tff(fact_1343_minus__le__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).
% minus_le_iff
tff(fact_1344_minus__le__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A) ) ).
% minus_le_iff
tff(fact_1345_le__imp__neg__le,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(A),B)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).
% le_imp_neg_le
tff(fact_1346_le__imp__neg__le,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).
% le_imp_neg_le
tff(fact_1347_le__imp__neg__le,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).
% le_imp_neg_le
tff(fact_1348_compl__mono,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(uminus_uminus_assn,Y)),aa_assn_assn(uminus_uminus_assn,X)) ) ).
% compl_mono
tff(fact_1349_compl__mono,axiom,
! [X: set_nat,Y: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)) ) ).
% compl_mono
tff(fact_1350_compl__le__swap1,axiom,
! [Y: assn,X: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),aa_assn_assn(uminus_uminus_assn,X))
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(uminus_uminus_assn,Y)) ) ).
% compl_le_swap1
tff(fact_1351_compl__le__swap1,axiom,
! [Y: set_nat,X: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),aa_set_nat_set_nat(uminus5710092332889474511et_nat,X))
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)) ) ).
% compl_le_swap1
tff(fact_1352_compl__le__swap2,axiom,
! [Y: assn,X: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(uminus_uminus_assn,Y)),X)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(uminus_uminus_assn,X)),Y) ) ).
% compl_le_swap2
tff(fact_1353_compl__le__swap2,axiom,
! [Y: set_nat,X: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)),X)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),Y) ) ).
% compl_le_swap2
tff(fact_1354_top_Oextremum__uniqueI,axiom,
! [A: set_o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,top_top_set_o),A)
=> ( A = top_top_set_o ) ) ).
% top.extremum_uniqueI
tff(fact_1355_top_Oextremum__uniqueI,axiom,
! [A: set_Product_unit] :
( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,top_to1996260823553986621t_unit),A)
=> ( A = top_to1996260823553986621t_unit ) ) ).
% top.extremum_uniqueI
tff(fact_1356_top_Oextremum__uniqueI,axiom,
! [A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,top_top_assn),A)
=> ( A = top_top_assn ) ) ).
% top.extremum_uniqueI
tff(fact_1357_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,top_top_set_nat),A)
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
tff(fact_1358_top_Oextremum__unique,axiom,
! [A: set_o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,top_top_set_o),A)
<=> ( A = top_top_set_o ) ) ).
% top.extremum_unique
tff(fact_1359_top_Oextremum__unique,axiom,
! [A: set_Product_unit] :
( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,top_to1996260823553986621t_unit),A)
<=> ( A = top_to1996260823553986621t_unit ) ) ).
% top.extremum_unique
tff(fact_1360_top_Oextremum__unique,axiom,
! [A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,top_top_assn),A)
<=> ( A = top_top_assn ) ) ).
% top.extremum_unique
tff(fact_1361_top_Oextremum__unique,axiom,
! [A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,top_top_set_nat),A)
<=> ( A = top_top_set_nat ) ) ).
% top.extremum_unique
tff(fact_1362_top__greatest,axiom,
! [A: set_o] : aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),top_top_set_o) ).
% top_greatest
tff(fact_1363_top__greatest,axiom,
! [A: set_Product_unit] : aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),top_to1996260823553986621t_unit) ).
% top_greatest
tff(fact_1364_top__greatest,axiom,
! [A: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),top_top_assn) ).
% top_greatest
tff(fact_1365_top__greatest,axiom,
! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),top_top_set_nat) ).
% top_greatest
tff(fact_1366_inf__sup__ord_I2_J,axiom,
! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Y) ).
% inf_sup_ord(2)
tff(fact_1367_inf__sup__ord_I2_J,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y) ).
% inf_sup_ord(2)
tff(fact_1368_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Y) ).
% inf_sup_ord(2)
tff(fact_1369_inf__sup__ord_I2_J,axiom,
! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),Y) ).
% inf_sup_ord(2)
tff(fact_1370_inf__sup__ord_I1_J,axiom,
! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),X) ).
% inf_sup_ord(1)
tff(fact_1371_inf__sup__ord_I1_J,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),X) ).
% inf_sup_ord(1)
tff(fact_1372_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),X) ).
% inf_sup_ord(1)
tff(fact_1373_inf__sup__ord_I1_J,axiom,
! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),X) ).
% inf_sup_ord(1)
tff(fact_1374_inf__le1,axiom,
! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),X) ).
% inf_le1
tff(fact_1375_inf__le1,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),X) ).
% inf_le1
tff(fact_1376_inf__le1,axiom,
! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),X) ).
% inf_le1
tff(fact_1377_inf__le1,axiom,
! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),X) ).
% inf_le1
tff(fact_1378_inf__le2,axiom,
! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Y) ).
% inf_le2
tff(fact_1379_inf__le2,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y) ).
% inf_le2
tff(fact_1380_inf__le2,axiom,
! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Y) ).
% inf_le2
tff(fact_1381_inf__le2,axiom,
! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),Y) ).
% inf_le2
tff(fact_1382_le__infE,axiom,
! [X: assn,A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B))
=> ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),A)
=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),B) ) ) ).
% le_infE
tff(fact_1383_le__infE,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B))
=> ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A)
=> ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),B) ) ) ).
% le_infE
tff(fact_1384_le__infE,axiom,
! [X: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B))
=> ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),B) ) ) ).
% le_infE
tff(fact_1385_le__infE,axiom,
! [X: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B))
=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),B) ) ) ).
% le_infE
tff(fact_1386_le__infI,axiom,
! [X: assn,A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),A)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),B)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) ) ) ).
% le_infI
tff(fact_1387_le__infI,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),B)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).
% le_infI
tff(fact_1388_le__infI,axiom,
! [X: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) ) ) ).
% le_infI
tff(fact_1389_le__infI,axiom,
! [X: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)) ) ) ).
% le_infI
tff(fact_1390_inf__mono,axiom,
! [A: assn,C: assn,B: assn,D2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),D2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,C),D2)) ) ) ).
% inf_mono
tff(fact_1391_inf__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),D2)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C),D2)) ) ) ).
% inf_mono
tff(fact_1392_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,C),D2)) ) ) ).
% inf_mono
tff(fact_1393_inf__mono,axiom,
! [A: int,C: int,B: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),aa_int_int(aa_int_fun_int_int(inf_inf_int,C),D2)) ) ) ).
% inf_mono
tff(fact_1394_le__infI1,axiom,
! [A: assn,X: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),X)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).
% le_infI1
tff(fact_1395_le__infI1,axiom,
! [A: set_nat,X: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).
% le_infI1
tff(fact_1396_le__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).
% le_infI1
tff(fact_1397_le__infI1,axiom,
! [A: int,X: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).
% le_infI1
tff(fact_1398_le__infI2,axiom,
! [B: assn,X: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),X)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).
% le_infI2
tff(fact_1399_le__infI2,axiom,
! [B: set_nat,X: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),X)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).
% le_infI2
tff(fact_1400_le__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),X)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).
% le_infI2
tff(fact_1401_le__infI2,axiom,
! [B: int,X: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),X)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).
% le_infI2
tff(fact_1402_inf_OorderE,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).
% inf.orderE
tff(fact_1403_inf_OorderE,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ).
% inf.orderE
tff(fact_1404_inf_OorderE,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ) ).
% inf.orderE
tff(fact_1405_inf_OorderE,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ) ) ).
% inf.orderE
tff(fact_1406_inf_OorderI,axiom,
! [A: assn,B: assn] :
( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B) ) ).
% inf.orderI
tff(fact_1407_inf_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) )
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ).
% inf.orderI
tff(fact_1408_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).
% inf.orderI
tff(fact_1409_inf_OorderI,axiom,
! [A: int,B: int] :
( ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% inf.orderI
tff(fact_1410_inf__unique,axiom,
! [F: fun_as3983628580203722482n_assn,X: assn,Y: assn] :
( ! [X2: assn,Y2: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(F,X2),Y2)),X2)
=> ( ! [X2: assn,Y2: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(F,X2),Y2)),Y2)
=> ( ! [X2: assn,Y2: assn,Z2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Z2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),aa_assn_assn(aa_ass954072293579886259n_assn(F,Y2),Z2)) ) )
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(F,X),Y) ) ) ) ) ).
% inf_unique
tff(fact_1411_inf__unique,axiom,
! [F: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y2)),X2)
=> ( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y2)),Y2)
=> ( ! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),Y2)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),Z2)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y2),Z2)) ) )
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X),Y) ) ) ) ) ).
% inf_unique
tff(fact_1412_inf__unique,axiom,
! [F: fun_nat_fun_nat_nat,X: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y2)),X2)
=> ( ! [X2: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y2)),Y2)
=> ( ! [X2: nat,Y2: nat,Z2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Z2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(F,Y2),Z2)) ) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F,X),Y) ) ) ) ) ).
% inf_unique
tff(fact_1413_inf__unique,axiom,
! [F: fun_int_fun_int_int,X: int,Y: int] :
( ! [X2: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X2),Y2)),X2)
=> ( ! [X2: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X2),Y2)),Y2)
=> ( ! [X2: int,Y2: int,Z2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Z2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),aa_int_int(aa_int_fun_int_int(F,Y2),Z2)) ) )
=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = aa_int_int(aa_int_fun_int_int(F,X),Y) ) ) ) ) ).
% inf_unique
tff(fact_1414_le__iff__inf,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
<=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = X ) ) ).
% le_iff_inf
tff(fact_1415_le__iff__inf,axiom,
! [X: set_nat,Y: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
<=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = X ) ) ).
% le_iff_inf
tff(fact_1416_le__iff__inf,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = X ) ) ).
% le_iff_inf
tff(fact_1417_le__iff__inf,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
<=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = X ) ) ).
% le_iff_inf
tff(fact_1418_inf_Oabsorb1,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).
% inf.absorb1
tff(fact_1419_inf_Oabsorb1,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).
% inf.absorb1
tff(fact_1420_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = A ) ) ).
% inf.absorb1
tff(fact_1421_inf_Oabsorb1,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = A ) ) ).
% inf.absorb1
tff(fact_1422_inf_Oabsorb2,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).
% inf.absorb2
tff(fact_1423_inf_Oabsorb2,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).
% inf.absorb2
tff(fact_1424_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = B ) ) ).
% inf.absorb2
tff(fact_1425_inf_Oabsorb2,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = B ) ) ).
% inf.absorb2
tff(fact_1426_inf__absorb1,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = X ) ) ).
% inf_absorb1
tff(fact_1427_inf__absorb1,axiom,
! [X: set_nat,Y: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = X ) ) ).
% inf_absorb1
tff(fact_1428_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = X ) ) ).
% inf_absorb1
tff(fact_1429_inf__absorb1,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = X ) ) ).
% inf_absorb1
tff(fact_1430_inf__absorb2,axiom,
! [Y: assn,X: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = Y ) ) ).
% inf_absorb2
tff(fact_1431_inf__absorb2,axiom,
! [Y: set_nat,X: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = Y ) ) ).
% inf_absorb2
tff(fact_1432_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = Y ) ) ).
% inf_absorb2
tff(fact_1433_inf__absorb2,axiom,
! [Y: int,X: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = Y ) ) ).
% inf_absorb2
tff(fact_1434_inf_OboundedE,axiom,
! [A: assn,B: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C))
=> ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C) ) ) ).
% inf.boundedE
tff(fact_1435_inf_OboundedE,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C))
=> ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
=> ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C) ) ) ).
% inf.boundedE
tff(fact_1436_inf_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C))
=> ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).
% inf.boundedE
tff(fact_1437_inf_OboundedE,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C))
=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).
% inf.boundedE
tff(fact_1438_inf_OboundedI,axiom,
! [A: assn,B: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C)) ) ) ).
% inf.boundedI
tff(fact_1439_inf_OboundedI,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) ) ) ).
% inf.boundedI
tff(fact_1440_inf_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C)) ) ) ).
% inf.boundedI
tff(fact_1441_inf_OboundedI,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C)) ) ) ).
% inf.boundedI
tff(fact_1442_inf__greatest,axiom,
! [X: assn,Y: assn,Z: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Z)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ) ) ).
% inf_greatest
tff(fact_1443_inf__greatest,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) ) ) ).
% inf_greatest
tff(fact_1444_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) ) ) ).
% inf_greatest
tff(fact_1445_inf__greatest,axiom,
! [X: int,Y: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)) ) ) ).
% inf_greatest
tff(fact_1446_inf_Oorder__iff,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
<=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).
% inf.order_iff
tff(fact_1447_inf_Oorder__iff,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
<=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ).
% inf.order_iff
tff(fact_1448_inf_Oorder__iff,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
<=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ) ).
% inf.order_iff
tff(fact_1449_inf_Oorder__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
<=> ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ) ) ).
% inf.order_iff
tff(fact_1450_inf_Ocobounded1,axiom,
! [A: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),A) ).
% inf.cobounded1
tff(fact_1451_inf_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),A) ).
% inf.cobounded1
tff(fact_1452_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),A) ).
% inf.cobounded1
tff(fact_1453_inf_Ocobounded1,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),A) ).
% inf.cobounded1
tff(fact_1454_inf_Ocobounded2,axiom,
! [A: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),B) ).
% inf.cobounded2
tff(fact_1455_inf_Ocobounded2,axiom,
! [A: set_nat,B: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),B) ).
% inf.cobounded2
tff(fact_1456_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),B) ).
% inf.cobounded2
tff(fact_1457_inf_Ocobounded2,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),B) ).
% inf.cobounded2
tff(fact_1458_inf_Oabsorb__iff1,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
<=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).
% inf.absorb_iff1
tff(fact_1459_inf_Oabsorb__iff1,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
<=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).
% inf.absorb_iff1
tff(fact_1460_inf_Oabsorb__iff1,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = A ) ) ).
% inf.absorb_iff1
tff(fact_1461_inf_Oabsorb__iff1,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
<=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = A ) ) ).
% inf.absorb_iff1
tff(fact_1462_inf_Oabsorb__iff2,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
<=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).
% inf.absorb_iff2
tff(fact_1463_inf_Oabsorb__iff2,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
<=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).
% inf.absorb_iff2
tff(fact_1464_inf_Oabsorb__iff2,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = B ) ) ).
% inf.absorb_iff2
tff(fact_1465_inf_Oabsorb__iff2,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
<=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = B ) ) ).
% inf.absorb_iff2
tff(fact_1466_inf_OcoboundedI1,axiom,
! [A: assn,C: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C) ) ).
% inf.coboundedI1
tff(fact_1467_inf_OcoboundedI1,axiom,
! [A: set_nat,C: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).
% inf.coboundedI1
tff(fact_1468_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).
% inf.coboundedI1
tff(fact_1469_inf_OcoboundedI1,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).
% inf.coboundedI1
tff(fact_1470_inf_OcoboundedI2,axiom,
! [B: assn,C: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C) ) ).
% inf.coboundedI2
tff(fact_1471_inf_OcoboundedI2,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).
% inf.coboundedI2
tff(fact_1472_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).
% inf.coboundedI2
tff(fact_1473_inf_OcoboundedI2,axiom,
! [B: int,C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).
% inf.coboundedI2
tff(fact_1474_inf__sup__ord_I4_J,axiom,
! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).
% inf_sup_ord(4)
tff(fact_1475_inf__sup__ord_I4_J,axiom,
! [Y: assn,X: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) ).
% inf_sup_ord(4)
tff(fact_1476_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).
% inf_sup_ord(4)
tff(fact_1477_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).
% inf_sup_ord(4)
tff(fact_1478_inf__sup__ord_I4_J,axiom,
! [Y: int,X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).
% inf_sup_ord(4)
tff(fact_1479_inf__sup__ord_I3_J,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).
% inf_sup_ord(3)
tff(fact_1480_inf__sup__ord_I3_J,axiom,
! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) ).
% inf_sup_ord(3)
tff(fact_1481_inf__sup__ord_I3_J,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).
% inf_sup_ord(3)
tff(fact_1482_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).
% inf_sup_ord(3)
tff(fact_1483_inf__sup__ord_I3_J,axiom,
! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).
% inf_sup_ord(3)
tff(fact_1484_le__supE,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),X)
=> ~ ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),X)
=> ~ aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),X) ) ) ).
% le_supE
tff(fact_1485_le__supE,axiom,
! [A: assn,B: assn,X: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),X)
=> ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),X)
=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),X) ) ) ).
% le_supE
tff(fact_1486_le__supE,axiom,
! [A: set_nat,B: set_nat,X: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),X)
=> ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X)
=> ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),X) ) ) ).
% le_supE
tff(fact_1487_le__supE,axiom,
! [A: nat,B: nat,X: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),X)
=> ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),X) ) ) ).
% le_supE
tff(fact_1488_le__supE,axiom,
! [A: int,B: int,X: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),X)
=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),X) ) ) ).
% le_supE
tff(fact_1489_le__supI,axiom,
! [A: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),X)
=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),X)
=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),X) ) ) ).
% le_supI
tff(fact_1490_le__supI,axiom,
! [A: assn,X: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),X)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),X)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),X) ) ) ).
% le_supI
tff(fact_1491_le__supI,axiom,
! [A: set_nat,X: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),X)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),X) ) ) ).
% le_supI
tff(fact_1492_le__supI,axiom,
! [A: nat,X: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),X)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),X) ) ) ).
% le_supI
tff(fact_1493_le__supI,axiom,
! [A: int,X: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),X)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),X) ) ) ).
% le_supI
tff(fact_1494_sup__ge1,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).
% sup_ge1
tff(fact_1495_sup__ge1,axiom,
! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) ).
% sup_ge1
tff(fact_1496_sup__ge1,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).
% sup_ge1
tff(fact_1497_sup__ge1,axiom,
! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).
% sup_ge1
tff(fact_1498_sup__ge1,axiom,
! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).
% sup_ge1
tff(fact_1499_sup__ge2,axiom,
! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).
% sup_ge2
tff(fact_1500_sup__ge2,axiom,
! [Y: assn,X: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) ).
% sup_ge2
tff(fact_1501_sup__ge2,axiom,
! [Y: set_nat,X: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).
% sup_ge2
tff(fact_1502_sup__ge2,axiom,
! [Y: nat,X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).
% sup_ge2
tff(fact_1503_sup__ge2,axiom,
! [Y: int,X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).
% sup_ge2
tff(fact_1504_le__supI1,axiom,
! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),A)
=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).
% le_supI1
tff(fact_1505_le__supI1,axiom,
! [X: assn,A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),A)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).
% le_supI1
tff(fact_1506_le__supI1,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).
% le_supI1
tff(fact_1507_le__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).
% le_supI1
tff(fact_1508_le__supI1,axiom,
! [X: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).
% le_supI1
tff(fact_1509_le__supI2,axiom,
! [X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),B)
=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).
% le_supI2
tff(fact_1510_le__supI2,axiom,
! [X: assn,B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),B)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).
% le_supI2
tff(fact_1511_le__supI2,axiom,
! [X: set_nat,B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),B)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).
% le_supI2
tff(fact_1512_le__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).
% le_supI2
tff(fact_1513_le__supI2,axiom,
! [X: int,B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).
% le_supI2
tff(fact_1514_sup_Omono,axiom,
! [C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),A)
=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(D2),B)
=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C),D2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ) ).
% sup.mono
tff(fact_1515_sup_Omono,axiom,
! [C: assn,A: assn,D2: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),A)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,D2),B)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,C),D2)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ) ).
% sup.mono
tff(fact_1516_sup_Omono,axiom,
! [C: set_nat,A: set_nat,D2: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,D2),B)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),D2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).
% sup.mono
tff(fact_1517_sup_Omono,axiom,
! [C: nat,A: nat,D2: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D2),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C),D2)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ) ).
% sup.mono
tff(fact_1518_sup_Omono,axiom,
! [C: int,A: int,D2: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D2),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,C),D2)),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ) ).
% sup.mono
tff(fact_1519_sup__mono,axiom,
! [A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),C)
=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),D2)
=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C),D2)) ) ) ).
% sup_mono
tff(fact_1520_sup__mono,axiom,
! [A: assn,C: assn,B: assn,D2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),D2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,C),D2)) ) ) ).
% sup_mono
tff(fact_1521_sup__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),D2)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),D2)) ) ) ).
% sup_mono
tff(fact_1522_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C),D2)) ) ) ).
% sup_mono
tff(fact_1523_sup__mono,axiom,
! [A: int,C: int,B: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),aa_int_int(aa_int_fun_int_int(sup_sup_int,C),D2)) ) ) ).
% sup_mono
tff(fact_1524_sup__least,axiom,
! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),X)
=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Z),X)
=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)),X) ) ) ).
% sup_least
tff(fact_1525_sup__least,axiom,
! [Y: assn,X: assn,Z: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Z),X)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)),X) ) ) ).
% sup_least
tff(fact_1526_sup__least,axiom,
! [Y: set_nat,X: set_nat,Z: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z),X)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),X) ) ) ).
% sup_least
tff(fact_1527_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)),X) ) ) ).
% sup_least
tff(fact_1528_sup__least,axiom,
! [Y: int,X: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)),X) ) ) ).
% sup_least
tff(fact_1529_le__iff__sup,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),Y)
<=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = Y ) ) ).
% le_iff_sup
tff(fact_1530_le__iff__sup,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
<=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = Y ) ) ).
% le_iff_sup
tff(fact_1531_le__iff__sup,axiom,
! [X: set_nat,Y: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
<=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = Y ) ) ).
% le_iff_sup
tff(fact_1532_le__iff__sup,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = Y ) ) ).
% le_iff_sup
tff(fact_1533_le__iff__sup,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
<=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = Y ) ) ).
% le_iff_sup
tff(fact_1534_sup_OorderE,axiom,
! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
=> ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ) ) ).
% sup.orderE
tff(fact_1535_sup_OorderE,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ) ) ).
% sup.orderE
tff(fact_1536_sup_OorderE,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).
% sup.orderE
tff(fact_1537_sup_OorderE,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ) ) ).
% sup.orderE
tff(fact_1538_sup_OorderE,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ) ) ).
% sup.orderE
tff(fact_1539_sup_OorderI,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) )
=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A) ) ).
% sup.orderI
tff(fact_1540_sup_OorderI,axiom,
! [A: assn,B: assn] :
( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A) ) ).
% sup.orderI
tff(fact_1541_sup_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A) ) ).
% sup.orderI
tff(fact_1542_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).
% sup.orderI
tff(fact_1543_sup_OorderI,axiom,
! [A: int,B: int] :
( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).
% sup.orderI
tff(fact_1544_sup__unique,axiom,
! [F: fun_se177437144654672861at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,X2),Y2))
=> ( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,X2),Y2))
=> ( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y2),X2)
=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Z2),X2)
=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,Y2),Z2)),X2) ) )
=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F,X),Y) ) ) ) ) ).
% sup_unique
tff(fact_1545_sup__unique,axiom,
! [F: fun_as3983628580203722482n_assn,X: assn,Y: assn] :
( ! [X2: assn,Y2: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),aa_assn_assn(aa_ass954072293579886259n_assn(F,X2),Y2))
=> ( ! [X2: assn,Y2: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y2),aa_assn_assn(aa_ass954072293579886259n_assn(F,X2),Y2))
=> ( ! [X2: assn,Y2: assn,Z2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y2),X2)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Z2),X2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(F,Y2),Z2)),X2) ) )
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(F,X),Y) ) ) ) ) ).
% sup_unique
tff(fact_1546_sup__unique,axiom,
! [F: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y2))
=> ( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y2))
=> ( ! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y2),X2)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z2),X2)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y2),Z2)),X2) ) )
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X),Y) ) ) ) ) ).
% sup_unique
tff(fact_1547_sup__unique,axiom,
! [F: fun_nat_fun_nat_nat,X: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y2))
=> ( ! [X2: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y2))
=> ( ! [X2: nat,Y2: nat,Z2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X2)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z2),X2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,Y2),Z2)),X2) ) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F,X),Y) ) ) ) ) ).
% sup_unique
tff(fact_1548_sup__unique,axiom,
! [F: fun_int_fun_int_int,X: int,Y: int] :
( ! [X2: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),aa_int_int(aa_int_fun_int_int(F,X2),Y2))
=> ( ! [X2: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),aa_int_int(aa_int_fun_int_int(F,X2),Y2))
=> ( ! [X2: int,Y2: int,Z2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),X2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,Y2),Z2)),X2) ) )
=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = aa_int_int(aa_int_fun_int_int(F,X),Y) ) ) ) ) ).
% sup_unique
tff(fact_1549_sup_Oabsorb1,axiom,
! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = A ) ) ).
% sup.absorb1
tff(fact_1550_sup_Oabsorb1,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = A ) ) ).
% sup.absorb1
tff(fact_1551_sup_Oabsorb1,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).
% sup.absorb1
tff(fact_1552_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).
% sup.absorb1
tff(fact_1553_sup_Oabsorb1,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).
% sup.absorb1
tff(fact_1554_sup_Oabsorb2,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),B)
=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = B ) ) ).
% sup.absorb2
tff(fact_1555_sup_Oabsorb2,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = B ) ) ).
% sup.absorb2
tff(fact_1556_sup_Oabsorb2,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).
% sup.absorb2
tff(fact_1557_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).
% sup.absorb2
tff(fact_1558_sup_Oabsorb2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).
% sup.absorb2
tff(fact_1559_sup__absorb1,axiom,
! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),X)
=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = X ) ) ).
% sup_absorb1
tff(fact_1560_sup__absorb1,axiom,
! [Y: assn,X: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = X ) ) ).
% sup_absorb1
tff(fact_1561_sup__absorb1,axiom,
! [Y: set_nat,X: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = X ) ) ).
% sup_absorb1
tff(fact_1562_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = X ) ) ).
% sup_absorb1
tff(fact_1563_sup__absorb1,axiom,
! [Y: int,X: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = X ) ) ).
% sup_absorb1
tff(fact_1564_sup__absorb2,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),Y)
=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = Y ) ) ).
% sup_absorb2
tff(fact_1565_sup__absorb2,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = Y ) ) ).
% sup_absorb2
tff(fact_1566_sup__absorb2,axiom,
! [X: set_nat,Y: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = Y ) ) ).
% sup_absorb2
tff(fact_1567_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = Y ) ) ).
% sup_absorb2
tff(fact_1568_sup__absorb2,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = Y ) ) ).
% sup_absorb2
tff(fact_1569_sup_OboundedE,axiom,
! [B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C)),A)
=> ~ ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
=> ~ aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),A) ) ) ).
% sup.boundedE
tff(fact_1570_sup_OboundedE,axiom,
! [B: assn,C: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C)),A)
=> ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),A) ) ) ).
% sup.boundedE
tff(fact_1571_sup_OboundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A)
=> ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
=> ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A) ) ) ).
% sup.boundedE
tff(fact_1572_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A)
=> ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).
% sup.boundedE
tff(fact_1573_sup_OboundedE,axiom,
! [B: int,C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A)
=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).
% sup.boundedE
tff(fact_1574_sup_OboundedI,axiom,
! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),A)
=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C)),A) ) ) ).
% sup.boundedI
tff(fact_1575_sup_OboundedI,axiom,
! [B: assn,A: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),A)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C)),A) ) ) ).
% sup.boundedI
tff(fact_1576_sup_OboundedI,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A) ) ) ).
% sup.boundedI
tff(fact_1577_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A) ) ) ).
% sup.boundedI
tff(fact_1578_sup_OboundedI,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A) ) ) ).
% sup.boundedI
tff(fact_1579_sup_Oorder__iff,axiom,
! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
<=> ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ) ) ).
% sup.order_iff
tff(fact_1580_sup_Oorder__iff,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
<=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ) ) ).
% sup.order_iff
tff(fact_1581_sup_Oorder__iff,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
<=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).
% sup.order_iff
tff(fact_1582_sup_Oorder__iff,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
<=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ) ) ).
% sup.order_iff
tff(fact_1583_sup_Oorder__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
<=> ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ) ) ).
% sup.order_iff
tff(fact_1584_sup_Ocobounded1,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ).
% sup.cobounded1
tff(fact_1585_sup_Ocobounded1,axiom,
! [A: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ).
% sup.cobounded1
tff(fact_1586_sup_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ).
% sup.cobounded1
tff(fact_1587_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ).
% sup.cobounded1
tff(fact_1588_sup_Ocobounded1,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ).
% sup.cobounded1
tff(fact_1589_sup_Ocobounded2,axiom,
! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ).
% sup.cobounded2
tff(fact_1590_sup_Ocobounded2,axiom,
! [B: assn,A: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ).
% sup.cobounded2
tff(fact_1591_sup_Ocobounded2,axiom,
! [B: set_nat,A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ).
% sup.cobounded2
tff(fact_1592_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ).
% sup.cobounded2
tff(fact_1593_sup_Ocobounded2,axiom,
! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ).
% sup.cobounded2
tff(fact_1594_sup_Oabsorb__iff1,axiom,
! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
<=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = A ) ) ).
% sup.absorb_iff1
tff(fact_1595_sup_Oabsorb__iff1,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
<=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = A ) ) ).
% sup.absorb_iff1
tff(fact_1596_sup_Oabsorb__iff1,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
<=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).
% sup.absorb_iff1
tff(fact_1597_sup_Oabsorb__iff1,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).
% sup.absorb_iff1
tff(fact_1598_sup_Oabsorb__iff1,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
<=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).
% sup.absorb_iff1
tff(fact_1599_sup_Oabsorb__iff2,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),B)
<=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = B ) ) ).
% sup.absorb_iff2
tff(fact_1600_sup_Oabsorb__iff2,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
<=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = B ) ) ).
% sup.absorb_iff2
tff(fact_1601_sup_Oabsorb__iff2,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
<=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).
% sup.absorb_iff2
tff(fact_1602_sup_Oabsorb__iff2,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).
% sup.absorb_iff2
tff(fact_1603_sup_Oabsorb__iff2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
<=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).
% sup.absorb_iff2
tff(fact_1604_sup_OcoboundedI1,axiom,
! [C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),A)
=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).
% sup.coboundedI1
tff(fact_1605_sup_OcoboundedI1,axiom,
! [C: assn,A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),A)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).
% sup.coboundedI1
tff(fact_1606_sup_OcoboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).
% sup.coboundedI1
tff(fact_1607_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).
% sup.coboundedI1
tff(fact_1608_sup_OcoboundedI1,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).
% sup.coboundedI1
tff(fact_1609_sup_OcoboundedI2,axiom,
! [C: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),B)
=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).
% sup.coboundedI2
tff(fact_1610_sup_OcoboundedI2,axiom,
! [C: assn,B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),B)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).
% sup.coboundedI2
tff(fact_1611_sup_OcoboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),B)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).
% sup.coboundedI2
tff(fact_1612_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).
% sup.coboundedI2
tff(fact_1613_sup_OcoboundedI2,axiom,
! [C: int,B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).
% sup.coboundedI2
tff(fact_1614_mult__not__zero,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) != zero_z3403309356797280102nteger )
=> ( ( A != zero_z3403309356797280102nteger )
& ( B != zero_z3403309356797280102nteger ) ) ) ).
% mult_not_zero
tff(fact_1615_mult__not__zero,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat )
=> ( ( A != zero_zero_rat )
& ( B != zero_zero_rat ) ) ) ).
% mult_not_zero
tff(fact_1616_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
tff(fact_1617_mult__not__zero,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
tff(fact_1618_divisors__zero,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = zero_z3403309356797280102nteger )
=> ( ( A = zero_z3403309356797280102nteger )
| ( B = zero_z3403309356797280102nteger ) ) ) ).
% divisors_zero
tff(fact_1619_divisors__zero,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
=> ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divisors_zero
tff(fact_1620_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
tff(fact_1621_divisors__zero,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
tff(fact_1622_no__zero__divisors,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( B != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) != zero_z3403309356797280102nteger ) ) ) ).
% no_zero_divisors
tff(fact_1623_no__zero__divisors,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat ) ) ) ).
% no_zero_divisors
tff(fact_1624_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat ) ) ) ).
% no_zero_divisors
tff(fact_1625_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int ) ) ) ).
% no_zero_divisors
tff(fact_1626_mult__left__cancel,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
<=> ( A = B ) ) ) ).
% mult_left_cancel
tff(fact_1627_mult__left__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
<=> ( A = B ) ) ) ).
% mult_left_cancel
tff(fact_1628_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
<=> ( A = B ) ) ) ).
% mult_left_cancel
tff(fact_1629_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
<=> ( A = B ) ) ) ).
% mult_left_cancel
tff(fact_1630_mult__right__cancel,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
<=> ( A = B ) ) ) ).
% mult_right_cancel
tff(fact_1631_mult__right__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
<=> ( A = B ) ) ) ).
% mult_right_cancel
tff(fact_1632_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
<=> ( A = B ) ) ) ).
% mult_right_cancel
tff(fact_1633_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
<=> ( A = B ) ) ) ).
% mult_right_cancel
tff(fact_1634_zero__neq__one,axiom,
zero_z3403309356797280102nteger != one_one_Code_integer ).
% zero_neq_one
tff(fact_1635_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
% zero_neq_one
tff(fact_1636_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
tff(fact_1637_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
tff(fact_1638_mult_Oabel__semigroup__axioms,axiom,
abel_s8772020223019846471nteger(times_3573771949741848930nteger) ).
% mult.abel_semigroup_axioms
tff(fact_1639_mult_Oabel__semigroup__axioms,axiom,
abel_semigroup_assn(times_times_assn) ).
% mult.abel_semigroup_axioms
tff(fact_1640_mult_Oabel__semigroup__axioms,axiom,
abel_semigroup_rat(times_times_rat) ).
% mult.abel_semigroup_axioms
tff(fact_1641_mult_Oabel__semigroup__axioms,axiom,
abel_semigroup_nat(times_times_nat) ).
% mult.abel_semigroup_axioms
tff(fact_1642_mult_Oabel__semigroup__axioms,axiom,
abel_semigroup_int(times_times_int) ).
% mult.abel_semigroup_axioms
tff(fact_1643_inf_Oabel__semigroup__axioms,axiom,
abel_s3907380067910933104et_nat(inf_inf_set_nat) ).
% inf.abel_semigroup_axioms
tff(fact_1644_inf_Oabel__semigroup__axioms,axiom,
abel_semigroup_assn(inf_inf_assn) ).
% inf.abel_semigroup_axioms
tff(fact_1645_sup_Oabel__semigroup__axioms,axiom,
abel_semigroup_assn(sup_sup_assn) ).
% sup.abel_semigroup_axioms
tff(fact_1646_sup_Oabel__semigroup__axioms,axiom,
abel_s3907380067910933104et_nat(sup_sup_set_nat) ).
% sup.abel_semigroup_axioms
tff(fact_1647_sup_Oabel__semigroup__axioms,axiom,
abel_s6408242399240741597at_nat(sup_su5525570899277871387at_nat) ).
% sup.abel_semigroup_axioms
tff(fact_1648_relH__subset,axiom,
! [Bs: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,As: set_nat] :
( aa_hea7994211679605476167unit_o(relH(Bs,H2),H5)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,As),Bs)
=> aa_hea7994211679605476167unit_o(relH(As,H2),H5) ) ) ).
% relH_subset
tff(fact_1649_inf_Osemilattice__axioms,axiom,
semilattice_set_nat(inf_inf_set_nat) ).
% inf.semilattice_axioms
tff(fact_1650_inf_Osemilattice__axioms,axiom,
semilattice_assn(inf_inf_assn) ).
% inf.semilattice_axioms
tff(fact_1651_sup_Osemilattice__axioms,axiom,
semilattice_assn(sup_sup_assn) ).
% sup.semilattice_axioms
tff(fact_1652_sup_Osemilattice__axioms,axiom,
semilattice_set_nat(sup_sup_set_nat) ).
% sup.semilattice_axioms
tff(fact_1653_sup_Osemilattice__axioms,axiom,
semila366160313143161959at_nat(sup_su5525570899277871387at_nat) ).
% sup.semilattice_axioms
tff(fact_1654_semilattice__neutr_Oaxioms_I1_J,axiom,
! [F: fun_nat_fun_nat_nat,Z: nat] :
( aa_nat_o(semila9081495762789891438tr_nat(F),Z)
=> semilattice_nat(F) ) ).
% semilattice_neutr.axioms(1)
tff(fact_1655_diff__eq,axiom,
! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)) ).
% diff_eq
tff(fact_1656_diff__eq,axiom,
! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(minus_minus_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(uminus_uminus_assn,Y)) ).
% diff_eq
tff(fact_1657_le__minus__one__simps_I2_J,axiom,
aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).
% le_minus_one_simps(2)
tff(fact_1658_le__minus__one__simps_I2_J,axiom,
aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).
% le_minus_one_simps(2)
tff(fact_1659_le__minus__one__simps_I2_J,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).
% le_minus_one_simps(2)
tff(fact_1660_le__minus__one__simps_I4_J,axiom,
~ aa_rat_o(ord_less_eq_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).
% le_minus_one_simps(4)
tff(fact_1661_le__minus__one__simps_I4_J,axiom,
~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% le_minus_one_simps(4)
tff(fact_1662_le__minus__one__simps_I4_J,axiom,
~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).
% le_minus_one_simps(4)
tff(fact_1663_distrib__sup__le,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z))) ).
% distrib_sup_le
tff(fact_1664_distrib__sup__le,axiom,
! [X: assn,Y: assn,Z: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z))),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z))) ).
% distrib_sup_le
tff(fact_1665_distrib__sup__le,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z))) ).
% distrib_sup_le
tff(fact_1666_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z))) ).
% distrib_sup_le
tff(fact_1667_distrib__sup__le,axiom,
! [X: int,Y: int,Z: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Z))) ).
% distrib_sup_le
tff(fact_1668_distrib__inf__le,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Z))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z))) ).
% distrib_inf_le
tff(fact_1669_distrib__inf__le,axiom,
! [X: assn,Y: assn,Z: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z))),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z))) ).
% distrib_inf_le
tff(fact_1670_distrib__inf__le,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z))) ).
% distrib_inf_le
tff(fact_1671_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z))) ).
% distrib_inf_le
tff(fact_1672_distrib__inf__le,axiom,
! [X: int,Y: int,Z: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z))) ).
% distrib_inf_le
tff(fact_1673_disjoint__mono,axiom,
! [A: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A),A2)
=> ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(B),B2)
=> ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat ) ) ) ) ).
% disjoint_mono
tff(fact_1674_disjoint__mono,axiom,
! [A: set_o,A2: set_o,B: set_o,B2: set_o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),A2)
=> ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B),B2)
=> ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A2),B2) = bot_bot_set_o )
=> ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,A),B) = bot_bot_set_o ) ) ) ) ).
% disjoint_mono
tff(fact_1675_disjoint__mono,axiom,
! [A: set_int,A2: set_int,B: set_int,B2: set_int] :
( aa_set_int_o(ord_less_eq_set_int(A),A2)
=> ( aa_set_int_o(ord_less_eq_set_int(B),B2)
=> ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int )
=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = bot_bot_set_int ) ) ) ) ).
% disjoint_mono
tff(fact_1676_disjoint__mono,axiom,
! [A: set_nat,A2: set_nat,B: set_nat,B2: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),A2)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),B2)
=> ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat ) ) ) ) ).
% disjoint_mono
tff(fact_1677_zero__neq__neg__one,axiom,
zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).
% zero_neq_neg_one
tff(fact_1678_zero__neq__neg__one,axiom,
zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% zero_neq_neg_one
tff(fact_1679_zero__neq__neg__one,axiom,
zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% zero_neq_neg_one
tff(fact_1680_subset__Compl__self__eq,axiom,
! [Aa: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Aa),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Aa))
<=> ( Aa = bot_bo2099793752762293965at_nat ) ) ).
% subset_Compl_self_eq
tff(fact_1681_subset__Compl__self__eq,axiom,
! [Aa: set_o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Aa),aa_set_o_set_o(uminus_uminus_set_o,Aa))
<=> ( Aa = bot_bot_set_o ) ) ).
% subset_Compl_self_eq
tff(fact_1682_subset__Compl__self__eq,axiom,
! [Aa: set_int] :
( aa_set_int_o(ord_less_eq_set_int(Aa),aa_set_int_set_int(uminus1532241313380277803et_int,Aa))
<=> ( Aa = bot_bot_set_int ) ) ).
% subset_Compl_self_eq
tff(fact_1683_subset__Compl__self__eq,axiom,
! [Aa: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Aa))
<=> ( Aa = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
tff(fact_1684_in__range__subset,axiom,
! [As: set_nat,As2: set_nat,H2: heap_e7401611519738050253t_unit] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,As),As2)
=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As2))
=> aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As)) ) ) ).
% in_range_subset
tff(fact_1685_inf__shunt,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = bot_bo2099793752762293965at_nat )
<=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(X),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Y)) ) ).
% inf_shunt
tff(fact_1686_inf__shunt,axiom,
! [X: set_o,Y: set_o] :
( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,X),Y) = bot_bot_set_o )
<=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,X),aa_set_o_set_o(uminus_uminus_set_o,Y)) ) ).
% inf_shunt
tff(fact_1687_inf__shunt,axiom,
! [X: set_int,Y: set_int] :
( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = bot_bot_set_int )
<=> aa_set_int_o(ord_less_eq_set_int(X),aa_set_int_set_int(uminus1532241313380277803et_int,Y)) ) ).
% inf_shunt
tff(fact_1688_inf__shunt,axiom,
! [X: assn,Y: assn] :
( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = bot_bot_assn )
<=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(uminus_uminus_assn,Y)) ) ).
% inf_shunt
tff(fact_1689_inf__shunt,axiom,
! [X: set_nat,Y: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = bot_bot_set_nat )
<=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)) ) ).
% inf_shunt
tff(fact_1690_sup__shunt,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = top_to6833984726390702231at_nat )
<=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,X)),Y) ) ).
% sup_shunt
tff(fact_1691_sup__shunt,axiom,
! [X: set_o,Y: set_o] :
( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),Y) = top_top_set_o )
<=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,aa_set_o_set_o(uminus_uminus_set_o,X)),Y) ) ).
% sup_shunt
tff(fact_1692_sup__shunt,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,X),Y) = top_to1996260823553986621t_unit )
<=> aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,aa_set8324764978855220220t_unit(uminus5944136376168626660t_unit,X)),Y) ) ).
% sup_shunt
tff(fact_1693_sup__shunt,axiom,
! [X: assn,Y: assn] :
( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = top_top_assn )
<=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(uminus_uminus_assn,X)),Y) ) ).
% sup_shunt
tff(fact_1694_sup__shunt,axiom,
! [X: set_nat,Y: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = top_top_set_nat )
<=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),Y) ) ).
% sup_shunt
tff(fact_1695_sup__neg__inf,axiom,
! [P5: set_Pr4329608150637261639at_nat,Q4: set_Pr4329608150637261639at_nat,R2: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(P5),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Q4),R2))
<=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,P5),aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,Q4))),R2) ) ).
% sup_neg_inf
tff(fact_1696_sup__neg__inf,axiom,
! [P5: assn,Q4: assn,R2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,P5),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Q4),R2))
<=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P5),aa_assn_assn(uminus_uminus_assn,Q4))),R2) ) ).
% sup_neg_inf
tff(fact_1697_sup__neg__inf,axiom,
! [P5: set_nat,Q4: set_nat,R2: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,P5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Q4),R2))
<=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,P5),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Q4))),R2) ) ).
% sup_neg_inf
tff(fact_1698_shunt2,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,Y))),Z)
<=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) ) ).
% shunt2
tff(fact_1699_shunt2,axiom,
! [X: assn,Y: assn,Z: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(uminus_uminus_assn,Y))),Z)
<=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ) ).
% shunt2
tff(fact_1700_shunt2,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y))),Z)
<=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) ) ).
% shunt2
tff(fact_1701_shunt1,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),Z)
<=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,Y)),Z)) ) ).
% shunt1
tff(fact_1702_shunt1,axiom,
! [X: assn,Y: assn,Z: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Z)
<=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(uminus_uminus_assn,Y)),Z)) ) ).
% shunt1
tff(fact_1703_shunt1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Z)
<=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)),Z)) ) ).
% shunt1
tff(fact_1704_subset__emptyI,axiom,
! [Aa: set_fu3555266933746557007_nat_o] :
( ! [X2: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),Aa)
=> aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(Aa),bot_bo8632255151191677115_nat_o) ) ).
% subset_emptyI
tff(fact_1705_subset__emptyI,axiom,
! [Aa: set_Pr1261947904930325089at_nat] :
( ! [X2: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),Aa)
=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Aa),bot_bo2099793752762293965at_nat) ) ).
% subset_emptyI
tff(fact_1706_subset__emptyI,axiom,
! [Aa: set_o] :
( ! [X2: $o] : ~ aa_set_o_o(member_o((X2)),Aa)
=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Aa),bot_bot_set_o) ) ).
% subset_emptyI
tff(fact_1707_subset__emptyI,axiom,
! [Aa: set_int] :
( ! [X2: int] : ~ aa_set_int_o(member_int(X2),Aa)
=> aa_set_int_o(ord_less_eq_set_int(Aa),bot_bot_set_int) ) ).
% subset_emptyI
tff(fact_1708_subset__emptyI,axiom,
! [Aa: set_nat] :
( ! [X2: nat] : ~ aa_set_nat_o(member_nat(X2),Aa)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),bot_bot_set_nat) ) ).
% subset_emptyI
tff(fact_1709_inf__period_I2_J,axiom,
! [P: fun_Code_integer_o,D3: code_integer,Q: fun_Code_integer_o] :
( ! [X2: code_integer,K2: code_integer] :
( aa_Code_integer_o(P,X2)
<=> aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D3))) )
=> ( ! [X2: code_integer,K2: code_integer] :
( aa_Code_integer_o(Q,X2)
<=> aa_Code_integer_o(Q,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D3))) )
=> ! [X4: code_integer,K3: code_integer] :
( ( aa_Code_integer_o(P,X4)
| aa_Code_integer_o(Q,X4) )
<=> ( aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X4),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D3)))
| aa_Code_integer_o(Q,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X4),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D3))) ) ) ) ) ).
% inf_period(2)
tff(fact_1710_inf__period_I2_J,axiom,
! [P: fun_rat_o,D3: rat,Q: fun_rat_o] :
( ! [X2: rat,K2: rat] :
( aa_rat_o(P,X2)
<=> aa_rat_o(P,minus_minus_rat(X2,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) )
=> ( ! [X2: rat,K2: rat] :
( aa_rat_o(Q,X2)
<=> aa_rat_o(Q,minus_minus_rat(X2,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) )
=> ! [X4: rat,K3: rat] :
( ( aa_rat_o(P,X4)
| aa_rat_o(Q,X4) )
<=> ( aa_rat_o(P,minus_minus_rat(X4,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D3)))
| aa_rat_o(Q,minus_minus_rat(X4,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D3))) ) ) ) ) ).
% inf_period(2)
tff(fact_1711_inf__period_I2_J,axiom,
! [P: fun_int_o,D3: int,Q: fun_int_o] :
( ! [X2: int,K2: int] :
( aa_int_o(P,X2)
<=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
=> ( ! [X2: int,K2: int] :
( aa_int_o(Q,X2)
<=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
=> ! [X4: int,K3: int] :
( ( aa_int_o(P,X4)
| aa_int_o(Q,X4) )
<=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3)))
| aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3))) ) ) ) ) ).
% inf_period(2)
tff(fact_1712_inf__period_I1_J,axiom,
! [P: fun_Code_integer_o,D3: code_integer,Q: fun_Code_integer_o] :
( ! [X2: code_integer,K2: code_integer] :
( aa_Code_integer_o(P,X2)
<=> aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D3))) )
=> ( ! [X2: code_integer,K2: code_integer] :
( aa_Code_integer_o(Q,X2)
<=> aa_Code_integer_o(Q,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D3))) )
=> ! [X4: code_integer,K3: code_integer] :
( ( aa_Code_integer_o(P,X4)
& aa_Code_integer_o(Q,X4) )
<=> ( aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X4),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D3)))
& aa_Code_integer_o(Q,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X4),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D3))) ) ) ) ) ).
% inf_period(1)
tff(fact_1713_inf__period_I1_J,axiom,
! [P: fun_rat_o,D3: rat,Q: fun_rat_o] :
( ! [X2: rat,K2: rat] :
( aa_rat_o(P,X2)
<=> aa_rat_o(P,minus_minus_rat(X2,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) )
=> ( ! [X2: rat,K2: rat] :
( aa_rat_o(Q,X2)
<=> aa_rat_o(Q,minus_minus_rat(X2,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) )
=> ! [X4: rat,K3: rat] :
( ( aa_rat_o(P,X4)
& aa_rat_o(Q,X4) )
<=> ( aa_rat_o(P,minus_minus_rat(X4,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D3)))
& aa_rat_o(Q,minus_minus_rat(X4,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D3))) ) ) ) ) ).
% inf_period(1)
tff(fact_1714_inf__period_I1_J,axiom,
! [P: fun_int_o,D3: int,Q: fun_int_o] :
( ! [X2: int,K2: int] :
( aa_int_o(P,X2)
<=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
=> ( ! [X2: int,K2: int] :
( aa_int_o(Q,X2)
<=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) )
=> ! [X4: int,K3: int] :
( ( aa_int_o(P,X4)
& aa_int_o(Q,X4) )
<=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3)))
& aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3))) ) ) ) ) ).
% inf_period(1)
tff(fact_1715_mod__star__trueE_H,axiom,
! [P: assn,H2: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),top_top_assn)),H2)
=> ~ ! [H4: produc3658429121746597890et_nat] :
( ( aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,H4) = aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,H2) )
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,H4)),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,H2))
=> ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),H4) ) ) ) ).
% mod_star_trueE'
tff(fact_1716_divides__aux__eq,axiom,
! [Q4: code_integer,R2: code_integer] :
( aa_Pro4926521453222654329eger_o(unique5706413561485394159nteger,produc1086072967326762835nteger(Q4,R2))
<=> ( R2 = zero_z3403309356797280102nteger ) ) ).
% divides_aux_eq
tff(fact_1717_divides__aux__eq,axiom,
! [Q4: nat,R2: nat] :
( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q4),R2))
<=> ( R2 = zero_zero_nat ) ) ).
% divides_aux_eq
tff(fact_1718_divides__aux__eq,axiom,
! [Q4: int,R2: int] :
( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2))
<=> ( R2 = zero_zero_int ) ) ).
% divides_aux_eq
tff(fact_1719_mult__le__cancel__left1,axiom,
! [C: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
<=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_eq_rat(one_one_rat),B) )
& ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> aa_rat_o(ord_less_eq_rat(B),one_one_rat) ) ) ) ).
% mult_le_cancel_left1
tff(fact_1720_mult__le__cancel__left1,axiom,
! [C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),B) )
& ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer) ) ) ) ).
% mult_le_cancel_left1
tff(fact_1721_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).
% mult_le_cancel_left1
tff(fact_1722_mult__le__cancel__left2,axiom,
! [C: rat,A: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C)
<=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_eq_rat(A),one_one_rat) )
& ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> aa_rat_o(ord_less_eq_rat(one_one_rat),A) ) ) ) ).
% mult_le_cancel_left2
tff(fact_1723_mult__le__cancel__left2,axiom,
! [C: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),C)
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer) )
& ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A) ) ) ) ).
% mult_le_cancel_left2
tff(fact_1724_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).
% mult_le_cancel_left2
tff(fact_1725_Diff__empty,axiom,
! [Aa: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(Aa,bot_bo2099793752762293965at_nat) = Aa ).
% Diff_empty
tff(fact_1726_Diff__empty,axiom,
! [Aa: set_o] : minus_minus_set_o(Aa,bot_bot_set_o) = Aa ).
% Diff_empty
tff(fact_1727_Diff__empty,axiom,
! [Aa: set_int] : minus_minus_set_int(Aa,bot_bot_set_int) = Aa ).
% Diff_empty
tff(fact_1728_Diff__empty,axiom,
! [Aa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),bot_bot_set_nat) = Aa ).
% Diff_empty
tff(fact_1729_empty__Diff,axiom,
! [Aa: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(bot_bo2099793752762293965at_nat,Aa) = bot_bo2099793752762293965at_nat ).
% empty_Diff
tff(fact_1730_empty__Diff,axiom,
! [Aa: set_o] : minus_minus_set_o(bot_bot_set_o,Aa) = bot_bot_set_o ).
% empty_Diff
tff(fact_1731_empty__Diff,axiom,
! [Aa: set_int] : minus_minus_set_int(bot_bot_set_int,Aa) = bot_bot_set_int ).
% empty_Diff
tff(fact_1732_empty__Diff,axiom,
! [Aa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,bot_bot_set_nat),Aa) = bot_bot_set_nat ).
% empty_Diff
tff(fact_1733_Diff__cancel,axiom,
! [Aa: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(Aa,Aa) = bot_bo2099793752762293965at_nat ).
% Diff_cancel
tff(fact_1734_Diff__cancel,axiom,
! [Aa: set_o] : minus_minus_set_o(Aa,Aa) = bot_bot_set_o ).
% Diff_cancel
tff(fact_1735_Diff__cancel,axiom,
! [Aa: set_int] : minus_minus_set_int(Aa,Aa) = bot_bot_set_int ).
% Diff_cancel
tff(fact_1736_Diff__cancel,axiom,
! [Aa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),Aa) = bot_bot_set_nat ).
% Diff_cancel
tff(fact_1737_not__gr__zero,axiom,
! [N: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
<=> ( N = zero_zero_nat ) ) ).
% not_gr_zero
tff(fact_1738_compl__less__compl__iff,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(uminus_uminus_assn,Y))
<=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X) ) ).
% compl_less_compl_iff
tff(fact_1739_neg__less__iff__less,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).
% neg_less_iff_less
tff(fact_1740_neg__less__iff__less,axiom,
! [B: rat,A: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
<=> aa_rat_o(ord_less_rat(A),B) ) ).
% neg_less_iff_less
tff(fact_1741_neg__less__iff__less,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).
% neg_less_iff_less
tff(fact_1742_Diff__eq__empty__iff,axiom,
! [Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat(Aa,Ba) = bot_bo2099793752762293965at_nat )
<=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Aa),Ba) ) ).
% Diff_eq_empty_iff
tff(fact_1743_Diff__eq__empty__iff,axiom,
! [Aa: set_o,Ba: set_o] :
( ( minus_minus_set_o(Aa,Ba) = bot_bot_set_o )
<=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Aa),Ba) ) ).
% Diff_eq_empty_iff
tff(fact_1744_Diff__eq__empty__iff,axiom,
! [Aa: set_int,Ba: set_int] :
( ( minus_minus_set_int(Aa,Ba) = bot_bot_set_int )
<=> aa_set_int_o(ord_less_eq_set_int(Aa),Ba) ) ).
% Diff_eq_empty_iff
tff(fact_1745_Diff__eq__empty__iff,axiom,
! [Aa: set_nat,Ba: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),Ba) = bot_bot_set_nat )
<=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),Ba) ) ).
% Diff_eq_empty_iff
tff(fact_1746_Diff__UNIV,axiom,
! [Aa: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(Aa,top_to4669805908274784177at_nat) = bot_bo2099793752762293965at_nat ).
% Diff_UNIV
tff(fact_1747_Diff__UNIV,axiom,
! [Aa: set_int] : minus_minus_set_int(Aa,top_top_set_int) = bot_bot_set_int ).
% Diff_UNIV
tff(fact_1748_Diff__UNIV,axiom,
! [Aa: set_o] : minus_minus_set_o(Aa,top_top_set_o) = bot_bot_set_o ).
% Diff_UNIV
tff(fact_1749_Diff__UNIV,axiom,
! [Aa: set_Product_unit] : minus_6452836326544984404t_unit(Aa,top_to1996260823553986621t_unit) = bot_bo3957492148770167129t_unit ).
% Diff_UNIV
tff(fact_1750_Diff__UNIV,axiom,
! [Aa: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),top_top_set_nat) = bot_bot_set_nat ).
% Diff_UNIV
tff(fact_1751_Diff__disjoint,axiom,
! [Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Aa),minus_1356011639430497352at_nat(Ba,Aa)) = bot_bo2099793752762293965at_nat ).
% Diff_disjoint
tff(fact_1752_Diff__disjoint,axiom,
! [Aa: set_o,Ba: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Aa),minus_minus_set_o(Ba,Aa)) = bot_bot_set_o ).
% Diff_disjoint
tff(fact_1753_Diff__disjoint,axiom,
! [Aa: set_int,Ba: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Aa),minus_minus_set_int(Ba,Aa)) = bot_bot_set_int ).
% Diff_disjoint
tff(fact_1754_Diff__disjoint,axiom,
! [Aa: set_nat,Ba: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Ba),Aa)) = bot_bot_set_nat ).
% Diff_disjoint
tff(fact_1755_diff__gt__0__iff__gt,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),minus_minus_rat(A,B))
<=> aa_rat_o(ord_less_rat(B),A) ) ).
% diff_gt_0_iff_gt
tff(fact_1756_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).
% diff_gt_0_iff_gt
tff(fact_1757_diff__gt__0__iff__gt,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ).
% diff_gt_0_iff_gt
tff(fact_1758_neg__less__0__iff__less,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).
% neg_less_0_iff_less
tff(fact_1759_neg__less__0__iff__less,axiom,
! [A: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
<=> aa_rat_o(ord_less_rat(zero_zero_rat),A) ) ).
% neg_less_0_iff_less
tff(fact_1760_neg__less__0__iff__less,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).
% neg_less_0_iff_less
tff(fact_1761_neg__0__less__iff__less,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).
% neg_0_less_iff_less
tff(fact_1762_neg__0__less__iff__less,axiom,
! [A: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
<=> aa_rat_o(ord_less_rat(A),zero_zero_rat) ) ).
% neg_0_less_iff_less
tff(fact_1763_neg__0__less__iff__less,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).
% neg_0_less_iff_less
tff(fact_1764_neg__less__pos,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),A)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).
% neg_less_pos
tff(fact_1765_neg__less__pos,axiom,
! [A: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,A)),A)
<=> aa_rat_o(ord_less_rat(zero_zero_rat),A) ) ).
% neg_less_pos
tff(fact_1766_neg__less__pos,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).
% neg_less_pos
tff(fact_1767_less__neg__neg,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).
% less_neg_neg
tff(fact_1768_less__neg__neg,axiom,
! [A: rat] :
( aa_rat_o(ord_less_rat(A),aa_rat_rat(uminus_uminus_rat,A))
<=> aa_rat_o(ord_less_rat(A),zero_zero_rat) ) ).
% less_neg_neg
tff(fact_1769_less__neg__neg,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).
% less_neg_neg
tff(fact_1770_prod_Ocollapse,axiom,
! [Prod: produc2453742332425399446nteger] : produc565448451787809096nteger(aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,Prod),aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,Prod)) = Prod ).
% prod.collapse
tff(fact_1771_prod_Ocollapse,axiom,
! [Prod: produc8799533983703829792et_nat] : produc6017135503889078682et_nat(aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,Prod),aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,Prod)) = Prod ).
% prod.collapse
tff(fact_1772_prod_Ocollapse,axiom,
! [Prod: produc3112576180687729086et_nat] : produc7291855603632349880et_nat(aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,Prod),aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,Prod)) = Prod ).
% prod.collapse
tff(fact_1773_prod_Ocollapse,axiom,
! [Prod: produc3043004577601930755nt_int] : produc7320969046608631221nt_int(aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,Prod),aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,Prod)) = Prod ).
% prod.collapse
tff(fact_1774_prod_Ocollapse,axiom,
! [Prod: produc4788131091073775919nt_int] : produc4729913460834754785nt_int(aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,Prod),aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,Prod)) = Prod ).
% prod.collapse
tff(fact_1775_prod_Ocollapse,axiom,
! [Prod: produc3658429121746597890et_nat] : produc7507926704131184380et_nat(aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Prod),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Prod)) = Prod ).
% prod.collapse
tff(fact_1776_prod_Ocollapse,axiom,
! [Prod: product_prod_int_int] : aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) = Prod ).
% prod.collapse
tff(fact_1777_prod_Ocollapse,axiom,
! [Prod: produc5835291356934675326atural] : aa_Pro7165575656943818355atural(produc6639722614265839536atural(produc497848011034438852atural(Prod)),aa_Pro3673693039631929973atural(produc6591912806276919810atural,Prod)) = Prod ).
% prod.collapse
tff(fact_1778_order__less__imp__not__less,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X) ) ).
% order_less_imp_not_less
tff(fact_1779_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).
% order_less_imp_not_less
tff(fact_1780_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).
% order_less_imp_not_less
tff(fact_1781_order__less__imp__not__less,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ).
% order_less_imp_not_less
tff(fact_1782_order__less__imp__not__less,axiom,
! [X: code_natural,Y: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
=> ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Y),X) ) ).
% order_less_imp_not_less
tff(fact_1783_order__less__imp__not__eq2,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
tff(fact_1784_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
tff(fact_1785_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
tff(fact_1786_order__less__imp__not__eq2,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
tff(fact_1787_order__less__imp__not__eq2,axiom,
! [X: code_natural,Y: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
tff(fact_1788_order__less__imp__not__eq,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
=> ( X != Y ) ) ).
% order_less_imp_not_eq
tff(fact_1789_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
=> ( X != Y ) ) ).
% order_less_imp_not_eq
tff(fact_1790_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
=> ( X != Y ) ) ).
% order_less_imp_not_eq
tff(fact_1791_order__less__imp__not__eq,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
=> ( X != Y ) ) ).
% order_less_imp_not_eq
tff(fact_1792_order__less__imp__not__eq,axiom,
! [X: code_natural,Y: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
=> ( X != Y ) ) ).
% order_less_imp_not_eq
tff(fact_1793_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
| ( X = Y )
| aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).
% linorder_less_linear
tff(fact_1794_linorder__less__linear,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
| ( X = Y )
| aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).
% linorder_less_linear
tff(fact_1795_linorder__less__linear,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
| ( X = Y )
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ).
% linorder_less_linear
tff(fact_1796_linorder__less__linear,axiom,
! [X: code_natural,Y: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
| ( X = Y )
| aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Y),X) ) ).
% linorder_less_linear
tff(fact_1797_order__less__imp__triv,axiom,
! [X: assn,Y: assn,P: $o] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X)
=> (P) ) ) ).
% order_less_imp_triv
tff(fact_1798_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
=> (P) ) ) ).
% order_less_imp_triv
tff(fact_1799_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
=> (P) ) ) ).
% order_less_imp_triv
tff(fact_1800_order__less__imp__triv,axiom,
! [X: code_integer,Y: code_integer,P: $o] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X)
=> (P) ) ) ).
% order_less_imp_triv
tff(fact_1801_order__less__imp__triv,axiom,
! [X: code_natural,Y: code_natural,P: $o] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Y),X)
=> (P) ) ) ).
% order_less_imp_triv
tff(fact_1802_order__less__not__sym,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X) ) ).
% order_less_not_sym
tff(fact_1803_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).
% order_less_not_sym
tff(fact_1804_order__less__not__sym,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).
% order_less_not_sym
tff(fact_1805_order__less__not__sym,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ).
% order_less_not_sym
tff(fact_1806_order__less__not__sym,axiom,
! [X: code_natural,Y: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
=> ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Y),X) ) ).
% order_less_not_sym
tff(fact_1807_order__less__subst2,axiom,
! [A: assn,B: assn,F: fun_assn_assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,B)),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_1808_order__less__subst2,axiom,
! [A: assn,B: assn,F: fun_assn_nat,C: nat] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,B)),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_1809_order__less__subst2,axiom,
! [A: assn,B: assn,F: fun_assn_int,C: int] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,B)),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_1810_order__less__subst2,axiom,
! [A: assn,B: assn,F: fun_as9138348466139350018nteger,C: code_integer] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,B)),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,X2)),aa_assn_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_1811_order__less__subst2,axiom,
! [A: assn,B: assn,F: fun_as4373426249795763277atural,C: code_natural] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_assn_Code_natural(F,B)),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_assn_Code_natural(F,X2)),aa_assn_Code_natural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_assn_Code_natural(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_1812_order__less__subst2,axiom,
! [A: nat,B: nat,F: fun_nat_assn,C: assn] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,B)),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_1813_order__less__subst2,axiom,
! [A: nat,B: nat,F: fun_nat_nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,B)),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_1814_order__less__subst2,axiom,
! [A: nat,B: nat,F: fun_nat_int,C: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,B)),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_1815_order__less__subst2,axiom,
! [A: nat,B: nat,F: fun_nat_Code_integer,C: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,B)),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,X2)),aa_nat_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_1816_order__less__subst2,axiom,
! [A: nat,B: nat,F: fun_nat_Code_natural,C: code_natural] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(F,B)),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(F,X2)),aa_nat_Code_natural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(F,A)),C) ) ) ) ).
% order_less_subst2
tff(fact_1817_order__less__subst1,axiom,
! [A: assn,F: fun_assn_assn,B: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(F,B))
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_1818_order__less__subst1,axiom,
! [A: assn,F: fun_nat_assn,B: nat,C: nat] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_nat_assn(F,B))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_nat_assn(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_1819_order__less__subst1,axiom,
! [A: assn,F: fun_int_assn,B: int,C: int] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_int_assn(F,B))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> ( ! [X2: int,Y2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,X2)),aa_int_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_int_assn(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_1820_order__less__subst1,axiom,
! [A: assn,F: fun_Co4490034186337852034r_assn,B: code_integer,C: code_integer] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_Code_integer_assn(F,B))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_Code_integer_assn(F,X2)),aa_Code_integer_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_Code_integer_assn(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_1821_order__less__subst1,axiom,
! [A: assn,F: fun_Co8393145496866358861l_assn,B: code_natural,C: code_natural] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_Code_natural_assn(F,B))
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),C)
=> ( ! [X2: code_natural,Y2: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_Code_natural_assn(F,X2)),aa_Code_natural_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_Code_natural_assn(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_1822_order__less__subst1,axiom,
! [A: nat,F: fun_assn_nat,B: assn,C: assn] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_assn_nat(F,B))
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_assn_nat(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_1823_order__less__subst1,axiom,
! [A: nat,F: fun_nat_nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(F,B))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_1824_order__less__subst1,axiom,
! [A: nat,F: fun_int_nat,B: int,C: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_int_nat(F,B))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> ( ! [X2: int,Y2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X2)),aa_int_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_int_nat(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_1825_order__less__subst1,axiom,
! [A: nat,F: fun_Code_integer_nat,B: code_integer,C: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_Code_integer_nat(F,B))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(F,X2)),aa_Code_integer_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_Code_integer_nat(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_1826_order__less__subst1,axiom,
! [A: nat,F: fun_Code_natural_nat,B: code_natural,C: code_natural] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_Code_natural_nat(F,B))
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),C)
=> ( ! [X2: code_natural,Y2: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_natural_nat(F,X2)),aa_Code_natural_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_Code_natural_nat(F,C)) ) ) ) ).
% order_less_subst1
tff(fact_1827_order__less__irrefl,axiom,
! [X: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),X) ).
% order_less_irrefl
tff(fact_1828_order__less__irrefl,axiom,
! [X: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X) ).
% order_less_irrefl
tff(fact_1829_order__less__irrefl,axiom,
! [X: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),X) ).
% order_less_irrefl
tff(fact_1830_order__less__irrefl,axiom,
! [X: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),X) ).
% order_less_irrefl
tff(fact_1831_order__less__irrefl,axiom,
! [X: code_natural] : ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),X) ).
% order_less_irrefl
tff(fact_1832_ord__less__eq__subst,axiom,
! [A: assn,B: assn,F: fun_assn_assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( ( aa_assn_assn(F,B) = C )
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_1833_ord__less__eq__subst,axiom,
! [A: assn,B: assn,F: fun_assn_nat,C: nat] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( ( aa_assn_nat(F,B) = C )
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_1834_ord__less__eq__subst,axiom,
! [A: assn,B: assn,F: fun_assn_int,C: int] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( ( aa_assn_int(F,B) = C )
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_1835_ord__less__eq__subst,axiom,
! [A: assn,B: assn,F: fun_as9138348466139350018nteger,C: code_integer] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( ( aa_assn_Code_integer(F,B) = C )
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,X2)),aa_assn_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_1836_ord__less__eq__subst,axiom,
! [A: assn,B: assn,F: fun_as4373426249795763277atural,C: code_natural] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( ( aa_assn_Code_natural(F,B) = C )
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_assn_Code_natural(F,X2)),aa_assn_Code_natural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_assn_Code_natural(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_1837_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: fun_nat_assn,C: assn] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( ( aa_nat_assn(F,B) = C )
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_1838_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: fun_nat_nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( ( aa_nat_nat(F,B) = C )
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_1839_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: fun_nat_int,C: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( ( aa_nat_int(F,B) = C )
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_1840_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: fun_nat_Code_integer,C: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( ( aa_nat_Code_integer(F,B) = C )
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,X2)),aa_nat_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_1841_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: fun_nat_Code_natural,C: code_natural] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( ( aa_nat_Code_natural(F,B) = C )
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(F,X2)),aa_nat_Code_natural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(F,A)),C) ) ) ) ).
% ord_less_eq_subst
tff(fact_1842_ord__eq__less__subst,axiom,
! [A: assn,F: fun_assn_assn,B: assn,C: assn] :
( ( A = aa_assn_assn(F,B) )
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_1843_ord__eq__less__subst,axiom,
! [A: nat,F: fun_assn_nat,B: assn,C: assn] :
( ( A = aa_assn_nat(F,B) )
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_assn_nat(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_1844_ord__eq__less__subst,axiom,
! [A: int,F: fun_assn_int,B: assn,C: assn] :
( ( A = aa_assn_int(F,B) )
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_assn_int(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_1845_ord__eq__less__subst,axiom,
! [A: code_integer,F: fun_as9138348466139350018nteger,B: assn,C: assn] :
( ( A = aa_assn_Code_integer(F,B) )
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,X2)),aa_assn_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_assn_Code_integer(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_1846_ord__eq__less__subst,axiom,
! [A: code_natural,F: fun_as4373426249795763277atural,B: assn,C: assn] :
( ( A = aa_assn_Code_natural(F,B) )
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_assn_Code_natural(F,X2)),aa_assn_Code_natural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_assn_Code_natural(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_1847_ord__eq__less__subst,axiom,
! [A: assn,F: fun_nat_assn,B: nat,C: nat] :
( ( A = aa_nat_assn(F,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X2)),aa_nat_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_nat_assn(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_1848_ord__eq__less__subst,axiom,
! [A: nat,F: fun_nat_nat,B: nat,C: nat] :
( ( A = aa_nat_nat(F,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X2)),aa_nat_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_1849_ord__eq__less__subst,axiom,
! [A: int,F: fun_nat_int,B: nat,C: nat] :
( ( A = aa_nat_int(F,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X2)),aa_nat_int(F,Y2)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_1850_ord__eq__less__subst,axiom,
! [A: code_integer,F: fun_nat_Code_integer,B: nat,C: nat] :
( ( A = aa_nat_Code_integer(F,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,X2)),aa_nat_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_nat_Code_integer(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_1851_ord__eq__less__subst,axiom,
! [A: code_natural,F: fun_nat_Code_natural,B: nat,C: nat] :
( ( A = aa_nat_Code_natural(F,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(F,X2)),aa_nat_Code_natural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_nat_Code_natural(F,C)) ) ) ) ).
% ord_eq_less_subst
tff(fact_1852_order__less__trans,axiom,
! [X: assn,Y: assn,Z: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),Z)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Z) ) ) ).
% order_less_trans
tff(fact_1853_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z) ) ) ).
% order_less_trans
tff(fact_1854_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z) ) ) ).
% order_less_trans
tff(fact_1855_order__less__trans,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Z)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z) ) ) ).
% order_less_trans
tff(fact_1856_order__less__trans,axiom,
! [X: code_natural,Y: code_natural,Z: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Y),Z)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Z) ) ) ).
% order_less_trans
tff(fact_1857_order__less__asym_H,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A) ) ).
% order_less_asym'
tff(fact_1858_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).
% order_less_asym'
tff(fact_1859_order__less__asym_H,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).
% order_less_asym'
tff(fact_1860_order__less__asym_H,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ).
% order_less_asym'
tff(fact_1861_order__less__asym_H,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),A) ) ).
% order_less_asym'
tff(fact_1862_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).
% linorder_neq_iff
tff(fact_1863_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
| aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).
% linorder_neq_iff
tff(fact_1864_linorder__neq__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( X != Y )
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ) ).
% linorder_neq_iff
tff(fact_1865_linorder__neq__iff,axiom,
! [X: code_natural,Y: code_natural] :
( ( X != Y )
<=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
| aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Y),X) ) ) ).
% linorder_neq_iff
tff(fact_1866_order__less__asym,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X) ) ).
% order_less_asym
tff(fact_1867_order__less__asym,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).
% order_less_asym
tff(fact_1868_order__less__asym,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).
% order_less_asym
tff(fact_1869_order__less__asym,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ).
% order_less_asym
tff(fact_1870_order__less__asym,axiom,
! [X: code_natural,Y: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
=> ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Y),X) ) ).
% order_less_asym
tff(fact_1871_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).
% linorder_neqE
tff(fact_1872_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).
% linorder_neqE
tff(fact_1873_linorder__neqE,axiom,
! [X: code_integer,Y: code_integer] :
( ( X != Y )
=> ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ) ).
% linorder_neqE
tff(fact_1874_linorder__neqE,axiom,
! [X: code_natural,Y: code_natural] :
( ( X != Y )
=> ( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Y),X) ) ) ).
% linorder_neqE
tff(fact_1875_dual__order_Ostrict__implies__not__eq,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_1876_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_1877_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_1878_dual__order_Ostrict__implies__not__eq,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_1879_dual__order_Ostrict__implies__not__eq,axiom,
! [B: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),A)
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_1880_order_Ostrict__implies__not__eq,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( A != B ) ) ).
% order.strict_implies_not_eq
tff(fact_1881_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( A != B ) ) ).
% order.strict_implies_not_eq
tff(fact_1882_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( A != B ) ) ).
% order.strict_implies_not_eq
tff(fact_1883_order_Ostrict__implies__not__eq,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ( A != B ) ) ).
% order.strict_implies_not_eq
tff(fact_1884_order_Ostrict__implies__not__eq,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ( A != B ) ) ).
% order.strict_implies_not_eq
tff(fact_1885_dual__order_Ostrict__trans,axiom,
! [B: assn,A: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),B)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),A) ) ) ).
% dual_order.strict_trans
tff(fact_1886_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).
% dual_order.strict_trans
tff(fact_1887_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).
% dual_order.strict_trans
tff(fact_1888_dual__order_Ostrict__trans,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A) ) ) ).
% dual_order.strict_trans
tff(fact_1889_dual__order_Ostrict__trans,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,C),B)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,C),A) ) ) ).
% dual_order.strict_trans
tff(fact_1890_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_1891_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_1892_not__less__iff__gr__or__eq,axiom,
! [X: code_integer,Y: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X)
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_1893_not__less__iff__gr__or__eq,axiom,
! [X: code_natural,Y: code_natural] :
( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
<=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Y),X)
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_1894_order_Ostrict__trans,axiom,
! [A: assn,B: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C) ) ) ).
% order.strict_trans
tff(fact_1895_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).
% order.strict_trans
tff(fact_1896_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).
% order.strict_trans
tff(fact_1897_order_Ostrict__trans,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),C) ) ) ).
% order.strict_trans
tff(fact_1898_order_Ostrict__trans,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),C) ) ) ).
% order.strict_trans
tff(fact_1899_linorder__less__wlog,axiom,
! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B3)
=> aa_nat_o(aa_nat_fun_nat_o(P,A3),B3) )
=> ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A3),A3)
=> ( ! [A3: nat,B3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(P,B3),A3)
=> aa_nat_o(aa_nat_fun_nat_o(P,A3),B3) )
=> aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).
% linorder_less_wlog
tff(fact_1900_linorder__less__wlog,axiom,
! [P: fun_int_fun_int_o,A: int,B: int] :
( ! [A3: int,B3: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B3)
=> aa_int_o(aa_int_fun_int_o(P,A3),B3) )
=> ( ! [A3: int] : aa_int_o(aa_int_fun_int_o(P,A3),A3)
=> ( ! [A3: int,B3: int] :
( aa_int_o(aa_int_fun_int_o(P,B3),A3)
=> aa_int_o(aa_int_fun_int_o(P,A3),B3) )
=> aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ) ).
% linorder_less_wlog
tff(fact_1901_linorder__less__wlog,axiom,
! [P: fun_Co2081214437782444048eger_o,A: code_integer,B: code_integer] :
( ! [A3: code_integer,B3: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A3),B3)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A3),B3) )
=> ( ! [A3: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A3),A3)
=> ( ! [A3: code_integer,B3: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,B3),A3)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A3),B3) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A),B) ) ) ) ).
% linorder_less_wlog
tff(fact_1902_linorder__less__wlog,axiom,
! [P: fun_Co7302518231560758906ural_o,A: code_natural,B: code_natural] :
( ! [A3: code_natural,B3: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A3),B3)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(P,A3),B3) )
=> ( ! [A3: code_natural] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(P,A3),A3)
=> ( ! [A3: code_natural,B3: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(P,B3),A3)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(P,A3),B3) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(P,A),B) ) ) ) ).
% linorder_less_wlog
tff(fact_1903_exists__least__iff,axiom,
! [P: fun_nat_o] :
( ? [X_12: nat] : aa_nat_o(P,X_12)
<=> ? [N2: nat] :
( aa_nat_o(P,N2)
& ! [M: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
=> ~ aa_nat_o(P,M) ) ) ) ).
% exists_least_iff
tff(fact_1904_dual__order_Oirrefl,axiom,
! [A: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),A) ).
% dual_order.irrefl
tff(fact_1905_dual__order_Oirrefl,axiom,
! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).
% dual_order.irrefl
tff(fact_1906_dual__order_Oirrefl,axiom,
! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).
% dual_order.irrefl
tff(fact_1907_dual__order_Oirrefl,axiom,
! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),A) ).
% dual_order.irrefl
tff(fact_1908_dual__order_Oirrefl,axiom,
! [A: code_natural] : ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),A) ).
% dual_order.irrefl
tff(fact_1909_dual__order_Oasym,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ).
% dual_order.asym
tff(fact_1910_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).
% dual_order.asym
tff(fact_1911_dual__order_Oasym,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).
% dual_order.asym
tff(fact_1912_dual__order_Oasym,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).
% dual_order.asym
tff(fact_1913_dual__order_Oasym,axiom,
! [B: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),A)
=> ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B) ) ).
% dual_order.asym
tff(fact_1914_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
=> ( ( X != Y )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).
% linorder_cases
tff(fact_1915_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
=> ( ( X != Y )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).
% linorder_cases
tff(fact_1916_linorder__cases,axiom,
! [X: code_integer,Y: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
=> ( ( X != Y )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ) ).
% linorder_cases
tff(fact_1917_linorder__cases,axiom,
! [X: code_natural,Y: code_natural] :
( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
=> ( ( X != Y )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Y),X) ) ) ).
% linorder_cases
tff(fact_1918_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
<=> ( X = Y ) ) ) ).
% antisym_conv3
tff(fact_1919_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
<=> ( X = Y ) ) ) ).
% antisym_conv3
tff(fact_1920_antisym__conv3,axiom,
! [Y: code_integer,X: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X)
=> ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
<=> ( X = Y ) ) ) ).
% antisym_conv3
tff(fact_1921_antisym__conv3,axiom,
! [Y: code_natural,X: code_natural] :
( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Y),X)
=> ( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
<=> ( X = Y ) ) ) ).
% antisym_conv3
tff(fact_1922_less__induct,axiom,
! [P: fun_nat_o,A: nat] :
( ! [X2: nat] :
( ! [Y3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),X2)
=> aa_nat_o(P,Y3) )
=> aa_nat_o(P,X2) )
=> aa_nat_o(P,A) ) ).
% less_induct
tff(fact_1923_ord__less__eq__trans,axiom,
! [A: assn,B: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( ( B = C )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C) ) ) ).
% ord_less_eq_trans
tff(fact_1924_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( ( B = C )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).
% ord_less_eq_trans
tff(fact_1925_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( ( B = C )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).
% ord_less_eq_trans
tff(fact_1926_ord__less__eq__trans,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ( ( B = C )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),C) ) ) ).
% ord_less_eq_trans
tff(fact_1927_ord__less__eq__trans,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ( ( B = C )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),C) ) ) ).
% ord_less_eq_trans
tff(fact_1928_ord__eq__less__trans,axiom,
! [A: assn,B: assn,C: assn] :
( ( A = B )
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C) ) ) ).
% ord_eq_less_trans
tff(fact_1929_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).
% ord_eq_less_trans
tff(fact_1930_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).
% ord_eq_less_trans
tff(fact_1931_ord__eq__less__trans,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A = B )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),C) ) ) ).
% ord_eq_less_trans
tff(fact_1932_ord__eq__less__trans,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( ( A = B )
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),C) ) ) ).
% ord_eq_less_trans
tff(fact_1933_order_Oasym,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A) ) ).
% order.asym
tff(fact_1934_order_Oasym,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).
% order.asym
tff(fact_1935_order_Oasym,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).
% order.asym
tff(fact_1936_order_Oasym,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ).
% order.asym
tff(fact_1937_order_Oasym,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),A) ) ).
% order.asym
tff(fact_1938_less__imp__neq,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
=> ( X != Y ) ) ).
% less_imp_neq
tff(fact_1939_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
=> ( X != Y ) ) ).
% less_imp_neq
tff(fact_1940_less__imp__neq,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
=> ( X != Y ) ) ).
% less_imp_neq
tff(fact_1941_less__imp__neq,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
=> ( X != Y ) ) ).
% less_imp_neq
tff(fact_1942_less__imp__neq,axiom,
! [X: code_natural,Y: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
=> ( X != Y ) ) ).
% less_imp_neq
tff(fact_1943_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X_1) ).
% gt_ex
tff(fact_1944_gt__ex,axiom,
! [X: int] :
? [X_1: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,X),X_1) ).
% gt_ex
tff(fact_1945_lt__ex,axiom,
! [X: int] :
? [Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X) ).
% lt_ex
tff(fact_1946_disjoint__alt__simp3,axiom,
! [Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(minus_1356011639430497352at_nat(Aa,Ba)),Aa)
<=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Aa),Ba) != bot_bo2099793752762293965at_nat ) ) ).
% disjoint_alt_simp3
tff(fact_1947_disjoint__alt__simp3,axiom,
! [Aa: set_o,Ba: set_o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,minus_minus_set_o(Aa,Ba)),Aa)
<=> ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Aa),Ba) != bot_bot_set_o ) ) ).
% disjoint_alt_simp3
tff(fact_1948_disjoint__alt__simp3,axiom,
! [Aa: set_int,Ba: set_int] :
( aa_set_int_o(ord_less_set_int(minus_minus_set_int(Aa,Ba)),Aa)
<=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Aa),Ba) != bot_bot_set_int ) ) ).
% disjoint_alt_simp3
tff(fact_1949_disjoint__alt__simp3,axiom,
! [Aa: set_nat,Ba: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),Ba)),Aa)
<=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),Ba) != bot_bot_set_nat ) ) ).
% disjoint_alt_simp3
tff(fact_1950_fstE,axiom,
! [X: produc2453742332425399446nteger,A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,P: fun_fu4736795817735658359term_o] :
( ( X = produc565448451787809096nteger(A,B) )
=> ( aa_fun7993445754732355552term_o(P,aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,X))
=> aa_fun7993445754732355552term_o(P,A) ) ) ).
% fstE
tff(fact_1951_fstE,axiom,
! [X: produc8799533983703829792et_nat,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,P: fun_fu5135969500543783852at_o_o] :
( ( X = produc6017135503889078682et_nat(A,B) )
=> ( aa_fun1675921005893230629at_o_o(P,aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,X))
=> aa_fun1675921005893230629at_o_o(P,A) ) ) ).
% fstE
tff(fact_1952_fstE,axiom,
! [X: produc3112576180687729086et_nat,A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,P: fun_fu5135969500543783852at_o_o] :
( ( X = produc7291855603632349880et_nat(A,B) )
=> ( aa_fun1675921005893230629at_o_o(P,aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,X))
=> aa_fun1675921005893230629at_o_o(P,A) ) ) ).
% fstE
tff(fact_1953_fstE,axiom,
! [X: produc3043004577601930755nt_int,A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,P: fun_fu6378075912484669126term_o] :
( ( X = produc7320969046608631221nt_int(A,B) )
=> ( aa_fun7650857933016935215term_o(P,aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,X))
=> aa_fun7650857933016935215term_o(P,A) ) ) ).
% fstE
tff(fact_1954_fstE,axiom,
! [X: produc4788131091073775919nt_int,A: fun_in5730379987311621695e_term,B: product_prod_int_int,P: fun_fu3265955482920998386term_o] :
( ( X = produc4729913460834754785nt_int(A,B) )
=> ( aa_fun3020711759848831579term_o(P,aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,X))
=> aa_fun3020711759848831579term_o(P,A) ) ) ).
% fstE
tff(fact_1955_fstE,axiom,
! [X: produc3658429121746597890et_nat,A: heap_e7401611519738050253t_unit,B: set_nat,P: fun_he8413527710729349838unit_o] :
( ( X = produc7507926704131184380et_nat(A,B) )
=> ( aa_hea7994211679605476167unit_o(P,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,X))
=> aa_hea7994211679605476167unit_o(P,A) ) ) ).
% fstE
tff(fact_1956_fstE,axiom,
! [X: product_prod_int_int,A: int,B: int,P: fun_int_o] :
( ( X = aa_int7462565939943104719nt_int(product_Pair_int_int(A),B) )
=> ( aa_int_o(P,aa_Pro6799258434465214121nt_int(product_fst_int_int,X))
=> aa_int_o(P,A) ) ) ).
% fstE
tff(fact_1957_fst__eqD,axiom,
! [X: fun_Pr5098669352780829690e_term,Y: produc8923325533196201883nteger,A: fun_Pr5098669352780829690e_term] :
( ( aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,produc565448451787809096nteger(X,Y)) = A )
=> ( X = A ) ) ).
% fst_eqD
tff(fact_1958_fst__eqD,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat,A: fun_Pr6460944699146269935_nat_o] :
( ( aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,produc6017135503889078682et_nat(X,Y)) = A )
=> ( X = A ) ) ).
% fst_eqD
tff(fact_1959_fst__eqD,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat,A: fun_Pr6460944699146269935_nat_o] :
( ( aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,produc7291855603632349880et_nat(X,Y)) = A )
=> ( X = A ) ) ).
% fst_eqD
tff(fact_1960_fst__eqD,axiom,
! [X: fun_Pr7763680713840462443e_term,Y: product_prod_int_int,A: fun_Pr7763680713840462443e_term] :
( ( aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,produc7320969046608631221nt_int(X,Y)) = A )
=> ( X = A ) ) ).
% fst_eqD
tff(fact_1961_fst__eqD,axiom,
! [X: fun_in5730379987311621695e_term,Y: product_prod_int_int,A: fun_in5730379987311621695e_term] :
( ( aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,produc4729913460834754785nt_int(X,Y)) = A )
=> ( X = A ) ) ).
% fst_eqD
tff(fact_1962_fst__eqD,axiom,
! [X: heap_e7401611519738050253t_unit,Y: set_nat,A: heap_e7401611519738050253t_unit] :
( ( aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,produc7507926704131184380et_nat(X,Y)) = A )
=> ( X = A ) ) ).
% fst_eqD
tff(fact_1963_fst__eqD,axiom,
! [X: int,Y: int,A: int] :
( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y)) = A )
=> ( X = A ) ) ).
% fst_eqD
tff(fact_1964_fst__conv,axiom,
! [X1: fun_Pr5098669352780829690e_term,X22: produc8923325533196201883nteger] : aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,produc565448451787809096nteger(X1,X22)) = X1 ).
% fst_conv
tff(fact_1965_fst__conv,axiom,
! [X1: fun_Pr6460944699146269935_nat_o,X22: produc3658429121746597890et_nat] : aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,produc6017135503889078682et_nat(X1,X22)) = X1 ).
% fst_conv
tff(fact_1966_fst__conv,axiom,
! [X1: fun_Pr6460944699146269935_nat_o,X22: produc8799533983703829792et_nat] : aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,produc7291855603632349880et_nat(X1,X22)) = X1 ).
% fst_conv
tff(fact_1967_fst__conv,axiom,
! [X1: fun_Pr7763680713840462443e_term,X22: product_prod_int_int] : aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,produc7320969046608631221nt_int(X1,X22)) = X1 ).
% fst_conv
tff(fact_1968_fst__conv,axiom,
! [X1: fun_in5730379987311621695e_term,X22: product_prod_int_int] : aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,produc4729913460834754785nt_int(X1,X22)) = X1 ).
% fst_conv
tff(fact_1969_fst__conv,axiom,
! [X1: heap_e7401611519738050253t_unit,X22: set_nat] : aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,produc7507926704131184380et_nat(X1,X22)) = X1 ).
% fst_conv
tff(fact_1970_fst__conv,axiom,
! [X1: int,X22: int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(product_Pair_int_int(X1),X22)) = X1 ).
% fst_conv
tff(fact_1971_fstI,axiom,
! [X: produc2453742332425399446nteger,Y: fun_Pr5098669352780829690e_term,Z: produc8923325533196201883nteger] :
( ( X = produc565448451787809096nteger(Y,Z) )
=> ( aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,X) = Y ) ) ).
% fstI
tff(fact_1972_fstI,axiom,
! [X: produc8799533983703829792et_nat,Y: fun_Pr6460944699146269935_nat_o,Z: produc3658429121746597890et_nat] :
( ( X = produc6017135503889078682et_nat(Y,Z) )
=> ( aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,X) = Y ) ) ).
% fstI
tff(fact_1973_fstI,axiom,
! [X: produc3112576180687729086et_nat,Y: fun_Pr6460944699146269935_nat_o,Z: produc8799533983703829792et_nat] :
( ( X = produc7291855603632349880et_nat(Y,Z) )
=> ( aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,X) = Y ) ) ).
% fstI
tff(fact_1974_fstI,axiom,
! [X: produc3043004577601930755nt_int,Y: fun_Pr7763680713840462443e_term,Z: product_prod_int_int] :
( ( X = produc7320969046608631221nt_int(Y,Z) )
=> ( aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,X) = Y ) ) ).
% fstI
tff(fact_1975_fstI,axiom,
! [X: produc4788131091073775919nt_int,Y: fun_in5730379987311621695e_term,Z: product_prod_int_int] :
( ( X = produc4729913460834754785nt_int(Y,Z) )
=> ( aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,X) = Y ) ) ).
% fstI
tff(fact_1976_fstI,axiom,
! [X: produc3658429121746597890et_nat,Y: heap_e7401611519738050253t_unit,Z: set_nat] :
( ( X = produc7507926704131184380et_nat(Y,Z) )
=> ( aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,X) = Y ) ) ).
% fstI
tff(fact_1977_fstI,axiom,
! [X: product_prod_int_int,Y: int,Z: int] :
( ( X = aa_int7462565939943104719nt_int(product_Pair_int_int(Y),Z) )
=> ( aa_Pro6799258434465214121nt_int(product_fst_int_int,X) = Y ) ) ).
% fstI
tff(fact_1978_eq__fst__iff,axiom,
! [A: fun_Pr5098669352780829690e_term,P5: produc2453742332425399446nteger] :
( ( A = aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,P5) )
<=> ? [B4: produc8923325533196201883nteger] : P5 = produc565448451787809096nteger(A,B4) ) ).
% eq_fst_iff
tff(fact_1979_eq__fst__iff,axiom,
! [A: fun_Pr6460944699146269935_nat_o,P5: produc8799533983703829792et_nat] :
( ( A = aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,P5) )
<=> ? [B4: produc3658429121746597890et_nat] : P5 = produc6017135503889078682et_nat(A,B4) ) ).
% eq_fst_iff
tff(fact_1980_eq__fst__iff,axiom,
! [A: fun_Pr6460944699146269935_nat_o,P5: produc3112576180687729086et_nat] :
( ( A = aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,P5) )
<=> ? [B4: produc8799533983703829792et_nat] : P5 = produc7291855603632349880et_nat(A,B4) ) ).
% eq_fst_iff
tff(fact_1981_eq__fst__iff,axiom,
! [A: fun_Pr7763680713840462443e_term,P5: produc3043004577601930755nt_int] :
( ( A = aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,P5) )
<=> ? [B4: product_prod_int_int] : P5 = produc7320969046608631221nt_int(A,B4) ) ).
% eq_fst_iff
tff(fact_1982_eq__fst__iff,axiom,
! [A: fun_in5730379987311621695e_term,P5: produc4788131091073775919nt_int] :
( ( A = aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,P5) )
<=> ? [B4: product_prod_int_int] : P5 = produc4729913460834754785nt_int(A,B4) ) ).
% eq_fst_iff
tff(fact_1983_eq__fst__iff,axiom,
! [A: heap_e7401611519738050253t_unit,P5: produc3658429121746597890et_nat] :
( ( A = aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,P5) )
<=> ? [B4: set_nat] : P5 = produc7507926704131184380et_nat(A,B4) ) ).
% eq_fst_iff
tff(fact_1984_eq__fst__iff,axiom,
! [A: int,P5: product_prod_int_int] :
( ( A = aa_Pro6799258434465214121nt_int(product_fst_int_int,P5) )
<=> ? [B4: int] : P5 = aa_int7462565939943104719nt_int(product_Pair_int_int(A),B4) ) ).
% eq_fst_iff
tff(fact_1985_order__le__imp__less__or__eq,axiom,
! [X: code_natural,Y: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,X),Y)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_1986_order__le__imp__less__or__eq,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_1987_order__le__imp__less__or__eq,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_1988_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_1989_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_1990_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_1991_linorder__le__less__linear,axiom,
! [X: code_natural,Y: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,X),Y)
| aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Y),X) ) ).
% linorder_le_less_linear
tff(fact_1992_linorder__le__less__linear,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ).
% linorder_le_less_linear
tff(fact_1993_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).
% linorder_le_less_linear
tff(fact_1994_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
| aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).
% linorder_le_less_linear
tff(fact_1995_order__less__le__subst2,axiom,
! [A: assn,B: assn,F: fun_as4373426249795763277atural,C: code_natural] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_assn_Code_natural(F,B)),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_assn_Code_natural(F,X2)),aa_assn_Code_natural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_assn_Code_natural(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_1996_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: fun_nat_Code_natural,C: code_natural] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(F,B)),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(F,X2)),aa_nat_Code_natural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_1997_order__less__le__subst2,axiom,
! [A: int,B: int,F: fun_int_Code_natural,C: code_natural] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_int_Code_natural(F,B)),C)
=> ( ! [X2: int,Y2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_int_Code_natural(F,X2)),aa_int_Code_natural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_int_Code_natural(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_1998_order__less__le__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Co3734316457315071362atural,C: code_natural] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod607569804290114243atural(F,B)),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod607569804290114243atural(F,X2)),aa_Cod607569804290114243atural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod607569804290114243atural(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_1999_order__less__le__subst2,axiom,
! [A: code_natural,B: code_natural,F: fun_Co7398788559141407565atural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(F,B)),C)
=> ( ! [X2: code_natural,Y2: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(F,X2)),aa_Cod4272041906116450446atural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_2000_order__less__le__subst2,axiom,
! [A: assn,B: assn,F: fun_as9138348466139350018nteger,C: code_integer] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_assn_Code_integer(F,B)),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,X2)),aa_assn_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_2001_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: fun_nat_Code_integer,C: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,B)),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,X2)),aa_nat_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_2002_order__less__le__subst2,axiom,
! [A: int,B: int,F: fun_int_Code_integer,C: code_integer] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,B)),C)
=> ( ! [X2: int,Y2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(F,X2)),aa_int_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_2003_order__less__le__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Co8499238673658658103nteger,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F,B)),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,X2)),aa_Cod1783974652533396728nteger(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_2004_order__less__le__subst2,axiom,
! [A: code_natural,B: code_natural,F: fun_Co2940338738630218498nteger,C: code_integer] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod5448446754359732931nteger(F,B)),C)
=> ( ! [X2: code_natural,Y2: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod5448446754359732931nteger(F,X2)),aa_Cod5448446754359732931nteger(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod5448446754359732931nteger(F,A)),C) ) ) ) ).
% order_less_le_subst2
tff(fact_2005_order__less__le__subst1,axiom,
! [A: code_natural,F: fun_Co3734316457315071362atural,B: code_integer,C: code_integer] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_Cod607569804290114243atural(F,B))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod607569804290114243atural(F,X2)),aa_Cod607569804290114243atural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_Cod607569804290114243atural(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_2006_order__less__le__subst1,axiom,
! [A: code_integer,F: fun_Co8499238673658658103nteger,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(F,B))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F,X2)),aa_Cod1783974652533396728nteger(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_2007_order__less__le__subst1,axiom,
! [A: assn,F: fun_Co4490034186337852034r_assn,B: code_integer,C: code_integer] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_Code_integer_assn(F,B))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_Code_integer_assn(F,X2)),aa_Code_integer_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_Code_integer_assn(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_2008_order__less__le__subst1,axiom,
! [A: nat,F: fun_Code_integer_nat,B: code_integer,C: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_Code_integer_nat(F,B))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(F,X2)),aa_Code_integer_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_Code_integer_nat(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_2009_order__less__le__subst1,axiom,
! [A: int,F: fun_Code_integer_int,B: code_integer,C: code_integer] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_Code_integer_int(F,B))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(F,X2)),aa_Code_integer_int(F,Y2)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_Code_integer_int(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_2010_order__less__le__subst1,axiom,
! [A: code_natural,F: fun_as4373426249795763277atural,B: assn,C: assn] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_assn_Code_natural(F,B))
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_assn_Code_natural(F,X2)),aa_assn_Code_natural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_assn_Code_natural(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_2011_order__less__le__subst1,axiom,
! [A: code_integer,F: fun_as9138348466139350018nteger,B: assn,C: assn] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_assn_Code_integer(F,B))
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_assn_Code_integer(F,X2)),aa_assn_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_assn_Code_integer(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_2012_order__less__le__subst1,axiom,
! [A: assn,F: fun_assn_assn,B: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(F,B))
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_2013_order__less__le__subst1,axiom,
! [A: nat,F: fun_assn_nat,B: assn,C: assn] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_assn_nat(F,B))
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_assn_nat(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_2014_order__less__le__subst1,axiom,
! [A: int,F: fun_assn_int,B: assn,C: assn] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_assn_int(F,B))
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_assn_int(F,C)) ) ) ) ).
% order_less_le_subst1
tff(fact_2015_order__le__less__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Co3734316457315071362atural,C: code_natural] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod607569804290114243atural(F,B)),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod607569804290114243atural(F,X2)),aa_Cod607569804290114243atural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod607569804290114243atural(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_2016_order__le__less__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Co8499238673658658103nteger,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,B)),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F,X2)),aa_Cod1783974652533396728nteger(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_2017_order__le__less__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Co4490034186337852034r_assn,C: assn] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_Code_integer_assn(F,B)),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_Code_integer_assn(F,X2)),aa_Code_integer_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_Code_integer_assn(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_2018_order__le__less__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Code_integer_nat,C: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(F,B)),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(F,X2)),aa_Code_integer_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_2019_order__le__less__subst2,axiom,
! [A: code_integer,B: code_integer,F: fun_Code_integer_int,C: int] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(F,B)),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Y2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(F,X2)),aa_Code_integer_int(F,Y2)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_2020_order__le__less__subst2,axiom,
! [A: assn,B: assn,F: fun_as4373426249795763277atural,C: code_natural] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_assn_Code_natural(F,B)),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_assn_Code_natural(F,X2)),aa_assn_Code_natural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_assn_Code_natural(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_2021_order__le__less__subst2,axiom,
! [A: assn,B: assn,F: fun_as9138348466139350018nteger,C: code_integer] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,B)),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_assn_Code_integer(F,X2)),aa_assn_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_2022_order__le__less__subst2,axiom,
! [A: assn,B: assn,F: fun_assn_assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,B)),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,X2)),aa_assn_assn(F,Y2)) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_2023_order__le__less__subst2,axiom,
! [A: assn,B: assn,F: fun_assn_nat,C: nat] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,B)),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_assn_nat(F,X2)),aa_assn_nat(F,Y2)) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_2024_order__le__less__subst2,axiom,
! [A: assn,B: assn,F: fun_assn_int,C: int] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,B)),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_assn_int(F,X2)),aa_assn_int(F,Y2)) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,A)),C) ) ) ) ).
% order_le_less_subst2
tff(fact_2025_order__le__less__subst1,axiom,
! [A: code_natural,F: fun_as4373426249795763277atural,B: assn,C: assn] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),aa_assn_Code_natural(F,B))
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_assn_Code_natural(F,X2)),aa_assn_Code_natural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_assn_Code_natural(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_2026_order__le__less__subst1,axiom,
! [A: code_natural,F: fun_nat_Code_natural,B: nat,C: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),aa_nat_Code_natural(F,B))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(F,X2)),aa_nat_Code_natural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_nat_Code_natural(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_2027_order__le__less__subst1,axiom,
! [A: code_natural,F: fun_int_Code_natural,B: int,C: int] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),aa_int_Code_natural(F,B))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> ( ! [X2: int,Y2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_int_Code_natural(F,X2)),aa_int_Code_natural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_int_Code_natural(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_2028_order__le__less__subst1,axiom,
! [A: code_natural,F: fun_Co3734316457315071362atural,B: code_integer,C: code_integer] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),aa_Cod607569804290114243atural(F,B))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod607569804290114243atural(F,X2)),aa_Cod607569804290114243atural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_Cod607569804290114243atural(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_2029_order__le__less__subst1,axiom,
! [A: code_natural,F: fun_Co7398788559141407565atural,B: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),aa_Cod4272041906116450446atural(F,B))
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),C)
=> ( ! [X2: code_natural,Y2: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X2),Y2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(F,X2)),aa_Cod4272041906116450446atural(F,Y2)) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_Cod4272041906116450446atural(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_2030_order__le__less__subst1,axiom,
! [A: code_integer,F: fun_as9138348466139350018nteger,B: assn,C: assn] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_assn_Code_integer(F,B))
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
=> ( ! [X2: assn,Y2: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,X2)),aa_assn_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_assn_Code_integer(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_2031_order__le__less__subst1,axiom,
! [A: code_integer,F: fun_nat_Code_integer,B: nat,C: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_nat_Code_integer(F,B))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> ( ! [X2: nat,Y2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,X2)),aa_nat_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_nat_Code_integer(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_2032_order__le__less__subst1,axiom,
! [A: code_integer,F: fun_int_Code_integer,B: int,C: int] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_int_Code_integer(F,B))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> ( ! [X2: int,Y2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(F,X2)),aa_int_Code_integer(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_int_Code_integer(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_2033_order__le__less__subst1,axiom,
! [A: code_integer,F: fun_Co8499238673658658103nteger,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(F,B))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
=> ( ! [X2: code_integer,Y2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,X2)),aa_Cod1783974652533396728nteger(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_2034_order__le__less__subst1,axiom,
! [A: code_integer,F: fun_Co2940338738630218498nteger,B: code_natural,C: code_natural] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod5448446754359732931nteger(F,B))
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),C)
=> ( ! [X2: code_natural,Y2: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X2),Y2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod5448446754359732931nteger(F,X2)),aa_Cod5448446754359732931nteger(F,Y2)) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod5448446754359732931nteger(F,C)) ) ) ) ).
% order_le_less_subst1
tff(fact_2035_order__less__le__trans,axiom,
! [X: code_natural,Y: code_natural,Z: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,Y),Z)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Z) ) ) ).
% order_less_le_trans
tff(fact_2036_order__less__le__trans,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),Z)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z) ) ) ).
% order_less_le_trans
tff(fact_2037_order__less__le__trans,axiom,
! [X: assn,Y: assn,Z: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),Z)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Z) ) ) ).
% order_less_le_trans
tff(fact_2038_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Z)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Z) ) ) ).
% order_less_le_trans
tff(fact_2039_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z) ) ) ).
% order_less_le_trans
tff(fact_2040_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z) ) ) ).
% order_less_le_trans
tff(fact_2041_order__le__less__trans,axiom,
! [X: code_natural,Y: code_natural,Z: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,X),Y)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Y),Z)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Z) ) ) ).
% order_le_less_trans
tff(fact_2042_order__le__less__trans,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Z)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z) ) ) ).
% order_le_less_trans
tff(fact_2043_order__le__less__trans,axiom,
! [X: assn,Y: assn,Z: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),Z)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Z) ) ) ).
% order_le_less_trans
tff(fact_2044_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Y),Z)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Z) ) ) ).
% order_le_less_trans
tff(fact_2045_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z) ) ) ).
% order_le_less_trans
tff(fact_2046_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z) ) ) ).
% order_le_less_trans
tff(fact_2047_order__neq__le__trans,axiom,
! [A: code_natural,B: code_natural] :
( ( A != B )
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B) ) ) ).
% order_neq_le_trans
tff(fact_2048_order__neq__le__trans,axiom,
! [A: code_integer,B: code_integer] :
( ( A != B )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).
% order_neq_le_trans
tff(fact_2049_order__neq__le__trans,axiom,
! [A: assn,B: assn] :
( ( A != B )
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ) ).
% order_neq_le_trans
tff(fact_2050_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B) ) ) ).
% order_neq_le_trans
tff(fact_2051_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).
% order_neq_le_trans
tff(fact_2052_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).
% order_neq_le_trans
tff(fact_2053_order__le__neq__trans,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B)
=> ( ( A != B )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B) ) ) ).
% order_le_neq_trans
tff(fact_2054_order__le__neq__trans,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( ( A != B )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).
% order_le_neq_trans
tff(fact_2055_order__le__neq__trans,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( ( A != B )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ) ).
% order_le_neq_trans
tff(fact_2056_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
=> ( ( A != B )
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B) ) ) ).
% order_le_neq_trans
tff(fact_2057_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( ( A != B )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).
% order_le_neq_trans
tff(fact_2058_order__le__neq__trans,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( ( A != B )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).
% order_le_neq_trans
tff(fact_2059_order__less__imp__le,axiom,
! [X: code_natural,Y: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,X),Y) ) ).
% order_less_imp_le
tff(fact_2060_order__less__imp__le,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y) ) ).
% order_less_imp_le
tff(fact_2061_order__less__imp__le,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y) ) ).
% order_less_imp_le
tff(fact_2062_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y) ) ).
% order_less_imp_le
tff(fact_2063_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).
% order_less_imp_le
tff(fact_2064_order__less__imp__le,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).
% order_less_imp_le
tff(fact_2065_linorder__not__less,axiom,
! [X: code_natural,Y: code_natural] :
( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
<=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,Y),X) ) ).
% linorder_not_less
tff(fact_2066_linorder__not__less,axiom,
! [X: code_integer,Y: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X) ) ).
% linorder_not_less
tff(fact_2067_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).
% linorder_not_less
tff(fact_2068_linorder__not__less,axiom,
! [X: int,Y: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).
% linorder_not_less
tff(fact_2069_linorder__not__le,axiom,
! [X: code_natural,Y: code_natural] :
( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,X),Y)
<=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Y),X) ) ).
% linorder_not_le
tff(fact_2070_linorder__not__le,axiom,
! [X: code_integer,Y: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ).
% linorder_not_le
tff(fact_2071_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).
% linorder_not_le
tff(fact_2072_linorder__not__le,axiom,
! [X: int,Y: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).
% linorder_not_le
tff(fact_2073_order__less__le,axiom,
! [X: code_natural,Y: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
<=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,X),Y)
& ( X != Y ) ) ) ).
% order_less_le
tff(fact_2074_order__less__le,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
& ( X != Y ) ) ) ).
% order_less_le
tff(fact_2075_order__less__le,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
<=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
& ( X != Y ) ) ) ).
% order_less_le
tff(fact_2076_order__less__le,axiom,
! [X: set_nat,Y: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
<=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
& ( X != Y ) ) ) ).
% order_less_le
tff(fact_2077_order__less__le,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
& ( X != Y ) ) ) ).
% order_less_le
tff(fact_2078_order__less__le,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
& ( X != Y ) ) ) ).
% order_less_le
tff(fact_2079_order__le__less,axiom,
! [X: code_natural,Y: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,X),Y)
<=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
| ( X = Y ) ) ) ).
% order_le_less
tff(fact_2080_order__le__less,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
| ( X = Y ) ) ) ).
% order_le_less
tff(fact_2081_order__le__less,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
<=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
| ( X = Y ) ) ) ).
% order_le_less
tff(fact_2082_order__le__less,axiom,
! [X: set_nat,Y: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
<=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
| ( X = Y ) ) ) ).
% order_le_less
tff(fact_2083_order__le__less,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
| ( X = Y ) ) ) ).
% order_le_less
tff(fact_2084_order__le__less,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
| ( X = Y ) ) ) ).
% order_le_less
tff(fact_2085_dual__order_Ostrict__implies__order,axiom,
! [B: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),A)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),A) ) ).
% dual_order.strict_implies_order
tff(fact_2086_dual__order_Ostrict__implies__order,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ).
% dual_order.strict_implies_order
tff(fact_2087_dual__order_Ostrict__implies__order,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A) ) ).
% dual_order.strict_implies_order
tff(fact_2088_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A) ) ).
% dual_order.strict_implies_order
tff(fact_2089_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).
% dual_order.strict_implies_order
tff(fact_2090_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).
% dual_order.strict_implies_order
tff(fact_2091_order_Ostrict__implies__order,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B) ) ).
% order.strict_implies_order
tff(fact_2092_order_Ostrict__implies__order,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).
% order.strict_implies_order
tff(fact_2093_order_Ostrict__implies__order,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B) ) ).
% order.strict_implies_order
tff(fact_2094_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ).
% order.strict_implies_order
tff(fact_2095_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).
% order.strict_implies_order
tff(fact_2096_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% order.strict_implies_order
tff(fact_2097_dual__order_Ostrict__iff__not,axiom,
! [B: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),A)
<=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),A)
& ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B) ) ) ).
% dual_order.strict_iff_not
tff(fact_2098_dual__order_Ostrict__iff__not,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
& ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).
% dual_order.strict_iff_not
tff(fact_2099_dual__order_Ostrict__iff__not,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
<=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
& ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B) ) ) ).
% dual_order.strict_iff_not
tff(fact_2100_dual__order_Ostrict__iff__not,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
<=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
& ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ) ).
% dual_order.strict_iff_not
tff(fact_2101_dual__order_Ostrict__iff__not,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
& ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).
% dual_order.strict_iff_not
tff(fact_2102_dual__order_Ostrict__iff__not,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
& ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).
% dual_order.strict_iff_not
tff(fact_2103_dual__order_Ostrict__trans2,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,C),B)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,C),A) ) ) ).
% dual_order.strict_trans2
tff(fact_2104_dual__order_Ostrict__trans2,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A) ) ) ).
% dual_order.strict_trans2
tff(fact_2105_dual__order_Ostrict__trans2,axiom,
! [B: assn,A: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),B)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),A) ) ) ).
% dual_order.strict_trans2
tff(fact_2106_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),B)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),A) ) ) ).
% dual_order.strict_trans2
tff(fact_2107_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).
% dual_order.strict_trans2
tff(fact_2108_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).
% dual_order.strict_trans2
tff(fact_2109_dual__order_Ostrict__trans1,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,C),B)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,C),A) ) ) ).
% dual_order.strict_trans1
tff(fact_2110_dual__order_Ostrict__trans1,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A) ) ) ).
% dual_order.strict_trans1
tff(fact_2111_dual__order_Ostrict__trans1,axiom,
! [B: assn,A: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),B)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),A) ) ) ).
% dual_order.strict_trans1
tff(fact_2112_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),B)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),A) ) ) ).
% dual_order.strict_trans1
tff(fact_2113_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).
% dual_order.strict_trans1
tff(fact_2114_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).
% dual_order.strict_trans1
tff(fact_2115_dual__order_Ostrict__iff__order,axiom,
! [B: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),A)
<=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),A)
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_2116_dual__order_Ostrict__iff__order,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_2117_dual__order_Ostrict__iff__order,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
<=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_2118_dual__order_Ostrict__iff__order,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
<=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_2119_dual__order_Ostrict__iff__order,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_2120_dual__order_Ostrict__iff__order,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
& ( A != B ) ) ) ).
% dual_order.strict_iff_order
tff(fact_2121_dual__order_Oorder__iff__strict,axiom,
! [B: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),A)
<=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),A)
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_2122_dual__order_Oorder__iff__strict,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_2123_dual__order_Oorder__iff__strict,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
<=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_2124_dual__order_Oorder__iff__strict,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
<=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_2125_dual__order_Oorder__iff__strict,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_2126_dual__order_Oorder__iff__strict,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
| ( A = B ) ) ) ).
% dual_order.order_iff_strict
tff(fact_2127_order_Ostrict__iff__not,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
<=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B)
& ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),A) ) ) ).
% order.strict_iff_not
tff(fact_2128_order_Ostrict__iff__not,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
& ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ).
% order.strict_iff_not
tff(fact_2129_order_Ostrict__iff__not,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
<=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
& ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A) ) ) ).
% order.strict_iff_not
tff(fact_2130_order_Ostrict__iff__not,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
<=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
& ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A) ) ) ).
% order.strict_iff_not
tff(fact_2131_order_Ostrict__iff__not,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
& ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).
% order.strict_iff_not
tff(fact_2132_order_Ostrict__iff__not,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
& ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).
% order.strict_iff_not
tff(fact_2133_order_Ostrict__trans2,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),C) ) ) ).
% order.strict_trans2
tff(fact_2134_order_Ostrict__trans2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),C) ) ) ).
% order.strict_trans2
tff(fact_2135_order_Ostrict__trans2,axiom,
! [A: assn,B: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C) ) ) ).
% order.strict_trans2
tff(fact_2136_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C) ) ) ).
% order.strict_trans2
tff(fact_2137_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).
% order.strict_trans2
tff(fact_2138_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).
% order.strict_trans2
tff(fact_2139_order_Ostrict__trans1,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),C) ) ) ).
% order.strict_trans1
tff(fact_2140_order_Ostrict__trans1,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),C) ) ) ).
% order.strict_trans1
tff(fact_2141_order_Ostrict__trans1,axiom,
! [A: assn,B: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C) ) ) ).
% order.strict_trans1
tff(fact_2142_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),C)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C) ) ) ).
% order.strict_trans1
tff(fact_2143_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).
% order.strict_trans1
tff(fact_2144_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).
% order.strict_trans1
tff(fact_2145_order_Ostrict__iff__order,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
<=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B)
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_2146_order_Ostrict__iff__order,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_2147_order_Ostrict__iff__order,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
<=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_2148_order_Ostrict__iff__order,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
<=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_2149_order_Ostrict__iff__order,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_2150_order_Ostrict__iff__order,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
& ( A != B ) ) ) ).
% order.strict_iff_order
tff(fact_2151_order_Oorder__iff__strict,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B)
<=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_2152_order_Oorder__iff__strict,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_2153_order_Oorder__iff__strict,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
<=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_2154_order_Oorder__iff__strict,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
<=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_2155_order_Oorder__iff__strict,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_2156_order_Oorder__iff__strict,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
| ( A = B ) ) ) ).
% order.order_iff_strict
tff(fact_2157_not__le__imp__less,axiom,
! [Y: code_natural,X: code_natural] :
( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,Y),X)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y) ) ).
% not_le_imp_less
tff(fact_2158_not__le__imp__less,axiom,
! [Y: code_integer,X: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y) ) ).
% not_le_imp_less
tff(fact_2159_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ).
% not_le_imp_less
tff(fact_2160_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ).
% not_le_imp_less
tff(fact_2161_less__le__not__le,axiom,
! [X: code_natural,Y: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
<=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,X),Y)
& ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,Y),X) ) ) ).
% less_le_not_le
tff(fact_2162_less__le__not__le,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
& ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X) ) ) ).
% less_le_not_le
tff(fact_2163_less__le__not__le,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
<=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
& ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X) ) ) ).
% less_le_not_le
tff(fact_2164_less__le__not__le,axiom,
! [X: set_nat,Y: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
<=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
& ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X) ) ) ).
% less_le_not_le
tff(fact_2165_less__le__not__le,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
& ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ) ).
% less_le_not_le
tff(fact_2166_less__le__not__le,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
& ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ) ).
% less_le_not_le
tff(fact_2167_antisym__conv2,axiom,
! [X: code_natural,Y: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,X),Y)
=> ( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
<=> ( X = Y ) ) ) ).
% antisym_conv2
tff(fact_2168_antisym__conv2,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
=> ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
<=> ( X = Y ) ) ) ).
% antisym_conv2
tff(fact_2169_antisym__conv2,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
=> ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
<=> ( X = Y ) ) ) ).
% antisym_conv2
tff(fact_2170_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
=> ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
<=> ( X = Y ) ) ) ).
% antisym_conv2
tff(fact_2171_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
<=> ( X = Y ) ) ) ).
% antisym_conv2
tff(fact_2172_antisym__conv2,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
<=> ( X = Y ) ) ) ).
% antisym_conv2
tff(fact_2173_antisym__conv1,axiom,
! [X: code_natural,Y: code_natural] :
( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,X),Y)
<=> ( X = Y ) ) ) ).
% antisym_conv1
tff(fact_2174_antisym__conv1,axiom,
! [X: code_integer,Y: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
<=> ( X = Y ) ) ) ).
% antisym_conv1
tff(fact_2175_antisym__conv1,axiom,
! [X: assn,Y: assn] :
( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
<=> ( X = Y ) ) ) ).
% antisym_conv1
tff(fact_2176_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
<=> ( X = Y ) ) ) ).
% antisym_conv1
tff(fact_2177_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
<=> ( X = Y ) ) ) ).
% antisym_conv1
tff(fact_2178_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
<=> ( X = Y ) ) ) ).
% antisym_conv1
tff(fact_2179_nless__le,axiom,
! [A: code_natural,B: code_natural] :
( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
<=> ( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B)
| ( A = B ) ) ) ).
% nless_le
tff(fact_2180_nless__le,axiom,
! [A: code_integer,B: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
<=> ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
| ( A = B ) ) ) ).
% nless_le
tff(fact_2181_nless__le,axiom,
! [A: assn,B: assn] :
( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
<=> ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
| ( A = B ) ) ) ).
% nless_le
tff(fact_2182_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
<=> ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
| ( A = B ) ) ) ).
% nless_le
tff(fact_2183_nless__le,axiom,
! [A: nat,B: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
<=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
| ( A = B ) ) ) ).
% nless_le
tff(fact_2184_nless__le,axiom,
! [A: int,B: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
<=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
| ( A = B ) ) ) ).
% nless_le
tff(fact_2185_leI,axiom,
! [X: code_natural,Y: code_natural] :
( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,Y),X) ) ).
% leI
tff(fact_2186_leI,axiom,
! [X: code_integer,Y: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X) ) ).
% leI
tff(fact_2187_leI,axiom,
! [X: nat,Y: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).
% leI
tff(fact_2188_leI,axiom,
! [X: int,Y: int] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).
% leI
tff(fact_2189_leD,axiom,
! [Y: code_natural,X: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,Y),X)
=> ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Y) ) ).
% leD
tff(fact_2190_leD,axiom,
! [Y: code_integer,X: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X)
=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y) ) ).
% leD
tff(fact_2191_leD,axiom,
! [Y: assn,X: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y) ) ).
% leD
tff(fact_2192_leD,axiom,
! [Y: set_nat,X: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
=> ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y) ) ).
% leD
tff(fact_2193_leD,axiom,
! [Y: nat,X: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ).
% leD
tff(fact_2194_leD,axiom,
! [Y: int,X: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ).
% leD
tff(fact_2195_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
<=> ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
tff(fact_2196_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
tff(fact_2197_not__less__zero,axiom,
! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),zero_zero_nat) ).
% not_less_zero
tff(fact_2198_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).
% gr_zeroI
tff(fact_2199_less__numeral__extra_I4_J,axiom,
~ aa_rat_o(ord_less_rat(one_one_rat),one_one_rat) ).
% less_numeral_extra(4)
tff(fact_2200_less__numeral__extra_I4_J,axiom,
~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),one_one_nat) ).
% less_numeral_extra(4)
tff(fact_2201_less__numeral__extra_I4_J,axiom,
~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),one_one_int) ).
% less_numeral_extra(4)
tff(fact_2202_less__numeral__extra_I4_J,axiom,
~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),one_one_Code_integer) ).
% less_numeral_extra(4)
tff(fact_2203_less__numeral__extra_I4_J,axiom,
~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),one_one_Code_natural) ).
% less_numeral_extra(4)
tff(fact_2204_diff__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_rat(A),B)
=> aa_rat_o(ord_less_rat(minus_minus_rat(A,C)),minus_minus_rat(B,C)) ) ).
% diff_strict_right_mono
tff(fact_2205_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).
% diff_strict_right_mono
tff(fact_2206_diff__strict__right__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),C)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) ) ).
% diff_strict_right_mono
tff(fact_2207_diff__strict__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(ord_less_rat(B),A)
=> aa_rat_o(ord_less_rat(minus_minus_rat(C,A)),minus_minus_rat(C,B)) ) ).
% diff_strict_left_mono
tff(fact_2208_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).
% diff_strict_left_mono
tff(fact_2209_diff__strict__left__mono,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(C),A)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(C),B)) ) ).
% diff_strict_left_mono
tff(fact_2210_diff__eq__diff__less,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( minus_minus_rat(A,B) = minus_minus_rat(C,D2) )
=> ( aa_rat_o(ord_less_rat(A),B)
<=> aa_rat_o(ord_less_rat(C),D2) ) ) ).
% diff_eq_diff_less
tff(fact_2211_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D2) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2) ) ) ).
% diff_eq_diff_less
tff(fact_2212_diff__eq__diff__less,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(C),D2) )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D2) ) ) ).
% diff_eq_diff_less
tff(fact_2213_diff__strict__mono,axiom,
! [A: rat,B: rat,D2: rat,C: rat] :
( aa_rat_o(ord_less_rat(A),B)
=> ( aa_rat_o(ord_less_rat(D2),C)
=> aa_rat_o(ord_less_rat(minus_minus_rat(A,C)),minus_minus_rat(B,D2)) ) ) ).
% diff_strict_mono
tff(fact_2214_diff__strict__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,D2),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D2)) ) ) ).
% diff_strict_mono
tff(fact_2215_diff__strict__mono,axiom,
! [A: code_integer,B: code_integer,D2: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,D2),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),C)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),D2)) ) ) ).
% diff_strict_mono
tff(fact_2216_bot_Oextremum__strict,axiom,
! [A: set_Pr1261947904930325089at_nat] : ~ aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(A),bot_bo2099793752762293965at_nat) ).
% bot.extremum_strict
tff(fact_2217_bot_Oextremum__strict,axiom,
! [A: set_o] : ~ aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,A),bot_bot_set_o) ).
% bot.extremum_strict
tff(fact_2218_bot_Oextremum__strict,axiom,
! [A: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),bot_bot_set_nat) ).
% bot.extremum_strict
tff(fact_2219_bot_Oextremum__strict,axiom,
! [A: set_int] : ~ aa_set_int_o(ord_less_set_int(A),bot_bot_set_int) ).
% bot.extremum_strict
tff(fact_2220_bot_Oextremum__strict,axiom,
! [A: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),bot_bot_assn) ).
% bot.extremum_strict
tff(fact_2221_bot_Oextremum__strict,axiom,
! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),bot_bot_nat) ).
% bot.extremum_strict
tff(fact_2222_bot_Onot__eq__extremum,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( A != bot_bo2099793752762293965at_nat )
<=> aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(bot_bo2099793752762293965at_nat),A) ) ).
% bot.not_eq_extremum
tff(fact_2223_bot_Onot__eq__extremum,axiom,
! [A: set_o] :
( ( A != bot_bot_set_o )
<=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,bot_bot_set_o),A) ) ).
% bot.not_eq_extremum
tff(fact_2224_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
<=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,bot_bot_set_nat),A) ) ).
% bot.not_eq_extremum
tff(fact_2225_bot_Onot__eq__extremum,axiom,
! [A: set_int] :
( ( A != bot_bot_set_int )
<=> aa_set_int_o(ord_less_set_int(bot_bot_set_int),A) ) ).
% bot.not_eq_extremum
tff(fact_2226_bot_Onot__eq__extremum,axiom,
! [A: assn] :
( ( A != bot_bot_assn )
<=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,bot_bot_assn),A) ) ).
% bot.not_eq_extremum
tff(fact_2227_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,bot_bot_nat),A) ) ).
% bot.not_eq_extremum
tff(fact_2228_not__psubset__empty,axiom,
! [Aa: set_Pr1261947904930325089at_nat] : ~ aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(Aa),bot_bo2099793752762293965at_nat) ).
% not_psubset_empty
tff(fact_2229_not__psubset__empty,axiom,
! [Aa: set_o] : ~ aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,Aa),bot_bot_set_o) ).
% not_psubset_empty
tff(fact_2230_not__psubset__empty,axiom,
! [Aa: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Aa),bot_bot_set_nat) ).
% not_psubset_empty
tff(fact_2231_not__psubset__empty,axiom,
! [Aa: set_int] : ~ aa_set_int_o(ord_less_set_int(Aa),bot_bot_set_int) ).
% not_psubset_empty
tff(fact_2232_compl__less__swap2,axiom,
! [Y: assn,X: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(uminus_uminus_assn,Y)),X)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(uminus_uminus_assn,X)),Y) ) ).
% compl_less_swap2
tff(fact_2233_compl__less__swap1,axiom,
! [Y: assn,X: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),aa_assn_assn(uminus_uminus_assn,X))
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),aa_assn_assn(uminus_uminus_assn,Y)) ) ).
% compl_less_swap1
tff(fact_2234_minus__less__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),A) ) ).
% minus_less_iff
tff(fact_2235_minus__less__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,A)),B)
<=> aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,B)),A) ) ).
% minus_less_iff
tff(fact_2236_minus__less__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).
% minus_less_iff
tff(fact_2237_less__minus__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(uminus_uminus_int,A)) ) ).
% less_minus_iff
tff(fact_2238_less__minus__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(A),aa_rat_rat(uminus_uminus_rat,B))
<=> aa_rat_o(ord_less_rat(B),aa_rat_rat(uminus_uminus_rat,A)) ) ).
% less_minus_iff
tff(fact_2239_less__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).
% less_minus_iff
tff(fact_2240_top_Oextremum__strict,axiom,
! [A: set_o] : ~ aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,top_top_set_o),A) ).
% top.extremum_strict
tff(fact_2241_top_Oextremum__strict,axiom,
! [A: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,top_top_set_nat),A) ).
% top.extremum_strict
tff(fact_2242_top_Oextremum__strict,axiom,
! [A: set_Product_unit] : ~ aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le8056459307392131481t_unit,top_to1996260823553986621t_unit),A) ).
% top.extremum_strict
tff(fact_2243_top_Oextremum__strict,axiom,
! [A: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,top_top_assn),A) ).
% top.extremum_strict
tff(fact_2244_top_Onot__eq__extremum,axiom,
! [A: set_o] :
( ( A != top_top_set_o )
<=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,A),top_top_set_o) ) ).
% top.not_eq_extremum
tff(fact_2245_top_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != top_top_set_nat )
<=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),top_top_set_nat) ) ).
% top.not_eq_extremum
tff(fact_2246_top_Onot__eq__extremum,axiom,
! [A: set_Product_unit] :
( ( A != top_to1996260823553986621t_unit )
<=> aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le8056459307392131481t_unit,A),top_to1996260823553986621t_unit) ) ).
% top.not_eq_extremum
tff(fact_2247_top_Onot__eq__extremum,axiom,
! [A: assn] :
( ( A != top_top_assn )
<=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),top_top_assn) ) ).
% top.not_eq_extremum
tff(fact_2248_inf_Ostrict__coboundedI2,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),C)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).
% inf.strict_coboundedI2
tff(fact_2249_inf_Ostrict__coboundedI2,axiom,
! [B: assn,C: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C) ) ).
% inf.strict_coboundedI2
tff(fact_2250_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).
% inf.strict_coboundedI2
tff(fact_2251_inf_Ostrict__coboundedI2,axiom,
! [B: int,C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).
% inf.strict_coboundedI2
tff(fact_2252_inf_Ostrict__coboundedI1,axiom,
! [A: set_nat,C: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).
% inf.strict_coboundedI1
tff(fact_2253_inf_Ostrict__coboundedI1,axiom,
! [A: assn,C: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C) ) ).
% inf.strict_coboundedI1
tff(fact_2254_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).
% inf.strict_coboundedI1
tff(fact_2255_inf_Ostrict__coboundedI1,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).
% inf.strict_coboundedI1
tff(fact_2256_inf_Ostrict__order__iff,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
<=> ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) )
& ( A != B ) ) ) ).
% inf.strict_order_iff
tff(fact_2257_inf_Ostrict__order__iff,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
<=> ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) )
& ( A != B ) ) ) ).
% inf.strict_order_iff
tff(fact_2258_inf_Ostrict__order__iff,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
<=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) )
& ( A != B ) ) ) ).
% inf.strict_order_iff
tff(fact_2259_inf_Ostrict__order__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
<=> ( ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) )
& ( A != B ) ) ) ).
% inf.strict_order_iff
tff(fact_2260_inf_Ostrict__boundedE,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C))
=> ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
=> ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C) ) ) ).
% inf.strict_boundedE
tff(fact_2261_inf_Ostrict__boundedE,axiom,
! [A: assn,B: assn,C: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C))
=> ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C) ) ) ).
% inf.strict_boundedE
tff(fact_2262_inf_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C))
=> ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).
% inf.strict_boundedE
tff(fact_2263_inf_Ostrict__boundedE,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C))
=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).
% inf.strict_boundedE
tff(fact_2264_inf_Oabsorb4,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).
% inf.absorb4
tff(fact_2265_inf_Oabsorb4,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).
% inf.absorb4
tff(fact_2266_inf_Oabsorb4,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = B ) ) ).
% inf.absorb4
tff(fact_2267_inf_Oabsorb4,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = B ) ) ).
% inf.absorb4
tff(fact_2268_inf_Oabsorb3,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).
% inf.absorb3
tff(fact_2269_inf_Oabsorb3,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).
% inf.absorb3
tff(fact_2270_inf_Oabsorb3,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = A ) ) ).
% inf.absorb3
tff(fact_2271_inf_Oabsorb3,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = A ) ) ).
% inf.absorb3
tff(fact_2272_less__infI2,axiom,
! [B: set_nat,X: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),X)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).
% less_infI2
tff(fact_2273_less__infI2,axiom,
! [B: assn,X: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),X)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).
% less_infI2
tff(fact_2274_less__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),X)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).
% less_infI2
tff(fact_2275_less__infI2,axiom,
! [B: int,X: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),X)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).
% less_infI2
tff(fact_2276_less__infI1,axiom,
! [A: set_nat,X: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),X)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).
% less_infI1
tff(fact_2277_less__infI1,axiom,
! [A: assn,X: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),X)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).
% less_infI1
tff(fact_2278_less__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),X)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).
% less_infI1
tff(fact_2279_less__infI1,axiom,
! [A: int,X: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),X)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).
% less_infI1
tff(fact_2280_sup_Ostrict__coboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),B)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).
% sup.strict_coboundedI2
tff(fact_2281_sup_Ostrict__coboundedI2,axiom,
! [C: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(C),B)
=> aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).
% sup.strict_coboundedI2
tff(fact_2282_sup_Ostrict__coboundedI2,axiom,
! [C: assn,B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),B)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).
% sup.strict_coboundedI2
tff(fact_2283_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).
% sup.strict_coboundedI2
tff(fact_2284_sup_Ostrict__coboundedI2,axiom,
! [C: int,B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).
% sup.strict_coboundedI2
tff(fact_2285_sup_Ostrict__coboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),A)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).
% sup.strict_coboundedI1
tff(fact_2286_sup_Ostrict__coboundedI1,axiom,
! [C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(C),A)
=> aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(C),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).
% sup.strict_coboundedI1
tff(fact_2287_sup_Ostrict__coboundedI1,axiom,
! [C: assn,A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),A)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).
% sup.strict_coboundedI1
tff(fact_2288_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).
% sup.strict_coboundedI1
tff(fact_2289_sup_Ostrict__coboundedI1,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).
% sup.strict_coboundedI1
tff(fact_2290_sup_Ostrict__order__iff,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
<=> ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
& ( A != B ) ) ) ).
% sup.strict_order_iff
tff(fact_2291_sup_Ostrict__order__iff,axiom,
! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(B),A)
<=> ( ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) )
& ( A != B ) ) ) ).
% sup.strict_order_iff
tff(fact_2292_sup_Ostrict__order__iff,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
<=> ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) )
& ( A != B ) ) ) ).
% sup.strict_order_iff
tff(fact_2293_sup_Ostrict__order__iff,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
<=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) )
& ( A != B ) ) ) ).
% sup.strict_order_iff
tff(fact_2294_sup_Ostrict__order__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
<=> ( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) )
& ( A != B ) ) ) ).
% sup.strict_order_iff
tff(fact_2295_sup_Ostrict__boundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A)
=> ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
=> ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),A) ) ) ).
% sup.strict_boundedE
tff(fact_2296_sup_Ostrict__boundedE,axiom,
! [B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C)),A)
=> ~ ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(B),A)
=> ~ aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(C),A) ) ) ).
% sup.strict_boundedE
tff(fact_2297_sup_Ostrict__boundedE,axiom,
! [B: assn,C: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C)),A)
=> ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),A) ) ) ).
% sup.strict_boundedE
tff(fact_2298_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A)
=> ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).
% sup.strict_boundedE
tff(fact_2299_sup_Ostrict__boundedE,axiom,
! [B: int,C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A)
=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).
% sup.strict_boundedE
tff(fact_2300_sup_Oabsorb4,axiom,
! [A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).
% sup.absorb4
tff(fact_2301_sup_Oabsorb4,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(A),B)
=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = B ) ) ).
% sup.absorb4
tff(fact_2302_sup_Oabsorb4,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = B ) ) ).
% sup.absorb4
tff(fact_2303_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).
% sup.absorb4
tff(fact_2304_sup_Oabsorb4,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).
% sup.absorb4
tff(fact_2305_sup_Oabsorb3,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).
% sup.absorb3
tff(fact_2306_sup_Oabsorb3,axiom,
! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(B),A)
=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = A ) ) ).
% sup.absorb3
tff(fact_2307_sup_Oabsorb3,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = A ) ) ).
% sup.absorb3
tff(fact_2308_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).
% sup.absorb3
tff(fact_2309_sup_Oabsorb3,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).
% sup.absorb3
tff(fact_2310_less__supI2,axiom,
! [X: set_nat,B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),B)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).
% less_supI2
tff(fact_2311_less__supI2,axiom,
! [X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(X),B)
=> aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).
% less_supI2
tff(fact_2312_less__supI2,axiom,
! [X: assn,B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),B)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).
% less_supI2
tff(fact_2313_less__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).
% less_supI2
tff(fact_2314_less__supI2,axiom,
! [X: int,B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).
% less_supI2
tff(fact_2315_less__supI1,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),A)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).
% less_supI1
tff(fact_2316_less__supI1,axiom,
! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(X),A)
=> aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).
% less_supI1
tff(fact_2317_less__supI1,axiom,
! [X: assn,A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),A)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).
% less_supI1
tff(fact_2318_less__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).
% less_supI1
tff(fact_2319_less__supI1,axiom,
! [X: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).
% less_supI1
tff(fact_2320_less__eq__assn__def,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
<=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).
% less_eq_assn_def
tff(fact_2321_prod_Oexhaust__sel,axiom,
! [Prod: produc2453742332425399446nteger] : Prod = produc565448451787809096nteger(aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,Prod),aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,Prod)) ).
% prod.exhaust_sel
tff(fact_2322_prod_Oexhaust__sel,axiom,
! [Prod: produc8799533983703829792et_nat] : Prod = produc6017135503889078682et_nat(aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,Prod),aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,Prod)) ).
% prod.exhaust_sel
tff(fact_2323_prod_Oexhaust__sel,axiom,
! [Prod: produc3112576180687729086et_nat] : Prod = produc7291855603632349880et_nat(aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,Prod),aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,Prod)) ).
% prod.exhaust_sel
tff(fact_2324_prod_Oexhaust__sel,axiom,
! [Prod: produc3043004577601930755nt_int] : Prod = produc7320969046608631221nt_int(aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,Prod),aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,Prod)) ).
% prod.exhaust_sel
tff(fact_2325_prod_Oexhaust__sel,axiom,
! [Prod: produc4788131091073775919nt_int] : Prod = produc4729913460834754785nt_int(aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,Prod),aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,Prod)) ).
% prod.exhaust_sel
tff(fact_2326_prod_Oexhaust__sel,axiom,
! [Prod: produc3658429121746597890et_nat] : Prod = produc7507926704131184380et_nat(aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Prod),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Prod)) ).
% prod.exhaust_sel
tff(fact_2327_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_int_int] : Prod = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ).
% prod.exhaust_sel
tff(fact_2328_prod_Oexhaust__sel,axiom,
! [Prod: produc5835291356934675326atural] : Prod = aa_Pro7165575656943818355atural(produc6639722614265839536atural(produc497848011034438852atural(Prod)),aa_Pro3673693039631929973atural(produc6591912806276919810atural,Prod)) ).
% prod.exhaust_sel
tff(fact_2329_surjective__pairing,axiom,
! [T: produc2453742332425399446nteger] : T = produc565448451787809096nteger(aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,T),aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,T)) ).
% surjective_pairing
tff(fact_2330_surjective__pairing,axiom,
! [T: produc8799533983703829792et_nat] : T = produc6017135503889078682et_nat(aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,T),aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,T)) ).
% surjective_pairing
tff(fact_2331_surjective__pairing,axiom,
! [T: produc3112576180687729086et_nat] : T = produc7291855603632349880et_nat(aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,T),aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,T)) ).
% surjective_pairing
tff(fact_2332_surjective__pairing,axiom,
! [T: produc3043004577601930755nt_int] : T = produc7320969046608631221nt_int(aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,T),aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,T)) ).
% surjective_pairing
tff(fact_2333_surjective__pairing,axiom,
! [T: produc4788131091073775919nt_int] : T = produc4729913460834754785nt_int(aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,T),aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,T)) ).
% surjective_pairing
tff(fact_2334_surjective__pairing,axiom,
! [T: produc3658429121746597890et_nat] : T = produc7507926704131184380et_nat(aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,T),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,T)) ).
% surjective_pairing
tff(fact_2335_surjective__pairing,axiom,
! [T: product_prod_int_int] : T = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_fst_int_int,T)),aa_Pro6799258434465214121nt_int(product_snd_int_int,T)) ).
% surjective_pairing
tff(fact_2336_surjective__pairing,axiom,
! [T: produc5835291356934675326atural] : T = aa_Pro7165575656943818355atural(produc6639722614265839536atural(produc497848011034438852atural(T)),aa_Pro3673693039631929973atural(produc6591912806276919810atural,T)) ).
% surjective_pairing
tff(fact_2337_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_rat(A),B)
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2338_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2339_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2340_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2341_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B)) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2342_mult__less__cancel__right__disj,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
<=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
& aa_rat_o(ord_less_rat(A),B) )
| ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
& aa_rat_o(ord_less_rat(B),A) ) ) ) ).
% mult_less_cancel_right_disj
tff(fact_2343_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
& aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).
% mult_less_cancel_right_disj
tff(fact_2344_mult__less__cancel__right__disj,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
| ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).
% mult_less_cancel_right_disj
tff(fact_2345_mult__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_rat(A),B)
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).
% mult_strict_right_mono
tff(fact_2346_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).
% mult_strict_right_mono
tff(fact_2347_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).
% mult_strict_right_mono
tff(fact_2348_mult__strict__right__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).
% mult_strict_right_mono
tff(fact_2349_mult__strict__right__mono,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)) ) ) ).
% mult_strict_right_mono
tff(fact_2350_mult__strict__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(ord_less_rat(B),A)
=> ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).
% mult_strict_right_mono_neg
tff(fact_2351_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).
% mult_strict_right_mono_neg
tff(fact_2352_mult__strict__right__mono__neg,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).
% mult_strict_right_mono_neg
tff(fact_2353_mult__less__cancel__left__disj,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
<=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
& aa_rat_o(ord_less_rat(A),B) )
| ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
& aa_rat_o(ord_less_rat(B),A) ) ) ) ).
% mult_less_cancel_left_disj
tff(fact_2354_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
& aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).
% mult_less_cancel_left_disj
tff(fact_2355_mult__less__cancel__left__disj,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
| ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).
% mult_less_cancel_left_disj
tff(fact_2356_mult__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_rat(A),B)
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).
% mult_strict_left_mono
tff(fact_2357_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).
% mult_strict_left_mono
tff(fact_2358_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).
% mult_strict_left_mono
tff(fact_2359_mult__strict__left__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).
% mult_strict_left_mono
tff(fact_2360_mult__strict__left__mono,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B)) ) ) ).
% mult_strict_left_mono
tff(fact_2361_mult__strict__left__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(ord_less_rat(B),A)
=> ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).
% mult_strict_left_mono_neg
tff(fact_2362_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).
% mult_strict_left_mono_neg
tff(fact_2363_mult__strict__left__mono__neg,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).
% mult_strict_left_mono_neg
tff(fact_2364_mult__less__cancel__left__pos,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
<=> aa_rat_o(ord_less_rat(A),B) ) ) ).
% mult_less_cancel_left_pos
tff(fact_2365_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).
% mult_less_cancel_left_pos
tff(fact_2366_mult__less__cancel__left__pos,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).
% mult_less_cancel_left_pos
tff(fact_2367_mult__less__cancel__left__neg,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
<=> aa_rat_o(ord_less_rat(B),A) ) ) ).
% mult_less_cancel_left_neg
tff(fact_2368_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).
% mult_less_cancel_left_neg
tff(fact_2369_mult__less__cancel__left__neg,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ).
% mult_less_cancel_left_neg
tff(fact_2370_zero__less__mult__pos2,axiom,
! [B: rat,A: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A))
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> aa_rat_o(ord_less_rat(zero_zero_rat),B) ) ) ).
% zero_less_mult_pos2
tff(fact_2371_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).
% zero_less_mult_pos2
tff(fact_2372_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).
% zero_less_mult_pos2
tff(fact_2373_zero__less__mult__pos2,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ).
% zero_less_mult_pos2
tff(fact_2374_zero__less__mult__pos2,axiom,
! [B: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),A))
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B) ) ) ).
% zero_less_mult_pos2
tff(fact_2375_zero__less__mult__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> aa_rat_o(ord_less_rat(zero_zero_rat),B) ) ) ).
% zero_less_mult_pos
tff(fact_2376_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).
% zero_less_mult_pos
tff(fact_2377_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).
% zero_less_mult_pos
tff(fact_2378_zero__less__mult__pos,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ).
% zero_less_mult_pos
tff(fact_2379_zero__less__mult__pos,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B))
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B) ) ) ).
% zero_less_mult_pos
tff(fact_2380_zero__less__mult__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
<=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
& aa_rat_o(ord_less_rat(zero_zero_rat),B) )
| ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
& aa_rat_o(ord_less_rat(B),zero_zero_rat) ) ) ) ).
% zero_less_mult_iff
tff(fact_2381_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
& aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) ) ) ) ).
% zero_less_mult_iff
tff(fact_2382_zero__less__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) )
| ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger) ) ) ) ).
% zero_less_mult_iff
tff(fact_2383_mult__pos__neg2,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_rat(B),zero_zero_rat)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat) ) ) ).
% mult_pos_neg2
tff(fact_2384_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).
% mult_pos_neg2
tff(fact_2385_mult__pos__neg2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).
% mult_pos_neg2
tff(fact_2386_mult__pos__neg2,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),zero_z3403309356797280102nteger) ) ) ).
% mult_pos_neg2
tff(fact_2387_mult__pos__neg2,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),zero_z2226904508553997617atural)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),A)),zero_z2226904508553997617atural) ) ) ).
% mult_pos_neg2
tff(fact_2388_mult__pos__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),B)
=> aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).
% mult_pos_pos
tff(fact_2389_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).
% mult_pos_pos
tff(fact_2390_mult__pos__pos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).
% mult_pos_pos
tff(fact_2391_mult__pos__pos,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ) ).
% mult_pos_pos
tff(fact_2392_mult__pos__pos,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)) ) ) ).
% mult_pos_pos
tff(fact_2393_mult__pos__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_rat(B),zero_zero_rat)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).
% mult_pos_neg
tff(fact_2394_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).
% mult_pos_neg
tff(fact_2395_mult__pos__neg,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).
% mult_pos_neg
tff(fact_2396_mult__pos__neg,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).
% mult_pos_neg
tff(fact_2397_mult__pos__neg,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),zero_z2226904508553997617atural)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),zero_z2226904508553997617atural) ) ) ).
% mult_pos_neg
tff(fact_2398_mult__neg__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(A),zero_zero_rat)
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),B)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).
% mult_neg_pos
tff(fact_2399_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).
% mult_neg_pos
tff(fact_2400_mult__neg__pos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).
% mult_neg_pos
tff(fact_2401_mult__neg__pos,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).
% mult_neg_pos
tff(fact_2402_mult__neg__pos,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),zero_z2226904508553997617atural)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),zero_z2226904508553997617atural) ) ) ).
% mult_neg_pos
tff(fact_2403_mult__less__0__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
<=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
& aa_rat_o(ord_less_rat(B),zero_zero_rat) )
| ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
& aa_rat_o(ord_less_rat(zero_zero_rat),B) ) ) ) ).
% mult_less_0_iff
tff(fact_2404_mult__less__0__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
& aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) )
| ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
& aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).
% mult_less_0_iff
tff(fact_2405_mult__less__0__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger)
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger) )
| ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ) ).
% mult_less_0_iff
tff(fact_2406_not__square__less__zero,axiom,
! [A: rat] : ~ aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),zero_zero_rat) ).
% not_square_less_zero
tff(fact_2407_not__square__less__zero,axiom,
! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int) ).
% not_square_less_zero
tff(fact_2408_not__square__less__zero,axiom,
! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A)),zero_z3403309356797280102nteger) ).
% not_square_less_zero
tff(fact_2409_mult__neg__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(A),zero_zero_rat)
=> ( aa_rat_o(ord_less_rat(B),zero_zero_rat)
=> aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).
% mult_neg_neg
tff(fact_2410_mult__neg__neg,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).
% mult_neg_neg
tff(fact_2411_mult__neg__neg,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ) ).
% mult_neg_neg
tff(fact_2412_less__numeral__extra_I1_J,axiom,
aa_rat_o(ord_less_rat(zero_zero_rat),one_one_rat) ).
% less_numeral_extra(1)
tff(fact_2413_less__numeral__extra_I1_J,axiom,
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).
% less_numeral_extra(1)
tff(fact_2414_less__numeral__extra_I1_J,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).
% less_numeral_extra(1)
tff(fact_2415_less__numeral__extra_I1_J,axiom,
aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer) ).
% less_numeral_extra(1)
tff(fact_2416_less__numeral__extra_I1_J,axiom,
aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),one_one_Code_natural) ).
% less_numeral_extra(1)
tff(fact_2417_zero__less__one,axiom,
aa_rat_o(ord_less_rat(zero_zero_rat),one_one_rat) ).
% zero_less_one
tff(fact_2418_zero__less__one,axiom,
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).
% zero_less_one
tff(fact_2419_zero__less__one,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).
% zero_less_one
tff(fact_2420_zero__less__one,axiom,
aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer) ).
% zero_less_one
tff(fact_2421_zero__less__one,axiom,
aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),one_one_Code_natural) ).
% zero_less_one
tff(fact_2422_not__one__less__zero,axiom,
~ aa_rat_o(ord_less_rat(one_one_rat),zero_zero_rat) ).
% not_one_less_zero
tff(fact_2423_not__one__less__zero,axiom,
~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),zero_zero_nat) ).
% not_one_less_zero
tff(fact_2424_not__one__less__zero,axiom,
~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),zero_zero_int) ).
% not_one_less_zero
tff(fact_2425_not__one__less__zero,axiom,
~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),zero_z3403309356797280102nteger) ).
% not_one_less_zero
tff(fact_2426_not__one__less__zero,axiom,
~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),zero_z2226904508553997617atural) ).
% not_one_less_zero
tff(fact_2427_less__iff__diff__less__0,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(A),B)
<=> aa_rat_o(ord_less_rat(minus_minus_rat(A,B)),zero_zero_rat) ) ).
% less_iff_diff_less_0
tff(fact_2428_less__iff__diff__less__0,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).
% less_iff_diff_less_0
tff(fact_2429_less__iff__diff__less__0,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),zero_z3403309356797280102nteger) ) ).
% less_iff_diff_less_0
tff(fact_2430_subset__minus__empty,axiom,
! [Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Aa),Ba)
=> ( minus_1356011639430497352at_nat(Aa,Ba) = bot_bo2099793752762293965at_nat ) ) ).
% subset_minus_empty
tff(fact_2431_subset__minus__empty,axiom,
! [Aa: set_o,Ba: set_o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Aa),Ba)
=> ( minus_minus_set_o(Aa,Ba) = bot_bot_set_o ) ) ).
% subset_minus_empty
tff(fact_2432_subset__minus__empty,axiom,
! [Aa: set_int,Ba: set_int] :
( aa_set_int_o(ord_less_eq_set_int(Aa),Ba)
=> ( minus_minus_set_int(Aa,Ba) = bot_bot_set_int ) ) ).
% subset_minus_empty
tff(fact_2433_subset__minus__empty,axiom,
! [Aa: set_nat,Ba: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),Ba)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),Ba) = bot_bot_set_nat ) ) ).
% subset_minus_empty
tff(fact_2434_less__1__mult,axiom,
! [M2: rat,N: rat] :
( aa_rat_o(ord_less_rat(one_one_rat),M2)
=> ( aa_rat_o(ord_less_rat(one_one_rat),N)
=> aa_rat_o(ord_less_rat(one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M2),N)) ) ) ).
% less_1_mult
tff(fact_2435_less__1__mult,axiom,
! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M2)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),N)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) ) ) ).
% less_1_mult
tff(fact_2436_less__1__mult,axiom,
! [M2: int,N: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),M2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),N)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M2),N)) ) ) ).
% less_1_mult
tff(fact_2437_less__1__mult,axiom,
! [M2: code_integer,N: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),M2)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),N)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,M2),N)) ) ) ).
% less_1_mult
tff(fact_2438_less__1__mult,axiom,
! [M2: code_natural,N: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),M2)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),N)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(M2),N)) ) ) ).
% less_1_mult
tff(fact_2439_disjoint__alt__simp2,axiom,
! [Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat(Aa,Ba) != Aa )
<=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Aa),Ba) != bot_bo2099793752762293965at_nat ) ) ).
% disjoint_alt_simp2
tff(fact_2440_disjoint__alt__simp2,axiom,
! [Aa: set_o,Ba: set_o] :
( ( minus_minus_set_o(Aa,Ba) != Aa )
<=> ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Aa),Ba) != bot_bot_set_o ) ) ).
% disjoint_alt_simp2
tff(fact_2441_disjoint__alt__simp2,axiom,
! [Aa: set_int,Ba: set_int] :
( ( minus_minus_set_int(Aa,Ba) != Aa )
<=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Aa),Ba) != bot_bot_set_int ) ) ).
% disjoint_alt_simp2
tff(fact_2442_disjoint__alt__simp2,axiom,
! [Aa: set_nat,Ba: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),Ba) != Aa )
<=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),Ba) != bot_bot_set_nat ) ) ).
% disjoint_alt_simp2
tff(fact_2443_disjoint__alt__simp1,axiom,
! [Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat(Aa,Ba) = Aa )
<=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Aa),Ba) = bot_bo2099793752762293965at_nat ) ) ).
% disjoint_alt_simp1
tff(fact_2444_disjoint__alt__simp1,axiom,
! [Aa: set_o,Ba: set_o] :
( ( minus_minus_set_o(Aa,Ba) = Aa )
<=> ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Aa),Ba) = bot_bot_set_o ) ) ).
% disjoint_alt_simp1
tff(fact_2445_disjoint__alt__simp1,axiom,
! [Aa: set_int,Ba: set_int] :
( ( minus_minus_set_int(Aa,Ba) = Aa )
<=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Aa),Ba) = bot_bot_set_int ) ) ).
% disjoint_alt_simp1
tff(fact_2446_disjoint__alt__simp1,axiom,
! [Aa: set_nat,Ba: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),Ba) = Aa )
<=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),Ba) = bot_bot_set_nat ) ) ).
% disjoint_alt_simp1
tff(fact_2447_Int__Diff__disjoint,axiom,
! [Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Aa),Ba)),minus_1356011639430497352at_nat(Aa,Ba)) = bot_bo2099793752762293965at_nat ).
% Int_Diff_disjoint
tff(fact_2448_Int__Diff__disjoint,axiom,
! [Aa: set_o,Ba: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Aa),Ba)),minus_minus_set_o(Aa,Ba)) = bot_bot_set_o ).
% Int_Diff_disjoint
tff(fact_2449_Int__Diff__disjoint,axiom,
! [Aa: set_int,Ba: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Aa),Ba)),minus_minus_set_int(Aa,Ba)) = bot_bot_set_int ).
% Int_Diff_disjoint
tff(fact_2450_Int__Diff__disjoint,axiom,
! [Aa: set_nat,Ba: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),Ba)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),Ba)) = bot_bot_set_nat ).
% Int_Diff_disjoint
tff(fact_2451_Diff__triv,axiom,
! [Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Aa),Ba) = bot_bo2099793752762293965at_nat )
=> ( minus_1356011639430497352at_nat(Aa,Ba) = Aa ) ) ).
% Diff_triv
tff(fact_2452_Diff__triv,axiom,
! [Aa: set_o,Ba: set_o] :
( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Aa),Ba) = bot_bot_set_o )
=> ( minus_minus_set_o(Aa,Ba) = Aa ) ) ).
% Diff_triv
tff(fact_2453_Diff__triv,axiom,
! [Aa: set_int,Ba: set_int] :
( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Aa),Ba) = bot_bot_set_int )
=> ( minus_minus_set_int(Aa,Ba) = Aa ) ) ).
% Diff_triv
tff(fact_2454_Diff__triv,axiom,
! [Aa: set_nat,Ba: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),Ba) = bot_bot_set_nat )
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),Ba) = Aa ) ) ).
% Diff_triv
tff(fact_2455_less__minus__one__simps_I4_J,axiom,
~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).
% less_minus_one_simps(4)
tff(fact_2456_less__minus__one__simps_I4_J,axiom,
~ aa_rat_o(ord_less_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).
% less_minus_one_simps(4)
tff(fact_2457_less__minus__one__simps_I4_J,axiom,
~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% less_minus_one_simps(4)
tff(fact_2458_less__minus__one__simps_I2_J,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).
% less_minus_one_simps(2)
tff(fact_2459_less__minus__one__simps_I2_J,axiom,
aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).
% less_minus_one_simps(2)
tff(fact_2460_less__minus__one__simps_I2_J,axiom,
aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).
% less_minus_one_simps(2)
tff(fact_2461_mult__le__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
<=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_eq_rat(A),B) )
& ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> aa_rat_o(ord_less_eq_rat(B),A) ) ) ) ).
% mult_le_cancel_left
tff(fact_2462_mult__le__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) )
& ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ) ).
% mult_le_cancel_left
tff(fact_2463_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).
% mult_le_cancel_left
tff(fact_2464_mult__le__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
<=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_eq_rat(A),B) )
& ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> aa_rat_o(ord_less_eq_rat(B),A) ) ) ) ).
% mult_le_cancel_right
tff(fact_2465_mult__le__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) )
& ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ) ).
% mult_le_cancel_right
tff(fact_2466_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).
% mult_le_cancel_right
tff(fact_2467_mult__left__less__imp__less,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_rat(A),B) ) ) ).
% mult_left_less_imp_less
tff(fact_2468_mult__left__less__imp__less,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B))
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B) ) ) ).
% mult_left_less_imp_less
tff(fact_2469_mult__left__less__imp__less,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).
% mult_left_less_imp_less
tff(fact_2470_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).
% mult_left_less_imp_less
tff(fact_2471_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).
% mult_left_less_imp_less
tff(fact_2472_mult__strict__mono,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( aa_rat_o(ord_less_rat(A),B)
=> ( aa_rat_o(ord_less_rat(C),D2)
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),B)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).
% mult_strict_mono
tff(fact_2473_mult__strict__mono,axiom,
! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,C),D2)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),D2)) ) ) ) ) ).
% mult_strict_mono
tff(fact_2474_mult__strict__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D2)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).
% mult_strict_mono
tff(fact_2475_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).
% mult_strict_mono
tff(fact_2476_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).
% mult_strict_mono
tff(fact_2477_mult__less__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
<=> ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_rat(A),B) )
& ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
=> aa_rat_o(ord_less_rat(B),A) ) ) ) ).
% mult_less_cancel_left
tff(fact_2478_mult__less__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
& ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).
% mult_less_cancel_left
tff(fact_2479_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).
% mult_less_cancel_left
tff(fact_2480_mult__right__less__imp__less,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_rat(A),B) ) ) ).
% mult_right_less_imp_less
tff(fact_2481_mult__right__less__imp__less,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C))
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B) ) ) ).
% mult_right_less_imp_less
tff(fact_2482_mult__right__less__imp__less,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).
% mult_right_less_imp_less
tff(fact_2483_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).
% mult_right_less_imp_less
tff(fact_2484_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).
% mult_right_less_imp_less
tff(fact_2485_mult__strict__mono_H,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( aa_rat_o(ord_less_rat(A),B)
=> ( aa_rat_o(ord_less_rat(C),D2)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).
% mult_strict_mono'
tff(fact_2486_mult__strict__mono_H,axiom,
! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,C),D2)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),D2)) ) ) ) ) ).
% mult_strict_mono'
tff(fact_2487_mult__strict__mono_H,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D2)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).
% mult_strict_mono'
tff(fact_2488_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).
% mult_strict_mono'
tff(fact_2489_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).
% mult_strict_mono'
tff(fact_2490_mult__less__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
<=> ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_rat(A),B) )
& ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
=> aa_rat_o(ord_less_rat(B),A) ) ) ) ).
% mult_less_cancel_right
tff(fact_2491_mult__less__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
& ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).
% mult_less_cancel_right
tff(fact_2492_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).
% mult_less_cancel_right
tff(fact_2493_mult__le__cancel__left__neg,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
<=> aa_rat_o(ord_less_eq_rat(B),A) ) ) ).
% mult_le_cancel_left_neg
tff(fact_2494_mult__le__cancel__left__neg,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ).
% mult_le_cancel_left_neg
tff(fact_2495_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).
% mult_le_cancel_left_neg
tff(fact_2496_mult__le__cancel__left__pos,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
<=> aa_rat_o(ord_less_eq_rat(A),B) ) ) ).
% mult_le_cancel_left_pos
tff(fact_2497_mult__le__cancel__left__pos,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).
% mult_le_cancel_left_pos
tff(fact_2498_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).
% mult_le_cancel_left_pos
tff(fact_2499_mult__left__le__imp__le,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_eq_rat(A),B) ) ) ).
% mult_left_le_imp_le
tff(fact_2500_mult__left__le__imp__le,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B))
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B) ) ) ).
% mult_left_le_imp_le
tff(fact_2501_mult__left__le__imp__le,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).
% mult_left_le_imp_le
tff(fact_2502_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).
% mult_left_le_imp_le
tff(fact_2503_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).
% mult_left_le_imp_le
tff(fact_2504_mult__right__le__imp__le,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_eq_rat(A),B) ) ) ).
% mult_right_le_imp_le
tff(fact_2505_mult__right__le__imp__le,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C))
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B) ) ) ).
% mult_right_le_imp_le
tff(fact_2506_mult__right__le__imp__le,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).
% mult_right_le_imp_le
tff(fact_2507_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).
% mult_right_le_imp_le
tff(fact_2508_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).
% mult_right_le_imp_le
tff(fact_2509_mult__le__less__imp__less,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( aa_rat_o(ord_less_eq_rat(A),B)
=> ( aa_rat_o(ord_less_rat(C),D2)
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_2510_mult__le__less__imp__less,axiom,
! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,C),D2)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),D2)) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_2511_mult__le__less__imp__less,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D2)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_2512_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_2513_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_2514_mult__less__le__imp__less,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( aa_rat_o(ord_less_rat(A),B)
=> ( aa_rat_o(ord_less_eq_rat(C),D2)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_2515_mult__less__le__imp__less,axiom,
! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,C),D2)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),D2)) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_2516_mult__less__le__imp__less,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D2)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_2517_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_2518_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_2519_less__minus__one__simps_I3_J,axiom,
~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).
% less_minus_one_simps(3)
tff(fact_2520_less__minus__one__simps_I3_J,axiom,
~ aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).
% less_minus_one_simps(3)
tff(fact_2521_less__minus__one__simps_I3_J,axiom,
~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% less_minus_one_simps(3)
tff(fact_2522_less__minus__one__simps_I1_J,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).
% less_minus_one_simps(1)
tff(fact_2523_less__minus__one__simps_I1_J,axiom,
aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).
% less_minus_one_simps(1)
tff(fact_2524_less__minus__one__simps_I1_J,axiom,
aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).
% less_minus_one_simps(1)
tff(fact_2525_ssubst__Pair__rhs,axiom,
! [R2: nat,S2: nat,R: set_Pr1261947904930325089at_nat,S3: nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(R2),S2)),R)
=> ( ( S3 = S2 )
=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(R2),S3)),R) ) ) ).
% ssubst_Pair_rhs
tff(fact_2526_ssubst__Pair__rhs,axiom,
! [R2: fun_Pr5098669352780829690e_term,S2: produc8923325533196201883nteger,R: set_Pr3486312109254157132nteger,S3: produc8923325533196201883nteger] :
( member8978266922497061677nteger(produc565448451787809096nteger(R2,S2),R)
=> ( ( S3 = S2 )
=> member8978266922497061677nteger(produc565448451787809096nteger(R2,S3),R) ) ) ).
% ssubst_Pair_rhs
tff(fact_2527_ssubst__Pair__rhs,axiom,
! [R2: fun_Pr6460944699146269935_nat_o,S2: produc3658429121746597890et_nat,R: set_Pr1312406793078187350et_nat,S3: produc3658429121746597890et_nat] :
( member4513763875858039863et_nat(produc6017135503889078682et_nat(R2,S2),R)
=> ( ( S3 = S2 )
=> member4513763875858039863et_nat(produc6017135503889078682et_nat(R2,S3),R) ) ) ).
% ssubst_Pair_rhs
tff(fact_2528_ssubst__Pair__rhs,axiom,
! [R2: fun_Pr6460944699146269935_nat_o,S2: produc8799533983703829792et_nat,R: set_Pr4040394263813967220et_nat,S3: produc8799533983703829792et_nat] :
( member9174161442226468181et_nat(produc7291855603632349880et_nat(R2,S2),R)
=> ( ( S3 = S2 )
=> member9174161442226468181et_nat(produc7291855603632349880et_nat(R2,S3),R) ) ) ).
% ssubst_Pair_rhs
tff(fact_2529_ssubst__Pair__rhs,axiom,
! [R2: fun_Pr7763680713840462443e_term,S2: product_prod_int_int,R: set_Pr6700895561065273913nt_int,S3: product_prod_int_int] :
( member753276645541486746nt_int(produc7320969046608631221nt_int(R2,S2),R)
=> ( ( S3 = S2 )
=> member753276645541486746nt_int(produc7320969046608631221nt_int(R2,S3),R) ) ) ).
% ssubst_Pair_rhs
tff(fact_2530_ssubst__Pair__rhs,axiom,
! [R2: fun_in5730379987311621695e_term,S2: product_prod_int_int,R: set_Pr4166680992137330533nt_int,S3: product_prod_int_int] :
( member5890090881338784710nt_int(produc4729913460834754785nt_int(R2,S2),R)
=> ( ( S3 = S2 )
=> member5890090881338784710nt_int(produc4729913460834754785nt_int(R2,S3),R) ) ) ).
% ssubst_Pair_rhs
tff(fact_2531_mult__less__cancel__right2,axiom,
! [A: rat,C: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C)
<=> ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_rat(A),one_one_rat) )
& ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
=> aa_rat_o(ord_less_rat(one_one_rat),A) ) ) ) ).
% mult_less_cancel_right2
tff(fact_2532_mult__less__cancel__right2,axiom,
! [A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),C)
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer) )
& ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A) ) ) ) ).
% mult_less_cancel_right2
tff(fact_2533_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).
% mult_less_cancel_right2
tff(fact_2534_mult__less__cancel__right1,axiom,
! [C: rat,B: rat] :
( aa_rat_o(ord_less_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
<=> ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_rat(one_one_rat),B) )
& ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
=> aa_rat_o(ord_less_rat(B),one_one_rat) ) ) ) ).
% mult_less_cancel_right1
tff(fact_2535_mult__less__cancel__right1,axiom,
! [C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B) )
& ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer) ) ) ) ).
% mult_less_cancel_right1
tff(fact_2536_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).
% mult_less_cancel_right1
tff(fact_2537_mult__less__cancel__left2,axiom,
! [C: rat,A: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C)
<=> ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_rat(A),one_one_rat) )
& ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
=> aa_rat_o(ord_less_rat(one_one_rat),A) ) ) ) ).
% mult_less_cancel_left2
tff(fact_2538_mult__less__cancel__left2,axiom,
! [C: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),C)
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer) )
& ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A) ) ) ) ).
% mult_less_cancel_left2
tff(fact_2539_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).
% mult_less_cancel_left2
tff(fact_2540_mult__less__cancel__left1,axiom,
! [C: rat,B: rat] :
( aa_rat_o(ord_less_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
<=> ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_rat(one_one_rat),B) )
& ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
=> aa_rat_o(ord_less_rat(B),one_one_rat) ) ) ) ).
% mult_less_cancel_left1
tff(fact_2541_mult__less__cancel__left1,axiom,
! [C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B) )
& ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer) ) ) ) ).
% mult_less_cancel_left1
tff(fact_2542_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).
% mult_less_cancel_left1
tff(fact_2543_mult__le__cancel__right2,axiom,
! [A: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C)
<=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_eq_rat(A),one_one_rat) )
& ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> aa_rat_o(ord_less_eq_rat(one_one_rat),A) ) ) ) ).
% mult_le_cancel_right2
tff(fact_2544_mult__le__cancel__right2,axiom,
! [A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),C)
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer) )
& ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A) ) ) ) ).
% mult_le_cancel_right2
tff(fact_2545_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).
% mult_le_cancel_right2
tff(fact_2546_mult__le__cancel__right1,axiom,
! [C: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
<=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> aa_rat_o(ord_less_eq_rat(one_one_rat),B) )
& ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> aa_rat_o(ord_less_eq_rat(B),one_one_rat) ) ) ) ).
% mult_le_cancel_right1
tff(fact_2547_mult__le__cancel__right1,axiom,
! [C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
<=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),B) )
& ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer) ) ) ) ).
% mult_le_cancel_right1
tff(fact_2548_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
<=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).
% mult_le_cancel_right1
tff(fact_2549_field__le__mult__one__interval,axiom,
! [X: rat,Y: rat] :
( ! [Z2: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),Z2)
=> ( aa_rat_o(ord_less_rat(Z2),one_one_rat)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z2),X)),Y) ) )
=> aa_rat_o(ord_less_eq_rat(X),Y) ) ).
% field_le_mult_one_interval
tff(fact_2550_mult__le__cancel__iff2,axiom,
! [Z: rat,X: rat,Y: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),Z)
=> ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
<=> aa_rat_o(ord_less_eq_rat(X),Y) ) ) ).
% mult_le_cancel_iff2
tff(fact_2551_mult__le__cancel__iff2,axiom,
! [Z: code_integer,X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),Y))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y) ) ) ).
% mult_le_cancel_iff2
tff(fact_2552_mult__le__cancel__iff2,axiom,
! [Z: int,X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ).
% mult_le_cancel_iff2
tff(fact_2553_mult__le__cancel__iff1,axiom,
! [Z: rat,X: rat,Y: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),Z)
=> ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))
<=> aa_rat_o(ord_less_eq_rat(X),Y) ) ) ).
% mult_le_cancel_iff1
tff(fact_2554_mult__le__cancel__iff1,axiom,
! [Z: code_integer,X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Z))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y) ) ) ).
% mult_le_cancel_iff1
tff(fact_2555_mult__le__cancel__iff1,axiom,
! [Z: int,X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ).
% mult_le_cancel_iff1
tff(fact_2556_exI__realizer,axiom,
! [P: fun_Pr4680327687137491167term_o,Y: produc8923325533196201883nteger,X: fun_Pr5098669352780829690e_term] :
( aa_fun7993445754732355552term_o(aa_Pro9035772811636298456term_o(P,Y),X)
=> aa_fun7993445754732355552term_o(aa_Pro9035772811636298456term_o(P,aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,produc565448451787809096nteger(X,Y))),aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,produc565448451787809096nteger(X,Y))) ) ).
% exI_realizer
tff(fact_2557_exI__realizer,axiom,
! [P: fun_Pr5843349257357397163at_o_o,Y: produc3658429121746597890et_nat,X: fun_Pr6460944699146269935_nat_o] :
( aa_fun1675921005893230629at_o_o(aa_Pro3588693547670092268at_o_o(P,Y),X)
=> aa_fun1675921005893230629at_o_o(aa_Pro3588693547670092268at_o_o(P,aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,produc6017135503889078682et_nat(X,Y))),aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,produc6017135503889078682et_nat(X,Y))) ) ).
% exI_realizer
tff(fact_2558_exI__realizer,axiom,
! [P: fun_Pr2647030239595059785at_o_o,Y: produc8799533983703829792et_nat,X: fun_Pr6460944699146269935_nat_o] :
( aa_fun1675921005893230629at_o_o(aa_Pro1702148226396177290at_o_o(P,Y),X)
=> aa_fun1675921005893230629at_o_o(aa_Pro1702148226396177290at_o_o(P,aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,produc7291855603632349880et_nat(X,Y))),aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,produc7291855603632349880et_nat(X,Y))) ) ).
% exI_realizer
tff(fact_2559_exI__realizer,axiom,
! [P: fun_Pr5526332658708858896term_o,Y: product_prod_int_int,X: fun_Pr7763680713840462443e_term] :
( aa_fun7650857933016935215term_o(aa_Pro1023942307601852809term_o(P,Y),X)
=> aa_fun7650857933016935215term_o(aa_Pro1023942307601852809term_o(P,aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,produc7320969046608631221nt_int(X,Y))),aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,produc7320969046608631221nt_int(X,Y))) ) ).
% exI_realizer
tff(fact_2560_exI__realizer,axiom,
! [P: fun_Pr7145817153658775868term_o,Y: product_prod_int_int,X: fun_in5730379987311621695e_term] :
( aa_fun3020711759848831579term_o(aa_Pro1235490124264145077term_o(P,Y),X)
=> aa_fun3020711759848831579term_o(aa_Pro1235490124264145077term_o(P,aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,produc4729913460834754785nt_int(X,Y))),aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,produc4729913460834754785nt_int(X,Y))) ) ).
% exI_realizer
tff(fact_2561_exI__realizer,axiom,
! [P: fun_se9038067516210351697unit_o,Y: set_nat,X: heap_e7401611519738050253t_unit] :
( aa_hea7994211679605476167unit_o(aa_set6435858302011484818unit_o(P,Y),X)
=> aa_hea7994211679605476167unit_o(aa_set6435858302011484818unit_o(P,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,produc7507926704131184380et_nat(X,Y))),aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,produc7507926704131184380et_nat(X,Y))) ) ).
% exI_realizer
tff(fact_2562_exI__realizer,axiom,
! [P: fun_int_fun_int_o,Y: int,X: int] :
( aa_int_o(aa_int_fun_int_o(P,Y),X)
=> aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y))),aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y))) ) ).
% exI_realizer
tff(fact_2563_exI__realizer,axiom,
! [P: fun_Pr3928691295121555099ural_o,Y: produc7822875418678951345atural,X: code_natural] :
( aa_Code_natural_o(aa_Pro3271585298577294228ural_o(P,Y),X)
=> aa_Code_natural_o(aa_Pro3271585298577294228ural_o(P,aa_Pro3673693039631929973atural(produc6591912806276919810atural,aa_Pro7165575656943818355atural(produc6639722614265839536atural(X),Y))),produc497848011034438852atural(aa_Pro7165575656943818355atural(produc6639722614265839536atural(X),Y))) ) ).
% exI_realizer
tff(fact_2564_conjI__realizer,axiom,
! [P: fun_fu4736795817735658359term_o,P5: fun_Pr5098669352780829690e_term,Q: fun_Pr7211993375558241536eger_o,Q4: produc8923325533196201883nteger] :
( aa_fun7993445754732355552term_o(P,P5)
=> ( aa_Pro4926521453222654329eger_o(Q,Q4)
=> ( aa_fun7993445754732355552term_o(P,aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,produc565448451787809096nteger(P5,Q4)))
& aa_Pro4926521453222654329eger_o(Q,aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,produc565448451787809096nteger(P5,Q4))) ) ) ) ).
% conjI_realizer
tff(fact_2565_conjI__realizer,axiom,
! [P: fun_fu5135969500543783852at_o_o,P5: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o,Q4: produc3658429121746597890et_nat] :
( aa_fun1675921005893230629at_o_o(P,P5)
=> ( aa_Pro653082453696054616_nat_o(Q,Q4)
=> ( aa_fun1675921005893230629at_o_o(P,aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,produc6017135503889078682et_nat(P5,Q4)))
& aa_Pro653082453696054616_nat_o(Q,aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,produc6017135503889078682et_nat(P5,Q4))) ) ) ) ).
% conjI_realizer
tff(fact_2566_conjI__realizer,axiom,
! [P: fun_fu5135969500543783852at_o_o,P5: fun_Pr6460944699146269935_nat_o,Q: fun_Pr2620521676128474833_nat_o,Q4: produc8799533983703829792et_nat] :
( aa_fun1675921005893230629at_o_o(P,P5)
=> ( aa_Pro5643009357358181178_nat_o(Q,Q4)
=> ( aa_fun1675921005893230629at_o_o(P,aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,produc7291855603632349880et_nat(P5,Q4)))
& aa_Pro5643009357358181178_nat_o(Q,aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,produc7291855603632349880et_nat(P5,Q4))) ) ) ) ).
% conjI_realizer
tff(fact_2567_conjI__realizer,axiom,
! [P: fun_fu6378075912484669126term_o,P5: fun_Pr7763680713840462443e_term,Q: fun_Pr3238374608701176162_int_o,Q4: product_prod_int_int] :
( aa_fun7650857933016935215term_o(P,P5)
=> ( aa_Pro4592068669816069595_int_o(Q,Q4)
=> ( aa_fun7650857933016935215term_o(P,aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,produc7320969046608631221nt_int(P5,Q4)))
& aa_Pro4592068669816069595_int_o(Q,aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,produc7320969046608631221nt_int(P5,Q4))) ) ) ) ).
% conjI_realizer
tff(fact_2568_conjI__realizer,axiom,
! [P: fun_fu3265955482920998386term_o,P5: fun_in5730379987311621695e_term,Q: fun_Pr3238374608701176162_int_o,Q4: product_prod_int_int] :
( aa_fun3020711759848831579term_o(P,P5)
=> ( aa_Pro4592068669816069595_int_o(Q,Q4)
=> ( aa_fun3020711759848831579term_o(P,aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,produc4729913460834754785nt_int(P5,Q4)))
& aa_Pro4592068669816069595_int_o(Q,aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,produc4729913460834754785nt_int(P5,Q4))) ) ) ) ).
% conjI_realizer
tff(fact_2569_conjI__realizer,axiom,
! [P: fun_he8413527710729349838unit_o,P5: heap_e7401611519738050253t_unit,Q: fun_set_nat_o,Q4: set_nat] :
( aa_hea7994211679605476167unit_o(P,P5)
=> ( aa_set_nat_o(Q,Q4)
=> ( aa_hea7994211679605476167unit_o(P,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,produc7507926704131184380et_nat(P5,Q4)))
& aa_set_nat_o(Q,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,produc7507926704131184380et_nat(P5,Q4))) ) ) ) ).
% conjI_realizer
tff(fact_2570_conjI__realizer,axiom,
! [P: fun_int_o,P5: int,Q: fun_int_o,Q4: int] :
( aa_int_o(P,P5)
=> ( aa_int_o(Q,Q4)
=> ( aa_int_o(P,aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(product_Pair_int_int(P5),Q4)))
& aa_int_o(Q,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(product_Pair_int_int(P5),Q4))) ) ) ) ).
% conjI_realizer
tff(fact_2571_conjI__realizer,axiom,
! [P: fun_Code_natural_o,P5: code_natural,Q: fun_Pr8359263388657976682ural_o,Q4: produc7822875418678951345atural] :
( aa_Code_natural_o(P,P5)
=> ( aa_Pro6073791466322389475ural_o(Q,Q4)
=> ( aa_Code_natural_o(P,produc497848011034438852atural(aa_Pro7165575656943818355atural(produc6639722614265839536atural(P5),Q4)))
& aa_Pro6073791466322389475ural_o(Q,aa_Pro3673693039631929973atural(produc6591912806276919810atural,aa_Pro7165575656943818355atural(produc6639722614265839536atural(P5),Q4))) ) ) ) ).
% conjI_realizer
tff(fact_2572_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: fun_fu7695273699059480439eger_o,X: fun_Pr5098669352780829690e_term,Y: produc8923325533196201883nteger,A: produc2453742332425399446nteger] :
( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(P,X),Y)
=> ( ( A = produc565448451787809096nteger(X,Y) )
=> aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(P,aa_Pro4276937008533432526e_term(produc2481150572478215260nteger,A)),aa_Pro4625778721341478855nteger(produc6589491691745714842nteger,A)) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_2573_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: fun_fu8419763957888848259_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat,A: produc8799533983703829792et_nat] :
( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,X),Y)
=> ( ( A = produc6017135503889078682et_nat(X,Y) )
=> aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,A)),aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,A)) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_2574_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: fun_fu2611741329380808165_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat,A: produc3112576180687729086et_nat] :
( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,X),Y)
=> ( ( A = produc7291855603632349880et_nat(X,Y) )
=> aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,A)),aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,A)) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_2575_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: fun_fu6865642474282033866_int_o,X: fun_Pr7763680713840462443e_term,Y: product_prod_int_int,A: produc3043004577601930755nt_int] :
( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(P,X),Y)
=> ( ( A = produc7320969046608631221nt_int(X,Y) )
=> aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(P,aa_Pro4010435699744810540e_term(produc3130796691802200265nt_int,A)),aa_Pro2675344531184151096nt_int(produc1557020426977177351nt_int,A)) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_2576_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: fun_fu2978539155105838238_int_o,X: fun_in5730379987311621695e_term,Y: product_prod_int_int,A: produc4788131091073775919nt_int] :
( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(P,X),Y)
=> ( ( A = produc4729913460834754785nt_int(X,Y) )
=> aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(P,aa_Pro2551989255418117932e_term(produc4463008262350210037nt_int,A)),aa_Pro4437156595432605708nt_int(produc7629796867588368947nt_int,A)) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_2577_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: fun_he5746958392356810273_nat_o,X: heap_e7401611519738050253t_unit,Y: set_nat,A: produc3658429121746597890et_nat] :
( aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,X),Y)
=> ( ( A = produc7507926704131184380et_nat(X,Y) )
=> aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,A)),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,A)) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_2578_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: fun_int_fun_int_o,X: int,Y: int,A: product_prod_int_int] :
( aa_int_o(aa_int_fun_int_o(P,X),Y)
=> ( ( A = aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y) )
=> aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_fst_int_int,A)),aa_Pro6799258434465214121nt_int(product_snd_int_int,A)) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_2579_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: fun_Co9066792663755620495ural_o,X: code_natural,Y: produc7822875418678951345atural,A: produc5835291356934675326atural] :
( aa_Pro6073791466322389475ural_o(aa_Cod1393124287678120144ural_o(P,X),Y)
=> ( ( A = aa_Pro7165575656943818355atural(produc6639722614265839536atural(X),Y) )
=> aa_Pro6073791466322389475ural_o(aa_Cod1393124287678120144ural_o(P,produc497848011034438852atural(A)),aa_Pro3673693039631929973atural(produc6591912806276919810atural,A)) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_2580_mult__less__iff1,axiom,
! [Z: rat,X: rat,Y: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),Z)
=> ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))
<=> aa_rat_o(ord_less_rat(X),Y) ) ) ).
% mult_less_iff1
tff(fact_2581_mult__less__iff1,axiom,
! [Z: int,X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ) ).
% mult_less_iff1
tff(fact_2582_mult__less__iff1,axiom,
! [Z: code_integer,X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Z))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y) ) ) ).
% mult_less_iff1
tff(fact_2583_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila2554085542299052326_set_o(inf_inf_set_o,top_top_set_o,ord_less_eq_set_o,ord_less_set_o) ).
% inf_top.semilattice_neutr_order_axioms
tff(fact_2584_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila4423693302617054363t_unit(inf_in4660618365625256667t_unit,top_to1996260823553986621t_unit,ord_le3507040750410214029t_unit,ord_le8056459307392131481t_unit) ).
% inf_top.semilattice_neutr_order_axioms
tff(fact_2585_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila8603258263270017530r_assn(inf_inf_assn,top_top_assn,ord_less_eq_assn,ord_less_assn) ).
% inf_top.semilattice_neutr_order_axioms
tff(fact_2586_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila1667268886620078168et_nat(inf_inf_set_nat,top_top_set_nat,ord_less_eq_set_nat,ord_less_set_nat) ).
% inf_top.semilattice_neutr_order_axioms
tff(fact_2587_sgn__neg,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
=> ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).
% sgn_neg
tff(fact_2588_sgn__neg,axiom,
! [A: rat] :
( aa_rat_o(ord_less_rat(A),zero_zero_rat)
=> ( sgn_sgn_rat(A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).
% sgn_neg
tff(fact_2589_sgn__neg,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
=> ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).
% sgn_neg
tff(fact_2590_top_Oordering__top__axioms,axiom,
aa_set_o_o(ordering_top_set_o(ord_less_eq_set_o,ord_less_set_o),top_top_set_o) ).
% top.ordering_top_axioms
tff(fact_2591_top_Oordering__top__axioms,axiom,
orderi2935705153647960990t_unit(ord_le3507040750410214029t_unit,ord_le8056459307392131481t_unit,top_to1996260823553986621t_unit) ).
% top.ordering_top_axioms
tff(fact_2592_top_Oordering__top__axioms,axiom,
aa_assn_o(ordering_top_assn(ord_less_eq_assn,ord_less_assn),top_top_assn) ).
% top.ordering_top_axioms
tff(fact_2593_top_Oordering__top__axioms,axiom,
aa_set_nat_o(ordering_top_set_nat(ord_less_eq_set_nat,ord_less_set_nat),top_top_set_nat) ).
% top.ordering_top_axioms
tff(fact_2594_sgn__1,axiom,
aa_int_int(sgn_sgn_int,one_one_int) = one_one_int ).
% sgn_1
tff(fact_2595_sgn__1,axiom,
sgn_sgn_rat(one_one_rat) = one_one_rat ).
% sgn_1
tff(fact_2596_sgn__1,axiom,
aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,one_one_Code_integer) = one_one_Code_integer ).
% sgn_1
tff(fact_2597_sgn__pos,axiom,
! [A: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( sgn_sgn_rat(A) = one_one_rat ) ) ).
% sgn_pos
tff(fact_2598_sgn__pos,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_int(sgn_sgn_int,A) = one_one_int ) ) ).
% sgn_pos
tff(fact_2599_sgn__pos,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer ) ) ).
% sgn_pos
tff(fact_2600_less__assn__def,axiom,
! [A: assn,B: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
<=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
& ( A != B ) ) ) ).
% less_assn_def
tff(fact_2601_sgn__mult,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B)) ).
% sgn_mult
tff(fact_2602_sgn__mult,axiom,
! [A: rat,B: rat] : sgn_sgn_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,sgn_sgn_rat(A)),sgn_sgn_rat(B)) ).
% sgn_mult
tff(fact_2603_sgn__mult,axiom,
! [A: int,B: int] : aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B)) ).
% sgn_mult
tff(fact_2604_sgn__minus__1,axiom,
aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).
% sgn_minus_1
tff(fact_2605_sgn__minus__1,axiom,
sgn_sgn_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% sgn_minus_1
tff(fact_2606_sgn__minus__1,axiom,
aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% sgn_minus_1
tff(fact_2607_semilattice__neutr__order_Oaxioms_I1_J,axiom,
! [F: fun_nat_fun_nat_nat,Z: nat,Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o] :
( semila1623282765462674594er_nat(F,Z,Less_eq,Less)
=> aa_nat_o(semila9081495762789891438tr_nat(F),Z) ) ).
% semilattice_neutr_order.axioms(1)
tff(fact_2608_sgn__1__pos,axiom,
! [A: rat] :
( ( sgn_sgn_rat(A) = one_one_rat )
<=> aa_rat_o(ord_less_rat(zero_zero_rat),A) ) ).
% sgn_1_pos
tff(fact_2609_sgn__1__pos,axiom,
! [A: int] :
( ( aa_int_int(sgn_sgn_int,A) = one_one_int )
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).
% sgn_1_pos
tff(fact_2610_sgn__1__pos,axiom,
! [A: code_integer] :
( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer )
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).
% sgn_1_pos
tff(fact_2611_sgn__if,axiom,
! [X: int] :
aa_int_int(sgn_sgn_int,X) = $ite(
X = zero_zero_int,
zero_zero_int,
$ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).
% sgn_if
tff(fact_2612_sgn__if,axiom,
! [X: rat] :
sgn_sgn_rat(X) = $ite(
X = zero_zero_rat,
zero_zero_rat,
$ite(aa_rat_o(ord_less_rat(zero_zero_rat),X),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).
% sgn_if
tff(fact_2613_sgn__if,axiom,
! [X: code_integer] :
aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X) = $ite(
X = zero_z3403309356797280102nteger,
zero_z3403309356797280102nteger,
$ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ) ).
% sgn_if
tff(fact_2614_sgn__1__neg,axiom,
! [A: int] :
( ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) )
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).
% sgn_1_neg
tff(fact_2615_sgn__1__neg,axiom,
! [A: rat] :
( ( sgn_sgn_rat(A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
<=> aa_rat_o(ord_less_rat(A),zero_zero_rat) ) ).
% sgn_1_neg
tff(fact_2616_sgn__1__neg,axiom,
! [A: code_integer] :
( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).
% sgn_1_neg
tff(fact_2617_convex__bound__lt,axiom,
! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
( aa_rat_o(ord_less_rat(X),A)
=> ( aa_rat_o(ord_less_rat(Y),A)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),U)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),V)
=> ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_2618_convex__bound__lt,axiom,
! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),V)
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,U),V) = one_one_Code_integer )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,U),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V),Y))),A) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_2619_convex__bound__lt,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,X),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_2620_le__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C)))
<=> $ite(
aa_rat_o(ord_less_rat(zero_zero_rat),C),
aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)),
$ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)) ) ) ).
% le_minus_divide_eq
tff(fact_2621_minus__divide__le__eq,axiom,
! [B: rat,C: rat,A: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C))),A)
<=> $ite(
aa_rat_o(ord_less_rat(zero_zero_rat),C),
aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
$ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)) ) ) ).
% minus_divide_le_eq
tff(fact_2622_neg__le__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> ( aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C)))
<=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).
% neg_le_minus_divide_eq
tff(fact_2623_neg__minus__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C))),A)
<=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).
% neg_minus_divide_le_eq
tff(fact_2624_pos__le__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> ( aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C)))
<=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).
% pos_le_minus_divide_eq
tff(fact_2625_pos__minus__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C))),A)
<=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).
% pos_minus_divide_le_eq
tff(fact_2626_prod_Oswap__def,axiom,
! [P5: produc592931078481472324e_term] : aa_Pro8970750471354738200nteger(produc1039113572222304414e_term,P5) = produc565448451787809096nteger(produc7766732615400916880e_term(P5),produc3658391496133417298e_term(P5)) ).
% prod.swap_def
tff(fact_2627_prod_Oswap__def,axiom,
! [P5: produc2739566037923759970_nat_o] : aa_Pro1287330115173305408et_nat(produc6424530459701215220_nat_o,P5) = produc6017135503889078682et_nat(produc5282040756795020774_nat_o(P5),produc5647884604762787496_nat_o(P5)) ).
% prod.swap_def
tff(fact_2628_prod_Oswap__def,axiom,
! [P5: produc6858501863712832324_nat_o] : produc1452984123473447766_nat_o(P5) = produc7291855603632349880et_nat(produc7941837674541168712_nat_o(P5),produc2938390969378414602_nat_o(P5)) ).
% prod.swap_def
tff(fact_2629_prod_Oswap__def,axiom,
! [P5: produc4155239895104388691e_term] : aa_Pro228862134518707348nt_int(produc4754075759091963949e_term,P5) = produc7320969046608631221nt_int(produc7082206506123769631e_term(P5),produc8655982770948792545e_term(P5)) ).
% prod.swap_def
tff(fact_2630_prod_Oswap__def,axiom,
! [P5: produc5464280974217902119e_term] : aa_Pro7711173378726590868nt_int(produc4587076998185972737e_term,P5) = produc4729913460834754785nt_int(produc1448379345012327155e_term(P5),produc7504962776628944053e_term(P5)) ).
% prod.swap_def
tff(fact_2631_prod_Oswap__def,axiom,
! [P5: produc3658429121746597890et_nat] : produc6300697153377358172et_nat(P5) = produc2471974422597015918t_unit(aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,P5),aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,P5)) ).
% prod.swap_def
tff(fact_2632_prod_Oswap__def,axiom,
! [P5: product_prod_int_int] : product_swap_int_int(P5) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,P5)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P5)) ).
% prod.swap_def
tff(fact_2633_prod_Oswap__def,axiom,
! [P5: produc5835291356934675326atural] : produc850777735424141840atural(P5) = produc1272419604612767874atural(aa_Pro3673693039631929973atural(produc6591912806276919810atural,P5),produc497848011034438852atural(P5)) ).
% prod.swap_def
tff(fact_2634_le__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(one_one_rat),divide_divide_rat(B,A))
<=> aa_rat_o(ord_less_eq_rat(A),B) ) ) ).
% le_divide_eq_1_pos
tff(fact_2635_add__right__cancel,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A) )
<=> ( B = C ) ) ).
% add_right_cancel
tff(fact_2636_add__right__cancel,axiom,
! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
( ( aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,B),A) = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,C),A) )
<=> ( B = C ) ) ).
% add_right_cancel
tff(fact_2637_add__right__cancel,axiom,
! [B: rat,A: rat,C: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
<=> ( B = C ) ) ).
% add_right_cancel
tff(fact_2638_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
<=> ( B = C ) ) ).
% add_right_cancel
tff(fact_2639_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
<=> ( B = C ) ) ).
% add_right_cancel
tff(fact_2640_add__left__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) )
<=> ( B = C ) ) ).
% add_left_cancel
tff(fact_2641_add__left__cancel,axiom,
! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
( ( aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),B) = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),C) )
<=> ( B = C ) ) ).
% add_left_cancel
tff(fact_2642_add__left__cancel,axiom,
! [A: rat,B: rat,C: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
<=> ( B = C ) ) ).
% add_left_cancel
tff(fact_2643_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
<=> ( B = C ) ) ).
% add_left_cancel
tff(fact_2644_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
<=> ( B = C ) ) ).
% add_left_cancel
tff(fact_2645_add__le__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
<=> aa_rat_o(ord_less_eq_rat(A),B) ) ).
% add_le_cancel_right
tff(fact_2646_add__le__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).
% add_le_cancel_right
tff(fact_2647_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).
% add_le_cancel_right
tff(fact_2648_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% add_le_cancel_right
tff(fact_2649_add__le__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
<=> aa_rat_o(ord_less_eq_rat(A),B) ) ).
% add_le_cancel_left
tff(fact_2650_add__le__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).
% add_le_cancel_left
tff(fact_2651_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).
% add_le_cancel_left
tff(fact_2652_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% add_le_cancel_left
tff(fact_2653_add__0,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A) = A ).
% add_0
tff(fact_2654_add__0,axiom,
! [A: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,zero_z1048942125864253310at_nat),A) = A ).
% add_0
tff(fact_2655_add__0,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).
% add_0
tff(fact_2656_add__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).
% add_0
tff(fact_2657_add__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).
% add_0
tff(fact_2658_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) )
<=> ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
tff(fact_2659_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
<=> ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
tff(fact_2660_add__cancel__right__right,axiom,
! [A: code_integer,B: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) )
<=> ( B = zero_z3403309356797280102nteger ) ) ).
% add_cancel_right_right
tff(fact_2661_add__cancel__right__right,axiom,
! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
( ( A = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),B) )
<=> ( B = zero_z1048942125864253310at_nat ) ) ).
% add_cancel_right_right
tff(fact_2662_add__cancel__right__right,axiom,
! [A: rat,B: rat] :
( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) )
<=> ( B = zero_zero_rat ) ) ).
% add_cancel_right_right
tff(fact_2663_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) )
<=> ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
tff(fact_2664_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) )
<=> ( B = zero_zero_int ) ) ).
% add_cancel_right_right
tff(fact_2665_add__cancel__right__left,axiom,
! [A: code_integer,B: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) )
<=> ( B = zero_z3403309356797280102nteger ) ) ).
% add_cancel_right_left
tff(fact_2666_add__cancel__right__left,axiom,
! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
( ( A = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,B),A) )
<=> ( B = zero_z1048942125864253310at_nat ) ) ).
% add_cancel_right_left
tff(fact_2667_add__cancel__right__left,axiom,
! [A: rat,B: rat] :
( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
<=> ( B = zero_zero_rat ) ) ).
% add_cancel_right_left
tff(fact_2668_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
<=> ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
tff(fact_2669_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
<=> ( B = zero_zero_int ) ) ).
% add_cancel_right_left
tff(fact_2670_add__cancel__left__right,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = A )
<=> ( B = zero_z3403309356797280102nteger ) ) ).
% add_cancel_left_right
tff(fact_2671_add__cancel__left__right,axiom,
! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
( ( aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),B) = A )
<=> ( B = zero_z1048942125864253310at_nat ) ) ).
% add_cancel_left_right
tff(fact_2672_add__cancel__left__right,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = A )
<=> ( B = zero_zero_rat ) ) ).
% add_cancel_left_right
tff(fact_2673_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = A )
<=> ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
tff(fact_2674_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = A )
<=> ( B = zero_zero_int ) ) ).
% add_cancel_left_right
tff(fact_2675_add__cancel__left__left,axiom,
! [B: code_integer,A: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) = A )
<=> ( B = zero_z3403309356797280102nteger ) ) ).
% add_cancel_left_left
tff(fact_2676_add__cancel__left__left,axiom,
! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat] :
( ( aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,B),A) = A )
<=> ( B = zero_z1048942125864253310at_nat ) ) ).
% add_cancel_left_left
tff(fact_2677_add__cancel__left__left,axiom,
! [B: rat,A: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = A )
<=> ( B = zero_zero_rat ) ) ).
% add_cancel_left_left
tff(fact_2678_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = A )
<=> ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
tff(fact_2679_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = A )
<=> ( B = zero_zero_int ) ) ).
% add_cancel_left_left
tff(fact_2680_double__zero__sym,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A) )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% double_zero_sym
tff(fact_2681_double__zero__sym,axiom,
! [A: rat] :
( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) )
<=> ( A = zero_zero_rat ) ) ).
% double_zero_sym
tff(fact_2682_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) )
<=> ( A = zero_zero_int ) ) ).
% double_zero_sym
tff(fact_2683_add_Oright__neutral,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_z3403309356797280102nteger) = A ).
% add.right_neutral
tff(fact_2684_add_Oright__neutral,axiom,
! [A: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),zero_z1048942125864253310at_nat) = A ).
% add.right_neutral
tff(fact_2685_add_Oright__neutral,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).
% add.right_neutral
tff(fact_2686_add_Oright__neutral,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).
% add.right_neutral
tff(fact_2687_add_Oright__neutral,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).
% add.right_neutral
tff(fact_2688_add__less__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
<=> aa_rat_o(ord_less_rat(A),B) ) ).
% add_less_cancel_right
tff(fact_2689_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).
% add_less_cancel_right
tff(fact_2690_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).
% add_less_cancel_right
tff(fact_2691_add__less__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).
% add_less_cancel_right
tff(fact_2692_add__less__cancel__right,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),C)),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(B),C))
<=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B) ) ).
% add_less_cancel_right
tff(fact_2693_add__less__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
<=> aa_rat_o(ord_less_rat(A),B) ) ).
% add_less_cancel_left
tff(fact_2694_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).
% add_less_cancel_left
tff(fact_2695_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).
% add_less_cancel_left
tff(fact_2696_add__less__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).
% add_less_cancel_left
tff(fact_2697_add__less__cancel__left,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(C),A)),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(C),B))
<=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B) ) ).
% add_less_cancel_left
tff(fact_2698_add__diff__cancel__right_H,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B) = A ).
% add_diff_cancel_right'
tff(fact_2699_add__diff__cancel__right_H,axiom,
! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),B),B) = A ).
% add_diff_cancel_right'
tff(fact_2700_add__diff__cancel__right_H,axiom,
! [A: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),B) = A ).
% add_diff_cancel_right'
tff(fact_2701_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = A ).
% add_diff_cancel_right'
tff(fact_2702_add__diff__cancel__right_H,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).
% add_diff_cancel_right'
tff(fact_2703_add__diff__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B) ).
% add_diff_cancel_right
tff(fact_2704_add__diff__cancel__right,axiom,
! [A: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),C),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,B),C)) = minus_4286766774447292334at_nat(A,B) ).
% add_diff_cancel_right
tff(fact_2705_add__diff__cancel__right,axiom,
! [A: rat,C: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = minus_minus_rat(A,B) ).
% add_diff_cancel_right
tff(fact_2706_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).
% add_diff_cancel_right
tff(fact_2707_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).
% add_diff_cancel_right
tff(fact_2708_add__diff__cancel__left_H,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),A) = B ).
% add_diff_cancel_left'
tff(fact_2709_add__diff__cancel__left_H,axiom,
! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),B),A) = B ).
% add_diff_cancel_left'
tff(fact_2710_add__diff__cancel__left_H,axiom,
! [A: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),A) = B ).
% add_diff_cancel_left'
tff(fact_2711_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),A) = B ).
% add_diff_cancel_left'
tff(fact_2712_add__diff__cancel__left_H,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),A) = B ).
% add_diff_cancel_left'
tff(fact_2713_add__diff__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B) ).
% add_diff_cancel_left
tff(fact_2714_add__diff__cancel__left,axiom,
! [C: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,C),A),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,C),B)) = minus_4286766774447292334at_nat(A,B) ).
% add_diff_cancel_left
tff(fact_2715_add__diff__cancel__left,axiom,
! [C: rat,A: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) = minus_minus_rat(A,B) ).
% add_diff_cancel_left
tff(fact_2716_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).
% add_diff_cancel_left
tff(fact_2717_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).
% add_diff_cancel_left
tff(fact_2718_diff__add__cancel,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),B) = A ).
% diff_add_cancel
tff(fact_2719_diff__add__cancel,axiom,
! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A,B)),B) = A ).
% diff_add_cancel
tff(fact_2720_diff__add__cancel,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ).
% diff_add_cancel
tff(fact_2721_add__diff__cancel,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B) = A ).
% add_diff_cancel
tff(fact_2722_add__diff__cancel,axiom,
! [A: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),B) = A ).
% add_diff_cancel
tff(fact_2723_add__diff__cancel,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).
% add_diff_cancel
tff(fact_2724_minus__add__distrib,axiom,
! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) ).
% minus_add_distrib
tff(fact_2725_minus__add__distrib,axiom,
! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ).
% minus_add_distrib
tff(fact_2726_minus__add__distrib,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).
% minus_add_distrib
tff(fact_2727_minus__add__cancel,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = B ).
% minus_add_cancel
tff(fact_2728_minus__add__cancel,axiom,
! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = B ).
% minus_add_cancel
tff(fact_2729_minus__add__cancel,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = B ).
% minus_add_cancel
tff(fact_2730_add__minus__cancel,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B)) = B ).
% add_minus_cancel
tff(fact_2731_add__minus__cancel,axiom,
! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) = B ).
% add_minus_cancel
tff(fact_2732_add__minus__cancel,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) = B ).
% add_minus_cancel
tff(fact_2733_times__divide__eq__right,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),divide_divide_rat(B,C)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),C) ).
% times_divide_eq_right
tff(fact_2734_divide__divide__eq__right,axiom,
! [A: rat,B: rat,C: rat] : divide_divide_rat(A,divide_divide_rat(B,C)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),B) ).
% divide_divide_eq_right
tff(fact_2735_divide__divide__eq__left,axiom,
! [A: rat,B: rat,C: rat] : divide_divide_rat(divide_divide_rat(A,B),C) = divide_divide_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).
% divide_divide_eq_left
tff(fact_2736_times__divide__eq__left,axiom,
! [B: rat,C: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,divide_divide_rat(B,C)),A) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A),C) ).
% times_divide_eq_left
tff(fact_2737_div__by__1,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).
% div_by_1
tff(fact_2738_div__by__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).
% div_by_1
tff(fact_2739_div__by__1,axiom,
! [A: rat] : divide_divide_rat(A,one_one_rat) = A ).
% div_by_1
tff(fact_2740_div__by__1,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),one_one_Code_integer) = A ).
% div_by_1
tff(fact_2741_div__by__1,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),one_one_Code_natural) = A ).
% div_by_1
tff(fact_2742_swap__simp,axiom,
! [X: produc8923325533196201883nteger,Y: fun_Pr5098669352780829690e_term] : aa_Pro8970750471354738200nteger(produc1039113572222304414e_term,produc1742689375443011134e_term(X,Y)) = produc565448451787809096nteger(Y,X) ).
% swap_simp
tff(fact_2743_swap__simp,axiom,
! [X: produc3658429121746597890et_nat,Y: fun_Pr6460944699146269935_nat_o] : aa_Pro1287330115173305408et_nat(produc6424530459701215220_nat_o,produc6339119935501530516_nat_o(X,Y)) = produc6017135503889078682et_nat(Y,X) ).
% swap_simp
tff(fact_2744_swap__simp,axiom,
! [X: produc8799533983703829792et_nat,Y: fun_Pr6460944699146269935_nat_o] : produc1452984123473447766_nat_o(produc6151749222514847990_nat_o(X,Y)) = produc7291855603632349880et_nat(Y,X) ).
% swap_simp
tff(fact_2745_swap__simp,axiom,
! [X: product_prod_int_int,Y: fun_Pr7763680713840462443e_term] : aa_Pro228862134518707348nt_int(produc4754075759091963949e_term,produc3622783088900447693e_term(X,Y)) = produc7320969046608631221nt_int(Y,X) ).
% swap_simp
tff(fact_2746_swap__simp,axiom,
! [X: product_prod_int_int,Y: fun_in5730379987311621695e_term] : aa_Pro7711173378726590868nt_int(produc4587076998185972737e_term,produc7771867975113488801e_term(X,Y)) = produc4729913460834754785nt_int(Y,X) ).
% swap_simp
tff(fact_2747_swap__simp,axiom,
! [X: fun_Pr5098669352780829690e_term,Y: produc8923325533196201883nteger] : aa_Pro4049104960294949400e_term(produc9085244685421878184nteger,produc565448451787809096nteger(X,Y)) = produc1742689375443011134e_term(Y,X) ).
% swap_simp
tff(fact_2748_swap__simp,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat] : aa_Pro9094726134102481216_nat_o(produc6102546028088763386et_nat,produc6017135503889078682et_nat(X,Y)) = produc6339119935501530516_nat_o(Y,X) ).
% swap_simp
tff(fact_2749_swap__simp,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat] : aa_Pro345323721036081728_nat_o(produc2593090504590949656et_nat,produc7291855603632349880et_nat(X,Y)) = produc6151749222514847990_nat_o(Y,X) ).
% swap_simp
tff(fact_2750_swap__simp,axiom,
! [X: fun_Pr7763680713840462443e_term,Y: product_prod_int_int] : aa_Pro2176780875811443860e_term(produc8452261716800147477nt_int,produc7320969046608631221nt_int(X,Y)) = produc3622783088900447693e_term(Y,X) ).
% swap_simp
tff(fact_2751_swap__simp,axiom,
! [X: fun_in5730379987311621695e_term,Y: product_prod_int_int] : aa_Pro1032260729317935508e_term(produc1545122483907238721nt_int,produc4729913460834754785nt_int(X,Y)) = produc7771867975113488801e_term(Y,X) ).
% swap_simp
tff(fact_2752_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A))
<=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),A) ) ).
% zero_le_double_add_iff_zero_le_single_add
tff(fact_2753_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).
% zero_le_double_add_iff_zero_le_single_add
tff(fact_2754_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).
% zero_le_double_add_iff_zero_le_single_add
tff(fact_2755_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)
<=> aa_rat_o(ord_less_eq_rat(A),zero_zero_rat) ) ).
% double_add_le_zero_iff_single_add_le_zero
tff(fact_2756_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A)),zero_z3403309356797280102nteger)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).
% double_add_le_zero_iff_single_add_le_zero
tff(fact_2757_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).
% double_add_le_zero_iff_single_add_le_zero
tff(fact_2758_le__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
<=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),B) ) ).
% le_add_same_cancel2
tff(fact_2759_le__add__same__cancel2,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) ).
% le_add_same_cancel2
tff(fact_2760_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).
% le_add_same_cancel2
tff(fact_2761_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).
% le_add_same_cancel2
tff(fact_2762_le__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
<=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),B) ) ).
% le_add_same_cancel1
tff(fact_2763_le__add__same__cancel1,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) ).
% le_add_same_cancel1
tff(fact_2764_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).
% le_add_same_cancel1
tff(fact_2765_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).
% le_add_same_cancel1
tff(fact_2766_add__le__same__cancel2,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B)
<=> aa_rat_o(ord_less_eq_rat(A),zero_zero_rat) ) ).
% add_le_same_cancel2
tff(fact_2767_add__le__same__cancel2,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).
% add_le_same_cancel2
tff(fact_2768_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).
% add_le_same_cancel2
tff(fact_2769_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).
% add_le_same_cancel2
tff(fact_2770_add__le__same__cancel1,axiom,
! [B: rat,A: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B)
<=> aa_rat_o(ord_less_eq_rat(A),zero_zero_rat) ) ).
% add_le_same_cancel1
tff(fact_2771_add__le__same__cancel1,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)),B)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).
% add_le_same_cancel1
tff(fact_2772_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).
% add_le_same_cancel1
tff(fact_2773_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).
% add_le_same_cancel1
tff(fact_2774_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A))
<=> aa_rat_o(ord_less_rat(zero_zero_rat),A) ) ).
% zero_less_double_add_iff_zero_less_single_add
tff(fact_2775_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).
% zero_less_double_add_iff_zero_less_single_add
tff(fact_2776_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).
% zero_less_double_add_iff_zero_less_single_add
tff(fact_2777_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)
<=> aa_rat_o(ord_less_rat(A),zero_zero_rat) ) ).
% double_add_less_zero_iff_single_add_less_zero
tff(fact_2778_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).
% double_add_less_zero_iff_single_add_less_zero
tff(fact_2779_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A)),zero_z3403309356797280102nteger)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).
% double_add_less_zero_iff_single_add_less_zero
tff(fact_2780_less__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
<=> aa_rat_o(ord_less_rat(zero_zero_rat),B) ) ).
% less_add_same_cancel2
tff(fact_2781_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).
% less_add_same_cancel2
tff(fact_2782_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).
% less_add_same_cancel2
tff(fact_2783_less__add__same__cancel2,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ).
% less_add_same_cancel2
tff(fact_2784_less__add__same__cancel2,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(B),A))
<=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B) ) ).
% less_add_same_cancel2
tff(fact_2785_less__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
<=> aa_rat_o(ord_less_rat(zero_zero_rat),B) ) ).
% less_add_same_cancel1
tff(fact_2786_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).
% less_add_same_cancel1
tff(fact_2787_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).
% less_add_same_cancel1
tff(fact_2788_less__add__same__cancel1,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ).
% less_add_same_cancel1
tff(fact_2789_less__add__same__cancel1,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),B))
<=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B) ) ).
% less_add_same_cancel1
tff(fact_2790_add__less__same__cancel2,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B)
<=> aa_rat_o(ord_less_rat(A),zero_zero_rat) ) ).
% add_less_same_cancel2
tff(fact_2791_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).
% add_less_same_cancel2
tff(fact_2792_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).
% add_less_same_cancel2
tff(fact_2793_add__less__same__cancel2,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).
% add_less_same_cancel2
tff(fact_2794_add__less__same__cancel2,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),B)),B)
<=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),zero_z2226904508553997617atural) ) ).
% add_less_same_cancel2
tff(fact_2795_add__less__same__cancel1,axiom,
! [B: rat,A: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B)
<=> aa_rat_o(ord_less_rat(A),zero_zero_rat) ) ).
% add_less_same_cancel1
tff(fact_2796_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).
% add_less_same_cancel1
tff(fact_2797_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).
% add_less_same_cancel1
tff(fact_2798_add__less__same__cancel1,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)),B)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).
% add_less_same_cancel1
tff(fact_2799_add__less__same__cancel1,axiom,
! [B: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(B),A)),B)
<=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),zero_z2226904508553997617atural) ) ).
% add_less_same_cancel1
tff(fact_2800_diff__add__zero,axiom,
! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(A,aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),B)) = zero_z1048942125864253310at_nat ).
% diff_add_zero
tff(fact_2801_diff__add__zero,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = zero_zero_nat ).
% diff_add_zero
tff(fact_2802_ab__left__minus,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).
% ab_left_minus
tff(fact_2803_ab__left__minus,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).
% ab_left_minus
tff(fact_2804_ab__left__minus,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A) = zero_z3403309356797280102nteger ).
% ab_left_minus
tff(fact_2805_add_Oright__inverse,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,A)) = zero_zero_int ).
% add.right_inverse
tff(fact_2806_add_Oright__inverse,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).
% add.right_inverse
tff(fact_2807_add_Oright__inverse,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).
% add.right_inverse
tff(fact_2808_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = A ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_2809_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = A ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_2810_nonzero__mult__div__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),B) = A ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_2811_nonzero__mult__div__cancel__right,axiom,
! [B: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),B) = A ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_2812_nonzero__mult__div__cancel__right,axiom,
! [B: code_natural,A: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),B) = A ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_2813_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),A) = B ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_2814_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),A) = B ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_2815_nonzero__mult__div__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),A) = B ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_2816_nonzero__mult__div__cancel__left,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),A) = B ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_2817_nonzero__mult__div__cancel__left,axiom,
! [A: code_natural,B: code_natural] :
( ( A != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),A) = B ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_2818_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = divide_divide_rat(A,B) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
tff(fact_2819_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) = divide_divide_rat(A,B) ) ) ).
% nonzero_mult_divide_mult_cancel_right
tff(fact_2820_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) = divide_divide_rat(A,B) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
tff(fact_2821_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = divide_divide_rat(A,B) ) ) ).
% nonzero_mult_divide_mult_cancel_left
tff(fact_2822_mult__divide__mult__cancel__left__if,axiom,
! [C: rat,A: rat,B: rat] :
divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = $ite(C = zero_zero_rat,zero_zero_rat,divide_divide_rat(A,B)) ).
% mult_divide_mult_cancel_left_if
tff(fact_2823_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),A) = one_one_nat ) ) ).
% div_self
tff(fact_2824_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),A) = one_one_int ) ) ).
% div_self
tff(fact_2825_div__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( divide_divide_rat(A,A) = one_one_rat ) ) ).
% div_self
tff(fact_2826_div__self,axiom,
! [A: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),A) = one_one_Code_integer ) ) ).
% div_self
tff(fact_2827_div__self,axiom,
! [A: code_natural] :
( ( A != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),A) = one_one_Code_natural ) ) ).
% div_self
tff(fact_2828_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
tff(fact_2829_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
tff(fact_2830_divide__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( divide_divide_rat(A,A) = one_one_rat ) ) ).
% divide_self
tff(fact_2831_divide__self__if,axiom,
! [A: rat] :
divide_divide_rat(A,A) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).
% divide_self_if
tff(fact_2832_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
tff(fact_2833_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
tff(fact_2834_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
tff(fact_2835_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
tff(fact_2836_uminus__add__conv__diff,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).
% uminus_add_conv_diff
tff(fact_2837_uminus__add__conv__diff,axiom,
! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = minus_minus_rat(B,A) ).
% uminus_add_conv_diff
tff(fact_2838_uminus__add__conv__diff,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A) ).
% uminus_add_conv_diff
tff(fact_2839_diff__minus__eq__add,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) ).
% diff_minus_eq_add
tff(fact_2840_diff__minus__eq__add,axiom,
! [A: rat,B: rat] : minus_minus_rat(A,aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) ).
% diff_minus_eq_add
tff(fact_2841_diff__minus__eq__add,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) ).
% diff_minus_eq_add
tff(fact_2842_divide__minus1,axiom,
! [X: rat] : divide_divide_rat(X,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,X) ).
% divide_minus1
tff(fact_2843_divide__sgn,axiom,
! [A: rat,B: rat] : divide_divide_rat(A,sgn_sgn_rat(B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),sgn_sgn_rat(B)) ).
% divide_sgn
tff(fact_2844_divide__le__0__1__iff,axiom,
! [A: rat] :
( aa_rat_o(ord_less_eq_rat(divide_divide_rat(one_one_rat,A)),zero_zero_rat)
<=> aa_rat_o(ord_less_eq_rat(A),zero_zero_rat) ) ).
% divide_le_0_1_iff
tff(fact_2845_zero__le__divide__1__iff,axiom,
! [A: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),divide_divide_rat(one_one_rat,A))
<=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),A) ) ).
% zero_le_divide_1_iff
tff(fact_2846_divide__less__0__1__iff,axiom,
! [A: rat] :
( aa_rat_o(ord_less_rat(divide_divide_rat(one_one_rat,A)),zero_zero_rat)
<=> aa_rat_o(ord_less_rat(A),zero_zero_rat) ) ).
% divide_less_0_1_iff
tff(fact_2847_divide__less__eq__1__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(A),zero_zero_rat)
=> ( aa_rat_o(ord_less_rat(divide_divide_rat(B,A)),one_one_rat)
<=> aa_rat_o(ord_less_rat(A),B) ) ) ).
% divide_less_eq_1_neg
tff(fact_2848_divide__less__eq__1__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_rat(divide_divide_rat(B,A)),one_one_rat)
<=> aa_rat_o(ord_less_rat(B),A) ) ) ).
% divide_less_eq_1_pos
tff(fact_2849_less__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(A),zero_zero_rat)
=> ( aa_rat_o(ord_less_rat(one_one_rat),divide_divide_rat(B,A))
<=> aa_rat_o(ord_less_rat(B),A) ) ) ).
% less_divide_eq_1_neg
tff(fact_2850_less__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_rat(one_one_rat),divide_divide_rat(B,A))
<=> aa_rat_o(ord_less_rat(A),B) ) ) ).
% less_divide_eq_1_pos
tff(fact_2851_zero__less__divide__1__iff,axiom,
! [A: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),divide_divide_rat(one_one_rat,A))
<=> aa_rat_o(ord_less_rat(zero_zero_rat),A) ) ).
% zero_less_divide_1_iff
tff(fact_2852_add__neg__numeral__special_I8_J,axiom,
aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = zero_zero_int ).
% add_neg_numeral_special(8)
tff(fact_2853_add__neg__numeral__special_I8_J,axiom,
aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).
% add_neg_numeral_special(8)
tff(fact_2854_add__neg__numeral__special_I8_J,axiom,
aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) = zero_z3403309356797280102nteger ).
% add_neg_numeral_special(8)
tff(fact_2855_add__neg__numeral__special_I7_J,axiom,
aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).
% add_neg_numeral_special(7)
tff(fact_2856_add__neg__numeral__special_I7_J,axiom,
aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).
% add_neg_numeral_special(7)
tff(fact_2857_add__neg__numeral__special_I7_J,axiom,
aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).
% add_neg_numeral_special(7)
tff(fact_2858_nonzero__divide__mult__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( divide_divide_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = divide_divide_rat(one_one_rat,B) ) ) ).
% nonzero_divide_mult_cancel_left
tff(fact_2859_nonzero__divide__mult__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( divide_divide_rat(B,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = divide_divide_rat(one_one_rat,A) ) ) ).
% nonzero_divide_mult_cancel_right
tff(fact_2860_divide__le__eq__1__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(A),zero_zero_rat)
=> ( aa_rat_o(ord_less_eq_rat(divide_divide_rat(B,A)),one_one_rat)
<=> aa_rat_o(ord_less_eq_rat(A),B) ) ) ).
% divide_le_eq_1_neg
tff(fact_2861_divide__le__eq__1__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(divide_divide_rat(B,A)),one_one_rat)
<=> aa_rat_o(ord_less_eq_rat(B),A) ) ) ).
% divide_le_eq_1_pos
tff(fact_2862_le__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(A),zero_zero_rat)
=> ( aa_rat_o(ord_less_eq_rat(one_one_rat),divide_divide_rat(B,A))
<=> aa_rat_o(ord_less_eq_rat(B),A) ) ) ).
% le_divide_eq_1_neg
tff(fact_2863_add__right__imp__eq,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A) )
=> ( B = C ) ) ).
% add_right_imp_eq
tff(fact_2864_add__right__imp__eq,axiom,
! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
( ( aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,B),A) = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,C),A) )
=> ( B = C ) ) ).
% add_right_imp_eq
tff(fact_2865_add__right__imp__eq,axiom,
! [B: rat,A: rat,C: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
=> ( B = C ) ) ).
% add_right_imp_eq
tff(fact_2866_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
=> ( B = C ) ) ).
% add_right_imp_eq
tff(fact_2867_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
=> ( B = C ) ) ).
% add_right_imp_eq
tff(fact_2868_add__left__imp__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) )
=> ( B = C ) ) ).
% add_left_imp_eq
tff(fact_2869_add__left__imp__eq,axiom,
! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
( ( aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),B) = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),C) )
=> ( B = C ) ) ).
% add_left_imp_eq
tff(fact_2870_add__left__imp__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
=> ( B = C ) ) ).
% add_left_imp_eq
tff(fact_2871_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
=> ( B = C ) ) ).
% add_left_imp_eq
tff(fact_2872_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
=> ( B = C ) ) ).
% add_left_imp_eq
tff(fact_2873_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [B: code_integer,A: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ).
% ab_semigroup_add_class.add.left_commute
tff(fact_2874_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,B),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),C)) = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,B),C)) ).
% ab_semigroup_add_class.add.left_commute
tff(fact_2875_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).
% ab_semigroup_add_class.add.left_commute
tff(fact_2876_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).
% ab_semigroup_add_class.add.left_commute
tff(fact_2877_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).
% ab_semigroup_add_class.add.left_commute
tff(fact_2878_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) ).
% ab_semigroup_add_class.add.commute
tff(fact_2879_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),B) = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,B),A) ).
% ab_semigroup_add_class.add.commute
tff(fact_2880_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) ).
% ab_semigroup_add_class.add.commute
tff(fact_2881_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ).
% ab_semigroup_add_class.add.commute
tff(fact_2882_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) ).
% ab_semigroup_add_class.add.commute
tff(fact_2883_add_Oright__cancel,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A) )
<=> ( B = C ) ) ).
% add.right_cancel
tff(fact_2884_add_Oright__cancel,axiom,
! [B: rat,A: rat,C: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
<=> ( B = C ) ) ).
% add.right_cancel
tff(fact_2885_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
<=> ( B = C ) ) ).
% add.right_cancel
tff(fact_2886_add_Oleft__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) )
<=> ( B = C ) ) ).
% add.left_cancel
tff(fact_2887_add_Oleft__cancel,axiom,
! [A: rat,B: rat,C: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
<=> ( B = C ) ) ).
% add.left_cancel
tff(fact_2888_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
<=> ( B = C ) ) ).
% add.left_cancel
tff(fact_2889_add_Oassoc,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ).
% add.assoc
tff(fact_2890_add_Oassoc,axiom,
! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),B)),C) = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,B),C)) ).
% add.assoc
tff(fact_2891_add_Oassoc,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).
% add.assoc
tff(fact_2892_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).
% add.assoc
tff(fact_2893_add_Oassoc,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).
% add.assoc
tff(fact_2894_add_Oright__commute,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B) ).
% add.right_commute
tff(fact_2895_add_Oright__commute,axiom,
! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),B)),C) = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),C)),B) ).
% add.right_commute
tff(fact_2896_add_Oright__commute,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ).
% add.right_commute
tff(fact_2897_add_Oright__commute,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ).
% add.right_commute
tff(fact_2898_add_Oright__commute,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).
% add.right_commute
tff(fact_2899_add_Oright__assoc,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ).
% add.right_assoc
tff(fact_2900_add_Oright__assoc,axiom,
! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),B)),C) = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,B),C)) ).
% add.right_assoc
tff(fact_2901_add_Oright__assoc,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).
% add.right_assoc
tff(fact_2902_add_Oright__assoc,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).
% add.right_assoc
tff(fact_2903_add_Oright__assoc,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).
% add.right_assoc
tff(fact_2904_group__cancel_Oadd2,axiom,
! [Ba: code_integer,K: code_integer,B: code_integer,A: code_integer] :
( ( Ba = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),B) )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).
% group_cancel.add2
tff(fact_2905_group__cancel_Oadd2,axiom,
! [Ba: multis2468970476368604999at_nat,K: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat] :
( ( Ba = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,K),B) )
=> ( aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),Ba) = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,K),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),B)) ) ) ).
% group_cancel.add2
tff(fact_2906_group__cancel_Oadd2,axiom,
! [Ba: rat,K: rat,B: rat,A: rat] :
( ( Ba = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),Ba) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).
% group_cancel.add2
tff(fact_2907_group__cancel_Oadd2,axiom,
! [Ba: nat,K: nat,B: nat,A: nat] :
( ( Ba = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).
% group_cancel.add2
tff(fact_2908_group__cancel_Oadd2,axiom,
! [Ba: int,K: int,B: int,A: int] :
( ( Ba = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),Ba) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).
% group_cancel.add2
tff(fact_2909_group__cancel_Oadd1,axiom,
! [Aa: code_integer,K: code_integer,A: code_integer,B: code_integer] :
( ( Aa = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),A) )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).
% group_cancel.add1
tff(fact_2910_group__cancel_Oadd1,axiom,
! [Aa: multis2468970476368604999at_nat,K: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
( ( Aa = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,K),A) )
=> ( aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Aa),B) = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,K),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),B)) ) ) ).
% group_cancel.add1
tff(fact_2911_group__cancel_Oadd1,axiom,
! [Aa: rat,K: rat,A: rat,B: rat] :
( ( Aa = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Aa),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).
% group_cancel.add1
tff(fact_2912_group__cancel_Oadd1,axiom,
! [Aa: nat,K: nat,A: nat,B: nat] :
( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).
% group_cancel.add1
tff(fact_2913_group__cancel_Oadd1,axiom,
! [Aa: int,K: int,A: int,B: int] :
( ( Aa = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).
% group_cancel.add1
tff(fact_2914_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( ( I = J )
& ( K = L ) )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L) ) ) ).
% add_mono_thms_linordered_semiring(4)
tff(fact_2915_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( K = L ) )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L) ) ) ).
% add_mono_thms_linordered_semiring(4)
tff(fact_2916_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).
% add_mono_thms_linordered_semiring(4)
tff(fact_2917_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).
% add_mono_thms_linordered_semiring(4)
tff(fact_2918_divide__add__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(X,Z)),Y) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)),Z) ) ) ).
% divide_add_eq_iff
tff(fact_2919_add__divide__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),divide_divide_rat(Y,Z)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),Y),Z) ) ) ).
% add_divide_eq_iff
tff(fact_2920_add__num__frac,axiom,
! [Y: rat,Z: rat,X: rat] :
( ( Y != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),divide_divide_rat(X,Y)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),Y) ) ) ).
% add_num_frac
tff(fact_2921_add__frac__num,axiom,
! [Y: rat,X: rat,Z: rat] :
( ( Y != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(X,Y)),Z) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),Y) ) ) ).
% add_frac_num
tff(fact_2922_add__frac__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(X,Y)),divide_divide_rat(W,Z)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).
% add_frac_eq
tff(fact_2923_add__divide__eq__if__simps_I1_J,axiom,
! [A: rat,B: rat,Z: rat] :
aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),divide_divide_rat(B,Z)) = $ite(Z = zero_zero_rat,A,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z)),B),Z)) ).
% add_divide_eq_if_simps(1)
tff(fact_2924_add__divide__eq__if__simps_I2_J,axiom,
! [A: rat,Z: rat,B: rat] :
aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(A,Z)),B) = $ite(Z = zero_zero_rat,B,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z)),Z)) ).
% add_divide_eq_if_simps(2)
tff(fact_2925_gt__half__sum,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(A),B)
=> aa_rat_o(ord_less_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))),B) ) ).
% gt_half_sum
tff(fact_2926_less__half__sum,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(A),B)
=> aa_rat_o(ord_less_rat(A),divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ) ).
% less_half_sum
tff(fact_2927_add__le__imp__le__right,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
=> aa_rat_o(ord_less_eq_rat(A),B) ) ).
% add_le_imp_le_right
tff(fact_2928_add__le__imp__le__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C))
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).
% add_le_imp_le_right
tff(fact_2929_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).
% add_le_imp_le_right
tff(fact_2930_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% add_le_imp_le_right
tff(fact_2931_add__le__imp__le__left,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
=> aa_rat_o(ord_less_eq_rat(A),B) ) ).
% add_le_imp_le_left
tff(fact_2932_add__le__imp__le__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B))
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).
% add_le_imp_le_left
tff(fact_2933_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).
% add_le_imp_le_left
tff(fact_2934_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% add_le_imp_le_left
tff(fact_2935_le__iff__add,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
<=> ? [C3: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).
% le_iff_add
tff(fact_2936_add__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(A),B)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).
% add_right_mono
tff(fact_2937_add__right__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ) ).
% add_right_mono
tff(fact_2938_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).
% add_right_mono
tff(fact_2939_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).
% add_right_mono
tff(fact_2940_less__eqE,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ~ ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ) ).
% less_eqE
tff(fact_2941_add__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(A),B)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).
% add_left_mono
tff(fact_2942_add__left__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) ) ).
% add_left_mono
tff(fact_2943_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).
% add_left_mono
tff(fact_2944_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).
% add_left_mono
tff(fact_2945_add__mono,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( aa_rat_o(ord_less_eq_rat(A),B)
=> ( aa_rat_o(ord_less_eq_rat(C),D2)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).
% add_mono
tff(fact_2946_add__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),D2)) ) ) ).
% add_mono
tff(fact_2947_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).
% add_mono
tff(fact_2948_add__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).
% add_mono
tff(fact_2949_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( aa_rat_o(ord_less_eq_rat(I),J)
& aa_rat_o(ord_less_eq_rat(K),L) )
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_2950_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,I),J)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),L) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_2951_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_2952_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_2953_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& aa_rat_o(ord_less_eq_rat(K),L) )
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_2954_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( ( I = J )
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),L) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_2955_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_2956_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_2957_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( aa_rat_o(ord_less_eq_rat(I),J)
& ( K = L ) )
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_2958_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,I),J)
& ( K = L ) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_2959_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
& ( K = L ) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_2960_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
& ( K = L ) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_2961_add_Ogroup__left__neutral,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A) = A ).
% add.group_left_neutral
tff(fact_2962_add_Ogroup__left__neutral,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).
% add.group_left_neutral
tff(fact_2963_add_Ogroup__left__neutral,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).
% add.group_left_neutral
tff(fact_2964_add_Ocomm__neutral,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_z3403309356797280102nteger) = A ).
% add.comm_neutral
tff(fact_2965_add_Ocomm__neutral,axiom,
! [A: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),zero_z1048942125864253310at_nat) = A ).
% add.comm_neutral
tff(fact_2966_add_Ocomm__neutral,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).
% add.comm_neutral
tff(fact_2967_add_Ocomm__neutral,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).
% add.comm_neutral
tff(fact_2968_add_Ocomm__neutral,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).
% add.comm_neutral
tff(fact_2969_comm__monoid__add__class_Oadd__0,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A) = A ).
% comm_monoid_add_class.add_0
tff(fact_2970_comm__monoid__add__class_Oadd__0,axiom,
! [A: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,zero_z1048942125864253310at_nat),A) = A ).
% comm_monoid_add_class.add_0
tff(fact_2971_comm__monoid__add__class_Oadd__0,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).
% comm_monoid_add_class.add_0
tff(fact_2972_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).
% comm_monoid_add_class.add_0
tff(fact_2973_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).
% comm_monoid_add_class.add_0
tff(fact_2974_divide__divide__eq__left_H,axiom,
! [A: rat,B: rat,C: rat] : divide_divide_rat(divide_divide_rat(A,B),C) = divide_divide_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ).
% divide_divide_eq_left'
tff(fact_2975_divide__divide__times__eq,axiom,
! [X: rat,Y: rat,Z: rat,W: rat] : divide_divide_rat(divide_divide_rat(X,Y),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),W),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ).
% divide_divide_times_eq
tff(fact_2976_times__divide__times__eq,axiom,
! [X: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,divide_divide_rat(X,Y)),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),W)) ).
% times_divide_times_eq
tff(fact_2977_add__less__imp__less__right,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
=> aa_rat_o(ord_less_rat(A),B) ) ).
% add_less_imp_less_right
tff(fact_2978_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).
% add_less_imp_less_right
tff(fact_2979_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).
% add_less_imp_less_right
tff(fact_2980_add__less__imp__less__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C))
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).
% add_less_imp_less_right
tff(fact_2981_add__less__imp__less__right,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),C)),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(B),C))
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B) ) ).
% add_less_imp_less_right
tff(fact_2982_add__less__imp__less__left,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
=> aa_rat_o(ord_less_rat(A),B) ) ).
% add_less_imp_less_left
tff(fact_2983_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).
% add_less_imp_less_left
tff(fact_2984_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).
% add_less_imp_less_left
tff(fact_2985_add__less__imp__less__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B))
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).
% add_less_imp_less_left
tff(fact_2986_add__less__imp__less__left,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(C),A)),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(C),B))
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B) ) ).
% add_less_imp_less_left
tff(fact_2987_add__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_rat(A),B)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).
% add_strict_right_mono
tff(fact_2988_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).
% add_strict_right_mono
tff(fact_2989_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).
% add_strict_right_mono
tff(fact_2990_add__strict__right__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ) ).
% add_strict_right_mono
tff(fact_2991_add__strict__right__mono,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),C)),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(B),C)) ) ).
% add_strict_right_mono
tff(fact_2992_add__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_rat(A),B)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).
% add_strict_left_mono
tff(fact_2993_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).
% add_strict_left_mono
tff(fact_2994_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).
% add_strict_left_mono
tff(fact_2995_add__strict__left__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) ) ).
% add_strict_left_mono
tff(fact_2996_add__strict__left__mono,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(C),A)),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(C),B)) ) ).
% add_strict_left_mono
tff(fact_2997_add__strict__mono,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( aa_rat_o(ord_less_rat(A),B)
=> ( aa_rat_o(ord_less_rat(C),D2)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).
% add_strict_mono
tff(fact_2998_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).
% add_strict_mono
tff(fact_2999_add__strict__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).
% add_strict_mono
tff(fact_3000_add__strict__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),D2)) ) ) ).
% add_strict_mono
tff(fact_3001_add__strict__mono,axiom,
! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,C),D2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),C)),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(B),D2)) ) ) ).
% add_strict_mono
tff(fact_3002_add__mono__thms__linordered__field_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( aa_rat_o(ord_less_rat(I),J)
& ( K = L ) )
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_3003_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
& ( K = L ) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_3004_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
& ( K = L ) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_3005_add__mono__thms__linordered__field_I1_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),J)
& ( K = L ) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_3006_add__mono__thms__linordered__field_I1_J,axiom,
! [I: code_natural,J: code_natural,K: code_natural,L: code_natural] :
( ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,I),J)
& ( K = L ) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(I),K)),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(J),L)) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_3007_add__mono__thms__linordered__field_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& aa_rat_o(ord_less_rat(K),L) )
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_3008_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_3009_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_3010_add__mono__thms__linordered__field_I2_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( ( I = J )
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),L) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_3011_add__mono__thms__linordered__field_I2_J,axiom,
! [I: code_natural,J: code_natural,K: code_natural,L: code_natural] :
( ( ( I = J )
& aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,K),L) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(I),K)),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(J),L)) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_3012_add__mono__thms__linordered__field_I5_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( aa_rat_o(ord_less_rat(I),J)
& aa_rat_o(ord_less_rat(K),L) )
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_3013_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_3014_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
& aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_3015_add__mono__thms__linordered__field_I5_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),J)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),L) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_3016_add__mono__thms__linordered__field_I5_J,axiom,
! [I: code_natural,J: code_natural,K: code_natural,L: code_natural] :
( ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,I),J)
& aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,K),L) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(I),K)),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(J),L)) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_3017_combine__common__factor,axiom,
! [A: code_integer,E: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),E)),C) ).
% combine_common_factor
tff(fact_3018_combine__common__factor,axiom,
! [A: rat,E: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),E)),C) ).
% combine_common_factor
tff(fact_3019_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E)),C) ).
% combine_common_factor
tff(fact_3020_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E)),C) ).
% combine_common_factor
tff(fact_3021_distrib__right,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).
% distrib_right
tff(fact_3022_distrib__right,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).
% distrib_right
tff(fact_3023_distrib__right,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).
% distrib_right
tff(fact_3024_distrib__right,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).
% distrib_right
tff(fact_3025_distrib__left,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ).
% distrib_left
tff(fact_3026_distrib__left,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).
% distrib_left
tff(fact_3027_distrib__left,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).
% distrib_left
tff(fact_3028_distrib__left,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).
% distrib_left
tff(fact_3029_comm__semiring__class_Odistrib,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).
% comm_semiring_class.distrib
tff(fact_3030_comm__semiring__class_Odistrib,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).
% comm_semiring_class.distrib
tff(fact_3031_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).
% comm_semiring_class.distrib
tff(fact_3032_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).
% comm_semiring_class.distrib
tff(fact_3033_ring__class_Oring__distribs_I1_J,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ).
% ring_class.ring_distribs(1)
tff(fact_3034_ring__class_Oring__distribs_I1_J,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).
% ring_class.ring_distribs(1)
tff(fact_3035_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).
% ring_class.ring_distribs(1)
tff(fact_3036_ring__class_Oring__distribs_I2_J,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).
% ring_class.ring_distribs(2)
tff(fact_3037_ring__class_Oring__distribs_I2_J,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).
% ring_class.ring_distribs(2)
tff(fact_3038_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).
% ring_class.ring_distribs(2)
tff(fact_3039_diff__diff__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ).
% diff_diff_eq
tff(fact_3040_diff__diff__eq,axiom,
! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(minus_4286766774447292334at_nat(A,B),C) = minus_4286766774447292334at_nat(A,aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,B),C)) ).
% diff_diff_eq
tff(fact_3041_diff__diff__eq,axiom,
! [A: rat,B: rat,C: rat] : minus_minus_rat(minus_minus_rat(A,B),C) = minus_minus_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).
% diff_diff_eq
tff(fact_3042_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).
% diff_diff_eq
tff(fact_3043_diff__diff__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).
% diff_diff_eq
tff(fact_3044_add__implies__diff,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B) = A )
=> ( C = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B) ) ) ).
% add_implies_diff
tff(fact_3045_add__implies__diff,axiom,
! [C: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat] :
( ( aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,C),B) = A )
=> ( C = minus_4286766774447292334at_nat(A,B) ) ) ).
% add_implies_diff
tff(fact_3046_add__implies__diff,axiom,
! [C: rat,B: rat,A: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) = A )
=> ( C = minus_minus_rat(A,B) ) ) ).
% add_implies_diff
tff(fact_3047_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B) = A )
=> ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).
% add_implies_diff
tff(fact_3048_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) = A )
=> ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).
% add_implies_diff
tff(fact_3049_diff__add__eq__diff__diff__swap,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),C)),B) ).
% diff_add_eq_diff_diff_swap
tff(fact_3050_diff__add__eq__diff__diff__swap,axiom,
! [A: rat,B: rat,C: rat] : minus_minus_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = minus_minus_rat(minus_minus_rat(A,C),B) ).
% diff_add_eq_diff_diff_swap
tff(fact_3051_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) ).
% diff_add_eq_diff_diff_swap
tff(fact_3052_diff__add__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B) ).
% diff_add_eq
tff(fact_3053_diff__add__eq,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C),B) ).
% diff_add_eq
tff(fact_3054_diff__add__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).
% diff_add_eq
tff(fact_3055_diff__diff__eq2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B) ).
% diff_diff_eq2
tff(fact_3056_diff__diff__eq2,axiom,
! [A: rat,B: rat,C: rat] : minus_minus_rat(A,minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C),B) ).
% diff_diff_eq2
tff(fact_3057_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).
% diff_diff_eq2
tff(fact_3058_add__diff__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) ).
% add_diff_eq
tff(fact_3059_add__diff__eq,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),C) ).
% add_diff_eq
tff(fact_3060_add__diff__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).
% add_diff_eq
tff(fact_3061_eq__diff__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(C),B) )
<=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = C ) ) ).
% eq_diff_eq
tff(fact_3062_eq__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( ( A = minus_minus_rat(C,B) )
<=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = C ) ) ).
% eq_diff_eq
tff(fact_3063_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B) )
<=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C ) ) ).
% eq_diff_eq
tff(fact_3064_diff__eq__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B) = C )
<=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B) ) ) ).
% diff_eq_eq
tff(fact_3065_diff__eq__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( minus_minus_rat(A,B) = C )
<=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) ) ) ).
% diff_eq_eq
tff(fact_3066_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C )
<=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) ) ) ).
% diff_eq_eq
tff(fact_3067_group__cancel_Osub1,axiom,
! [Aa: code_integer,K: code_integer,A: code_integer,B: code_integer] :
( ( Aa = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),A) )
=> ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Aa),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)) ) ) ).
% group_cancel.sub1
tff(fact_3068_group__cancel_Osub1,axiom,
! [Aa: rat,K: rat,A: rat,B: rat] :
( ( Aa = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
=> ( minus_minus_rat(Aa,B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),minus_minus_rat(A,B)) ) ) ).
% group_cancel.sub1
tff(fact_3069_group__cancel_Osub1,axiom,
! [Aa: int,K: int,A: int,B: int] :
( ( Aa = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).
% group_cancel.sub1
tff(fact_3070_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).
% add.inverse_distrib_swap
tff(fact_3071_add_Oinverse__distrib__swap,axiom,
! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).
% add.inverse_distrib_swap
tff(fact_3072_add_Oinverse__distrib__swap,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).
% add.inverse_distrib_swap
tff(fact_3073_group__cancel_Oneg1,axiom,
! [Aa: int,K: int,A: int] :
( ( Aa = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
=> ( aa_int_int(uminus_uminus_int,Aa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A)) ) ) ).
% group_cancel.neg1
tff(fact_3074_group__cancel_Oneg1,axiom,
! [Aa: rat,K: rat,A: rat] :
( ( Aa = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
=> ( aa_rat_rat(uminus_uminus_rat,Aa) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).
% group_cancel.neg1
tff(fact_3075_group__cancel_Oneg1,axiom,
! [Aa: code_integer,K: code_integer,A: code_integer] :
( ( Aa = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),A) )
=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).
% group_cancel.neg1
tff(fact_3076_add_Oabel__semigroup__axioms,axiom,
abel_s8772020223019846471nteger(plus_p5714425477246183910nteger) ).
% add.abel_semigroup_axioms
tff(fact_3077_add_Oabel__semigroup__axioms,axiom,
abel_s8335990944309670109at_nat(plus_p7104986032573967614at_nat) ).
% add.abel_semigroup_axioms
tff(fact_3078_add_Oabel__semigroup__axioms,axiom,
abel_semigroup_rat(plus_plus_rat) ).
% add.abel_semigroup_axioms
tff(fact_3079_add_Oabel__semigroup__axioms,axiom,
abel_semigroup_nat(plus_plus_nat) ).
% add.abel_semigroup_axioms
tff(fact_3080_add_Oabel__semigroup__axioms,axiom,
abel_semigroup_int(plus_plus_int) ).
% add.abel_semigroup_axioms
tff(fact_3081_add_Osemigroup__axioms,axiom,
semigr7036097719740861967nteger(plus_p5714425477246183910nteger) ).
% add.semigroup_axioms
tff(fact_3082_add_Osemigroup__axioms,axiom,
semigr4652511051630528533at_nat(plus_p7104986032573967614at_nat) ).
% add.semigroup_axioms
tff(fact_3083_add_Osemigroup__axioms,axiom,
semigroup_rat(plus_plus_rat) ).
% add.semigroup_axioms
tff(fact_3084_add_Osemigroup__axioms,axiom,
semigroup_nat(plus_plus_nat) ).
% add.semigroup_axioms
tff(fact_3085_add_Osemigroup__axioms,axiom,
semigroup_int(plus_plus_int) ).
% add.semigroup_axioms
tff(fact_3086_minus__divide__add__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,divide_divide_rat(X,Z))),Y) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)),Z) ) ) ).
% minus_divide_add_eq_iff
tff(fact_3087_add__divide__eq__if__simps_I3_J,axiom,
! [A: rat,Z: rat,B: rat] :
aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,divide_divide_rat(A,Z))),B) = $ite(Z = zero_zero_rat,B,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z)),Z)) ).
% add_divide_eq_if_simps(3)
tff(fact_3088_add_Oac__operator__axioms,axiom,
syntax414191934074104506nteger(plus_p5714425477246183910nteger) ).
% add.ac_operator_axioms
tff(fact_3089_add_Oac__operator__axioms,axiom,
syntax4222486904863270186at_nat(plus_p7104986032573967614at_nat) ).
% add.ac_operator_axioms
tff(fact_3090_add_Oac__operator__axioms,axiom,
syntax8165409007564224565or_rat(plus_plus_rat) ).
% add.ac_operator_axioms
tff(fact_3091_add_Oac__operator__axioms,axiom,
syntax8800539067650720301or_nat(plus_plus_nat) ).
% add.ac_operator_axioms
tff(fact_3092_add_Oac__operator__axioms,axiom,
syntax8798048597141670025or_int(plus_plus_int) ).
% add.ac_operator_axioms
tff(fact_3093_add_Osafe__commute,axiom,
! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(syntax380677951125133566nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y)),A)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) ) ) ).
% add.safe_commute
tff(fact_3094_add_Osafe__commute,axiom,
! [X: multis2468970476368604999at_nat,Y: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
( syntax5084084251851430274at_nat(aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,X),Y),A)
=> ( aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A),B) = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,B),A) ) ) ).
% add.safe_commute
tff(fact_3095_add_Osafe__commute,axiom,
! [X: rat,Y: rat,A: rat,B: rat] :
( aa_rat_o(syntax3730441303064801268at_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),A)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) ) ) ).
% add.safe_commute
tff(fact_3096_add_Osafe__commute,axiom,
! [X: nat,Y: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(syntax4682126007086162916at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ) ) ).
% add.safe_commute
tff(fact_3097_add_Osafe__commute,axiom,
! [X: int,Y: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(syntax5678989248478167196nt_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),A)
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) ) ) ).
% add.safe_commute
tff(fact_3098_scaling__mono,axiom,
! [U: rat,V: rat,R2: rat,S2: rat] :
( aa_rat_o(ord_less_eq_rat(U),V)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),R2)
=> ( aa_rat_o(ord_less_eq_rat(R2),S2)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),minus_minus_rat(V,U)),S2))),V) ) ) ) ).
% scaling_mono
tff(fact_3099_nonzero__eq__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( A = divide_divide_rat(B,C) )
<=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B ) ) ) ).
% nonzero_eq_divide_eq
tff(fact_3100_nonzero__divide__eq__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat(B,C) = A )
<=> ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) ) ) ) ).
% nonzero_divide_eq_eq
tff(fact_3101_eq__divide__imp,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B )
=> ( A = divide_divide_rat(B,C) ) ) ) ).
% eq_divide_imp
tff(fact_3102_divide__eq__imp,axiom,
! [C: rat,B: rat,A: rat] :
( ( C != zero_zero_rat )
=> ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) )
=> ( divide_divide_rat(B,C) = A ) ) ) ).
% divide_eq_imp
tff(fact_3103_eq__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = divide_divide_rat(B,C) )
<=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B,A = zero_zero_rat) ) ).
% eq_divide_eq
tff(fact_3104_divide__eq__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( divide_divide_rat(B,C) = A )
<=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),A = zero_zero_rat) ) ).
% divide_eq_eq
tff(fact_3105_frac__eq__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( divide_divide_rat(X,Y) = divide_divide_rat(W,Z) )
<=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y) ) ) ) ) ).
% frac_eq_eq
tff(fact_3106_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
tff(fact_3107_add__nonpos__eq__0__iff,axiom,
! [X: rat,Y: rat] :
( aa_rat_o(ord_less_eq_rat(X),zero_zero_rat)
=> ( aa_rat_o(ord_less_eq_rat(Y),zero_zero_rat)
=> ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
<=> ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_3108_add__nonpos__eq__0__iff,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),zero_z3403309356797280102nteger)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),zero_z3403309356797280102nteger)
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y) = zero_z3403309356797280102nteger )
<=> ( ( X = zero_z3403309356797280102nteger )
& ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_3109_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),zero_zero_nat)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
<=> ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_3110_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
<=> ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_3111_add__nonneg__eq__0__iff,axiom,
! [X: rat,Y: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),X)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),Y)
=> ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
<=> ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_3112_add__nonneg__eq__0__iff,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y) = zero_z3403309356797280102nteger )
<=> ( ( X = zero_z3403309356797280102nteger )
& ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_3113_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
<=> ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_3114_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
<=> ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_3115_add__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
=> ( aa_rat_o(ord_less_eq_rat(B),zero_zero_rat)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).
% add_nonpos_nonpos
tff(fact_3116_add__nonpos__nonpos,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).
% add_nonpos_nonpos
tff(fact_3117_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).
% add_nonpos_nonpos
tff(fact_3118_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).
% add_nonpos_nonpos
tff(fact_3119_add__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),B)
=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).
% add_nonneg_nonneg
tff(fact_3120_add__nonneg__nonneg,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).
% add_nonneg_nonneg
tff(fact_3121_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).
% add_nonneg_nonneg
tff(fact_3122_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).
% add_nonneg_nonneg
tff(fact_3123_add__increasing2,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> ( aa_rat_o(ord_less_eq_rat(B),A)
=> aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).
% add_increasing2
tff(fact_3124_add__increasing2,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).
% add_increasing2
tff(fact_3125_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).
% add_increasing2
tff(fact_3126_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).
% add_increasing2
tff(fact_3127_add__decreasing2,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
=> ( aa_rat_o(ord_less_eq_rat(A),B)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ) ).
% add_decreasing2
tff(fact_3128_add__decreasing2,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B) ) ) ).
% add_decreasing2
tff(fact_3129_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).
% add_decreasing2
tff(fact_3130_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).
% add_decreasing2
tff(fact_3131_add__increasing,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(B),C)
=> aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).
% add_increasing
tff(fact_3132_add__increasing,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).
% add_increasing
tff(fact_3133_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).
% add_increasing
tff(fact_3134_add__increasing,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).
% add_increasing
tff(fact_3135_add__decreasing,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
=> ( aa_rat_o(ord_less_eq_rat(C),B)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ) ).
% add_decreasing
tff(fact_3136_add__decreasing,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B) ) ) ).
% add_decreasing
tff(fact_3137_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).
% add_decreasing
tff(fact_3138_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).
% add_decreasing
tff(fact_3139_add__less__le__mono,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( aa_rat_o(ord_less_rat(A),B)
=> ( aa_rat_o(ord_less_eq_rat(C),D2)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).
% add_less_le_mono
tff(fact_3140_add__less__le__mono,axiom,
! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,C),D2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),C)),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(B),D2)) ) ) ).
% add_less_le_mono
tff(fact_3141_add__less__le__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),D2)) ) ) ).
% add_less_le_mono
tff(fact_3142_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).
% add_less_le_mono
tff(fact_3143_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).
% add_less_le_mono
tff(fact_3144_add__le__less__mono,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( aa_rat_o(ord_less_eq_rat(A),B)
=> ( aa_rat_o(ord_less_rat(C),D2)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).
% add_le_less_mono
tff(fact_3145_add__le__less__mono,axiom,
! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,C),D2)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),C)),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(B),D2)) ) ) ).
% add_le_less_mono
tff(fact_3146_add__le__less__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),D2)) ) ) ).
% add_le_less_mono
tff(fact_3147_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).
% add_le_less_mono
tff(fact_3148_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).
% add_le_less_mono
tff(fact_3149_add__mono__thms__linordered__field_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( aa_rat_o(ord_less_rat(I),J)
& aa_rat_o(ord_less_eq_rat(K),L) )
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_3150_add__mono__thms__linordered__field_I3_J,axiom,
! [I: code_natural,J: code_natural,K: code_natural,L: code_natural] :
( ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,I),J)
& aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,K),L) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(I),K)),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(J),L)) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_3151_add__mono__thms__linordered__field_I3_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),J)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),L) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_3152_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_3153_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_3154_add__mono__thms__linordered__field_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( aa_rat_o(ord_less_eq_rat(I),J)
& aa_rat_o(ord_less_rat(K),L) )
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_3155_add__mono__thms__linordered__field_I4_J,axiom,
! [I: code_natural,J: code_natural,K: code_natural,L: code_natural] :
( ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,I),J)
& aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,K),L) )
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(I),K)),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(J),L)) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_3156_add__mono__thms__linordered__field_I4_J,axiom,
! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,I),J)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),L) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_3157_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_3158_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
& aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_3159_pos__add__strict,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_rat(B),C)
=> aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).
% pos_add_strict
tff(fact_3160_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).
% pos_add_strict
tff(fact_3161_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).
% pos_add_strict
tff(fact_3162_pos__add__strict,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).
% pos_add_strict
tff(fact_3163_pos__add__strict,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),C)) ) ) ).
% pos_add_strict
tff(fact_3164_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ~ ! [C2: nat] :
( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
tff(fact_3165_add__pos__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),B)
=> aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).
% add_pos_pos
tff(fact_3166_add__pos__pos,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).
% add_pos_pos
tff(fact_3167_add__pos__pos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).
% add_pos_pos
tff(fact_3168_add__pos__pos,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).
% add_pos_pos
tff(fact_3169_add__pos__pos,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),B)) ) ) ).
% add_pos_pos
tff(fact_3170_add__neg__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(A),zero_zero_rat)
=> ( aa_rat_o(ord_less_rat(B),zero_zero_rat)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).
% add_neg_neg
tff(fact_3171_add__neg__neg,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).
% add_neg_neg
tff(fact_3172_add__neg__neg,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).
% add_neg_neg
tff(fact_3173_add__neg__neg,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).
% add_neg_neg
tff(fact_3174_add__neg__neg,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),zero_z2226904508553997617atural)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),zero_z2226904508553997617atural)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),B)),zero_z2226904508553997617atural) ) ) ).
% add_neg_neg
tff(fact_3175_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
<=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_3176_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_3177_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_3178_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_3179_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_3180_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_3181_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_3182_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_3183_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A)) ) ).
% le_add_diff
tff(fact_3184_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_3185_le__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(A),minus_minus_rat(C,B))
<=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ) ).
% le_diff_eq
tff(fact_3186_le__diff__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(C),B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) ) ).
% le_diff_eq
tff(fact_3187_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).
% le_diff_eq
tff(fact_3188_diff__le__eq,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(minus_minus_rat(A,B)),C)
<=> aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).
% diff_le_eq
tff(fact_3189_diff__le__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) ) ).
% diff_le_eq
tff(fact_3190_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).
% diff_le_eq
tff(fact_3191_less__add__one,axiom,
! [A: rat] : aa_rat_o(ord_less_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)) ).
% less_add_one
tff(fact_3192_less__add__one,axiom,
! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)) ).
% less_add_one
tff(fact_3193_less__add__one,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)) ).
% less_add_one
tff(fact_3194_less__add__one,axiom,
! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)) ).
% less_add_one
tff(fact_3195_less__add__one,axiom,
! [A: code_natural] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),one_one_Code_natural)) ).
% less_add_one
tff(fact_3196_add__mono1,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(A),B)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),one_one_rat)) ) ).
% add_mono1
tff(fact_3197_add__mono1,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat)) ) ).
% add_mono1
tff(fact_3198_add__mono1,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)) ) ).
% add_mono1
tff(fact_3199_add__mono1,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),one_one_Code_integer)) ) ).
% add_mono1
tff(fact_3200_add__mono1,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),one_one_Code_natural)),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(B),one_one_Code_natural)) ) ).
% add_mono1
tff(fact_3201_less__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(ord_less_rat(A),minus_minus_rat(C,B))
<=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ) ).
% less_diff_eq
tff(fact_3202_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).
% less_diff_eq
tff(fact_3203_less__diff__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(C),B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) ) ).
% less_diff_eq
tff(fact_3204_diff__less__eq,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_rat(minus_minus_rat(A,B)),C)
<=> aa_rat_o(ord_less_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).
% diff_less_eq
tff(fact_3205_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).
% diff_less_eq
tff(fact_3206_diff__less__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) ) ).
% diff_less_eq
tff(fact_3207_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( aa_int_int(uminus_uminus_int,A) = B )
<=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
tff(fact_3208_neg__eq__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(uminus_uminus_rat,A) = B )
<=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).
% neg_eq_iff_add_eq_0
tff(fact_3209_neg__eq__iff__add__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
<=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger ) ) ).
% neg_eq_iff_add_eq_0
tff(fact_3210_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A = aa_int_int(uminus_uminus_int,B) )
<=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
tff(fact_3211_eq__neg__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( A = aa_rat_rat(uminus_uminus_rat,B) )
<=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).
% eq_neg_iff_add_eq_0
tff(fact_3212_eq__neg__iff__add__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
<=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger ) ) ).
% eq_neg_iff_add_eq_0
tff(fact_3213_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
=> ( aa_int_int(uminus_uminus_int,A) = B ) ) ).
% add.inverse_unique
tff(fact_3214_add_Oinverse__unique,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
=> ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).
% add.inverse_unique
tff(fact_3215_add_Oinverse__unique,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B ) ) ).
% add.inverse_unique
tff(fact_3216_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).
% ab_group_add_class.ab_left_minus
tff(fact_3217_ab__group__add__class_Oab__left__minus,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).
% ab_group_add_class.ab_left_minus
tff(fact_3218_ab__group__add__class_Oab__left__minus,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A) = zero_z3403309356797280102nteger ).
% ab_group_add_class.ab_left_minus
tff(fact_3219_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
<=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).
% add_eq_0_iff
tff(fact_3220_add__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
<=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).
% add_eq_0_iff
tff(fact_3221_add__eq__0__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger )
<=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).
% add_eq_0_iff
tff(fact_3222_square__diff__square__factored,axiom,
! [X: code_integer,Y: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),Y)) ).
% square_diff_square_factored
tff(fact_3223_square__diff__square__factored,axiom,
! [X: rat,Y: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),minus_minus_rat(X,Y)) ).
% square_diff_square_factored
tff(fact_3224_square__diff__square__factored,axiom,
! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) ).
% square_diff_square_factored
tff(fact_3225_eq__add__iff2,axiom,
! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D2) )
<=> ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A)),E)),D2) ) ) ).
% eq_add_iff2
tff(fact_3226_eq__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D2) )
<=> ( C = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(B,A)),E)),D2) ) ) ).
% eq_add_iff2
tff(fact_3227_eq__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D2) )
<=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D2) ) ) ).
% eq_add_iff2
tff(fact_3228_eq__add__iff1,axiom,
! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D2) )
<=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),E)),C) = D2 ) ) ).
% eq_add_iff1
tff(fact_3229_eq__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D2) )
<=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),E)),C) = D2 ) ) ).
% eq_add_iff1
tff(fact_3230_eq__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D2) )
<=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C) = D2 ) ) ).
% eq_add_iff1
tff(fact_3231_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).
% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3232_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: rat,B: rat] : minus_minus_rat(A,B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).
% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3233_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).
% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3234_diff__conv__add__uminus,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).
% diff_conv_add_uminus
tff(fact_3235_diff__conv__add__uminus,axiom,
! [A: rat,B: rat] : minus_minus_rat(A,B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).
% diff_conv_add_uminus
tff(fact_3236_diff__conv__add__uminus,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).
% diff_conv_add_uminus
tff(fact_3237_group__cancel_Osub2,axiom,
! [Ba: int,K: int,B: int,A: int] :
( ( Ba = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),Ba) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).
% group_cancel.sub2
tff(fact_3238_group__cancel_Osub2,axiom,
! [Ba: rat,K: rat,B: rat,A: rat] :
( ( Ba = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
=> ( minus_minus_rat(A,Ba) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),minus_minus_rat(A,B)) ) ) ).
% group_cancel.sub2
tff(fact_3239_group__cancel_Osub2,axiom,
! [Ba: code_integer,K: code_integer,B: code_integer,A: code_integer] :
( ( Ba = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),B) )
=> ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)) ) ) ).
% group_cancel.sub2
tff(fact_3240_add_Ocomm__monoid__axioms,axiom,
aa_Code_integer_o(comm_m1654649897431166824nteger(plus_p5714425477246183910nteger),zero_z3403309356797280102nteger) ).
% add.comm_monoid_axioms
tff(fact_3241_add_Ocomm__monoid__axioms,axiom,
comm_m1862543879040115196at_nat(plus_p7104986032573967614at_nat,zero_z1048942125864253310at_nat) ).
% add.comm_monoid_axioms
tff(fact_3242_add_Ocomm__monoid__axioms,axiom,
aa_rat_o(comm_monoid_rat(plus_plus_rat),zero_zero_rat) ).
% add.comm_monoid_axioms
tff(fact_3243_add_Ocomm__monoid__axioms,axiom,
aa_nat_o(comm_monoid_nat(plus_plus_nat),zero_zero_nat) ).
% add.comm_monoid_axioms
tff(fact_3244_add_Ocomm__monoid__axioms,axiom,
aa_int_o(comm_monoid_int(plus_plus_int),zero_zero_int) ).
% add.comm_monoid_axioms
tff(fact_3245_add_Omonoid__axioms,axiom,
aa_Code_integer_o(monoid_Code_integer(plus_p5714425477246183910nteger),zero_z3403309356797280102nteger) ).
% add.monoid_axioms
tff(fact_3246_add_Omonoid__axioms,axiom,
monoid66256789251010760at_nat(plus_p7104986032573967614at_nat,zero_z1048942125864253310at_nat) ).
% add.monoid_axioms
tff(fact_3247_add_Omonoid__axioms,axiom,
aa_rat_o(monoid_rat(plus_plus_rat),zero_zero_rat) ).
% add.monoid_axioms
tff(fact_3248_add_Omonoid__axioms,axiom,
aa_nat_o(monoid_nat(plus_plus_nat),zero_zero_nat) ).
% add.monoid_axioms
tff(fact_3249_add_Omonoid__axioms,axiom,
aa_int_o(monoid_int(plus_plus_int),zero_zero_int) ).
% add.monoid_axioms
tff(fact_3250_sum__list_Omonoid__list__axioms,axiom,
aa_Code_integer_o(groups5100778238730182072nteger(plus_p5714425477246183910nteger),zero_z3403309356797280102nteger) ).
% sum_list.monoid_list_axioms
tff(fact_3251_sum__list_Omonoid__list__axioms,axiom,
groups5598625667918239660at_nat(plus_p7104986032573967614at_nat,zero_z1048942125864253310at_nat) ).
% sum_list.monoid_list_axioms
tff(fact_3252_sum__list_Omonoid__list__axioms,axiom,
aa_rat_o(groups7826448149055759155st_rat(plus_plus_rat),zero_zero_rat) ).
% sum_list.monoid_list_axioms
tff(fact_3253_sum__list_Omonoid__list__axioms,axiom,
aa_nat_o(groups8461578209142254891st_nat(plus_plus_nat),zero_zero_nat) ).
% sum_list.monoid_list_axioms
tff(fact_3254_sum__list_Omonoid__list__axioms,axiom,
aa_int_o(groups8459087738633204615st_int(plus_plus_int),zero_zero_int) ).
% sum_list.monoid_list_axioms
tff(fact_3255_sum__list_Ocomm__monoid__list__axioms,axiom,
aa_Code_integer_o(groups2567640417189887108nteger(plus_p5714425477246183910nteger),zero_z3403309356797280102nteger) ).
% sum_list.comm_monoid_list_axioms
tff(fact_3256_sum__list_Ocomm__monoid__list__axioms,axiom,
groups7159712745983395808at_nat(plus_p7104986032573967614at_nat,zero_z1048942125864253310at_nat) ).
% sum_list.comm_monoid_list_axioms
tff(fact_3257_sum__list_Ocomm__monoid__list__axioms,axiom,
aa_rat_o(groups593308882113688063st_rat(plus_plus_rat),zero_zero_rat) ).
% sum_list.comm_monoid_list_axioms
tff(fact_3258_sum__list_Ocomm__monoid__list__axioms,axiom,
aa_nat_o(groups1228438942200183799st_nat(plus_plus_nat),zero_zero_nat) ).
% sum_list.comm_monoid_list_axioms
tff(fact_3259_sum__list_Ocomm__monoid__list__axioms,axiom,
aa_int_o(groups1225948471691133523st_int(plus_plus_int),zero_zero_int) ).
% sum_list.comm_monoid_list_axioms
tff(fact_3260_sum_Ocomm__monoid__list__set__axioms,axiom,
aa_Code_integer_o(groups7165323350619936122nteger(plus_p5714425477246183910nteger),zero_z3403309356797280102nteger) ).
% sum.comm_monoid_list_set_axioms
tff(fact_3261_sum_Ocomm__monoid__list__set__axioms,axiom,
groups4109576622710581866at_nat(plus_p7104986032573967614at_nat,zero_z1048942125864253310at_nat) ).
% sum.comm_monoid_list_set_axioms
tff(fact_3262_sum_Ocomm__monoid__list__set__axioms,axiom,
aa_rat_o(groups2612948464029442805et_rat(plus_plus_rat),zero_zero_rat) ).
% sum.comm_monoid_list_set_axioms
tff(fact_3263_sum_Ocomm__monoid__list__set__axioms,axiom,
aa_nat_o(groups3248078524115938541et_nat(plus_plus_nat),zero_zero_nat) ).
% sum.comm_monoid_list_set_axioms
tff(fact_3264_sum_Ocomm__monoid__list__set__axioms,axiom,
aa_int_o(groups3245588053606888265et_int(plus_plus_int),zero_zero_int) ).
% sum.comm_monoid_list_set_axioms
tff(fact_3265_dbl__inc__def,axiom,
! [X: code_integer] : aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,X) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),X)),one_one_Code_integer) ).
% dbl_inc_def
tff(fact_3266_dbl__inc__def,axiom,
! [X: rat] : neg_nu5219082963157363817nc_rat(X) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X)),one_one_rat) ).
% dbl_inc_def
tff(fact_3267_dbl__inc__def,axiom,
! [X: int] : aa_int_int(neg_nu5851722552734809277nc_int,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).
% dbl_inc_def
tff(fact_3268_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3269_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3270_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3271_divide__strict__left__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_rat(A),B)
=> ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
=> aa_rat_o(ord_less_rat(divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).
% divide_strict_left_mono_neg
tff(fact_3272_divide__strict__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(ord_less_rat(B),A)
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
=> aa_rat_o(ord_less_rat(divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).
% divide_strict_left_mono
tff(fact_3273_mult__imp__less__div__pos,axiom,
! [Y: rat,Z: rat,X: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),Y)
=> ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),X)
=> aa_rat_o(ord_less_rat(Z),divide_divide_rat(X,Y)) ) ) ).
% mult_imp_less_div_pos
tff(fact_3274_mult__imp__div__pos__less,axiom,
! [Y: rat,X: rat,Z: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),Y)
=> ( aa_rat_o(ord_less_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
=> aa_rat_o(ord_less_rat(divide_divide_rat(X,Y)),Z) ) ) ).
% mult_imp_div_pos_less
tff(fact_3275_pos__less__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> ( aa_rat_o(ord_less_rat(A),divide_divide_rat(B,C))
<=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).
% pos_less_divide_eq
tff(fact_3276_pos__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> ( aa_rat_o(ord_less_rat(divide_divide_rat(B,C)),A)
<=> aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).
% pos_divide_less_eq
tff(fact_3277_neg__less__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> ( aa_rat_o(ord_less_rat(A),divide_divide_rat(B,C))
<=> aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).
% neg_less_divide_eq
tff(fact_3278_neg__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> ( aa_rat_o(ord_less_rat(divide_divide_rat(B,C)),A)
<=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).
% neg_divide_less_eq
tff(fact_3279_less__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_rat(A),divide_divide_rat(B,C))
<=> $ite(
aa_rat_o(ord_less_rat(zero_zero_rat),C),
aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),
$ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(ord_less_rat(A),zero_zero_rat)) ) ) ).
% less_divide_eq
tff(fact_3280_divide__less__eq,axiom,
! [B: rat,C: rat,A: rat] :
( aa_rat_o(ord_less_rat(divide_divide_rat(B,C)),A)
<=> $ite(
aa_rat_o(ord_less_rat(zero_zero_rat),C),
aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
$ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),aa_rat_o(ord_less_rat(zero_zero_rat),A)) ) ) ).
% divide_less_eq
tff(fact_3281_divide__less__eq__1,axiom,
! [B: rat,A: rat] :
( aa_rat_o(ord_less_rat(divide_divide_rat(B,A)),one_one_rat)
<=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
& aa_rat_o(ord_less_rat(B),A) )
| ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
& aa_rat_o(ord_less_rat(A),B) )
| ( A = zero_zero_rat ) ) ) ).
% divide_less_eq_1
tff(fact_3282_less__divide__eq__1,axiom,
! [B: rat,A: rat] :
( aa_rat_o(ord_less_rat(one_one_rat),divide_divide_rat(B,A))
<=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
& aa_rat_o(ord_less_rat(A),B) )
| ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
& aa_rat_o(ord_less_rat(B),A) ) ) ) ).
% less_divide_eq_1
tff(fact_3283_divide__diff__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( minus_minus_rat(divide_divide_rat(X,Z),Y) = divide_divide_rat(minus_minus_rat(X,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)),Z) ) ) ).
% divide_diff_eq_iff
tff(fact_3284_diff__divide__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( minus_minus_rat(X,divide_divide_rat(Y,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z),Y),Z) ) ) ).
% diff_divide_eq_iff
tff(fact_3285_diff__frac__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( minus_minus_rat(divide_divide_rat(X,Y),divide_divide_rat(W,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).
% diff_frac_eq
tff(fact_3286_add__divide__eq__if__simps_I4_J,axiom,
! [A: rat,B: rat,Z: rat] :
minus_minus_rat(A,divide_divide_rat(B,Z)) = $ite(Z = zero_zero_rat,A,divide_divide_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z),B),Z)) ).
% add_divide_eq_if_simps(4)
tff(fact_3287_add__strict__increasing2,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_rat(B),C)
=> aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).
% add_strict_increasing2
tff(fact_3288_add__strict__increasing2,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),C)) ) ) ).
% add_strict_increasing2
tff(fact_3289_add__strict__increasing2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).
% add_strict_increasing2
tff(fact_3290_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).
% add_strict_increasing2
tff(fact_3291_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).
% add_strict_increasing2
tff(fact_3292_add__strict__increasing,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(B),C)
=> aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).
% add_strict_increasing
tff(fact_3293_add__strict__increasing,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),C)) ) ) ).
% add_strict_increasing
tff(fact_3294_add__strict__increasing,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).
% add_strict_increasing
tff(fact_3295_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).
% add_strict_increasing
tff(fact_3296_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).
% add_strict_increasing
tff(fact_3297_add__pos__nonneg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),B)
=> aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).
% add_pos_nonneg
tff(fact_3298_add__pos__nonneg,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),B)) ) ) ).
% add_pos_nonneg
tff(fact_3299_add__pos__nonneg,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).
% add_pos_nonneg
tff(fact_3300_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).
% add_pos_nonneg
tff(fact_3301_add__pos__nonneg,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).
% add_pos_nonneg
tff(fact_3302_add__nonpos__neg,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
=> ( aa_rat_o(ord_less_rat(B),zero_zero_rat)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).
% add_nonpos_neg
tff(fact_3303_add__nonpos__neg,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A),zero_z2226904508553997617atural)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),zero_z2226904508553997617atural)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),B)),zero_z2226904508553997617atural) ) ) ).
% add_nonpos_neg
tff(fact_3304_add__nonpos__neg,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).
% add_nonpos_neg
tff(fact_3305_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).
% add_nonpos_neg
tff(fact_3306_add__nonpos__neg,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).
% add_nonpos_neg
tff(fact_3307_add__nonneg__pos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),B)
=> aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).
% add_nonneg_pos
tff(fact_3308_add__nonneg__pos,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),B)) ) ) ).
% add_nonneg_pos
tff(fact_3309_add__nonneg__pos,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).
% add_nonneg_pos
tff(fact_3310_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).
% add_nonneg_pos
tff(fact_3311_add__nonneg__pos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).
% add_nonneg_pos
tff(fact_3312_add__neg__nonpos,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(A),zero_zero_rat)
=> ( aa_rat_o(ord_less_eq_rat(B),zero_zero_rat)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).
% add_neg_nonpos
tff(fact_3313_add__neg__nonpos,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),zero_z2226904508553997617atural)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),zero_z2226904508553997617atural)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),B)),zero_z2226904508553997617atural) ) ) ).
% add_neg_nonpos
tff(fact_3314_add__neg__nonpos,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).
% add_neg_nonpos
tff(fact_3315_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).
% add_neg_nonpos
tff(fact_3316_add__neg__nonpos,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).
% add_neg_nonpos
tff(fact_3317_sum__squares__ge__zero,axiom,
! [X: rat,Y: rat] : aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))) ).
% sum_squares_ge_zero
tff(fact_3318_sum__squares__ge__zero,axiom,
! [X: code_integer,Y: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y))) ).
% sum_squares_ge_zero
tff(fact_3319_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))) ).
% sum_squares_ge_zero
tff(fact_3320_nonzero__neg__divide__eq__eq2,axiom,
! [B: rat,C: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( C = aa_rat_rat(uminus_uminus_rat,divide_divide_rat(A,B)) )
<=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).
% nonzero_neg_divide_eq_eq2
tff(fact_3321_nonzero__neg__divide__eq__eq,axiom,
! [B: rat,A: rat,C: rat] :
( ( B != zero_zero_rat )
=> ( ( aa_rat_rat(uminus_uminus_rat,divide_divide_rat(A,B)) = C )
<=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) ) ) ) ).
% nonzero_neg_divide_eq_eq
tff(fact_3322_minus__divide__eq__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C)) = A )
<=> $ite(C != zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),A = zero_zero_rat) ) ).
% minus_divide_eq_eq
tff(fact_3323_eq__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C)) )
<=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(uminus_uminus_rat,B),A = zero_zero_rat) ) ).
% eq_minus_divide_eq
tff(fact_3324_divide__eq__minus__1__iff,axiom,
! [A: rat,B: rat] :
( ( divide_divide_rat(A,B) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
<=> ( ( B != zero_zero_rat )
& ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).
% divide_eq_minus_1_iff
tff(fact_3325_not__sum__squares__lt__zero,axiom,
! [X: rat,Y: rat] : ~ aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat) ).
% not_sum_squares_lt_zero
tff(fact_3326_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int) ).
% not_sum_squares_lt_zero
tff(fact_3327_not__sum__squares__lt__zero,axiom,
! [X: code_integer,Y: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y))),zero_z3403309356797280102nteger) ).
% not_sum_squares_lt_zero
tff(fact_3328_discrete,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),B) ) ).
% discrete
tff(fact_3329_discrete,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B) ) ).
% discrete
tff(fact_3330_discrete,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B) ) ).
% discrete
tff(fact_3331_zero__less__two,axiom,
aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)) ).
% zero_less_two
tff(fact_3332_zero__less__two,axiom,
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)) ).
% zero_less_two
tff(fact_3333_zero__less__two,axiom,
aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)) ).
% zero_less_two
tff(fact_3334_zero__less__two,axiom,
aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),one_one_Code_integer)) ).
% zero_less_two
tff(fact_3335_zero__less__two,axiom,
aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),one_one_Code_natural)) ).
% zero_less_two
tff(fact_3336_le__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D2))
<=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),E)),C)),D2) ) ).
% le_add_iff1
tff(fact_3337_le__add__iff1,axiom,
! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D2))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),E)),C)),D2) ) ).
% le_add_iff1
tff(fact_3338_le__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D2))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D2) ) ).
% le_add_iff1
tff(fact_3339_le__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D2))
<=> aa_rat_o(ord_less_eq_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(B,A)),E)),D2)) ) ).
% le_add_iff2
tff(fact_3340_le__add__iff2,axiom,
! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D2))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A)),E)),D2)) ) ).
% le_add_iff2
tff(fact_3341_le__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D2))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D2)) ) ).
% le_add_iff2
tff(fact_3342_less__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D2))
<=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),E)),C)),D2) ) ).
% less_add_iff1
tff(fact_3343_less__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D2))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D2) ) ).
% less_add_iff1
tff(fact_3344_less__add__iff1,axiom,
! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D2))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),E)),C)),D2) ) ).
% less_add_iff1
tff(fact_3345_less__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D2))
<=> aa_rat_o(ord_less_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(B,A)),E)),D2)) ) ).
% less_add_iff2
tff(fact_3346_less__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D2))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D2)) ) ).
% less_add_iff2
tff(fact_3347_less__add__iff2,axiom,
! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E)),D2))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A)),E)),D2)) ) ).
% less_add_iff2
tff(fact_3348_square__diff__one__factored,axiom,
! [X: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),one_one_Code_integer)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),one_one_Code_integer)) ).
% square_diff_one_factored
tff(fact_3349_square__diff__one__factored,axiom,
! [X: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)),minus_minus_rat(X,one_one_rat)) ).
% square_diff_one_factored
tff(fact_3350_square__diff__one__factored,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)) ).
% square_diff_one_factored
tff(fact_3351_dbl__dec__def,axiom,
! [X: code_integer] : aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,X) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),X)),one_one_Code_integer) ).
% dbl_dec_def
tff(fact_3352_dbl__dec__def,axiom,
! [X: rat] : neg_nu3179335615603231917ec_rat(X) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X),one_one_rat) ).
% dbl_dec_def
tff(fact_3353_dbl__dec__def,axiom,
! [X: int] : aa_int_int(neg_nu3811975205180677377ec_int,X) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).
% dbl_dec_def
tff(fact_3354_divide__left__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(A),B)
=> ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
=> aa_rat_o(ord_less_eq_rat(divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).
% divide_left_mono_neg
tff(fact_3355_mult__imp__le__div__pos,axiom,
! [Y: rat,Z: rat,X: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),Y)
=> ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),X)
=> aa_rat_o(ord_less_eq_rat(Z),divide_divide_rat(X,Y)) ) ) ).
% mult_imp_le_div_pos
tff(fact_3356_mult__imp__div__pos__le,axiom,
! [Y: rat,X: rat,Z: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),Y)
=> ( aa_rat_o(ord_less_eq_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
=> aa_rat_o(ord_less_eq_rat(divide_divide_rat(X,Y)),Z) ) ) ).
% mult_imp_div_pos_le
tff(fact_3357_pos__le__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> ( aa_rat_o(ord_less_eq_rat(A),divide_divide_rat(B,C))
<=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).
% pos_le_divide_eq
tff(fact_3358_pos__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> ( aa_rat_o(ord_less_eq_rat(divide_divide_rat(B,C)),A)
<=> aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).
% pos_divide_le_eq
tff(fact_3359_neg__le__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> ( aa_rat_o(ord_less_eq_rat(A),divide_divide_rat(B,C))
<=> aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).
% neg_le_divide_eq
tff(fact_3360_neg__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> ( aa_rat_o(ord_less_eq_rat(divide_divide_rat(B,C)),A)
<=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).
% neg_divide_le_eq
tff(fact_3361_divide__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(B),A)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
=> aa_rat_o(ord_less_eq_rat(divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).
% divide_left_mono
tff(fact_3362_le__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(A),divide_divide_rat(B,C))
<=> $ite(
aa_rat_o(ord_less_rat(zero_zero_rat),C),
aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),
$ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)) ) ) ).
% le_divide_eq
tff(fact_3363_divide__le__eq,axiom,
! [B: rat,C: rat,A: rat] :
( aa_rat_o(ord_less_eq_rat(divide_divide_rat(B,C)),A)
<=> $ite(
aa_rat_o(ord_less_rat(zero_zero_rat),C),
aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
$ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)) ) ) ).
% divide_le_eq
tff(fact_3364_divide__le__eq__1,axiom,
! [B: rat,A: rat] :
( aa_rat_o(ord_less_eq_rat(divide_divide_rat(B,A)),one_one_rat)
<=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
& aa_rat_o(ord_less_eq_rat(B),A) )
| ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
& aa_rat_o(ord_less_eq_rat(A),B) )
| ( A = zero_zero_rat ) ) ) ).
% divide_le_eq_1
tff(fact_3365_le__divide__eq__1,axiom,
! [B: rat,A: rat] :
( aa_rat_o(ord_less_eq_rat(one_one_rat),divide_divide_rat(B,A))
<=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
& aa_rat_o(ord_less_eq_rat(A),B) )
| ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
& aa_rat_o(ord_less_eq_rat(B),A) ) ) ) ).
% le_divide_eq_1
tff(fact_3366_frac__le__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( aa_rat_o(ord_less_eq_rat(divide_divide_rat(X,Y)),divide_divide_rat(W,Z))
<=> aa_rat_o(ord_less_eq_rat(divide_divide_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat) ) ) ) ).
% frac_le_eq
tff(fact_3367_frac__less__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( aa_rat_o(ord_less_rat(divide_divide_rat(X,Y)),divide_divide_rat(W,Z))
<=> aa_rat_o(ord_less_rat(divide_divide_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat) ) ) ) ).
% frac_less_eq
tff(fact_3368_pos__minus__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> ( aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C))),A)
<=> aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).
% pos_minus_divide_less_eq
tff(fact_3369_pos__less__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),C)
=> ( aa_rat_o(ord_less_rat(A),aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C)))
<=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).
% pos_less_minus_divide_eq
tff(fact_3370_neg__minus__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> ( aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C))),A)
<=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).
% neg_minus_divide_less_eq
tff(fact_3371_neg__less__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(C),zero_zero_rat)
=> ( aa_rat_o(ord_less_rat(A),aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C)))
<=> aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).
% neg_less_minus_divide_eq
tff(fact_3372_minus__divide__less__eq,axiom,
! [B: rat,C: rat,A: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C))),A)
<=> $ite(
aa_rat_o(ord_less_rat(zero_zero_rat),C),
aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
$ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(ord_less_rat(zero_zero_rat),A)) ) ) ).
% minus_divide_less_eq
tff(fact_3373_less__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(ord_less_rat(A),aa_rat_rat(uminus_uminus_rat,divide_divide_rat(B,C)))
<=> $ite(
aa_rat_o(ord_less_rat(zero_zero_rat),C),
aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B)),
$ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(ord_less_rat(A),zero_zero_rat)) ) ) ).
% less_minus_divide_eq
tff(fact_3374_minus__divide__diff__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( minus_minus_rat(aa_rat_rat(uminus_uminus_rat,divide_divide_rat(X,Z)),Y) = divide_divide_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)),Z) ) ) ).
% minus_divide_diff_eq_iff
tff(fact_3375_add__divide__eq__if__simps_I5_J,axiom,
! [A: rat,Z: rat,B: rat] :
minus_minus_rat(divide_divide_rat(A,Z),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),divide_divide_rat(minus_minus_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z)),Z)) ).
% add_divide_eq_if_simps(5)
tff(fact_3376_add__divide__eq__if__simps_I6_J,axiom,
! [A: rat,Z: rat,B: rat] :
minus_minus_rat(aa_rat_rat(uminus_uminus_rat,divide_divide_rat(A,Z)),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),divide_divide_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z)),Z)) ).
% add_divide_eq_if_simps(6)
tff(fact_3377_convex__bound__le,axiom,
! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
( aa_rat_o(ord_less_eq_rat(X),A)
=> ( aa_rat_o(ord_less_eq_rat(Y),A)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),U)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),V)
=> ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A) ) ) ) ) ) ).
% convex_bound_le
tff(fact_3378_convex__bound__le,axiom,
! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),V)
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,U),V) = one_one_Code_integer )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,U),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V),Y))),A) ) ) ) ) ) ).
% convex_bound_le
tff(fact_3379_convex__bound__le,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).
% convex_bound_le
tff(fact_3380_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).
% div_mult_self4
tff(fact_3381_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).
% div_mult_self4
tff(fact_3382_div__mult__self4,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ) ) ).
% div_mult_self4
tff(fact_3383_div__mult__self4,axiom,
! [B: code_natural,C: code_natural,A: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)),A)),B) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(C),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)) ) ) ).
% div_mult_self4
tff(fact_3384_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).
% div_mult_self3
tff(fact_3385_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).
% div_mult_self3
tff(fact_3386_div__mult__self3,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ) ) ).
% div_mult_self3
tff(fact_3387_div__mult__self3,axiom,
! [B: code_natural,C: code_natural,A: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B)),A)),B) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(C),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)) ) ) ).
% div_mult_self3
tff(fact_3388_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).
% div_mult_self2
tff(fact_3389_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).
% div_mult_self2
tff(fact_3390_div__mult__self2,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ) ) ).
% div_mult_self2
tff(fact_3391_div__mult__self2,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C))),B) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(C),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)) ) ) ).
% div_mult_self2
tff(fact_3392_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).
% div_mult_self1
tff(fact_3393_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).
% div_mult_self1
tff(fact_3394_div__mult__self1,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ) ) ).
% div_mult_self1
tff(fact_3395_div__mult__self1,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B))),B) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(C),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)) ) ) ).
% div_mult_self1
tff(fact_3396_div__minus1__right,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A) ).
% div_minus1_right
tff(fact_3397_div__minus1__right,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).
% div_minus1_right
tff(fact_3398_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = $ite(C = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).
% div_mult_mult1_if
tff(fact_3399_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = $ite(C = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).
% div_mult_mult1_if
tff(fact_3400_div__mult__mult1__if,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) = $ite(C = zero_z3403309356797280102nteger,zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ).
% div_mult_mult1_if
tff(fact_3401_div__mult__mult1__if,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B)) = $ite(C = zero_z2226904508553997617atural,zero_z2226904508553997617atural,aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)) ).
% div_mult_mult1_if
tff(fact_3402_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).
% div_mult_mult2
tff(fact_3403_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).
% div_mult_mult2
tff(fact_3404_div__mult__mult2,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) ) ) ).
% div_mult_mult2
tff(fact_3405_div__mult__mult2,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( ( C != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B) ) ) ).
% div_mult_mult2
tff(fact_3406_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).
% div_mult_mult1
tff(fact_3407_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).
% div_mult_mult1
tff(fact_3408_div__mult__mult1,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) ) ) ).
% div_mult_mult1
tff(fact_3409_div__mult__mult1,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( ( C != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B)) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B) ) ) ).
% div_mult_mult1
tff(fact_3410_bits__div__by__1,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).
% bits_div_by_1
tff(fact_3411_bits__div__by__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).
% bits_div_by_1
tff(fact_3412_bits__div__by__1,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),one_one_Code_integer) = A ).
% bits_div_by_1
tff(fact_3413_bits__div__by__1,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),one_one_Code_natural) = A ).
% bits_div_by_1
tff(fact_3414_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).
% div_add_self2
tff(fact_3415_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).
% div_add_self2
tff(fact_3416_div__add__self2,axiom,
! [B: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),one_one_Code_integer) ) ) ).
% div_add_self2
tff(fact_3417_div__add__self2,axiom,
! [B: code_natural,A: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),B)),B) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),one_one_Code_natural) ) ) ).
% div_add_self2
tff(fact_3418_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).
% div_add_self1
tff(fact_3419_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).
% div_add_self1
tff(fact_3420_div__add__self1,axiom,
! [B: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),one_one_Code_integer) ) ) ).
% div_add_self1
tff(fact_3421_div__add__self1,axiom,
! [B: code_natural,A: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(B),A)),B) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),one_one_Code_natural) ) ) ).
% div_add_self1
tff(fact_3422_sum__squares__gt__zero__iff,axiom,
! [X: rat,Y: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)))
<=> ( ( X != zero_zero_rat )
| ( Y != zero_zero_rat ) ) ) ).
% sum_squares_gt_zero_iff
tff(fact_3423_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))
<=> ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
tff(fact_3424_sum__squares__gt__zero__iff,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y)))
<=> ( ( X != zero_z3403309356797280102nteger )
| ( Y != zero_z3403309356797280102nteger ) ) ) ).
% sum_squares_gt_zero_iff
tff(fact_3425_sum__squares__le__zero__iff,axiom,
! [X: rat,Y: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat)
<=> ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_le_zero_iff
tff(fact_3426_sum__squares__le__zero__iff,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y))),zero_z3403309356797280102nteger)
<=> ( ( X = zero_z3403309356797280102nteger )
& ( Y = zero_z3403309356797280102nteger ) ) ) ).
% sum_squares_le_zero_iff
tff(fact_3427_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)
<=> ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
tff(fact_3428_sum__squares__eq__zero__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y)) = zero_z3403309356797280102nteger )
<=> ( ( X = zero_z3403309356797280102nteger )
& ( Y = zero_z3403309356797280102nteger ) ) ) ).
% sum_squares_eq_zero_iff
tff(fact_3429_sum__squares__eq__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = zero_zero_rat )
<=> ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_eq_zero_iff
tff(fact_3430_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
<=> ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
tff(fact_3431_add__scale__eq__noteq,axiom,
! [R2: code_integer,A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( R2 != zero_z3403309356797280102nteger )
=> ( ( ( A = B )
& ( C != D2 ) )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),C)) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),D2)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_3432_add__scale__eq__noteq,axiom,
! [R2: rat,A: rat,B: rat,C: rat,D2: rat] :
( ( R2 != zero_zero_rat )
=> ( ( ( A = B )
& ( C != D2 ) )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),C)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),D2)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_3433_add__scale__eq__noteq,axiom,
! [R2: nat,A: nat,B: nat,C: nat,D2: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D2 ) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D2)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_3434_add__scale__eq__noteq,axiom,
! [R2: int,A: int,B: int,C: int,D2: int] :
( ( R2 != zero_zero_int )
=> ( ( ( A = B )
& ( C != D2 ) )
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D2)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_3435_crossproduct__eq,axiom,
! [W: code_integer,Y: code_integer,X: code_integer,Z: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,W),Y)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,W),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y)) )
<=> ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
tff(fact_3436_crossproduct__eq,axiom,
! [W: rat,Y: rat,X: rat,Z: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)) )
<=> ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
tff(fact_3437_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
<=> ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
tff(fact_3438_crossproduct__eq,axiom,
! [W: int,Y: int,X: int,Z: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)) )
<=> ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
tff(fact_3439_crossproduct__noteq,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ( A != B )
& ( C != D2 ) )
<=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).
% crossproduct_noteq
tff(fact_3440_crossproduct__noteq,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( ( A != B )
& ( C != D2 ) )
<=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),D2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).
% crossproduct_noteq
tff(fact_3441_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( A != B )
& ( C != D2 ) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).
% crossproduct_noteq
tff(fact_3442_crossproduct__noteq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( A != B )
& ( C != D2 ) )
<=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).
% crossproduct_noteq
tff(fact_3443_divide__le__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( aa_rat_o(ord_less_eq_rat(divide_divide_rat(B,C)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))
<=> $ite(
aa_rat_o(ord_less_rat(zero_zero_rat),C),
aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),
$ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),B),aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))) ) ) ).
% divide_le_eq_numeral(2)
tff(fact_3444_le__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),divide_divide_rat(B,C))
<=> $ite(
aa_rat_o(ord_less_rat(zero_zero_rat),C),
aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),B),
$ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),zero_zero_rat)) ) ) ).
% le_divide_eq_numeral(2)
tff(fact_3445_right__diff__distrib__NO__MATCH,axiom,
! [X: nat,Y: nat,A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).
% right_diff_distrib_NO_MATCH
tff(fact_3446_right__diff__distrib__NO__MATCH,axiom,
! [X: nat,Y: nat,A: rat,B: rat,C: rat] :
( aa_rat_o(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).
% right_diff_distrib_NO_MATCH
tff(fact_3447_right__diff__distrib__NO__MATCH,axiom,
! [X: nat,Y: nat,A: int,B: int,C: int] :
( aa_int_o(nO_MATCH_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).
% right_diff_distrib_NO_MATCH
tff(fact_3448_right__diff__distrib__NO__MATCH,axiom,
! [X: int,Y: int,A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).
% right_diff_distrib_NO_MATCH
tff(fact_3449_right__diff__distrib__NO__MATCH,axiom,
! [X: int,Y: int,A: rat,B: rat,C: rat] :
( aa_rat_o(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).
% right_diff_distrib_NO_MATCH
tff(fact_3450_right__diff__distrib__NO__MATCH,axiom,
! [X: int,Y: int,A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).
% right_diff_distrib_NO_MATCH
tff(fact_3451_right__diff__distrib__NO__MATCH,axiom,
! [X: rat,Y: rat,A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(nO_MAT243659986023775650nteger(divide_divide_rat(X,Y)),A)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).
% right_diff_distrib_NO_MATCH
tff(fact_3452_right__diff__distrib__NO__MATCH,axiom,
! [X: rat,Y: rat,A: rat,B: rat,C: rat] :
( aa_rat_o(nO_MATCH_rat_rat(divide_divide_rat(X,Y)),A)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).
% right_diff_distrib_NO_MATCH
tff(fact_3453_right__diff__distrib__NO__MATCH,axiom,
! [X: rat,Y: rat,A: int,B: int,C: int] :
( aa_int_o(nO_MATCH_rat_int(divide_divide_rat(X,Y)),A)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).
% right_diff_distrib_NO_MATCH
tff(fact_3454_right__diff__distrib__NO__MATCH,axiom,
! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(nO_MAT8252062027627875367nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X),Y)),A)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).
% right_diff_distrib_NO_MATCH
tff(fact_3455_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(times_times_num(V,W))),Z) ).
% mult_numeral_left_semiring_numeral
tff(fact_3456_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(W)),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(times_times_num(V,W))),Z) ).
% mult_numeral_left_semiring_numeral
tff(fact_3457_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(times_times_num(V,W))),Z) ).
% mult_numeral_left_semiring_numeral
tff(fact_3458_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(times_times_num(V,W))),Z) ).
% mult_numeral_left_semiring_numeral
tff(fact_3459_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: code_natural] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(V)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(W)),Z)) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(times_times_num(V,W))),Z) ).
% mult_numeral_left_semiring_numeral
tff(fact_3460_numeral__times__numeral,axiom,
! [M2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(M2)),numeral_numeral_nat(N)) = numeral_numeral_nat(times_times_num(M2,N)) ).
% numeral_times_numeral
tff(fact_3461_numeral__times__numeral,axiom,
! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(M2)),numeral_numeral_int(N)) = numeral_numeral_int(times_times_num(M2,N)) ).
% numeral_times_numeral
tff(fact_3462_numeral__times__numeral,axiom,
! [M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(M2)),numeral_numeral_rat(N)) = numeral_numeral_rat(times_times_num(M2,N)) ).
% numeral_times_numeral
tff(fact_3463_numeral__times__numeral,axiom,
! [M2: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(M2)),numera6620942414471956472nteger(N)) = numera6620942414471956472nteger(times_times_num(M2,N)) ).
% numeral_times_numeral
tff(fact_3464_numeral__times__numeral,axiom,
! [M2: num,N: num] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(M2)),numera5444537566228673987atural(N)) = numera5444537566228673987atural(times_times_num(M2,N)) ).
% numeral_times_numeral
tff(fact_3465_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),C)) ).
% distrib_left_numeral
tff(fact_3466_distrib__left__numeral,axiom,
! [V: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),C)) ).
% distrib_left_numeral
tff(fact_3467_distrib__left__numeral,axiom,
! [V: num,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),C)) ).
% distrib_left_numeral
tff(fact_3468_distrib__left__numeral,axiom,
! [V: num,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V)),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V)),C)) ).
% distrib_left_numeral
tff(fact_3469_distrib__left__numeral,axiom,
! [V: num,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(V)),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(B),C)) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(V)),B)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(V)),C)) ).
% distrib_left_numeral
tff(fact_3470_distrib__right__numeral,axiom,
! [A: nat,B: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),numeral_numeral_nat(V)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),numeral_numeral_nat(V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),numeral_numeral_nat(V))) ).
% distrib_right_numeral
tff(fact_3471_distrib__right__numeral,axiom,
! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),numeral_numeral_int(V)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),numeral_numeral_int(V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),numeral_numeral_int(V))) ).
% distrib_right_numeral
tff(fact_3472_distrib__right__numeral,axiom,
! [A: rat,B: rat,V: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),numeral_numeral_rat(V)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),numeral_numeral_rat(V))) ).
% distrib_right_numeral
tff(fact_3473_distrib__right__numeral,axiom,
! [A: code_integer,B: code_integer,V: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),numera6620942414471956472nteger(V)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),numera6620942414471956472nteger(V))) ).
% distrib_right_numeral
tff(fact_3474_distrib__right__numeral,axiom,
! [A: code_natural,B: code_natural,V: num] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),B)),numera5444537566228673987atural(V)) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),numera5444537566228673987atural(V))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),numera5444537566228673987atural(V))) ).
% distrib_right_numeral
tff(fact_3475_right__diff__distrib__numeral,axiom,
! [V: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),C)) ).
% right_diff_distrib_numeral
tff(fact_3476_right__diff__distrib__numeral,axiom,
! [V: num,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),C)) ).
% right_diff_distrib_numeral
tff(fact_3477_right__diff__distrib__numeral,axiom,
! [V: num,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V)),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V)),C)) ).
% right_diff_distrib_numeral
tff(fact_3478_left__diff__distrib__numeral,axiom,
! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),numeral_numeral_int(V)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),numeral_numeral_int(V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),numeral_numeral_int(V))) ).
% left_diff_distrib_numeral
tff(fact_3479_left__diff__distrib__numeral,axiom,
! [A: rat,B: rat,V: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),numeral_numeral_rat(V)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),numeral_numeral_rat(V))) ).
% left_diff_distrib_numeral
tff(fact_3480_left__diff__distrib__numeral,axiom,
! [A: code_integer,B: code_integer,V: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),numera6620942414471956472nteger(V)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),numera6620942414471956472nteger(V))) ).
% left_diff_distrib_numeral
tff(fact_3481_mult__neg__numeral__simps_I1_J,axiom,
! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = numeral_numeral_int(times_times_num(M2,N)) ).
% mult_neg_numeral_simps(1)
tff(fact_3482_mult__neg__numeral__simps_I1_J,axiom,
! [M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M2))),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = numeral_numeral_rat(times_times_num(M2,N)) ).
% mult_neg_numeral_simps(1)
tff(fact_3483_mult__neg__numeral__simps_I1_J,axiom,
! [M2: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(times_times_num(M2,N)) ).
% mult_neg_numeral_simps(1)
tff(fact_3484_mult__neg__numeral__simps_I2_J,axiom,
! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),numeral_numeral_int(N)) = aa_int_int(uminus_uminus_int,numeral_numeral_int(times_times_num(M2,N))) ).
% mult_neg_numeral_simps(2)
tff(fact_3485_mult__neg__numeral__simps_I2_J,axiom,
! [M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M2))),numeral_numeral_rat(N)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(times_times_num(M2,N))) ).
% mult_neg_numeral_simps(2)
tff(fact_3486_mult__neg__numeral__simps_I2_J,axiom,
! [M2: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),numera6620942414471956472nteger(N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(times_times_num(M2,N))) ).
% mult_neg_numeral_simps(2)
tff(fact_3487_mult__neg__numeral__simps_I3_J,axiom,
! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = aa_int_int(uminus_uminus_int,numeral_numeral_int(times_times_num(M2,N))) ).
% mult_neg_numeral_simps(3)
tff(fact_3488_mult__neg__numeral__simps_I3_J,axiom,
! [M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(M2)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(times_times_num(M2,N))) ).
% mult_neg_numeral_simps(3)
tff(fact_3489_mult__neg__numeral__simps_I3_J,axiom,
! [M2: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(times_times_num(M2,N))) ).
% mult_neg_numeral_simps(3)
tff(fact_3490_divide__le__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( aa_rat_o(ord_less_eq_rat(divide_divide_rat(B,numeral_numeral_rat(W))),A)
<=> aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W))) ) ).
% divide_le_eq_numeral1(1)
tff(fact_3491_le__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( aa_rat_o(ord_less_eq_rat(A),divide_divide_rat(B,numeral_numeral_rat(W)))
<=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W))),B) ) ).
% le_divide_eq_numeral1(1)
tff(fact_3492_divide__eq__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( ( divide_divide_rat(B,numeral_numeral_rat(W)) = A )
<=> $ite(numeral_numeral_rat(W) != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W)),A = zero_zero_rat) ) ).
% divide_eq_eq_numeral1(1)
tff(fact_3493_eq__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( ( A = divide_divide_rat(B,numeral_numeral_rat(W)) )
<=> $ite(numeral_numeral_rat(W) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W)) = B,A = zero_zero_rat) ) ).
% eq_divide_eq_numeral1(1)
tff(fact_3494_divide__less__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( aa_rat_o(ord_less_rat(divide_divide_rat(B,numeral_numeral_rat(W))),A)
<=> aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W))) ) ).
% divide_less_eq_numeral1(1)
tff(fact_3495_less__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( aa_rat_o(ord_less_rat(A),divide_divide_rat(B,numeral_numeral_rat(W)))
<=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W))),B) ) ).
% less_divide_eq_numeral1(1)
tff(fact_3496_le__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( aa_rat_o(ord_less_eq_rat(A),divide_divide_rat(B,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))))
<=> aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))) ) ).
% le_divide_eq_numeral1(2)
tff(fact_3497_divide__le__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( aa_rat_o(ord_less_eq_rat(divide_divide_rat(B,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))),A)
<=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))),B) ) ).
% divide_le_eq_numeral1(2)
tff(fact_3498_eq__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( ( A = divide_divide_rat(B,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))) )
<=> $ite(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))) = B,A = zero_zero_rat) ) ).
% eq_divide_eq_numeral1(2)
tff(fact_3499_divide__eq__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( ( divide_divide_rat(B,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))) = A )
<=> $ite(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),A = zero_zero_rat) ) ).
% divide_eq_eq_numeral1(2)
tff(fact_3500_divide__less__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( aa_rat_o(ord_less_rat(divide_divide_rat(B,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))),A)
<=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))),B) ) ).
% divide_less_eq_numeral1(2)
tff(fact_3501_less__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( aa_rat_o(ord_less_rat(A),divide_divide_rat(B,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))))
<=> aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))) ) ).
% less_divide_eq_numeral1(2)
tff(fact_3502_one__le__numeral,axiom,
! [N: num] : aa_rat_o(ord_less_eq_rat(one_one_rat),numeral_numeral_rat(N)) ).
% one_le_numeral
tff(fact_3503_one__le__numeral,axiom,
! [N: num] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),numera5444537566228673987atural(N)) ).
% one_le_numeral
tff(fact_3504_one__le__numeral,axiom,
! [N: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),numera6620942414471956472nteger(N)) ).
% one_le_numeral
tff(fact_3505_one__le__numeral,axiom,
! [N: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),numeral_numeral_nat(N)) ).
% one_le_numeral
tff(fact_3506_one__le__numeral,axiom,
! [N: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),numeral_numeral_int(N)) ).
% one_le_numeral
tff(fact_3507_not__numeral__less__one,axiom,
! [N: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,numeral_numeral_nat(N)),one_one_nat) ).
% not_numeral_less_one
tff(fact_3508_not__numeral__less__one,axiom,
! [N: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,numeral_numeral_int(N)),one_one_int) ).
% not_numeral_less_one
tff(fact_3509_not__numeral__less__one,axiom,
! [N: num] : ~ aa_rat_o(ord_less_rat(numeral_numeral_rat(N)),one_one_rat) ).
% not_numeral_less_one
tff(fact_3510_not__numeral__less__one,axiom,
! [N: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(N)),one_one_Code_integer) ).
% not_numeral_less_one
tff(fact_3511_not__numeral__less__one,axiom,
! [N: num] : ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,numera5444537566228673987atural(N)),one_one_Code_natural) ).
% not_numeral_less_one
tff(fact_3512_one__plus__numeral__commute,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),numeral_numeral_nat(X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(X)),one_one_nat) ).
% one_plus_numeral_commute
tff(fact_3513_one__plus__numeral__commute,axiom,
! [X: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),numeral_numeral_int(X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(X)),one_one_int) ).
% one_plus_numeral_commute
tff(fact_3514_one__plus__numeral__commute,axiom,
! [X: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),numeral_numeral_rat(X)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(X)),one_one_rat) ).
% one_plus_numeral_commute
tff(fact_3515_one__plus__numeral__commute,axiom,
! [X: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),numera6620942414471956472nteger(X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,numera6620942414471956472nteger(X)),one_one_Code_integer) ).
% one_plus_numeral_commute
tff(fact_3516_one__plus__numeral__commute,axiom,
! [X: num] : aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),numera5444537566228673987atural(X)) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(numera5444537566228673987atural(X)),one_one_Code_natural) ).
% one_plus_numeral_commute
tff(fact_3517_numeral__neq__neg__one,axiom,
! [N: num] : numeral_numeral_int(N) != aa_int_int(uminus_uminus_int,one_one_int) ).
% numeral_neq_neg_one
tff(fact_3518_numeral__neq__neg__one,axiom,
! [N: num] : numeral_numeral_rat(N) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% numeral_neq_neg_one
tff(fact_3519_numeral__neq__neg__one,axiom,
! [N: num] : numera6620942414471956472nteger(N) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% numeral_neq_neg_one
tff(fact_3520_one__neq__neg__numeral,axiom,
! [N: num] : one_one_int != aa_int_int(uminus_uminus_int,numeral_numeral_int(N)) ).
% one_neq_neg_numeral
tff(fact_3521_one__neq__neg__numeral,axiom,
! [N: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N)) ).
% one_neq_neg_numeral
tff(fact_3522_one__neq__neg__numeral,axiom,
! [N: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) ).
% one_neq_neg_numeral
tff(fact_3523_neg__numeral__le__one,axiom,
! [M2: num] : aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M2))),one_one_rat) ).
% neg_numeral_le_one
tff(fact_3524_neg__numeral__le__one,axiom,
! [M2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),one_one_Code_integer) ).
% neg_numeral_le_one
tff(fact_3525_neg__numeral__le__one,axiom,
! [M2: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),one_one_int) ).
% neg_numeral_le_one
tff(fact_3526_neg__one__le__numeral,axiom,
! [M2: num] : aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),numeral_numeral_rat(M2)) ).
% neg_one_le_numeral
tff(fact_3527_neg__one__le__numeral,axiom,
! [M2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M2)) ).
% neg_one_le_numeral
tff(fact_3528_neg__one__le__numeral,axiom,
! [M2: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),numeral_numeral_int(M2)) ).
% neg_one_le_numeral
tff(fact_3529_neg__numeral__le__neg__one,axiom,
! [M2: num] : aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M2))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).
% neg_numeral_le_neg_one
tff(fact_3530_neg__numeral__le__neg__one,axiom,
! [M2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% neg_numeral_le_neg_one
tff(fact_3531_neg__numeral__le__neg__one,axiom,
! [M2: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_int,one_one_int)) ).
% neg_numeral_le_neg_one
tff(fact_3532_not__numeral__le__neg__one,axiom,
! [M2: num] : ~ aa_rat_o(ord_less_eq_rat(numeral_numeral_rat(M2)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).
% not_numeral_le_neg_one
tff(fact_3533_not__numeral__le__neg__one,axiom,
! [M2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% not_numeral_le_neg_one
tff(fact_3534_not__numeral__le__neg__one,axiom,
! [M2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,one_one_int)) ).
% not_numeral_le_neg_one
tff(fact_3535_not__one__le__neg__numeral,axiom,
! [M2: num] : ~ aa_rat_o(ord_less_eq_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M2))) ).
% not_one_le_neg_numeral
tff(fact_3536_not__one__le__neg__numeral,axiom,
! [M2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))) ).
% not_one_le_neg_numeral
tff(fact_3537_not__one__le__neg__numeral,axiom,
! [M2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))) ).
% not_one_le_neg_numeral
tff(fact_3538_divide__eq__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( ( divide_divide_rat(B,C) = numeral_numeral_rat(W) )
<=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C),numeral_numeral_rat(W) = zero_zero_rat) ) ).
% divide_eq_eq_numeral(1)
tff(fact_3539_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( ( numeral_numeral_rat(W) = divide_divide_rat(B,C) )
<=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C) = B,numeral_numeral_rat(W) = zero_zero_rat) ) ).
% eq_divide_eq_numeral(1)
tff(fact_3540_neg__numeral__less__one,axiom,
! [M2: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),one_one_int) ).
% neg_numeral_less_one
tff(fact_3541_neg__numeral__less__one,axiom,
! [M2: num] : aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M2))),one_one_rat) ).
% neg_numeral_less_one
tff(fact_3542_neg__numeral__less__one,axiom,
! [M2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),one_one_Code_integer) ).
% neg_numeral_less_one
tff(fact_3543_neg__one__less__numeral,axiom,
! [M2: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),numeral_numeral_int(M2)) ).
% neg_one_less_numeral
tff(fact_3544_neg__one__less__numeral,axiom,
! [M2: num] : aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),numeral_numeral_rat(M2)) ).
% neg_one_less_numeral
tff(fact_3545_neg__one__less__numeral,axiom,
! [M2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M2)) ).
% neg_one_less_numeral
tff(fact_3546_not__numeral__less__neg__one,axiom,
! [M2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,one_one_int)) ).
% not_numeral_less_neg_one
tff(fact_3547_not__numeral__less__neg__one,axiom,
! [M2: num] : ~ aa_rat_o(ord_less_rat(numeral_numeral_rat(M2)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).
% not_numeral_less_neg_one
tff(fact_3548_not__numeral__less__neg__one,axiom,
! [M2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% not_numeral_less_neg_one
tff(fact_3549_not__one__less__neg__numeral,axiom,
! [M2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))) ).
% not_one_less_neg_numeral
tff(fact_3550_not__one__less__neg__numeral,axiom,
! [M2: num] : ~ aa_rat_o(ord_less_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M2))) ).
% not_one_less_neg_numeral
tff(fact_3551_not__one__less__neg__numeral,axiom,
! [M2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))) ).
% not_one_less_neg_numeral
tff(fact_3552_not__neg__one__less__neg__numeral,axiom,
! [M2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))) ).
% not_neg_one_less_neg_numeral
tff(fact_3553_not__neg__one__less__neg__numeral,axiom,
! [M2: num] : ~ aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M2))) ).
% not_neg_one_less_neg_numeral
tff(fact_3554_not__neg__one__less__neg__numeral,axiom,
! [M2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))) ).
% not_neg_one_less_neg_numeral
tff(fact_3555_divide__less__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( aa_rat_o(ord_less_rat(divide_divide_rat(B,C)),numeral_numeral_rat(W))
<=> $ite(
aa_rat_o(ord_less_rat(zero_zero_rat),C),
aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),
$ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),aa_rat_o(ord_less_rat(zero_zero_rat),numeral_numeral_rat(W))) ) ) ).
% divide_less_eq_numeral(1)
tff(fact_3556_less__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( aa_rat_o(ord_less_rat(numeral_numeral_rat(W)),divide_divide_rat(B,C))
<=> $ite(
aa_rat_o(ord_less_rat(zero_zero_rat),C),
aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),
$ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),aa_rat_o(ord_less_rat(numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).
% less_divide_eq_numeral(1)
tff(fact_3557_divide__eq__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( ( divide_divide_rat(B,C) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) )
<=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) = zero_zero_rat) ) ).
% divide_eq_eq_numeral(2)
tff(fact_3558_eq__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( ( aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) = divide_divide_rat(B,C) )
<=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C) = B,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)) = zero_zero_rat) ) ).
% eq_divide_eq_numeral(2)
tff(fact_3559_divide__le__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( aa_rat_o(ord_less_eq_rat(divide_divide_rat(B,C)),numeral_numeral_rat(W))
<=> $ite(
aa_rat_o(ord_less_rat(zero_zero_rat),C),
aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),
$ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),aa_rat_o(ord_less_eq_rat(zero_zero_rat),numeral_numeral_rat(W))) ) ) ).
% divide_le_eq_numeral(1)
tff(fact_3560_le__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( aa_rat_o(ord_less_eq_rat(numeral_numeral_rat(W)),divide_divide_rat(B,C))
<=> $ite(
aa_rat_o(ord_less_rat(zero_zero_rat),C),
aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),
$ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),aa_rat_o(ord_less_eq_rat(numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).
% le_divide_eq_numeral(1)
tff(fact_3561_less__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),divide_divide_rat(B,C))
<=> $ite(
aa_rat_o(ord_less_rat(zero_zero_rat),C),
aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),B),
$ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),zero_zero_rat)) ) ) ).
% less_divide_eq_numeral(2)
tff(fact_3562_divide__less__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( aa_rat_o(ord_less_rat(divide_divide_rat(B,C)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))
<=> $ite(
aa_rat_o(ord_less_rat(zero_zero_rat),C),
aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),
$ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),C)),B),aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W)))) ) ) ).
% divide_less_eq_numeral(2)
tff(fact_3563_distrib__left__NO__MATCH,axiom,
! [X: nat,Y: nat,A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).
% distrib_left_NO_MATCH
tff(fact_3564_distrib__left__NO__MATCH,axiom,
! [X: nat,Y: nat,A: rat,B: rat,C: rat] :
( aa_rat_o(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).
% distrib_left_NO_MATCH
tff(fact_3565_distrib__left__NO__MATCH,axiom,
! [X: nat,Y: nat,A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(nO_MATCH_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ) ) ).
% distrib_left_NO_MATCH
tff(fact_3566_distrib__left__NO__MATCH,axiom,
! [X: nat,Y: nat,A: int,B: int,C: int] :
( aa_int_o(nO_MATCH_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).
% distrib_left_NO_MATCH
tff(fact_3567_distrib__left__NO__MATCH,axiom,
! [X: int,Y: int,A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).
% distrib_left_NO_MATCH
tff(fact_3568_distrib__left__NO__MATCH,axiom,
! [X: int,Y: int,A: rat,B: rat,C: rat] :
( aa_rat_o(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).
% distrib_left_NO_MATCH
tff(fact_3569_distrib__left__NO__MATCH,axiom,
! [X: int,Y: int,A: nat,B: nat,C: nat] :
( aa_nat_o(nO_MATCH_int_nat(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ) ) ).
% distrib_left_NO_MATCH
tff(fact_3570_distrib__left__NO__MATCH,axiom,
! [X: int,Y: int,A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).
% distrib_left_NO_MATCH
tff(fact_3571_distrib__left__NO__MATCH,axiom,
! [X: rat,Y: rat,A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(nO_MAT243659986023775650nteger(divide_divide_rat(X,Y)),A)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).
% distrib_left_NO_MATCH
tff(fact_3572_distrib__left__NO__MATCH,axiom,
! [X: rat,Y: rat,A: rat,B: rat,C: rat] :
( aa_rat_o(nO_MATCH_rat_rat(divide_divide_rat(X,Y)),A)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).
% distrib_left_NO_MATCH
tff(fact_3573_distrib__right__NO__MATCH,axiom,
! [X: nat,Y: nat,C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).
% distrib_right_NO_MATCH
tff(fact_3574_distrib__right__NO__MATCH,axiom,
! [X: nat,Y: nat,C: rat,A: rat,B: rat] :
( aa_rat_o(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).
% distrib_right_NO_MATCH
tff(fact_3575_distrib__right__NO__MATCH,axiom,
! [X: nat,Y: nat,C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(nO_MATCH_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).
% distrib_right_NO_MATCH
tff(fact_3576_distrib__right__NO__MATCH,axiom,
! [X: nat,Y: nat,C: int,A: int,B: int] :
( aa_int_o(nO_MATCH_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).
% distrib_right_NO_MATCH
tff(fact_3577_distrib__right__NO__MATCH,axiom,
! [X: int,Y: int,C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).
% distrib_right_NO_MATCH
tff(fact_3578_distrib__right__NO__MATCH,axiom,
! [X: int,Y: int,C: rat,A: rat,B: rat] :
( aa_rat_o(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).
% distrib_right_NO_MATCH
tff(fact_3579_distrib__right__NO__MATCH,axiom,
! [X: int,Y: int,C: nat,A: nat,B: nat] :
( aa_nat_o(nO_MATCH_int_nat(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).
% distrib_right_NO_MATCH
tff(fact_3580_distrib__right__NO__MATCH,axiom,
! [X: int,Y: int,C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).
% distrib_right_NO_MATCH
tff(fact_3581_distrib__right__NO__MATCH,axiom,
! [X: rat,Y: rat,C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(nO_MAT243659986023775650nteger(divide_divide_rat(X,Y)),C)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).
% distrib_right_NO_MATCH
tff(fact_3582_distrib__right__NO__MATCH,axiom,
! [X: rat,Y: rat,C: rat,A: rat,B: rat] :
( aa_rat_o(nO_MATCH_rat_rat(divide_divide_rat(X,Y)),C)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).
% distrib_right_NO_MATCH
tff(fact_3583_left__diff__distrib__NO__MATCH,axiom,
! [X: nat,Y: nat,C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).
% left_diff_distrib_NO_MATCH
tff(fact_3584_left__diff__distrib__NO__MATCH,axiom,
! [X: nat,Y: nat,C: rat,A: rat,B: rat] :
( aa_rat_o(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).
% left_diff_distrib_NO_MATCH
tff(fact_3585_left__diff__distrib__NO__MATCH,axiom,
! [X: nat,Y: nat,C: int,A: int,B: int] :
( aa_int_o(nO_MATCH_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).
% left_diff_distrib_NO_MATCH
tff(fact_3586_left__diff__distrib__NO__MATCH,axiom,
! [X: int,Y: int,C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).
% left_diff_distrib_NO_MATCH
tff(fact_3587_left__diff__distrib__NO__MATCH,axiom,
! [X: int,Y: int,C: rat,A: rat,B: rat] :
( aa_rat_o(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).
% left_diff_distrib_NO_MATCH
tff(fact_3588_left__diff__distrib__NO__MATCH,axiom,
! [X: int,Y: int,C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).
% left_diff_distrib_NO_MATCH
tff(fact_3589_left__diff__distrib__NO__MATCH,axiom,
! [X: rat,Y: rat,C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(nO_MAT243659986023775650nteger(divide_divide_rat(X,Y)),C)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).
% left_diff_distrib_NO_MATCH
tff(fact_3590_left__diff__distrib__NO__MATCH,axiom,
! [X: rat,Y: rat,C: rat,A: rat,B: rat] :
( aa_rat_o(nO_MATCH_rat_rat(divide_divide_rat(X,Y)),C)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).
% left_diff_distrib_NO_MATCH
tff(fact_3591_left__diff__distrib__NO__MATCH,axiom,
! [X: rat,Y: rat,C: int,A: int,B: int] :
( aa_int_o(nO_MATCH_rat_int(divide_divide_rat(X,Y)),C)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).
% left_diff_distrib_NO_MATCH
tff(fact_3592_left__diff__distrib__NO__MATCH,axiom,
! [X: code_integer,Y: code_integer,C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(nO_MAT8252062027627875367nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(X),Y)),C)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)),C) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).
% left_diff_distrib_NO_MATCH
tff(fact_3593_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(V))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(W)),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(times_times_num(V,W)))),Y) ).
% semiring_norm(170)
tff(fact_3594_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(times_times_num(V,W)))),Y) ).
% semiring_norm(170)
tff(fact_3595_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(times_times_num(V,W)))),Y) ).
% semiring_norm(170)
tff(fact_3596_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(times_times_num(V,W)))),Y) ).
% semiring_norm(171)
tff(fact_3597_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(times_times_num(V,W)))),Y) ).
% semiring_norm(171)
tff(fact_3598_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(times_times_num(V,W)))),Y) ).
% semiring_norm(171)
tff(fact_3599_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(times_times_num(V,W))),Y) ).
% semiring_norm(172)
tff(fact_3600_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(times_times_num(V,W))),Y) ).
% semiring_norm(172)
tff(fact_3601_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(times_times_num(V,W))),Y) ).
% semiring_norm(172)
tff(fact_3602_numeral__times__minus__swap,axiom,
! [W: num,X: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(W)),aa_int_int(uminus_uminus_int,X)) = aa_int_int(aa_int_fun_int_int(times_times_int,X),aa_int_int(uminus_uminus_int,numeral_numeral_int(W))) ).
% numeral_times_minus_swap
tff(fact_3603_numeral__times__minus__swap,axiom,
! [W: num,X: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),aa_rat_rat(uminus_uminus_rat,X)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))) ).
% numeral_times_minus_swap
tff(fact_3604_numeral__times__minus__swap,axiom,
! [W: num,X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))) ).
% numeral_times_minus_swap
tff(fact_3605_diff__numeral__special_I6_J,axiom,
! [M2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,one_one_int)) = numeral_numeral_int(inc(M2)) ).
% diff_numeral_special(6)
tff(fact_3606_diff__numeral__special_I6_J,axiom,
! [M2: num] : minus_minus_rat(numeral_numeral_rat(M2),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = numeral_numeral_rat(inc(M2)) ).
% diff_numeral_special(6)
tff(fact_3607_diff__numeral__special_I6_J,axiom,
! [M2: num] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(inc(M2)) ).
% diff_numeral_special(6)
tff(fact_3608_diff__numeral__special_I5_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),numeral_numeral_int(N)) = aa_int_int(uminus_uminus_int,numeral_numeral_int(inc(N))) ).
% diff_numeral_special(5)
tff(fact_3609_diff__numeral__special_I5_J,axiom,
! [N: num] : minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat),numeral_numeral_rat(N)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(inc(N))) ).
% diff_numeral_special(5)
tff(fact_3610_diff__numeral__special_I5_J,axiom,
! [N: num] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(inc(N))) ).
% diff_numeral_special(5)
tff(fact_3611_add__neg__numeral__special_I6_J,axiom,
! [M2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,numeral_numeral_int(inc(M2))) ).
% add_neg_numeral_special(6)
tff(fact_3612_add__neg__numeral__special_I6_J,axiom,
! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M2))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(inc(M2))) ).
% add_neg_numeral_special(6)
tff(fact_3613_add__neg__numeral__special_I6_J,axiom,
! [M2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(inc(M2))) ).
% add_neg_numeral_special(6)
tff(fact_3614_add__neg__numeral__special_I5_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = aa_int_int(uminus_uminus_int,numeral_numeral_int(inc(N))) ).
% add_neg_numeral_special(5)
tff(fact_3615_add__neg__numeral__special_I5_J,axiom,
! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(inc(N))) ).
% add_neg_numeral_special(5)
tff(fact_3616_add__neg__numeral__special_I5_J,axiom,
! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(inc(N))) ).
% add_neg_numeral_special(5)
tff(fact_3617_numeral__inc,axiom,
! [X: num] : numeral_numeral_nat(inc(X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(X)),one_one_nat) ).
% numeral_inc
tff(fact_3618_numeral__inc,axiom,
! [X: num] : numeral_numeral_int(inc(X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(X)),one_one_int) ).
% numeral_inc
tff(fact_3619_numeral__inc,axiom,
! [X: num] : numeral_numeral_rat(inc(X)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(X)),one_one_rat) ).
% numeral_inc
tff(fact_3620_numeral__inc,axiom,
! [X: num] : numera6620942414471956472nteger(inc(X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,numera6620942414471956472nteger(X)),one_one_Code_integer) ).
% numeral_inc
tff(fact_3621_numeral__inc,axiom,
! [X: num] : numera5444537566228673987atural(inc(X)) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(numera5444537566228673987atural(X)),one_one_Code_natural) ).
% numeral_inc
tff(fact_3622_diff__numeral__special_I3_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = numeral_numeral_int(plus_plus_num(one,N)) ).
% diff_numeral_special(3)
tff(fact_3623_diff__numeral__special_I3_J,axiom,
! [N: num] : minus_minus_rat(one_one_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = numeral_numeral_rat(plus_plus_num(one,N)) ).
% diff_numeral_special(3)
tff(fact_3624_diff__numeral__special_I3_J,axiom,
! [N: num] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(plus_plus_num(one,N)) ).
% diff_numeral_special(3)
tff(fact_3625_diff__numeral__special_I4_J,axiom,
! [M2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),one_one_int) = aa_int_int(uminus_uminus_int,numeral_numeral_int(plus_plus_num(M2,one))) ).
% diff_numeral_special(4)
tff(fact_3626_diff__numeral__special_I4_J,axiom,
! [M2: num] : minus_minus_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M2)),one_one_rat) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(plus_plus_num(M2,one))) ).
% diff_numeral_special(4)
tff(fact_3627_diff__numeral__special_I4_J,axiom,
! [M2: num] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(plus_plus_num(M2,one))) ).
% diff_numeral_special(4)
tff(fact_3628_neg__numeral__less__neg__one__iff,axiom,
! [M2: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_int,one_one_int))
<=> ( M2 != one ) ) ).
% neg_numeral_less_neg_one_iff
tff(fact_3629_neg__numeral__less__neg__one__iff,axiom,
! [M2: num] :
( aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M2))),aa_rat_rat(uminus_uminus_rat,one_one_rat))
<=> ( M2 != one ) ) ).
% neg_numeral_less_neg_one_iff
tff(fact_3630_neg__numeral__less__neg__one__iff,axiom,
! [M2: num] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))
<=> ( M2 != one ) ) ).
% neg_numeral_less_neg_one_iff
tff(fact_3631_not__neg__one__le__neg__numeral__iff,axiom,
! [M2: num] :
( ~ aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M2)))
<=> ( M2 != one ) ) ).
% not_neg_one_le_neg_numeral_iff
tff(fact_3632_not__neg__one__le__neg__numeral__iff,axiom,
! [M2: num] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)))
<=> ( M2 != one ) ) ).
% not_neg_one_le_neg_numeral_iff
tff(fact_3633_not__neg__one__le__neg__numeral__iff,axiom,
! [M2: num] :
( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,numeral_numeral_int(M2)))
<=> ( M2 != one ) ) ).
% not_neg_one_le_neg_numeral_iff
tff(fact_3634_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3635_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3636_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3637_power__decreasing__iff,axiom,
! [B: rat,M2: nat,N: nat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),B)
=> ( aa_rat_o(ord_less_rat(B),one_one_rat)
=> ( aa_rat_o(ord_less_eq_rat(aa_nat_rat(power_power_rat(B),M2)),aa_nat_rat(power_power_rat(B),N))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ) ) ) ).
% power_decreasing_iff
tff(fact_3638_power__decreasing__iff,axiom,
! [B: code_natural,M2: nat,N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),one_one_Code_natural)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(B),M2)),aa_nat_Code_natural(power_7079662738309270450atural(B),N))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ) ) ) ).
% power_decreasing_iff
tff(fact_3639_power__decreasing__iff,axiom,
! [B: code_integer,M2: nat,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),M2)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ) ) ) ).
% power_decreasing_iff
tff(fact_3640_power__decreasing__iff,axiom,
! [B: nat,M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ) ) ) ).
% power_decreasing_iff
tff(fact_3641_power__decreasing__iff,axiom,
! [B: int,M2: nat,N: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),M2)),aa_nat_int(power_power_int(B),N))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ) ) ) ).
% power_decreasing_iff
tff(fact_3642_add_Ogroup__axioms,axiom,
group_int(plus_plus_int,zero_zero_int,uminus_uminus_int) ).
% add.group_axioms
tff(fact_3643_add_Ogroup__axioms,axiom,
group_rat(plus_plus_rat,zero_zero_rat,uminus_uminus_rat) ).
% add.group_axioms
tff(fact_3644_add_Ogroup__axioms,axiom,
group_Code_integer(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger,uminus1351360451143612070nteger) ).
% add.group_axioms
tff(fact_3645_inverse__eq__divide__neg__numeral,axiom,
! [W: num] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))) = divide_divide_rat(one_one_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(W))) ).
% inverse_eq_divide_neg_numeral
tff(fact_3646_power__minus_H,axiom,
! [X: int,N: nat] :
( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,one_one_int),X)
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,X)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(power_power_int(X),N)) ) ) ).
% power_minus'
tff(fact_3647_power__minus_H,axiom,
! [X: rat,N: nat] :
( aa_rat_o(nO_MATCH_rat_rat(one_one_rat),X)
=> ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,X)),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(X),N)) ) ) ).
% power_minus'
tff(fact_3648_power__minus_H,axiom,
! [X: code_integer,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(nO_MAT8252062027627875367nteger,one_one_Code_integer),X)
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(X),N)) ) ) ).
% power_minus'
tff(fact_3649_power__one,axiom,
! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),N) = one_one_Code_integer ).
% power_one
tff(fact_3650_power__one,axiom,
! [N: nat] : aa_nat_assn(power_power_assn(one_one_assn),N) = one_one_assn ).
% power_one
tff(fact_3651_power__one,axiom,
! [N: nat] : aa_nat_rat(power_power_rat(one_one_rat),N) = one_one_rat ).
% power_one
tff(fact_3652_power__one,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),N) = one_one_nat ).
% power_one
tff(fact_3653_power__one,axiom,
! [N: nat] : aa_nat_int(power_power_int(one_one_int),N) = one_one_int ).
% power_one
tff(fact_3654_inverse__mult__distrib,axiom,
! [A: rat,B: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) ).
% inverse_mult_distrib
tff(fact_3655_inverse__eq__1__iff,axiom,
! [X: rat] :
( ( aa_rat_rat(inverse_inverse_rat,X) = one_one_rat )
<=> ( X = one_one_rat ) ) ).
% inverse_eq_1_iff
tff(fact_3656_inverse__1,axiom,
aa_rat_rat(inverse_inverse_rat,one_one_rat) = one_one_rat ).
% inverse_1
tff(fact_3657_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat = numeral_numeral_nat(N) )
<=> ( one = N ) ) ).
% one_eq_numeral_iff
tff(fact_3658_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int = numeral_numeral_int(N) )
<=> ( one = N ) ) ).
% one_eq_numeral_iff
tff(fact_3659_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_rat = numeral_numeral_rat(N) )
<=> ( one = N ) ) ).
% one_eq_numeral_iff
tff(fact_3660_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_Code_integer = numera6620942414471956472nteger(N) )
<=> ( one = N ) ) ).
% one_eq_numeral_iff
tff(fact_3661_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_Code_natural = numera5444537566228673987atural(N) )
<=> ( one = N ) ) ).
% one_eq_numeral_iff
tff(fact_3662_numeral__eq__one__iff,axiom,
! [N: num] :
( ( numeral_numeral_nat(N) = one_one_nat )
<=> ( N = one ) ) ).
% numeral_eq_one_iff
tff(fact_3663_numeral__eq__one__iff,axiom,
! [N: num] :
( ( numeral_numeral_int(N) = one_one_int )
<=> ( N = one ) ) ).
% numeral_eq_one_iff
tff(fact_3664_numeral__eq__one__iff,axiom,
! [N: num] :
( ( numeral_numeral_rat(N) = one_one_rat )
<=> ( N = one ) ) ).
% numeral_eq_one_iff
tff(fact_3665_numeral__eq__one__iff,axiom,
! [N: num] :
( ( numera6620942414471956472nteger(N) = one_one_Code_integer )
<=> ( N = one ) ) ).
% numeral_eq_one_iff
tff(fact_3666_numeral__eq__one__iff,axiom,
! [N: num] :
( ( numera5444537566228673987atural(N) = one_one_Code_natural )
<=> ( N = one ) ) ).
% numeral_eq_one_iff
tff(fact_3667_power__inject__exp,axiom,
! [A: rat,M2: nat,N: nat] :
( aa_rat_o(ord_less_rat(one_one_rat),A)
=> ( ( aa_nat_rat(power_power_rat(A),M2) = aa_nat_rat(power_power_rat(A),N) )
<=> ( M2 = N ) ) ) ).
% power_inject_exp
tff(fact_3668_power__inject__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) )
<=> ( M2 = N ) ) ) ).
% power_inject_exp
tff(fact_3669_power__inject__exp,axiom,
! [A: int,M2: nat,N: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
=> ( ( aa_nat_int(power_power_int(A),M2) = aa_nat_int(power_power_int(A),N) )
<=> ( M2 = N ) ) ) ).
% power_inject_exp
tff(fact_3670_power__inject__exp,axiom,
! [A: code_integer,M2: nat,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),M2) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) )
<=> ( M2 = N ) ) ) ).
% power_inject_exp
tff(fact_3671_power__inject__exp,axiom,
! [A: code_natural,M2: nat,N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A)
=> ( ( aa_nat_Code_natural(power_7079662738309270450atural(A),M2) = aa_nat_Code_natural(power_7079662738309270450atural(A),N) )
<=> ( M2 = N ) ) ) ).
% power_inject_exp
tff(fact_3672_power__strict__increasing__iff,axiom,
! [B: rat,X: nat,Y: nat] :
( aa_rat_o(ord_less_rat(one_one_rat),B)
=> ( aa_rat_o(ord_less_rat(aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).
% power_strict_increasing_iff
tff(fact_3673_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).
% power_strict_increasing_iff
tff(fact_3674_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).
% power_strict_increasing_iff
tff(fact_3675_power__strict__increasing__iff,axiom,
! [B: code_integer,X: nat,Y: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),X)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).
% power_strict_increasing_iff
tff(fact_3676_power__strict__increasing__iff,axiom,
! [B: code_natural,X: nat,Y: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(B),X)),aa_nat_Code_natural(power_7079662738309270450atural(B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).
% power_strict_increasing_iff
tff(fact_3677_mod__mult__self2__is__0,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = zero_zero_nat ).
% mod_mult_self2_is_0
tff(fact_3678_mod__mult__self2__is__0,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = zero_zero_int ).
% mod_mult_self2_is_0
tff(fact_3679_mod__mult__self2__is__0,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),B) = zero_z3403309356797280102nteger ).
% mod_mult_self2_is_0
tff(fact_3680_mod__mult__self2__is__0,axiom,
! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),B) = zero_z2226904508553997617atural ).
% mod_mult_self2_is_0
tff(fact_3681_mod__mult__self1__is__0,axiom,
! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),B) = zero_zero_nat ).
% mod_mult_self1_is_0
tff(fact_3682_mod__mult__self1__is__0,axiom,
! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),B) = zero_zero_int ).
% mod_mult_self1_is_0
tff(fact_3683_mod__mult__self1__is__0,axiom,
! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),B) = zero_z3403309356797280102nteger ).
% mod_mult_self1_is_0
tff(fact_3684_mod__mult__self1__is__0,axiom,
! [B: code_natural,A: code_natural] : aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),A)),B) = zero_z2226904508553997617atural ).
% mod_mult_self1_is_0
tff(fact_3685_mod__by__1,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).
% mod_by_1
tff(fact_3686_mod__by__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).
% mod_by_1
tff(fact_3687_mod__by__1,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),one_one_Code_integer) = zero_z3403309356797280102nteger ).
% mod_by_1
tff(fact_3688_mod__by__1,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),one_one_Code_natural) = zero_z2226904508553997617atural ).
% mod_by_1
tff(fact_3689_bits__mod__by__1,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).
% bits_mod_by_1
tff(fact_3690_bits__mod__by__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).
% bits_mod_by_1
tff(fact_3691_bits__mod__by__1,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),one_one_Code_integer) = zero_z3403309356797280102nteger ).
% bits_mod_by_1
tff(fact_3692_bits__mod__by__1,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),one_one_Code_natural) = zero_z2226904508553997617atural ).
% bits_mod_by_1
tff(fact_3693_mod__mult__self4,axiom,
! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).
% mod_mult_self4
tff(fact_3694_mod__mult__self4,axiom,
! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% mod_mult_self4
tff(fact_3695_mod__mult__self4,axiom,
! [B: code_integer,C: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).
% mod_mult_self4
tff(fact_3696_mod__mult__self4,axiom,
! [B: code_natural,C: code_natural,A: code_natural] : aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)),A)),B) = aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B) ).
% mod_mult_self4
tff(fact_3697_mod__mult__self3,axiom,
! [C: nat,B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).
% mod_mult_self3
tff(fact_3698_mod__mult__self3,axiom,
! [C: int,B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% mod_mult_self3
tff(fact_3699_mod__mult__self3,axiom,
! [C: code_integer,B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),A)),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).
% mod_mult_self3
tff(fact_3700_mod__mult__self3,axiom,
! [C: code_natural,B: code_natural,A: code_natural] : aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B)),A)),B) = aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B) ).
% mod_mult_self3
tff(fact_3701_mod__mult__self2,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).
% mod_mult_self2
tff(fact_3702_mod__mult__self2,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% mod_mult_self2
tff(fact_3703_mod__mult__self2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).
% mod_mult_self2
tff(fact_3704_mod__mult__self2,axiom,
! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C))),B) = aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B) ).
% mod_mult_self2
tff(fact_3705_mod__mult__self1,axiom,
! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).
% mod_mult_self1
tff(fact_3706_mod__mult__self1,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% mod_mult_self1
tff(fact_3707_mod__mult__self1,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).
% mod_mult_self1
tff(fact_3708_mod__mult__self1,axiom,
! [A: code_natural,C: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B))),B) = aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B) ).
% mod_mult_self1
tff(fact_3709_power__add__numeral,axiom,
! [A: code_integer,M2: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(M2))),aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(N))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(plus_plus_num(M2,N))) ).
% power_add_numeral
tff(fact_3710_power__add__numeral,axiom,
! [A: assn,M2: num,N: num] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),numeral_numeral_nat(M2))),aa_nat_assn(power_power_assn(A),numeral_numeral_nat(N))) = aa_nat_assn(power_power_assn(A),numeral_numeral_nat(plus_plus_num(M2,N))) ).
% power_add_numeral
tff(fact_3711_power__add__numeral,axiom,
! [A: rat,M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),numeral_numeral_nat(M2))),aa_nat_rat(power_power_rat(A),numeral_numeral_nat(N))) = aa_nat_rat(power_power_rat(A),numeral_numeral_nat(plus_plus_num(M2,N))) ).
% power_add_numeral
tff(fact_3712_power__add__numeral,axiom,
! [A: nat,M2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(plus_plus_num(M2,N))) ).
% power_add_numeral
tff(fact_3713_power__add__numeral,axiom,
! [A: int,M2: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),numeral_numeral_nat(M2))),aa_nat_int(power_power_int(A),numeral_numeral_nat(N))) = aa_nat_int(power_power_int(A),numeral_numeral_nat(plus_plus_num(M2,N))) ).
% power_add_numeral
tff(fact_3714_power__add__numeral2,axiom,
! [A: code_integer,M2: num,N: num,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(M2))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(N))),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(plus_plus_num(M2,N)))),B) ).
% power_add_numeral2
tff(fact_3715_power__add__numeral2,axiom,
! [A: assn,M2: num,N: num,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),numeral_numeral_nat(M2))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),numeral_numeral_nat(N))),B)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),numeral_numeral_nat(plus_plus_num(M2,N)))),B) ).
% power_add_numeral2
tff(fact_3716_power__add__numeral2,axiom,
! [A: rat,M2: num,N: num,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),numeral_numeral_nat(M2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),numeral_numeral_nat(N))),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),numeral_numeral_nat(plus_plus_num(M2,N)))),B) ).
% power_add_numeral2
tff(fact_3717_power__add__numeral2,axiom,
! [A: nat,M2: num,N: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(M2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(N))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(plus_plus_num(M2,N)))),B) ).
% power_add_numeral2
tff(fact_3718_power__add__numeral2,axiom,
! [A: int,M2: num,N: num,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),numeral_numeral_nat(M2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),numeral_numeral_nat(N))),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),numeral_numeral_nat(plus_plus_num(M2,N)))),B) ).
% power_add_numeral2
tff(fact_3719_numeral__le__one__iff,axiom,
! [N: num] :
( aa_rat_o(ord_less_eq_rat(numeral_numeral_rat(N)),one_one_rat)
<=> ord_less_eq_num(N,one) ) ).
% numeral_le_one_iff
tff(fact_3720_numeral__le__one__iff,axiom,
! [N: num] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,numera5444537566228673987atural(N)),one_one_Code_natural)
<=> ord_less_eq_num(N,one) ) ).
% numeral_le_one_iff
tff(fact_3721_numeral__le__one__iff,axiom,
! [N: num] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(N)),one_one_Code_integer)
<=> ord_less_eq_num(N,one) ) ).
% numeral_le_one_iff
tff(fact_3722_numeral__le__one__iff,axiom,
! [N: num] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(N)),one_one_nat)
<=> ord_less_eq_num(N,one) ) ).
% numeral_le_one_iff
tff(fact_3723_numeral__le__one__iff,axiom,
! [N: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(N)),one_one_int)
<=> ord_less_eq_num(N,one) ) ).
% numeral_le_one_iff
tff(fact_3724_one__less__numeral__iff,axiom,
! [N: num] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),numeral_numeral_nat(N))
<=> ord_less_num(one,N) ) ).
% one_less_numeral_iff
tff(fact_3725_one__less__numeral__iff,axiom,
! [N: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),numeral_numeral_int(N))
<=> ord_less_num(one,N) ) ).
% one_less_numeral_iff
tff(fact_3726_one__less__numeral__iff,axiom,
! [N: num] :
( aa_rat_o(ord_less_rat(one_one_rat),numeral_numeral_rat(N))
<=> ord_less_num(one,N) ) ).
% one_less_numeral_iff
tff(fact_3727_one__less__numeral__iff,axiom,
! [N: num] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),numera6620942414471956472nteger(N))
<=> ord_less_num(one,N) ) ).
% one_less_numeral_iff
tff(fact_3728_one__less__numeral__iff,axiom,
! [N: num] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),numera5444537566228673987atural(N))
<=> ord_less_num(one,N) ) ).
% one_less_numeral_iff
tff(fact_3729_power__strict__decreasing__iff,axiom,
! [B: rat,M2: nat,N: nat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),B)
=> ( aa_rat_o(ord_less_rat(B),one_one_rat)
=> ( aa_rat_o(ord_less_rat(aa_nat_rat(power_power_rat(B),M2)),aa_nat_rat(power_power_rat(B),N))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_3730_power__strict__decreasing__iff,axiom,
! [B: nat,M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_3731_power__strict__decreasing__iff,axiom,
! [B: int,M2: nat,N: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),M2)),aa_nat_int(power_power_int(B),N))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_3732_power__strict__decreasing__iff,axiom,
! [B: code_integer,M2: nat,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),M2)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_3733_power__strict__decreasing__iff,axiom,
! [B: code_natural,M2: nat,N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),one_one_Code_natural)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(B),M2)),aa_nat_Code_natural(power_7079662738309270450atural(B),N))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_3734_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,numeral_numeral_int(N)) )
<=> ( N = one ) ) ).
% neg_one_eq_numeral_iff
tff(fact_3735_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N)) )
<=> ( N = one ) ) ).
% neg_one_eq_numeral_iff
tff(fact_3736_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) )
<=> ( N = one ) ) ).
% neg_one_eq_numeral_iff
tff(fact_3737_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( aa_int_int(uminus_uminus_int,numeral_numeral_int(N)) = aa_int_int(uminus_uminus_int,one_one_int) )
<=> ( N = one ) ) ).
% numeral_eq_neg_one_iff
tff(fact_3738_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
<=> ( N = one ) ) ).
% numeral_eq_neg_one_iff
tff(fact_3739_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
<=> ( N = one ) ) ).
% numeral_eq_neg_one_iff
tff(fact_3740_mod__minus1__right,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).
% mod_minus1_right
tff(fact_3741_mod__minus1__right,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).
% mod_minus1_right
tff(fact_3742_minus__one__mult__self,axiom,
! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)) = one_one_int ).
% minus_one_mult_self
tff(fact_3743_minus__one__mult__self,axiom,
! [N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)) = one_one_rat ).
% minus_one_mult_self
tff(fact_3744_minus__one__mult__self,axiom,
! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)) = one_one_Code_integer ).
% minus_one_mult_self
tff(fact_3745_left__minus__one__mult__self,axiom,
! [N: nat,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),A)) = A ).
% left_minus_one_mult_self
tff(fact_3746_left__minus__one__mult__self,axiom,
! [N: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),A)) = A ).
% left_minus_one_mult_self
tff(fact_3747_left__minus__one__mult__self,axiom,
! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),A)) = A ).
% left_minus_one_mult_self
tff(fact_3748_left__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).
% left_inverse
tff(fact_3749_right__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(inverse_inverse_rat,A)) = one_one_rat ) ) ).
% right_inverse
tff(fact_3750_inverse__eq__divide__numeral,axiom,
! [W: num] : aa_rat_rat(inverse_inverse_rat,numeral_numeral_rat(W)) = divide_divide_rat(one_one_rat,numeral_numeral_rat(W)) ).
% inverse_eq_divide_numeral
tff(fact_3751_power__increasing__iff,axiom,
! [B: rat,X: nat,Y: nat] :
( aa_rat_o(ord_less_rat(one_one_rat),B)
=> ( aa_rat_o(ord_less_eq_rat(aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).
% power_increasing_iff
tff(fact_3752_power__increasing__iff,axiom,
! [B: code_natural,X: nat,Y: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(B),X)),aa_nat_Code_natural(power_7079662738309270450atural(B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).
% power_increasing_iff
tff(fact_3753_power__increasing__iff,axiom,
! [B: code_integer,X: nat,Y: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),X)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).
% power_increasing_iff
tff(fact_3754_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).
% power_increasing_iff
tff(fact_3755_power__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).
% power_increasing_iff
tff(fact_3756_one__plus__numeral,axiom,
! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),numeral_numeral_nat(N)) = numeral_numeral_nat(plus_plus_num(one,N)) ).
% one_plus_numeral
tff(fact_3757_one__plus__numeral,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),numeral_numeral_int(N)) = numeral_numeral_int(plus_plus_num(one,N)) ).
% one_plus_numeral
tff(fact_3758_one__plus__numeral,axiom,
! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),numeral_numeral_rat(N)) = numeral_numeral_rat(plus_plus_num(one,N)) ).
% one_plus_numeral
tff(fact_3759_one__plus__numeral,axiom,
! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),numera6620942414471956472nteger(N)) = numera6620942414471956472nteger(plus_plus_num(one,N)) ).
% one_plus_numeral
tff(fact_3760_one__plus__numeral,axiom,
! [N: num] : aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),numera5444537566228673987atural(N)) = numera5444537566228673987atural(plus_plus_num(one,N)) ).
% one_plus_numeral
tff(fact_3761_numeral__plus__one,axiom,
! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(N)),one_one_nat) = numeral_numeral_nat(plus_plus_num(N,one)) ).
% numeral_plus_one
tff(fact_3762_numeral__plus__one,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(N)),one_one_int) = numeral_numeral_int(plus_plus_num(N,one)) ).
% numeral_plus_one
tff(fact_3763_numeral__plus__one,axiom,
! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(N)),one_one_rat) = numeral_numeral_rat(plus_plus_num(N,one)) ).
% numeral_plus_one
tff(fact_3764_numeral__plus__one,axiom,
! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,numera6620942414471956472nteger(N)),one_one_Code_integer) = numera6620942414471956472nteger(plus_plus_num(N,one)) ).
% numeral_plus_one
tff(fact_3765_numeral__plus__one,axiom,
! [N: num] : aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(numera5444537566228673987atural(N)),one_one_Code_natural) = numera5444537566228673987atural(plus_plus_num(N,one)) ).
% numeral_plus_one
tff(fact_3766_mult__commute__imp__mult__inverse__commute,axiom,
! [Y: rat,X: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y) )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,Y)),X) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),aa_rat_rat(inverse_inverse_rat,Y)) ) ) ).
% mult_commute_imp_mult_inverse_commute
tff(fact_3767_mod__mult__eq,axiom,
! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).
% mod_mult_eq
tff(fact_3768_mod__mult__eq,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).
% mod_mult_eq
tff(fact_3769_mod__mult__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C))),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) ).
% mod_mult_eq
tff(fact_3770_mod__mult__eq,axiom,
! [A: code_natural,C: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),C)),aa_Cod4272041906116450446atural(modulo8411746178871703098atural(B),C))),C) = aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),C) ).
% mod_mult_eq
tff(fact_3771_mod__mult__cong,axiom,
! [A: nat,C: nat,A2: nat,B: nat,B2: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C) )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C) ) ) ) ).
% mod_mult_cong
tff(fact_3772_mod__mult__cong,axiom,
! [A: int,C: int,A2: int,B: int,B2: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C) )
=> ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C) )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C) ) ) ) ).
% mod_mult_cong
tff(fact_3773_mod__mult__cong,axiom,
! [A: code_integer,C: code_integer,A2: code_integer,B: code_integer,B2: code_integer] :
( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C) )
=> ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C) )
=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),C) ) ) ) ).
% mod_mult_cong
tff(fact_3774_mod__mult__cong,axiom,
! [A: code_natural,C: code_natural,A2: code_natural,B: code_natural,B2: code_natural] :
( ( aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),C) = aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A2),C) )
=> ( ( aa_Cod4272041906116450446atural(modulo8411746178871703098atural(B),C) = aa_Cod4272041906116450446atural(modulo8411746178871703098atural(B2),C) )
=> ( aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),C) = aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A2),B2)),C) ) ) ) ).
% mod_mult_cong
tff(fact_3775_mod__mult__mult2,axiom,
! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) ).
% mod_mult_mult2
tff(fact_3776_mod__mult__mult2,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),C) ).
% mod_mult_mult2
tff(fact_3777_mod__mult__mult2,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)),C) ).
% mod_mult_mult2
tff(fact_3778_mod__mult__mult2,axiom,
! [A: code_natural,C: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B)),C) ).
% mod_mult_mult2
tff(fact_3779_mult__mod__right,axiom,
! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ).
% mult_mod_right
tff(fact_3780_mult__mod__right,axiom,
! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ).
% mult_mod_right
tff(fact_3781_mult__mod__right,axiom,
! [C: code_integer,A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ).
% mult_mod_right
tff(fact_3782_mult__mod__right,axiom,
! [C: code_natural,A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B)) = aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B)) ).
% mult_mod_right
tff(fact_3783_mod__mult__left__eq,axiom,
! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).
% mod_mult_left_eq
tff(fact_3784_mod__mult__left__eq,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).
% mod_mult_left_eq
tff(fact_3785_mod__mult__left__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C)),B)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) ).
% mod_mult_left_eq
tff(fact_3786_mod__mult__left__eq,axiom,
! [A: code_natural,C: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),C)),B)),C) = aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),C) ).
% mod_mult_left_eq
tff(fact_3787_mod__mult__right__eq,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).
% mod_mult_right_eq
tff(fact_3788_mod__mult__right__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).
% mod_mult_right_eq
tff(fact_3789_mod__mult__right__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C))),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) ).
% mod_mult_right_eq
tff(fact_3790_mod__mult__right__eq,axiom,
! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),aa_Cod4272041906116450446atural(modulo8411746178871703098atural(B),C))),C) = aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),C) ).
% mod_mult_right_eq
tff(fact_3791_power__commutes,axiom,
! [A: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).
% power_commutes
tff(fact_3792_power__commutes,axiom,
! [A: assn,N: nat] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),N)),A) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_nat_assn(power_power_assn(A),N)) ).
% power_commutes
tff(fact_3793_power__commutes,axiom,
! [A: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),N)),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N)) ).
% power_commutes
tff(fact_3794_power__commutes,axiom,
! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).
% power_commutes
tff(fact_3795_power__commutes,axiom,
! [A: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)) ).
% power_commutes
tff(fact_3796_power__mult__distrib,axiom,
! [A: code_integer,B: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)) ).
% power_mult_distrib
tff(fact_3797_power__mult__distrib,axiom,
! [A: assn,B: assn,N: nat] : aa_nat_assn(power_power_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),N) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),N)),aa_nat_assn(power_power_assn(B),N)) ).
% power_mult_distrib
tff(fact_3798_power__mult__distrib,axiom,
! [A: rat,B: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)) ).
% power_mult_distrib
tff(fact_3799_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)) ).
% power_mult_distrib
tff(fact_3800_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)) ).
% power_mult_distrib
tff(fact_3801_power__commuting__commutes,axiom,
! [X: code_integer,Y: code_integer,N: nat] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X) )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),N)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_nat_Code_integer(power_8256067586552552935nteger(X),N)) ) ) ).
% power_commuting_commutes
tff(fact_3802_power__commuting__commutes,axiom,
! [X: assn,Y: assn,N: nat] :
( ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Y),X) )
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(X),N)),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Y),aa_nat_assn(power_power_assn(X),N)) ) ) ).
% power_commuting_commutes
tff(fact_3803_power__commuting__commutes,axiom,
! [X: rat,Y: rat,N: nat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X) )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),N)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(power_power_rat(X),N)) ) ) ).
% power_commuting_commutes
tff(fact_3804_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),X) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)) ) ) ).
% power_commuting_commutes
tff(fact_3805_power__commuting__commutes,axiom,
! [X: int,Y: int,N: nat] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),X) )
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),N)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(power_power_int(X),N)) ) ) ).
% power_commuting_commutes
tff(fact_3806_power__minus__mult,axiom,
! [N: nat,A: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) ) ) ).
% power_minus_mult
tff(fact_3807_power__minus__mult,axiom,
! [N: nat,A: assn] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_assn(power_power_assn(A),N) ) ) ).
% power_minus_mult
tff(fact_3808_power__minus__mult,axiom,
! [N: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),N) ) ) ).
% power_minus_mult
tff(fact_3809_power__minus__mult,axiom,
! [N: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) ) ) ).
% power_minus_mult
tff(fact_3810_power__minus__mult,axiom,
! [N: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_int(power_power_int(A),N) ) ) ).
% power_minus_mult
tff(fact_3811_power__add,axiom,
! [A: code_integer,M2: nat,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M2)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).
% power_add
tff(fact_3812_power__add,axiom,
! [A: assn,M2: nat,N: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),M2)),aa_nat_assn(power_power_assn(A),N)) ).
% power_add
tff(fact_3813_power__add,axiom,
! [A: rat,M2: nat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),M2)),aa_nat_rat(power_power_rat(A),N)) ).
% power_add
tff(fact_3814_power__add,axiom,
! [A: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).
% power_add
tff(fact_3815_power__add,axiom,
! [A: int,M2: nat,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N)) ).
% power_add
tff(fact_3816_power__0,axiom,
! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),zero_zero_nat) = one_one_Code_integer ).
% power_0
tff(fact_3817_power__0,axiom,
! [A: assn] : aa_nat_assn(power_power_assn(A),zero_zero_nat) = one_one_assn ).
% power_0
tff(fact_3818_power__0,axiom,
! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).
% power_0
tff(fact_3819_power__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),zero_zero_nat) = one_one_nat ).
% power_0
tff(fact_3820_power__0,axiom,
! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).
% power_0
tff(fact_3821_nonzero__inverse__mult__distrib,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ) ).
% nonzero_inverse_mult_distrib
tff(fact_3822_inverse__unique,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = one_one_rat )
=> ( aa_rat_rat(inverse_inverse_rat,A) = B ) ) ).
% inverse_unique
tff(fact_3823_field__class_Ofield__divide__inverse,axiom,
! [A: rat,B: rat] : divide_divide_rat(A,B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(inverse_inverse_rat,B)) ).
% field_class.field_divide_inverse
tff(fact_3824_divide__inverse,axiom,
! [A: rat,B: rat] : divide_divide_rat(A,B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(inverse_inverse_rat,B)) ).
% divide_inverse
tff(fact_3825_divide__inverse__commute,axiom,
! [A: rat,B: rat] : divide_divide_rat(A,B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,B)),A) ).
% divide_inverse_commute
tff(fact_3826_inverse__eq__divide,axiom,
! [A: rat] : aa_rat_rat(inverse_inverse_rat,A) = divide_divide_rat(one_one_rat,A) ).
% inverse_eq_divide
tff(fact_3827_mod__eqE,axiom,
! [A: int,C: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
=> ~ ! [D: int] : B != aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),D)) ) ).
% mod_eqE
tff(fact_3828_mod__eqE,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C) )
=> ~ ! [D: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D)) ) ).
% mod_eqE
tff(fact_3829_self__le__power,axiom,
! [A: rat,N: nat] :
( aa_rat_o(ord_less_eq_rat(one_one_rat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> aa_rat_o(ord_less_eq_rat(A),aa_nat_rat(power_power_rat(A),N)) ) ) ).
% self_le_power
tff(fact_3830_self__le__power,axiom,
! [A: code_integer,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ).
% self_le_power
tff(fact_3831_self__le__power,axiom,
! [A: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ).
% self_le_power
tff(fact_3832_self__le__power,axiom,
! [A: int,N: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(A),N)) ) ) ).
% self_le_power
tff(fact_3833_one__le__power,axiom,
! [A: rat,N: nat] :
( aa_rat_o(ord_less_eq_rat(one_one_rat),A)
=> aa_rat_o(ord_less_eq_rat(one_one_rat),aa_nat_rat(power_power_rat(A),N)) ) ).
% one_le_power
tff(fact_3834_one__le__power,axiom,
! [A: code_integer,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ).
% one_le_power
tff(fact_3835_one__le__power,axiom,
! [A: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ).
% one_le_power
tff(fact_3836_one__le__power,axiom,
! [A: int,N: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),N)) ) ).
% one_le_power
tff(fact_3837_power__increasing,axiom,
! [N: nat,N3: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
=> ( aa_rat_o(ord_less_eq_rat(one_one_rat),A)
=> aa_rat_o(ord_less_eq_rat(aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N3)) ) ) ).
% power_increasing
tff(fact_3838_power__increasing,axiom,
! [N: nat,N3: nat,A: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N3)) ) ) ).
% power_increasing
tff(fact_3839_power__increasing,axiom,
! [N: nat,N3: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)) ) ) ).
% power_increasing
tff(fact_3840_power__increasing,axiom,
! [N: nat,N3: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N3)) ) ) ).
% power_increasing
tff(fact_3841_power__0__left,axiom,
! [N: nat] :
aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N) = $ite(N = zero_zero_nat,one_one_Code_integer,zero_z3403309356797280102nteger) ).
% power_0_left
tff(fact_3842_power__0__left,axiom,
! [N: nat] :
aa_nat_rat(power_power_rat(zero_zero_rat),N) = $ite(N = zero_zero_nat,one_one_rat,zero_zero_rat) ).
% power_0_left
tff(fact_3843_power__0__left,axiom,
! [N: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = $ite(N = zero_zero_nat,one_one_nat,zero_zero_nat) ).
% power_0_left
tff(fact_3844_power__0__left,axiom,
! [N: nat] :
aa_nat_int(power_power_int(zero_zero_int),N) = $ite(N = zero_zero_nat,one_one_int,zero_zero_int) ).
% power_0_left
tff(fact_3845_one__less__power,axiom,
! [A: rat,N: nat] :
( aa_rat_o(ord_less_rat(one_one_rat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> aa_rat_o(ord_less_rat(one_one_rat),aa_nat_rat(power_power_rat(A),N)) ) ) ).
% one_less_power
tff(fact_3846_one__less__power,axiom,
! [A: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ).
% one_less_power
tff(fact_3847_one__less__power,axiom,
! [A: int,N: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),N)) ) ) ).
% one_less_power
tff(fact_3848_one__less__power,axiom,
! [A: code_integer,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ).
% one_less_power
tff(fact_3849_one__less__power,axiom,
! [A: code_natural,N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(A),N)) ) ) ).
% one_less_power
tff(fact_3850_power__less__imp__less__exp,axiom,
! [A: rat,M2: nat,N: nat] :
( aa_rat_o(ord_less_rat(one_one_rat),A)
=> ( aa_rat_o(ord_less_rat(aa_nat_rat(power_power_rat(A),M2)),aa_nat_rat(power_power_rat(A),N))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N) ) ) ).
% power_less_imp_less_exp
tff(fact_3851_power__less__imp__less__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N) ) ) ).
% power_less_imp_less_exp
tff(fact_3852_power__less__imp__less__exp,axiom,
! [A: int,M2: nat,N: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N) ) ) ).
% power_less_imp_less_exp
tff(fact_3853_power__less__imp__less__exp,axiom,
! [A: code_integer,M2: nat,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M2)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N) ) ) ).
% power_less_imp_less_exp
tff(fact_3854_power__less__imp__less__exp,axiom,
! [A: code_natural,M2: nat,N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(A),M2)),aa_nat_Code_natural(power_7079662738309270450atural(A),N))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N) ) ) ).
% power_less_imp_less_exp
tff(fact_3855_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N3)
=> ( aa_rat_o(ord_less_rat(one_one_rat),A)
=> aa_rat_o(ord_less_rat(aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N3)) ) ) ).
% power_strict_increasing
tff(fact_3856_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)) ) ) ).
% power_strict_increasing
tff(fact_3857_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N3)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N3)) ) ) ).
% power_strict_increasing
tff(fact_3858_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N3)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N3)) ) ) ).
% power_strict_increasing
tff(fact_3859_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: code_natural] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N3)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(A),N)),aa_nat_Code_natural(power_7079662738309270450atural(A),N3)) ) ) ).
% power_strict_increasing
tff(fact_3860_power__eq__if,axiom,
! [P5: code_integer,M2: nat] :
aa_nat_Code_integer(power_8256067586552552935nteger(P5),M2) = $ite(M2 = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,P5),aa_nat_Code_integer(power_8256067586552552935nteger(P5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).
% power_eq_if
tff(fact_3861_power__eq__if,axiom,
! [P5: assn,M2: nat] :
aa_nat_assn(power_power_assn(P5),M2) = $ite(M2 = zero_zero_nat,one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P5),aa_nat_assn(power_power_assn(P5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).
% power_eq_if
tff(fact_3862_power__eq__if,axiom,
! [P5: rat,M2: nat] :
aa_nat_rat(power_power_rat(P5),M2) = $ite(M2 = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P5),aa_nat_rat(power_power_rat(P5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).
% power_eq_if
tff(fact_3863_power__eq__if,axiom,
! [P5: nat,M2: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P5),M2) = $ite(M2 = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).
% power_eq_if
tff(fact_3864_power__eq__if,axiom,
! [P5: int,M2: nat] :
aa_nat_int(power_power_int(P5),M2) = $ite(M2 = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P5),aa_nat_int(power_power_int(P5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).
% power_eq_if
tff(fact_3865_left__right__inverse__power,axiom,
! [X: code_integer,Y: code_integer,N: nat] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y) = one_one_Code_integer )
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),N)) = one_one_Code_integer ) ) ).
% left_right_inverse_power
tff(fact_3866_left__right__inverse__power,axiom,
! [X: assn,Y: assn,N: nat] :
( ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),Y) = one_one_assn )
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(X),N)),aa_nat_assn(power_power_assn(Y),N)) = one_one_assn ) ) ).
% left_right_inverse_power
tff(fact_3867_left__right__inverse__power,axiom,
! [X: rat,Y: rat,N: nat] :
( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y) = one_one_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(Y),N)) = one_one_rat ) ) ).
% left_right_inverse_power
tff(fact_3868_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = one_one_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N)) = one_one_nat ) ) ).
% left_right_inverse_power
tff(fact_3869_left__right__inverse__power,axiom,
! [X: int,Y: int,N: nat] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y) = one_one_int )
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N)) = one_one_int ) ) ).
% left_right_inverse_power
tff(fact_3870_power__one__over,axiom,
! [A: rat,N: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(one_one_rat,A)),N) = divide_divide_rat(one_one_rat,aa_nat_rat(power_power_rat(A),N)) ).
% power_one_over
tff(fact_3871_mult__numeral__1,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(one)),A) = A ).
% mult_numeral_1
tff(fact_3872_mult__numeral__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(one)),A) = A ).
% mult_numeral_1
tff(fact_3873_mult__numeral__1,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(one)),A) = A ).
% mult_numeral_1
tff(fact_3874_mult__numeral__1,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(one)),A) = A ).
% mult_numeral_1
tff(fact_3875_mult__numeral__1,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(one)),A) = A ).
% mult_numeral_1
tff(fact_3876_mult__numeral__1__right,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),numeral_numeral_nat(one)) = A ).
% mult_numeral_1_right
tff(fact_3877_mult__numeral__1__right,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),numeral_numeral_int(one)) = A ).
% mult_numeral_1_right
tff(fact_3878_mult__numeral__1__right,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(one)) = A ).
% mult_numeral_1_right
tff(fact_3879_mult__numeral__1__right,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),numera6620942414471956472nteger(one)) = A ).
% mult_numeral_1_right
tff(fact_3880_mult__numeral__1__right,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),numera5444537566228673987atural(one)) = A ).
% mult_numeral_1_right
tff(fact_3881_numeral__One,axiom,
numeral_numeral_nat(one) = one_one_nat ).
% numeral_One
tff(fact_3882_numeral__One,axiom,
numeral_numeral_int(one) = one_one_int ).
% numeral_One
tff(fact_3883_numeral__One,axiom,
numeral_numeral_rat(one) = one_one_rat ).
% numeral_One
tff(fact_3884_numeral__One,axiom,
numera6620942414471956472nteger(one) = one_one_Code_integer ).
% numeral_One
tff(fact_3885_numeral__One,axiom,
numera5444537566228673987atural(one) = one_one_Code_natural ).
% numeral_One
tff(fact_3886_inverse__le__1__iff,axiom,
! [X: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(inverse_inverse_rat,X)),one_one_rat)
<=> ( aa_rat_o(ord_less_eq_rat(X),zero_zero_rat)
| aa_rat_o(ord_less_eq_rat(one_one_rat),X) ) ) ).
% inverse_le_1_iff
tff(fact_3887_one__less__inverse,axiom,
! [A: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_rat(A),one_one_rat)
=> aa_rat_o(ord_less_rat(one_one_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).
% one_less_inverse
tff(fact_3888_one__less__inverse__iff,axiom,
! [X: rat] :
( aa_rat_o(ord_less_rat(one_one_rat),aa_rat_rat(inverse_inverse_rat,X))
<=> ( aa_rat_o(ord_less_rat(zero_zero_rat),X)
& aa_rat_o(ord_less_rat(X),one_one_rat) ) ) ).
% one_less_inverse_iff
tff(fact_3889_division__ring__inverse__add,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).
% division_ring_inverse_add
tff(fact_3890_inverse__add,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(inverse_inverse_rat,A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).
% inverse_add
tff(fact_3891_field__class_Ofield__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).
% field_class.field_inverse
tff(fact_3892_division__ring__inverse__diff,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( minus_minus_rat(aa_rat_rat(inverse_inverse_rat,A),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),minus_minus_rat(B,A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).
% division_ring_inverse_diff
tff(fact_3893_nonzero__inverse__eq__divide,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( aa_rat_rat(inverse_inverse_rat,A) = divide_divide_rat(one_one_rat,A) ) ) ).
% nonzero_inverse_eq_divide
tff(fact_3894_cancel__div__mod__rules_I2_J,axiom,
! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).
% cancel_div_mod_rules(2)
tff(fact_3895_cancel__div__mod__rules_I2_J,axiom,
! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).
% cancel_div_mod_rules(2)
tff(fact_3896_cancel__div__mod__rules_I2_J,axiom,
! [B: code_integer,A: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) ).
% cancel_div_mod_rules(2)
tff(fact_3897_cancel__div__mod__rules_I2_J,axiom,
! [B: code_natural,A: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B))),aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B))),C) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),C) ).
% cancel_div_mod_rules(2)
tff(fact_3898_cancel__div__mod__rules_I1_J,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).
% cancel_div_mod_rules(1)
tff(fact_3899_cancel__div__mod__rules_I1_J,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).
% cancel_div_mod_rules(1)
tff(fact_3900_cancel__div__mod__rules_I1_J,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),B)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) ).
% cancel_div_mod_rules(1)
tff(fact_3901_cancel__div__mod__rules_I1_J,axiom,
! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),B)),aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B))),C) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),C) ).
% cancel_div_mod_rules(1)
tff(fact_3902_mod__div__decomp,axiom,
! [A: nat,B: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ).
% mod_div_decomp
tff(fact_3903_mod__div__decomp,axiom,
! [A: int,B: int] : A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).
% mod_div_decomp
tff(fact_3904_mod__div__decomp,axiom,
! [A: code_integer,B: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),B)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) ).
% mod_div_decomp
tff(fact_3905_mod__div__decomp,axiom,
! [A: code_natural,B: code_natural] : A = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),B)),aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B)) ).
% mod_div_decomp
tff(fact_3906_div__mult__mod__eq,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).
% div_mult_mod_eq
tff(fact_3907_div__mult__mod__eq,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).
% div_mult_mod_eq
tff(fact_3908_div__mult__mod__eq,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),B)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) = A ).
% div_mult_mod_eq
tff(fact_3909_div__mult__mod__eq,axiom,
! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),B)),aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B)) = A ).
% div_mult_mod_eq
tff(fact_3910_mod__div__mult__eq,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = A ).
% mod_div_mult_eq
tff(fact_3911_mod__div__mult__eq,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = A ).
% mod_div_mult_eq
tff(fact_3912_mod__div__mult__eq,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),B)) = A ).
% mod_div_mult_eq
tff(fact_3913_mod__div__mult__eq,axiom,
! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),B)) = A ).
% mod_div_mult_eq
tff(fact_3914_mod__mult__div__eq,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = A ).
% mod_mult_div_eq
tff(fact_3915_mod__mult__div__eq,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = A ).
% mod_mult_div_eq
tff(fact_3916_mod__mult__div__eq,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B))) = A ).
% mod_mult_div_eq
tff(fact_3917_mod__mult__div__eq,axiom,
! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B))) = A ).
% mod_mult_div_eq
tff(fact_3918_mult__div__mod__eq,axiom,
! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).
% mult_div_mod_eq
tff(fact_3919_mult__div__mod__eq,axiom,
! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).
% mult_div_mod_eq
tff(fact_3920_mult__div__mod__eq,axiom,
! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) = A ).
% mult_div_mod_eq
tff(fact_3921_mult__div__mod__eq,axiom,
! [B: code_natural,A: code_natural] : aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B))),aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B)) = A ).
% mult_div_mod_eq
tff(fact_3922_div__mult1__eq,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).
% div_mult1_eq
tff(fact_3923_div__mult1__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).
% div_mult1_eq
tff(fact_3924_div__mult1__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B),C))),C)) ).
% div_mult1_eq
tff(fact_3925_div__mult1__eq,axiom,
! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),C) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),aa_Cod4272041906116450446atural(divide5121882707175180666atural(B),C))),aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),aa_Cod4272041906116450446atural(modulo8411746178871703098atural(B),C))),C)) ).
% div_mult1_eq
tff(fact_3926_minus__div__mult__eq__mod,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).
% minus_div_mult_eq_mod
tff(fact_3927_minus__div__mult__eq__mod,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% minus_div_mult_eq_mod
tff(fact_3928_minus__div__mult__eq__mod,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),B)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).
% minus_div_mult_eq_mod
tff(fact_3929_minus__div__mult__eq__mod,axiom,
! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(minus_7197305767214868737atural(A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),B)) = aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B) ).
% minus_div_mult_eq_mod
tff(fact_3930_minus__mod__eq__div__mult,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B) ).
% minus_mod_eq_div_mult
tff(fact_3931_minus__mod__eq__div__mult,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B) ).
% minus_mod_eq_div_mult
tff(fact_3932_minus__mod__eq__div__mult,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),B) ).
% minus_mod_eq_div_mult
tff(fact_3933_minus__mod__eq__div__mult,axiom,
! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(minus_7197305767214868737atural(A),aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B)) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),B) ).
% minus_mod_eq_div_mult
tff(fact_3934_minus__mod__eq__mult__div,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).
% minus_mod_eq_mult_div
tff(fact_3935_minus__mod__eq__mult__div,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).
% minus_mod_eq_mult_div
tff(fact_3936_minus__mod__eq__mult__div,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)) ).
% minus_mod_eq_mult_div
tff(fact_3937_minus__mod__eq__mult__div,axiom,
! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(minus_7197305767214868737atural(A),aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B)) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)) ).
% minus_mod_eq_mult_div
tff(fact_3938_minus__mult__div__eq__mod,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).
% minus_mult_div_eq_mod
tff(fact_3939_minus__mult__div__eq__mod,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% minus_mult_div_eq_mod
tff(fact_3940_minus__mult__div__eq__mod,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B))) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ).
% minus_mult_div_eq_mod
tff(fact_3941_minus__mult__div__eq__mod,axiom,
! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(minus_7197305767214868737atural(A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B))) = aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B) ).
% minus_mult_div_eq_mod
tff(fact_3942_power__le__one,axiom,
! [A: rat,N: nat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(A),one_one_rat)
=> aa_rat_o(ord_less_eq_rat(aa_nat_rat(power_power_rat(A),N)),one_one_rat) ) ) ).
% power_le_one
tff(fact_3943_power__le__one,axiom,
! [A: code_integer,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),one_one_Code_integer) ) ) ).
% power_le_one
tff(fact_3944_power__le__one,axiom,
! [A: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat) ) ) ).
% power_le_one
tff(fact_3945_power__le__one,axiom,
! [A: int,N: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),one_one_int) ) ) ).
% power_le_one
tff(fact_3946_power__decreasing,axiom,
! [N: nat,N3: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(A),one_one_rat)
=> aa_rat_o(ord_less_eq_rat(aa_nat_rat(power_power_rat(A),N3)),aa_nat_rat(power_power_rat(A),N)) ) ) ) ).
% power_decreasing
tff(fact_3947_power__decreasing,axiom,
! [N: nat,N3: nat,A: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N3)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ) ).
% power_decreasing
tff(fact_3948_power__decreasing,axiom,
! [N: nat,N3: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).
% power_decreasing
tff(fact_3949_power__decreasing,axiom,
! [N: nat,N3: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N3)),aa_nat_int(power_power_int(A),N)) ) ) ) ).
% power_decreasing
tff(fact_3950_power__le__imp__le__exp,axiom,
! [A: rat,M2: nat,N: nat] :
( aa_rat_o(ord_less_rat(one_one_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(aa_nat_rat(power_power_rat(A),M2)),aa_nat_rat(power_power_rat(A),N))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).
% power_le_imp_le_exp
tff(fact_3951_power__le__imp__le__exp,axiom,
! [A: code_natural,M2: nat,N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(A),M2)),aa_nat_Code_natural(power_7079662738309270450atural(A),N))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).
% power_le_imp_le_exp
tff(fact_3952_power__le__imp__le__exp,axiom,
! [A: code_integer,M2: nat,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M2)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).
% power_le_imp_le_exp
tff(fact_3953_power__le__imp__le__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).
% power_le_imp_le_exp
tff(fact_3954_power__le__imp__le__exp,axiom,
! [A: int,M2: nat,N: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N))
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).
% power_le_imp_le_exp
tff(fact_3955_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N3)
=> ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_rat(A),one_one_rat)
=> aa_rat_o(ord_less_rat(aa_nat_rat(power_power_rat(A),N3)),aa_nat_rat(power_power_rat(A),N)) ) ) ) ).
% power_strict_decreasing
tff(fact_3956_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N3)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).
% power_strict_decreasing
tff(fact_3957_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N3)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N3)),aa_nat_int(power_power_int(A),N)) ) ) ) ).
% power_strict_decreasing
tff(fact_3958_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N3)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N3)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ) ).
% power_strict_decreasing
tff(fact_3959_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A: code_natural] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N3)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),one_one_Code_natural)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(A),N3)),aa_nat_Code_natural(power_7079662738309270450atural(A),N)) ) ) ) ).
% power_strict_decreasing
tff(fact_3960_power__gt1__lemma,axiom,
! [A: rat,N: nat] :
( aa_rat_o(ord_less_rat(one_one_rat),A)
=> aa_rat_o(ord_less_rat(one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N))) ) ).
% power_gt1_lemma
tff(fact_3961_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).
% power_gt1_lemma
tff(fact_3962_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N))) ) ).
% power_gt1_lemma
tff(fact_3963_power__gt1__lemma,axiom,
! [A: code_integer,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).
% power_gt1_lemma
tff(fact_3964_power__gt1__lemma,axiom,
! [A: code_natural,N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),aa_nat_Code_natural(power_7079662738309270450atural(A),N))) ) ).
% power_gt1_lemma
tff(fact_3965_power__less__power__Suc,axiom,
! [A: rat,N: nat] :
( aa_rat_o(ord_less_rat(one_one_rat),A)
=> aa_rat_o(ord_less_rat(aa_nat_rat(power_power_rat(A),N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N))) ) ).
% power_less_power_Suc
tff(fact_3966_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).
% power_less_power_Suc
tff(fact_3967_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N))) ) ).
% power_less_power_Suc
tff(fact_3968_power__less__power__Suc,axiom,
! [A: code_integer,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).
% power_less_power_Suc
tff(fact_3969_power__less__power__Suc,axiom,
! [A: code_natural,N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(A),N)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),aa_nat_Code_natural(power_7079662738309270450atural(A),N))) ) ).
% power_less_power_Suc
tff(fact_3970_power__diff__power__eq,axiom,
! [A: nat,M2: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ).
% power_diff_power_eq
tff(fact_3971_power__diff__power__eq,axiom,
! [A: int,M2: nat,N: nat] :
( ( A != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ).
% power_diff_power_eq
tff(fact_3972_power__diff__power__eq,axiom,
! [A: code_integer,M2: nat,N: nat] :
( ( A != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M2)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ).
% power_diff_power_eq
tff(fact_3973_power__diff__power__eq,axiom,
! [A: code_natural,M2: nat,N: nat] :
( ( A != zero_z2226904508553997617atural )
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_nat_Code_natural(power_7079662738309270450atural(A),M2)),aa_nat_Code_natural(power_7079662738309270450atural(A),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2),aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),aa_Cod4272041906116450446atural(divide5121882707175180666atural(one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ).
% power_diff_power_eq
tff(fact_3974_power__minus,axiom,
! [A: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(power_power_int(A),N)) ).
% power_minus
tff(fact_3975_power__minus,axiom,
! [A: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(A),N)) ).
% power_minus
tff(fact_3976_power__minus,axiom,
! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).
% power_minus
tff(fact_3977_mult__1s__ring__1_I1_J,axiom,
! [B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(one))),B) = aa_int_int(uminus_uminus_int,B) ).
% mult_1s_ring_1(1)
tff(fact_3978_mult__1s__ring__1_I1_J,axiom,
! [B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(one))),B) = aa_rat_rat(uminus_uminus_rat,B) ).
% mult_1s_ring_1(1)
tff(fact_3979_mult__1s__ring__1_I1_J,axiom,
! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).
% mult_1s_ring_1(1)
tff(fact_3980_mult__1s__ring__1_I2_J,axiom,
! [B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(uminus_uminus_int,numeral_numeral_int(one))) = aa_int_int(uminus_uminus_int,B) ).
% mult_1s_ring_1(2)
tff(fact_3981_mult__1s__ring__1_I2_J,axiom,
! [B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(one))) = aa_rat_rat(uminus_uminus_rat,B) ).
% mult_1s_ring_1(2)
tff(fact_3982_mult__1s__ring__1_I2_J,axiom,
! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).
% mult_1s_ring_1(2)
tff(fact_3983_uminus__numeral__One,axiom,
aa_int_int(uminus_uminus_int,numeral_numeral_int(one)) = aa_int_int(uminus_uminus_int,one_one_int) ).
% uminus_numeral_One
tff(fact_3984_uminus__numeral__One,axiom,
aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(one)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% uminus_numeral_One
tff(fact_3985_uminus__numeral__One,axiom,
aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% uminus_numeral_One
tff(fact_3986_inverse__less__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
<=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
=> aa_rat_o(ord_less_rat(B),A) )
& ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
=> aa_rat_o(ord_less_rat(A),B) ) ) ) ).
% inverse_less_iff
tff(fact_3987_inverse__le__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
<=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
=> aa_rat_o(ord_less_eq_rat(B),A) )
& ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
=> aa_rat_o(ord_less_eq_rat(A),B) ) ) ) ).
% inverse_le_iff
tff(fact_3988_one__le__inverse,axiom,
! [A: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(A),one_one_rat)
=> aa_rat_o(ord_less_eq_rat(one_one_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).
% one_le_inverse
tff(fact_3989_inverse__less__1__iff,axiom,
! [X: rat] :
( aa_rat_o(ord_less_rat(aa_rat_rat(inverse_inverse_rat,X)),one_one_rat)
<=> ( aa_rat_o(ord_less_eq_rat(X),zero_zero_rat)
| aa_rat_o(ord_less_rat(one_one_rat),X) ) ) ).
% inverse_less_1_iff
tff(fact_3990_one__le__inverse__iff,axiom,
! [X: rat] :
( aa_rat_o(ord_less_eq_rat(one_one_rat),aa_rat_rat(inverse_inverse_rat,X))
<=> ( aa_rat_o(ord_less_rat(zero_zero_rat),X)
& aa_rat_o(ord_less_eq_rat(X),one_one_rat) ) ) ).
% one_le_inverse_iff
tff(fact_3991_power__Suc__less,axiom,
! [A: rat,N: nat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_rat(A),one_one_rat)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N))),aa_nat_rat(power_power_rat(A),N)) ) ) ).
% power_Suc_less
tff(fact_3992_power__Suc__less,axiom,
! [A: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ).
% power_Suc_less
tff(fact_3993_power__Suc__less,axiom,
! [A: int,N: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N))),aa_nat_int(power_power_int(A),N)) ) ) ).
% power_Suc_less
tff(fact_3994_power__Suc__less,axiom,
! [A: code_integer,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ).
% power_Suc_less
tff(fact_3995_power__Suc__less,axiom,
! [A: code_natural,N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),one_one_Code_natural)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),aa_nat_Code_natural(power_7079662738309270450atural(A),N))),aa_nat_Code_natural(power_7079662738309270450atural(A),N)) ) ) ).
% power_Suc_less
tff(fact_3996_power__diff__conv__inverse,axiom,
! [X: rat,M2: nat,N: nat] :
( ( X != zero_zero_rat )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_nat_rat(power_power_rat(X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,X)),M2)) ) ) ) ).
% power_diff_conv_inverse
tff(fact_3997_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_3998_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
=> ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_3999_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4000_power__mult__inverse__distrib,axiom,
! [X: rat,M2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),M2)),aa_rat_rat(inverse_inverse_rat,X)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,X)),aa_nat_rat(power_power_rat(X),M2)) ).
% power_mult_inverse_distrib
tff(fact_4001_power__mult__power__inverse__commute,axiom,
! [X: rat,M2: nat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),M2)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,X)),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,X)),N)),aa_nat_rat(power_power_rat(X),M2)) ).
% power_mult_power_inverse_commute
tff(fact_4002_divmod__digit__1_I1_J,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)))),one_one_Code_integer) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) ) ) ) ) ).
% divmod_digit_1(1)
tff(fact_4003_divmod__digit__1_I1_J,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B)))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ) ).
% divmod_digit_1(1)
tff(fact_4004_divmod__digit__1_I1_J,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),B)))
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),B)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ) ).
% divmod_digit_1(1)
tff(fact_4005_one__mod__numeral,axiom,
! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),numeral_numeral_nat(N)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,N)) ).
% one_mod_numeral
tff(fact_4006_one__mod__numeral,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),numeral_numeral_int(N)) = aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,N)) ).
% one_mod_numeral
tff(fact_4007_one__mod__numeral,axiom,
! [N: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(one_one_Code_integer),numera6620942414471956472nteger(N)) = produc6174133586879617921nteger(unique3479559517661332726nteger(one,N)) ).
% one_mod_numeral
tff(fact_4008_one__div__numeral,axiom,
! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),numeral_numeral_nat(N)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,N)) ).
% one_div_numeral
tff(fact_4009_one__div__numeral,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),numeral_numeral_int(N)) = aa_Pro6799258434465214121nt_int(product_fst_int_int,unique5052692396658037445od_int(one,N)) ).
% one_div_numeral
tff(fact_4010_one__div__numeral,axiom,
! [N: num] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),numera6620942414471956472nteger(N)) = produc8508995932063986495nteger(unique3479559517661332726nteger(one,N)) ).
% one_div_numeral
tff(fact_4011_divmod__digit__1_I2_J,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)))
=> ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ) ) ) ) ).
% divmod_digit_1(2)
tff(fact_4012_divmod__digit__1_I2_J,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B)))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).
% divmod_digit_1(2)
tff(fact_4013_divmod__digit__1_I2_J,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),B)))
=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ) ).
% divmod_digit_1(2)
tff(fact_4014_mod__double__modulus,axiom,
! [M2: code_integer,X: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),M2)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
=> ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(X),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),M2)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(X),M2) )
| ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(X),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),M2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(X),M2)),M2) ) ) ) ) ).
% mod_double_modulus
tff(fact_4015_mod__double__modulus,axiom,
! [M2: nat,X: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M2) )
| ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M2)),M2) ) ) ) ) ).
% mod_double_modulus
tff(fact_4016_mod__double__modulus,axiom,
! [M2: int,X: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M2)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
=> ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),M2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),M2) )
| ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),M2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),M2)),M2) ) ) ) ) ).
% mod_double_modulus
tff(fact_4017_one__add__one,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = numeral_numeral_nat(aa_num_num(bit0,one)) ).
% one_add_one
tff(fact_4018_one__add__one,axiom,
aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = numeral_numeral_int(aa_num_num(bit0,one)) ).
% one_add_one
tff(fact_4019_one__add__one,axiom,
aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).
% one_add_one
tff(fact_4020_one__add__one,axiom,
aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).
% one_add_one
tff(fact_4021_one__add__one,axiom,
aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),one_one_Code_natural) = numera5444537566228673987atural(aa_num_num(bit0,one)) ).
% one_add_one
tff(fact_4022_one__mod__two__eq__one,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_nat ).
% one_mod_two_eq_one
tff(fact_4023_one__mod__two__eq__one,axiom,
aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),numeral_numeral_int(aa_num_num(bit0,one))) = one_one_int ).
% one_mod_two_eq_one
tff(fact_4024_one__mod__two__eq__one,axiom,
aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).
% one_mod_two_eq_one
tff(fact_4025_one__mod__two__eq__one,axiom,
aa_Cod4272041906116450446atural(modulo8411746178871703098atural(one_one_Code_natural),numera5444537566228673987atural(aa_num_num(bit0,one))) = one_one_Code_natural ).
% one_mod_two_eq_one
tff(fact_4026_bits__one__mod__two__eq__one,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_nat ).
% bits_one_mod_two_eq_one
tff(fact_4027_bits__one__mod__two__eq__one,axiom,
aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),numeral_numeral_int(aa_num_num(bit0,one))) = one_one_int ).
% bits_one_mod_two_eq_one
tff(fact_4028_bits__one__mod__two__eq__one,axiom,
aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).
% bits_one_mod_two_eq_one
tff(fact_4029_bits__one__mod__two__eq__one,axiom,
aa_Cod4272041906116450446atural(modulo8411746178871703098atural(one_one_Code_natural),numera5444537566228673987atural(aa_num_num(bit0,one))) = one_one_Code_natural ).
% bits_one_mod_two_eq_one
tff(fact_4030_divmod__algorithm__code_I2_J,axiom,
! [M2: num] : unique5055182867167087721od_nat(M2,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(numeral_numeral_nat(M2)),zero_zero_nat) ).
% divmod_algorithm_code(2)
tff(fact_4031_divmod__algorithm__code_I2_J,axiom,
! [M2: num] : unique5052692396658037445od_int(M2,one) = aa_int7462565939943104719nt_int(product_Pair_int_int(numeral_numeral_int(M2)),zero_zero_int) ).
% divmod_algorithm_code(2)
tff(fact_4032_divmod__algorithm__code_I2_J,axiom,
! [M2: num] : unique3479559517661332726nteger(M2,one) = produc1086072967326762835nteger(numera6620942414471956472nteger(M2),zero_z3403309356797280102nteger) ).
% divmod_algorithm_code(2)
tff(fact_4033_bits__1__div__2,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit0,one))) = zero_zero_nat ).
% bits_1_div_2
tff(fact_4034_bits__1__div__2,axiom,
aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),numeral_numeral_int(aa_num_num(bit0,one))) = zero_zero_int ).
% bits_1_div_2
tff(fact_4035_bits__1__div__2,axiom,
aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).
% bits_1_div_2
tff(fact_4036_bits__1__div__2,axiom,
aa_Cod4272041906116450446atural(divide5121882707175180666atural(one_one_Code_natural),numera5444537566228673987atural(aa_num_num(bit0,one))) = zero_z2226904508553997617atural ).
% bits_1_div_2
tff(fact_4037_one__div__two__eq__zero,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit0,one))) = zero_zero_nat ).
% one_div_two_eq_zero
tff(fact_4038_one__div__two__eq__zero,axiom,
aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),numeral_numeral_int(aa_num_num(bit0,one))) = zero_zero_int ).
% one_div_two_eq_zero
tff(fact_4039_one__div__two__eq__zero,axiom,
aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).
% one_div_two_eq_zero
tff(fact_4040_one__div__two__eq__zero,axiom,
aa_Cod4272041906116450446atural(divide5121882707175180666atural(one_one_Code_natural),numera5444537566228673987atural(aa_num_num(bit0,one))) = zero_z2226904508553997617atural ).
% one_div_two_eq_zero
tff(fact_4041_add__neg__numeral__special_I9_J,axiom,
aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,numeral_numeral_int(aa_num_num(bit0,one))) ).
% add_neg_numeral_special(9)
tff(fact_4042_add__neg__numeral__special_I9_J,axiom,
aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(aa_num_num(bit0,one))) ).
% add_neg_numeral_special(9)
tff(fact_4043_add__neg__numeral__special_I9_J,axiom,
aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% add_neg_numeral_special(9)
tff(fact_4044_diff__numeral__special_I10_J,axiom,
aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_int_int(uminus_uminus_int,numeral_numeral_int(aa_num_num(bit0,one))) ).
% diff_numeral_special(10)
tff(fact_4045_diff__numeral__special_I10_J,axiom,
minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat),one_one_rat) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(aa_num_num(bit0,one))) ).
% diff_numeral_special(10)
tff(fact_4046_diff__numeral__special_I10_J,axiom,
aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% diff_numeral_special(10)
tff(fact_4047_diff__numeral__special_I11_J,axiom,
aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = numeral_numeral_int(aa_num_num(bit0,one)) ).
% diff_numeral_special(11)
tff(fact_4048_diff__numeral__special_I11_J,axiom,
minus_minus_rat(one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = numeral_numeral_rat(aa_num_num(bit0,one)) ).
% diff_numeral_special(11)
tff(fact_4049_diff__numeral__special_I11_J,axiom,
aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).
% diff_numeral_special(11)
tff(fact_4050_not__mod__2__eq__0__eq__1,axiom,
! [A: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) != zero_zero_nat )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_nat ) ) ).
% not_mod_2_eq_0_eq_1
tff(fact_4051_not__mod__2__eq__0__eq__1,axiom,
! [A: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one))) != zero_zero_int )
<=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one))) = one_one_int ) ) ).
% not_mod_2_eq_0_eq_1
tff(fact_4052_not__mod__2__eq__0__eq__1,axiom,
! [A: code_integer] :
( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
<=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).
% not_mod_2_eq_0_eq_1
tff(fact_4053_not__mod__2__eq__0__eq__1,axiom,
! [A: code_natural] :
( ( aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))) != zero_z2226904508553997617atural )
<=> ( aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))) = one_one_Code_natural ) ) ).
% not_mod_2_eq_0_eq_1
tff(fact_4054_not__mod__2__eq__1__eq__0,axiom,
! [A: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) != one_one_nat )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) = zero_zero_nat ) ) ).
% not_mod_2_eq_1_eq_0
tff(fact_4055_not__mod__2__eq__1__eq__0,axiom,
! [A: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one))) != one_one_int )
<=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one))) = zero_zero_int ) ) ).
% not_mod_2_eq_1_eq_0
tff(fact_4056_not__mod__2__eq__1__eq__0,axiom,
! [A: code_integer] :
( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer )
<=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).
% not_mod_2_eq_1_eq_0
tff(fact_4057_not__mod__2__eq__1__eq__0,axiom,
! [A: code_natural] :
( ( aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))) != one_one_Code_natural )
<=> ( aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))) = zero_z2226904508553997617atural ) ) ).
% not_mod_2_eq_1_eq_0
tff(fact_4058_minus__1__div__2__eq,axiom,
aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),numeral_numeral_int(aa_num_num(bit0,one))) = aa_int_int(uminus_uminus_int,one_one_int) ).
% minus_1_div_2_eq
tff(fact_4059_minus__1__div__2__eq,axiom,
aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% minus_1_div_2_eq
tff(fact_4060_minus__1__mod__2__eq,axiom,
aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),numeral_numeral_int(aa_num_num(bit0,one))) = one_one_int ).
% minus_1_mod_2_eq
tff(fact_4061_minus__1__mod__2__eq,axiom,
aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).
% minus_1_mod_2_eq
tff(fact_4062_bits__minus__1__mod__2__eq,axiom,
aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),numeral_numeral_int(aa_num_num(bit0,one))) = one_one_int ).
% bits_minus_1_mod_2_eq
tff(fact_4063_bits__minus__1__mod__2__eq,axiom,
aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).
% bits_minus_1_mod_2_eq
tff(fact_4064_power__minus1__even,axiom,
! [N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = one_one_int ).
% power_minus1_even
tff(fact_4065_power__minus1__even,axiom,
! [N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = one_one_rat ).
% power_minus1_even
tff(fact_4066_power__minus1__even,axiom,
! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = one_one_Code_integer ).
% power_minus1_even
tff(fact_4067_divmod__algorithm__code_I3_J,axiom,
! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),numeral_numeral_nat(one)) ).
% divmod_algorithm_code(3)
tff(fact_4068_divmod__algorithm__code_I3_J,axiom,
! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,N)) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),numeral_numeral_int(one)) ).
% divmod_algorithm_code(3)
tff(fact_4069_divmod__algorithm__code_I3_J,axiom,
! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).
% divmod_algorithm_code(3)
tff(fact_4070_divmod_H__nat__def,axiom,
! [M2: num,N: num] : unique5055182867167087721od_nat(M2,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,numeral_numeral_nat(M2)),numeral_numeral_nat(N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(M2)),numeral_numeral_nat(N))) ).
% divmod'_nat_def
tff(fact_4071_power4__eq__xxxx,axiom,
! [X: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),X)),X) ).
% power4_eq_xxxx
tff(fact_4072_power4__eq__xxxx,axiom,
! [X: assn] : aa_nat_assn(power_power_assn(X),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),X)),X)),X) ).
% power4_eq_xxxx
tff(fact_4073_power4__eq__xxxx,axiom,
! [X: rat] : aa_nat_rat(power_power_rat(X),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),X)),X) ).
% power4_eq_xxxx
tff(fact_4074_power4__eq__xxxx,axiom,
! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),X)),X)),X) ).
% power4_eq_xxxx
tff(fact_4075_power4__eq__xxxx,axiom,
! [X: int] : aa_nat_int(power_power_int(X),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),X)),X) ).
% power4_eq_xxxx
tff(fact_4076_power2__eq__square,axiom,
! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) ).
% power2_eq_square
tff(fact_4077_power2__eq__square,axiom,
! [A: assn] : aa_nat_assn(power_power_assn(A),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),A) ).
% power2_eq_square
tff(fact_4078_power2__eq__square,axiom,
! [A: rat] : aa_nat_rat(power_power_rat(A),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) ).
% power2_eq_square
tff(fact_4079_power2__eq__square,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A) ).
% power2_eq_square
tff(fact_4080_power2__eq__square,axiom,
! [A: int] : aa_nat_int(power_power_int(A),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).
% power2_eq_square
tff(fact_4081_one__power2,axiom,
aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_Code_integer ).
% one_power2
tff(fact_4082_one__power2,axiom,
aa_nat_rat(power_power_rat(one_one_rat),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_rat ).
% one_power2
tff(fact_4083_one__power2,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_nat ).
% one_power2
tff(fact_4084_one__power2,axiom,
aa_nat_int(power_power_int(one_one_int),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_int ).
% one_power2
tff(fact_4085_mult__2,axiom,
! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).
% mult_2
tff(fact_4086_mult__2,axiom,
! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),Z) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).
% mult_2
tff(fact_4087_mult__2,axiom,
! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),Z) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ).
% mult_2
tff(fact_4088_mult__2,axiom,
! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Z) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),Z) ).
% mult_2
tff(fact_4089_mult__2,axiom,
! [Z: code_natural] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),Z) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(Z),Z) ).
% mult_2
tff(fact_4090_mult__2__right,axiom,
! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).
% mult_2_right
tff(fact_4091_mult__2__right,axiom,
! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),numeral_numeral_int(aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).
% mult_2_right
tff(fact_4092_mult__2__right,axiom,
! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),numeral_numeral_rat(aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ).
% mult_2_right
tff(fact_4093_mult__2__right,axiom,
! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),Z) ).
% mult_2_right
tff(fact_4094_mult__2__right,axiom,
! [Z: code_natural] : aa_Cod4272041906116450446atural(times_2397367101498566445atural(Z),numera5444537566228673987atural(aa_num_num(bit0,one))) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(Z),Z) ).
% mult_2_right
tff(fact_4095_left__add__twice,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)),B) ).
% left_add_twice
tff(fact_4096_left__add__twice,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),A)),B) ).
% left_add_twice
tff(fact_4097_left__add__twice,axiom,
! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),A)),B) ).
% left_add_twice
tff(fact_4098_left__add__twice,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)),B) ).
% left_add_twice
tff(fact_4099_left__add__twice,axiom,
! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),B)) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),A)),B) ).
% left_add_twice
tff(fact_4100_minus__power__mult__self,axiom,
! [A: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ).
% minus_power_mult_self
tff(fact_4101_minus__power__mult__self,axiom,
! [A: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ).
% minus_power_mult_self
tff(fact_4102_minus__power__mult__self,axiom,
! [A: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ).
% minus_power_mult_self
tff(fact_4103_power2__eq__1__iff,axiom,
! [A: int] :
( ( aa_nat_int(power_power_int(A),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_int )
<=> ( ( A = one_one_int )
| ( A = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).
% power2_eq_1_iff
tff(fact_4104_power2__eq__1__iff,axiom,
! [A: rat] :
( ( aa_nat_rat(power_power_rat(A),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_rat )
<=> ( ( A = one_one_rat )
| ( A = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).
% power2_eq_1_iff
tff(fact_4105_power2__eq__1__iff,axiom,
! [A: code_integer] :
( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_Code_integer )
<=> ( ( A = one_one_Code_integer )
| ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).
% power2_eq_1_iff
tff(fact_4106_power2__sum,axiom,
! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),X)),Y)) ).
% power2_sum
tff(fact_4107_power2__sum,axiom,
! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),X)),Y)) ).
% power2_sum
tff(fact_4108_power2__sum,axiom,
! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),X)),Y)) ).
% power2_sum
tff(fact_4109_power2__sum,axiom,
! [X: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),X)),Y)) ).
% power2_sum
tff(fact_4110_power2__sum,axiom,
! [X: code_natural,Y: code_natural] : aa_nat_Code_natural(power_7079662738309270450atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(X),Y)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_nat_Code_natural(power_7079662738309270450atural(X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_Code_natural(power_7079662738309270450atural(Y),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),X)),Y)) ).
% power2_sum
tff(fact_4111_square__le__1,axiom,
! [X: rat] :
( aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),X)
=> ( aa_rat_o(ord_less_eq_rat(X),one_one_rat)
=> aa_rat_o(ord_less_eq_rat(aa_nat_rat(power_power_rat(X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_rat) ) ) ).
% square_le_1
tff(fact_4112_square__le__1,axiom,
! [X: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).
% square_le_1
tff(fact_4113_square__le__1,axiom,
! [X: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),X)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_int) ) ) ).
% square_le_1
tff(fact_4114_divmod__def,axiom,
! [M2: num,N: num] : unique5055182867167087721od_nat(M2,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,numeral_numeral_nat(M2)),numeral_numeral_nat(N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(M2)),numeral_numeral_nat(N))) ).
% divmod_def
tff(fact_4115_divmod__def,axiom,
! [M2: num,N: num] : unique5052692396658037445od_int(M2,N) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,numeral_numeral_int(M2)),numeral_numeral_int(N))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,numeral_numeral_int(M2)),numeral_numeral_int(N))) ).
% divmod_def
tff(fact_4116_divmod__def,axiom,
! [M2: num,N: num] : unique3479559517661332726nteger(M2,N) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(M2)),numera6620942414471956472nteger(N)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(M2)),numera6620942414471956472nteger(N))) ).
% divmod_def
tff(fact_4117_divmod__digit__0_I2_J,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B))),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).
% divmod_digit_0(2)
tff(fact_4118_divmod__digit__0_I2_J,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),B))),B)
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ).
% divmod_digit_0(2)
tff(fact_4119_divmod__digit__0_I2_J,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B)
=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) ) ) ) ).
% divmod_digit_0(2)
tff(fact_4120_power2__diff,axiom,
! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),X)),Y)) ).
% power2_diff
tff(fact_4121_power2__diff,axiom,
! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(minus_minus_rat(X,Y)),numeral_numeral_nat(aa_num_num(bit0,one))) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),X)),Y)) ).
% power2_diff
tff(fact_4122_power2__diff,axiom,
! [X: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),Y)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(aa_num_num(bit0,one))))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),X)),Y)) ).
% power2_diff
tff(fact_4123_mult__exp__mod__exp__eq,axiom,
! [M2: nat,N: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M2)) ) ) ).
% mult_exp_mod_exp_eq
tff(fact_4124_mult__exp__mod__exp__eq,axiom,
! [M2: nat,N: nat,A: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),M2))),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),M2)) ) ) ).
% mult_exp_mod_exp_eq
tff(fact_4125_mult__exp__mod__exp__eq,axiom,
! [M2: nat,N: nat,A: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M2))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M2)) ) ) ).
% mult_exp_mod_exp_eq
tff(fact_4126_mult__exp__mod__exp__eq,axiom,
! [M2: nat,N: nat,A: code_natural] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M2))),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M2)) ) ) ).
% mult_exp_mod_exp_eq
tff(fact_4127_divmod__digit__0_I1_J,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B))),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).
% divmod_digit_0(1)
tff(fact_4128_divmod__digit__0_I1_J,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),B))),B)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).
% divmod_digit_0(1)
tff(fact_4129_divmod__digit__0_I1_J,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) ) ) ) ).
% divmod_digit_0(1)
tff(fact_4130_divmod__step__eq,axiom,
! [L: num,Q4: int,R2: int] :
unique5024387138958732305ep_int(L,aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(L)),R2),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),Q4)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),numeral_numeral_int(L))),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),Q4)),R2)) ).
% divmod_step_eq
tff(fact_4131_divmod__step__eq,axiom,
! [L: num,Q4: nat,R2: nat] :
unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q4),R2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(L)),R2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Q4)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R2),numeral_numeral_nat(L))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Q4)),R2)) ).
% divmod_step_eq
tff(fact_4132_divmod__step__eq,axiom,
! [L: num,Q4: code_integer,R2: code_integer] :
unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q4,R2)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q4)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(R2),numera6620942414471956472nteger(L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q4),R2)) ).
% divmod_step_eq
tff(fact_4133_set__bit__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))))) ).
% set_bit_0
tff(fact_4134_set__bit__0,axiom,
! [A: int] : aa_int_int(bit_se7879613467334960850it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(aa_num_num(bit0,one))))) ).
% set_bit_0
tff(fact_4135_set__bit__0,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(zero_zero_nat),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).
% set_bit_0
tff(fact_4136_set__bit__0,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(bit_se1617098188084679374atural(zero_zero_nat),A) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))))) ).
% set_bit_0
tff(fact_4137_unset__bit__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))) ).
% unset_bit_0
tff(fact_4138_unset__bit__0,axiom,
! [A: int] : aa_int_int(bit_se4203085406695923979it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(aa_num_num(bit0,one)))) ).
% unset_bit_0
tff(fact_4139_unset__bit__0,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(zero_zero_nat),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).
% unset_bit_0
tff(fact_4140_unset__bit__0,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(bit_se7083795435491715335atural(zero_zero_nat),A) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),numera5444537566228673987atural(aa_num_num(bit0,one)))) ).
% unset_bit_0
tff(fact_4141_divmod__divmod__step,axiom,
! [M2: num,N: num] :
unique5052692396658037445od_int(M2,N) = $ite(ord_less_num(M2,N),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),numeral_numeral_int(M2)),unique5024387138958732305ep_int(N,unique5052692396658037445od_int(M2,aa_num_num(bit0,N)))) ).
% divmod_divmod_step
tff(fact_4142_divmod__divmod__step,axiom,
! [M2: num,N: num] :
unique5055182867167087721od_nat(M2,N) = $ite(ord_less_num(M2,N),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),numeral_numeral_nat(M2)),unique5026877609467782581ep_nat(N,unique5055182867167087721od_nat(M2,aa_num_num(bit0,N)))) ).
% divmod_divmod_step
tff(fact_4143_divmod__divmod__step,axiom,
! [M2: num,N: num] :
unique3479559517661332726nteger(M2,N) = $ite(ord_less_num(M2,N),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(M2)),unique4921790084139445826nteger(N,unique3479559517661332726nteger(M2,aa_num_num(bit0,N)))) ).
% divmod_divmod_step
tff(fact_4144_signed__take__bit__rec,axiom,
! [N: nat,A: int] :
aa_int_int(bit_ri631733984087533419it_int(N),A) = $ite(N = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(aa_num_num(bit0,one))))))) ).
% signed_take_bit_rec
tff(fact_4145_signed__take__bit__rec,axiom,
! [N: nat,A: code_integer] :
aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(N),A) = $ite(N = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))))))) ).
% signed_take_bit_rec
tff(fact_4146_dbl__simps_I4_J,axiom,
aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,numeral_numeral_int(aa_num_num(bit0,one))) ).
% dbl_simps(4)
tff(fact_4147_dbl__simps_I4_J,axiom,
neg_numeral_dbl_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(aa_num_num(bit0,one))) ).
% dbl_simps(4)
tff(fact_4148_dbl__simps_I4_J,axiom,
aa_Cod1783974652533396728nteger(neg_nu8804712462038260780nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% dbl_simps(4)
tff(fact_4149_dbl__simps_I3_J,axiom,
aa_int_int(neg_numeral_dbl_int,one_one_int) = numeral_numeral_int(aa_num_num(bit0,one)) ).
% dbl_simps(3)
tff(fact_4150_dbl__simps_I3_J,axiom,
neg_numeral_dbl_rat(one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).
% dbl_simps(3)
tff(fact_4151_dbl__simps_I3_J,axiom,
aa_Cod1783974652533396728nteger(neg_nu8804712462038260780nteger,one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).
% dbl_simps(3)
tff(fact_4152_even__succ__mod__exp,axiom,
! [A: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) ) ) ) ).
% even_succ_mod_exp
tff(fact_4153_even__succ__mod__exp,axiom,
! [A: int,N: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N))) ) ) ) ).
% even_succ_mod_exp
tff(fact_4154_even__succ__mod__exp,axiom,
! [A: code_integer,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))) ) ) ) ).
% even_succ_mod_exp
tff(fact_4155_even__succ__mod__exp,axiom,
! [A: code_natural,N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),A)),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N))) ) ) ) ).
% even_succ_mod_exp
tff(fact_4156_dvd__times__right__cancel__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).
% dvd_times_right_cancel_iff
tff(fact_4157_dvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).
% dvd_times_right_cancel_iff
tff(fact_4158_dvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).
% dvd_times_right_cancel_iff
tff(fact_4159_dvd__times__left__cancel__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).
% dvd_times_left_cancel_iff
tff(fact_4160_dvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).
% dvd_times_left_cancel_iff
tff(fact_4161_dvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).
% dvd_times_left_cancel_iff
tff(fact_4162_dvd__mult__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
<=> ( ( C = zero_z3403309356797280102nteger )
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).
% dvd_mult_cancel_right
tff(fact_4163_dvd__mult__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(dvd_dvd_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
<=> ( ( C = zero_zero_rat )
| aa_rat_o(dvd_dvd_rat(A),B) ) ) ).
% dvd_mult_cancel_right
tff(fact_4164_dvd__mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
<=> ( ( C = zero_zero_int )
| aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).
% dvd_mult_cancel_right
tff(fact_4165_dvd__mult__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
<=> ( ( C = zero_z3403309356797280102nteger )
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).
% dvd_mult_cancel_left
tff(fact_4166_dvd__mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( aa_rat_o(dvd_dvd_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
<=> ( ( C = zero_zero_rat )
| aa_rat_o(dvd_dvd_rat(A),B) ) ) ).
% dvd_mult_cancel_left
tff(fact_4167_dvd__mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
<=> ( ( C = zero_zero_int )
| aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).
% dvd_mult_cancel_left
tff(fact_4168_dvd__add__times__triv__left__iff,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).
% dvd_add_times_triv_left_iff
tff(fact_4169_dvd__add__times__triv__left__iff,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(dvd_dvd_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),B))
<=> aa_rat_o(dvd_dvd_rat(A),B) ) ).
% dvd_add_times_triv_left_iff
tff(fact_4170_dvd__add__times__triv__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),B))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).
% dvd_add_times_triv_left_iff
tff(fact_4171_dvd__add__times__triv__left__iff,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),B))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).
% dvd_add_times_triv_left_iff
tff(fact_4172_dvd__add__times__triv__right__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).
% dvd_add_times_triv_right_iff
tff(fact_4173_dvd__add__times__triv__right__iff,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(dvd_dvd_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)))
<=> aa_rat_o(dvd_dvd_rat(A),B) ) ).
% dvd_add_times_triv_right_iff
tff(fact_4174_dvd__add__times__triv__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).
% dvd_add_times_triv_right_iff
tff(fact_4175_dvd__add__times__triv__right__iff,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).
% dvd_add_times_triv_right_iff
tff(fact_4176_unit__prod,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer) ) ) ).
% unit_prod
tff(fact_4177_unit__prod,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ).
% unit_prod
tff(fact_4178_unit__prod,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ).
% unit_prod
tff(fact_4179_dvd__div__mult__self,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).
% dvd_div_mult_self
tff(fact_4180_dvd__div__mult__self,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).
% dvd_div_mult_self
tff(fact_4181_dvd__div__mult__self,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A)),A) = B ) ) ).
% dvd_div_mult_self
tff(fact_4182_dvd__div__mult__self,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),B)
=> ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(B),A)),A) = B ) ) ).
% dvd_div_mult_self
tff(fact_4183_dvd__mult__div__cancel,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)) = B ) ) ).
% dvd_mult_div_cancel
tff(fact_4184_dvd__mult__div__cancel,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)) = B ) ) ).
% dvd_mult_div_cancel
tff(fact_4185_dvd__mult__div__cancel,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A)) = B ) ) ).
% dvd_mult_div_cancel
tff(fact_4186_dvd__mult__div__cancel,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),B)
=> ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),aa_Cod4272041906116450446atural(divide5121882707175180666atural(B),A)) = B ) ) ).
% dvd_mult_div_cancel
tff(fact_4187_unit__div__1__div__1,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = A ) ) ).
% unit_div_1_div_1
tff(fact_4188_unit__div__1__div__1,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = A ) ) ).
% unit_div_1_div_1
tff(fact_4189_unit__div__1__div__1,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),A)) = A ) ) ).
% unit_div_1_div_1
tff(fact_4190_unit__div__1__div__1,axiom,
! [A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(one_one_Code_natural),aa_Cod4272041906116450446atural(divide5121882707175180666atural(one_one_Code_natural),A)) = A ) ) ).
% unit_div_1_div_1
tff(fact_4191_unit__div__1__unit,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)),one_one_nat) ) ).
% unit_div_1_unit
tff(fact_4192_unit__div__1__unit,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)),one_one_int) ) ).
% unit_div_1_unit
tff(fact_4193_unit__div__1__unit,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),A)),one_one_Code_integer) ) ).
% unit_div_1_unit
tff(fact_4194_unit__div__1__unit,axiom,
! [A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(divide5121882707175180666atural(one_one_Code_natural),A)),one_one_Code_natural) ) ).
% unit_div_1_unit
tff(fact_4195_unit__div,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).
% unit_div
tff(fact_4196_unit__div,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).
% unit_div
tff(fact_4197_unit__div,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),one_one_Code_integer) ) ) ).
% unit_div
tff(fact_4198_unit__div,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),one_one_Code_natural) ) ) ).
% unit_div
tff(fact_4199_signed__take__bit__of__minus__1,axiom,
! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).
% signed_take_bit_of_minus_1
tff(fact_4200_signed__take__bit__of__minus__1,axiom,
! [N: nat] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% signed_take_bit_of_minus_1
tff(fact_4201_signed__take__bit__numeral__of__1,axiom,
! [K: num] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(numeral_numeral_nat(K)),one_one_Code_integer) = one_one_Code_integer ).
% signed_take_bit_numeral_of_1
tff(fact_4202_signed__take__bit__numeral__of__1,axiom,
! [K: num] : aa_int_int(bit_ri631733984087533419it_int(numeral_numeral_nat(K)),one_one_int) = one_one_int ).
% signed_take_bit_numeral_of_1
tff(fact_4203_unit__mult__div__div,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) ) ) ).
% unit_mult_div_div
tff(fact_4204_unit__mult__div__div,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).
% unit_mult_div_div
tff(fact_4205_unit__mult__div__div,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),A)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A) ) ) ).
% unit_mult_div_div
tff(fact_4206_unit__mult__div__div,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
=> ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),aa_Cod4272041906116450446atural(divide5121882707175180666atural(one_one_Code_natural),A)) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(B),A) ) ) ).
% unit_mult_div_div
tff(fact_4207_unit__div__mult__self,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).
% unit_div_mult_self
tff(fact_4208_unit__div__mult__self,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).
% unit_div_mult_self
tff(fact_4209_unit__div__mult__self,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A)),A) = B ) ) ).
% unit_div_mult_self
tff(fact_4210_unit__div__mult__self,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
=> ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(B),A)),A) = B ) ) ).
% unit_div_mult_self
tff(fact_4211_even__mult__iff,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
| aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B) ) ) ).
% even_mult_iff
tff(fact_4212_even__mult__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A)
| aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),B) ) ) ).
% even_mult_iff
tff(fact_4213_even__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).
% even_mult_iff
tff(fact_4214_even__mult__iff,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B))
<=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A)
| aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),B) ) ) ).
% even_mult_iff
tff(fact_4215_even__plus__one__iff,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))
<=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A) ) ).
% even_plus_one_iff
tff(fact_4216_even__plus__one__iff,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))
<=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A) ) ).
% even_plus_one_iff
tff(fact_4217_even__plus__one__iff,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer))
<=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).
% even_plus_one_iff
tff(fact_4218_even__plus__one__iff,axiom,
! [A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),one_one_Code_natural))
<=> ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A) ) ).
% even_plus_one_iff
tff(fact_4219_neg__one__even__power,axiom,
! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = one_one_int ) ) ).
% neg_one_even_power
tff(fact_4220_neg__one__even__power,axiom,
! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
=> ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) = one_one_rat ) ) ).
% neg_one_even_power
tff(fact_4221_neg__one__even__power,axiom,
! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = one_one_Code_integer ) ) ).
% neg_one_even_power
tff(fact_4222_neg__one__odd__power,axiom,
! [N: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).
% neg_one_odd_power
tff(fact_4223_neg__one__odd__power,axiom,
! [N: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
=> ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).
% neg_one_odd_power
tff(fact_4224_neg__one__odd__power,axiom,
! [N: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).
% neg_one_odd_power
tff(fact_4225_even__succ__div__2,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) ) ) ).
% even_succ_div_2
tff(fact_4226_even__succ__div__2,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A)
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),numeral_numeral_int(aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(aa_num_num(bit0,one))) ) ) ).
% even_succ_div_2
tff(fact_4227_even__succ__div__2,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).
% even_succ_div_2
tff(fact_4228_even__succ__div__2,axiom,
! [A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A)
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),A)),numera5444537566228673987atural(aa_num_num(bit0,one))) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))) ) ) ).
% even_succ_div_2
tff(fact_4229_even__succ__div__two,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) ) ) ).
% even_succ_div_two
tff(fact_4230_even__succ__div__two,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A)
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),numeral_numeral_int(aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(aa_num_num(bit0,one))) ) ) ).
% even_succ_div_two
tff(fact_4231_even__succ__div__two,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).
% even_succ_div_two
tff(fact_4232_even__succ__div__two,axiom,
! [A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A)
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),one_one_Code_natural)),numera5444537566228673987atural(aa_num_num(bit0,one))) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))) ) ) ).
% even_succ_div_two
tff(fact_4233_odd__succ__div__two,axiom,
! [A: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_nat) ) ) ).
% odd_succ_div_two
tff(fact_4234_odd__succ__div__two,axiom,
! [A: int] :
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A)
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),numeral_numeral_int(aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(aa_num_num(bit0,one)))),one_one_int) ) ) ).
% odd_succ_div_two
tff(fact_4235_odd__succ__div__two,axiom,
! [A: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).
% odd_succ_div_two
tff(fact_4236_odd__succ__div__two,axiom,
! [A: code_natural] :
( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A)
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),one_one_Code_natural)),numera5444537566228673987atural(aa_num_num(bit0,one))) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),numera5444537566228673987atural(aa_num_num(bit0,one)))),one_one_Code_natural) ) ) ).
% odd_succ_div_two
tff(fact_4237_odd__two__times__div__two__succ,axiom,
! [A: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))))),one_one_nat) = A ) ) ).
% odd_two_times_div_two_succ
tff(fact_4238_odd__two__times__div__two__succ,axiom,
! [A: int] :
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A)
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(aa_num_num(bit0,one))))),one_one_int) = A ) ) ).
% odd_two_times_div_two_succ
tff(fact_4239_odd__two__times__div__two__succ,axiom,
! [A: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))))),one_one_Code_integer) = A ) ) ).
% odd_two_times_div_two_succ
tff(fact_4240_odd__two__times__div__two__succ,axiom,
! [A: code_natural] :
( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A)
=> ( aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))))),one_one_Code_natural) = A ) ) ).
% odd_two_times_div_two_succ
tff(fact_4241_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),one_one_nat))
<=> ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
tff(fact_4242_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)),one_one_int))
<=> ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
tff(fact_4243_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)),one_one_Code_integer))
<=> ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
tff(fact_4244_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)),one_one_Code_natural))
<=> ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
tff(fact_4245_even__succ__div__exp,axiom,
! [A: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ) ) ) ).
% even_succ_div_exp
tff(fact_4246_even__succ__div__exp,axiom,
! [A: int,N: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)) ) ) ) ).
% even_succ_div_exp
tff(fact_4247_even__succ__div__exp,axiom,
! [A: code_integer,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ) ) ) ).
% even_succ_div_exp
tff(fact_4248_even__succ__div__exp,axiom,
! [A: code_natural,N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),A)),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) ) ) ) ).
% even_succ_div_exp
tff(fact_4249_dvd__triv__right,axiom,
! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) ).
% dvd_triv_right
tff(fact_4250_dvd__triv__right,axiom,
! [A: assn,B: assn] : aa_assn_o(dvd_dvd_assn(A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A)) ).
% dvd_triv_right
tff(fact_4251_dvd__triv__right,axiom,
! [A: rat,B: rat] : aa_rat_o(dvd_dvd_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)) ).
% dvd_triv_right
tff(fact_4252_dvd__triv__right,axiom,
! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) ).
% dvd_triv_right
tff(fact_4253_dvd__triv__right,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) ).
% dvd_triv_right
tff(fact_4254_dvd__mult__right,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ).
% dvd_mult_right
tff(fact_4255_dvd__mult__right,axiom,
! [A: assn,B: assn,C: assn] :
( aa_assn_o(dvd_dvd_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C)
=> aa_assn_o(dvd_dvd_assn(B),C) ) ).
% dvd_mult_right
tff(fact_4256_dvd__mult__right,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(dvd_dvd_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C)
=> aa_rat_o(dvd_dvd_rat(B),C) ) ).
% dvd_mult_right
tff(fact_4257_dvd__mult__right,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ).
% dvd_mult_right
tff(fact_4258_dvd__mult__right,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ).
% dvd_mult_right
tff(fact_4259_mult__dvd__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ).
% mult_dvd_mono
tff(fact_4260_mult__dvd__mono,axiom,
! [A: assn,B: assn,C: assn,D2: assn] :
( aa_assn_o(dvd_dvd_assn(A),B)
=> ( aa_assn_o(dvd_dvd_assn(C),D2)
=> aa_assn_o(dvd_dvd_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),D2)) ) ) ).
% mult_dvd_mono
tff(fact_4261_mult__dvd__mono,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( aa_rat_o(dvd_dvd_rat(A),B)
=> ( aa_rat_o(dvd_dvd_rat(C),D2)
=> aa_rat_o(dvd_dvd_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ).
% mult_dvd_mono
tff(fact_4262_mult__dvd__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D2)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ).
% mult_dvd_mono
tff(fact_4263_mult__dvd__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D2)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ).
% mult_dvd_mono
tff(fact_4264_dvd__triv__left,axiom,
! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).
% dvd_triv_left
tff(fact_4265_dvd__triv__left,axiom,
! [A: assn,B: assn] : aa_assn_o(dvd_dvd_assn(A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)) ).
% dvd_triv_left
tff(fact_4266_dvd__triv__left,axiom,
! [A: rat,B: rat] : aa_rat_o(dvd_dvd_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).
% dvd_triv_left
tff(fact_4267_dvd__triv__left,axiom,
! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ).
% dvd_triv_left
tff(fact_4268_dvd__triv__left,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).
% dvd_triv_left
tff(fact_4269_dvd__mult__left,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ).
% dvd_mult_left
tff(fact_4270_dvd__mult__left,axiom,
! [A: assn,B: assn,C: assn] :
( aa_assn_o(dvd_dvd_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C)
=> aa_assn_o(dvd_dvd_assn(A),C) ) ).
% dvd_mult_left
tff(fact_4271_dvd__mult__left,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(dvd_dvd_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C)
=> aa_rat_o(dvd_dvd_rat(A),C) ) ).
% dvd_mult_left
tff(fact_4272_dvd__mult__left,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ).
% dvd_mult_left
tff(fact_4273_dvd__mult__left,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ).
% dvd_mult_left
tff(fact_4274_dvd__mult2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).
% dvd_mult2
tff(fact_4275_dvd__mult2,axiom,
! [A: assn,B: assn,C: assn] :
( aa_assn_o(dvd_dvd_assn(A),B)
=> aa_assn_o(dvd_dvd_assn(A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ) ).
% dvd_mult2
tff(fact_4276_dvd__mult2,axiom,
! [A: rat,B: rat,C: rat] :
( aa_rat_o(dvd_dvd_rat(A),B)
=> aa_rat_o(dvd_dvd_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).
% dvd_mult2
tff(fact_4277_dvd__mult2,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).
% dvd_mult2
tff(fact_4278_dvd__mult2,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).
% dvd_mult2
tff(fact_4279_dvd__mult,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).
% dvd_mult
tff(fact_4280_dvd__mult,axiom,
! [A: assn,C: assn,B: assn] :
( aa_assn_o(dvd_dvd_assn(A),C)
=> aa_assn_o(dvd_dvd_assn(A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ) ).
% dvd_mult
tff(fact_4281_dvd__mult,axiom,
! [A: rat,C: rat,B: rat] :
( aa_rat_o(dvd_dvd_rat(A),C)
=> aa_rat_o(dvd_dvd_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).
% dvd_mult
tff(fact_4282_dvd__mult,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).
% dvd_mult
tff(fact_4283_dvd__mult,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).
% dvd_mult
tff(fact_4284_dvd__def,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
<=> ? [K4: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K4) ) ).
% dvd_def
tff(fact_4285_dvd__def,axiom,
! [B: assn,A: assn] :
( aa_assn_o(dvd_dvd_assn(B),A)
<=> ? [K4: assn] : A = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),K4) ) ).
% dvd_def
tff(fact_4286_dvd__def,axiom,
! [B: rat,A: rat] :
( aa_rat_o(dvd_dvd_rat(B),A)
<=> ? [K4: rat] : A = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K4) ) ).
% dvd_def
tff(fact_4287_dvd__def,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
<=> ? [K4: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K4) ) ).
% dvd_def
tff(fact_4288_dvd__def,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
<=> ? [K4: int] : A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K4) ) ).
% dvd_def
tff(fact_4289_dvdI,axiom,
! [A: code_integer,B: code_integer,K: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).
% dvdI
tff(fact_4290_dvdI,axiom,
! [A: assn,B: assn,K: assn] :
( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),K) )
=> aa_assn_o(dvd_dvd_assn(B),A) ) ).
% dvdI
tff(fact_4291_dvdI,axiom,
! [A: rat,B: rat,K: rat] :
( ( A = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K) )
=> aa_rat_o(dvd_dvd_rat(B),A) ) ).
% dvdI
tff(fact_4292_dvdI,axiom,
! [A: nat,B: nat,K: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) )
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).
% dvdI
tff(fact_4293_dvdI,axiom,
! [A: int,B: int,K: int] :
( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K) )
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).
% dvdI
tff(fact_4294_dvdE,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
=> ~ ! [K2: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K2) ) ).
% dvdE
tff(fact_4295_dvdE,axiom,
! [B: assn,A: assn] :
( aa_assn_o(dvd_dvd_assn(B),A)
=> ~ ! [K2: assn] : A != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),K2) ) ).
% dvdE
tff(fact_4296_dvdE,axiom,
! [B: rat,A: rat] :
( aa_rat_o(dvd_dvd_rat(B),A)
=> ~ ! [K2: rat] : A != aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K2) ) ).
% dvdE
tff(fact_4297_dvdE,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
=> ~ ! [K2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) ) ).
% dvdE
tff(fact_4298_dvdE,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
=> ~ ! [K2: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,B),K2) ) ).
% dvdE
tff(fact_4299_one__dvd,axiom,
! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,one_one_Code_integer),A) ).
% one_dvd
tff(fact_4300_one__dvd,axiom,
! [A: assn] : aa_assn_o(dvd_dvd_assn(one_one_assn),A) ).
% one_dvd
tff(fact_4301_one__dvd,axiom,
! [A: rat] : aa_rat_o(dvd_dvd_rat(one_one_rat),A) ).
% one_dvd
tff(fact_4302_one__dvd,axiom,
! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A) ).
% one_dvd
tff(fact_4303_one__dvd,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A) ).
% one_dvd
tff(fact_4304_unit__imp__dvd,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).
% unit_imp_dvd
tff(fact_4305_unit__imp__dvd,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).
% unit_imp_dvd
tff(fact_4306_unit__imp__dvd,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).
% unit_imp_dvd
tff(fact_4307_dvd__unit__imp__unit,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer) ) ) ).
% dvd_unit_imp_unit
tff(fact_4308_dvd__unit__imp__unit,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat) ) ) ).
% dvd_unit_imp_unit
tff(fact_4309_dvd__unit__imp__unit,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int) ) ) ).
% dvd_unit_imp_unit
tff(fact_4310_not__is__unit__0,axiom,
~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer) ).
% not_is_unit_0
tff(fact_4311_not__is__unit__0,axiom,
~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),one_one_nat) ).
% not_is_unit_0
tff(fact_4312_not__is__unit__0,axiom,
~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),one_one_int) ).
% not_is_unit_0
tff(fact_4313_is__unit__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer) ) ) ).
% is_unit_mult_iff
tff(fact_4314_is__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
& aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat) ) ) ).
% is_unit_mult_iff
tff(fact_4315_is__unit__mult__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
& aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int) ) ) ).
% is_unit_mult_iff
tff(fact_4316_dvd__mult__unit__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).
% dvd_mult_unit_iff
tff(fact_4317_dvd__mult__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).
% dvd_mult_unit_iff
tff(fact_4318_dvd__mult__unit__iff,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).
% dvd_mult_unit_iff
tff(fact_4319_mult__unit__dvd__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).
% mult_unit_dvd_iff
tff(fact_4320_mult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).
% mult_unit_dvd_iff
tff(fact_4321_mult__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).
% mult_unit_dvd_iff
tff(fact_4322_dvd__mult__unit__iff_H,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).
% dvd_mult_unit_iff'
tff(fact_4323_dvd__mult__unit__iff_H,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).
% dvd_mult_unit_iff'
tff(fact_4324_dvd__mult__unit__iff_H,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).
% dvd_mult_unit_iff'
tff(fact_4325_mult__unit__dvd__iff_H,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).
% mult_unit_dvd_iff'
tff(fact_4326_mult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).
% mult_unit_dvd_iff'
tff(fact_4327_mult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).
% mult_unit_dvd_iff'
tff(fact_4328_unit__mult__left__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) )
<=> ( B = C ) ) ) ).
% unit_mult_left_cancel
tff(fact_4329_unit__mult__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) )
<=> ( B = C ) ) ) ).
% unit_mult_left_cancel
tff(fact_4330_unit__mult__left__cancel,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),C) )
<=> ( B = C ) ) ) ).
% unit_mult_left_cancel
tff(fact_4331_unit__mult__right__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) )
<=> ( B = C ) ) ) ).
% unit_mult_right_cancel
tff(fact_4332_unit__mult__right__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) )
<=> ( B = C ) ) ) ).
% unit_mult_right_cancel
tff(fact_4333_unit__mult__right__cancel,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( ( aa_int_int(aa_int_fun_int_int(times_times_int,B),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) )
<=> ( B = C ) ) ) ).
% unit_mult_right_cancel
tff(fact_4334_dvd__div__mult,axiom,
! [C: nat,B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) ) ) ).
% dvd_div_mult
tff(fact_4335_dvd__div__mult,axiom,
! [C: int,B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C) ) ) ).
% dvd_div_mult
tff(fact_4336_dvd__div__mult,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)),A) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),C) ) ) ).
% dvd_div_mult
tff(fact_4337_dvd__div__mult,axiom,
! [C: code_natural,B: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),B)
=> ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(B),C)),A) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),A)),C) ) ) ).
% dvd_div_mult
tff(fact_4338_div__mult__swap,axiom,
! [C: nat,B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).
% div_mult_swap
tff(fact_4339_div__mult__swap,axiom,
! [C: int,B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).
% div_mult_swap
tff(fact_4340_div__mult__swap,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) ) ) ).
% div_mult_swap
tff(fact_4341_div__mult__swap,axiom,
! [C: code_natural,B: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),B)
=> ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),aa_Cod4272041906116450446atural(divide5121882707175180666atural(B),C)) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),C) ) ) ).
% div_mult_swap
tff(fact_4342_div__div__eq__right,axiom,
! [C: nat,B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).
% div_div_eq_right
tff(fact_4343_div__div__eq__right,axiom,
! [C: int,B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).
% div_div_eq_right
tff(fact_4344_div__div__eq__right,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C) ) ) ) ).
% div_div_eq_right
tff(fact_4345_div__div__eq__right,axiom,
! [C: code_natural,B: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),A)
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),aa_Cod4272041906116450446atural(divide5121882707175180666atural(B),C)) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),C) ) ) ) ).
% div_div_eq_right
tff(fact_4346_dvd__div__mult2__eq,axiom,
! [B: nat,C: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).
% dvd_div_mult2_eq
tff(fact_4347_dvd__div__mult2__eq,axiom,
! [B: int,C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).
% dvd_div_mult2_eq
tff(fact_4348_dvd__div__mult2__eq,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C) ) ) ).
% dvd_div_mult2_eq
tff(fact_4349_dvd__div__mult2__eq,axiom,
! [B: code_natural,C: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)),A)
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),C) ) ) ).
% dvd_div_mult2_eq
tff(fact_4350_dvd__mult__imp__div,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ).
% dvd_mult_imp_div
tff(fact_4351_dvd__mult__imp__div,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ).
% dvd_mult_imp_div
tff(fact_4352_dvd__mult__imp__div,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)) ) ).
% dvd_mult_imp_div
tff(fact_4353_dvd__mult__imp__div,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),B)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),aa_Cod4272041906116450446atural(divide5121882707175180666atural(B),C)) ) ).
% dvd_mult_imp_div
tff(fact_4354_div__mult__div__if__dvd,axiom,
! [B: nat,A: nat,D2: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),C)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ).
% div_mult_div_if_dvd
tff(fact_4355_div__mult__div__if__dvd,axiom,
! [B: int,A: int,D2: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),C)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),D2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ).
% div_mult_div_if_dvd
tff(fact_4356_div__mult__div__if__dvd,axiom,
! [B: code_integer,A: code_integer,D2: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),C)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),D2)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ).
% div_mult_div_if_dvd
tff(fact_4357_div__mult__div__if__dvd,axiom,
! [B: code_natural,A: code_natural,D2: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,D2),C)
=> ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),aa_Cod4272041906116450446atural(divide5121882707175180666atural(C),D2)) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),D2)) ) ) ) ).
% div_mult_div_if_dvd
tff(fact_4358_dvd__div__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).
% dvd_div_unit_iff
tff(fact_4359_dvd__div__unit__iff,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).
% dvd_div_unit_iff
tff(fact_4360_dvd__div__unit__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),B))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).
% dvd_div_unit_iff
tff(fact_4361_dvd__div__unit__iff,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),aa_Cod4272041906116450446atural(divide5121882707175180666atural(C),B))
<=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),C) ) ) ).
% dvd_div_unit_iff
tff(fact_4362_div__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).
% div_unit_dvd_iff
tff(fact_4363_div__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C)
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).
% div_unit_dvd_iff
tff(fact_4364_div__unit__dvd__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).
% div_unit_dvd_iff
tff(fact_4365_div__unit__dvd__iff,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),C)
<=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),C) ) ) ).
% div_unit_dvd_iff
tff(fact_4366_unit__div__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
<=> ( B = C ) ) ) ).
% unit_div_cancel
tff(fact_4367_unit__div__cancel,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) )
<=> ( B = C ) ) ) ).
% unit_div_cancel
tff(fact_4368_unit__div__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),A) )
<=> ( B = C ) ) ) ).
% unit_div_cancel
tff(fact_4369_unit__div__cancel,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
=> ( ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(B),A) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(C),A) )
<=> ( B = C ) ) ) ).
% unit_div_cancel
tff(fact_4370_unity__coeff__ex,axiom,
! [P: fun_Code_integer_o,L: code_integer] :
( ? [X3: code_integer] : aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X3))
<=> ? [X3: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X3),zero_z3403309356797280102nteger))
& aa_Code_integer_o(P,X3) ) ) ).
% unity_coeff_ex
tff(fact_4371_unity__coeff__ex,axiom,
! [P: fun_rat_o,L: rat] :
( ? [X3: rat] : aa_rat_o(P,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,L),X3))
<=> ? [X3: rat] :
( aa_rat_o(dvd_dvd_rat(L),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),zero_zero_rat))
& aa_rat_o(P,X3) ) ) ).
% unity_coeff_ex
tff(fact_4372_unity__coeff__ex,axiom,
! [P: fun_nat_o,L: nat] :
( ? [X3: nat] : aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X3))
<=> ? [X3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),zero_zero_nat))
& aa_nat_o(P,X3) ) ) ).
% unity_coeff_ex
tff(fact_4373_unity__coeff__ex,axiom,
! [P: fun_int_o,L: int] :
( ? [X3: int] : aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X3))
<=> ? [X3: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),zero_zero_int))
& aa_int_o(P,X3) ) ) ).
% unity_coeff_ex
tff(fact_4374_unit__dvdE,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ~ ( ( A != zero_z3403309356797280102nteger )
=> ! [C2: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2) ) ) ).
% unit_dvdE
tff(fact_4375_unit__dvdE,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ~ ( ( A != zero_zero_nat )
=> ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) ) ) ).
% unit_dvdE
tff(fact_4376_unit__dvdE,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ~ ( ( A != zero_zero_int )
=> ! [C2: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) ) ) ).
% unit_dvdE
tff(fact_4377_inf__period_I3_J,axiom,
! [D2: code_integer,D3: code_integer,T: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),D3)
=> ! [X4: code_integer,K3: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X4),T))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X4),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D3))),T)) ) ) ).
% inf_period(3)
tff(fact_4378_inf__period_I3_J,axiom,
! [D2: rat,D3: rat,T: rat] :
( aa_rat_o(dvd_dvd_rat(D2),D3)
=> ! [X4: rat,K3: rat] :
( aa_rat_o(dvd_dvd_rat(D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),T))
<=> aa_rat_o(dvd_dvd_rat(D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(X4,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D3))),T)) ) ) ).
% inf_period(3)
tff(fact_4379_inf__period_I3_J,axiom,
! [D2: int,D3: int,T: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D3)
=> ! [X4: int,K3: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),T))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3))),T)) ) ) ).
% inf_period(3)
tff(fact_4380_inf__period_I4_J,axiom,
! [D2: code_integer,D3: code_integer,T: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),D3)
=> ! [X4: code_integer,K3: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X4),T))
<=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X4),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D3))),T)) ) ) ).
% inf_period(4)
tff(fact_4381_inf__period_I4_J,axiom,
! [D2: rat,D3: rat,T: rat] :
( aa_rat_o(dvd_dvd_rat(D2),D3)
=> ! [X4: rat,K3: rat] :
( ~ aa_rat_o(dvd_dvd_rat(D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),T))
<=> ~ aa_rat_o(dvd_dvd_rat(D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(X4,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D3))),T)) ) ) ).
% inf_period(4)
tff(fact_4382_inf__period_I4_J,axiom,
! [D2: int,D3: int,T: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D3)
=> ! [X4: int,K3: int] :
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),T))
<=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3))),T)) ) ) ).
% inf_period(4)
tff(fact_4383_dvd__div__eq__mult,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C )
<=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).
% dvd_div_eq_mult
tff(fact_4384_dvd__div__eq__mult,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = C )
<=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).
% dvd_div_eq_mult
tff(fact_4385_dvd__div__eq__mult,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
=> ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A) = C )
<=> ( B = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) ) ) ) ) ).
% dvd_div_eq_mult
tff(fact_4386_dvd__div__eq__mult,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( ( A != zero_z2226904508553997617atural )
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),B)
=> ( ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(B),A) = C )
<=> ( B = aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),A) ) ) ) ) ).
% dvd_div_eq_mult
tff(fact_4387_div__dvd__iff__mult,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ) ).
% div_dvd_iff_mult
tff(fact_4388_div__dvd__iff__mult,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C)
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ) ).
% div_dvd_iff_mult
tff(fact_4389_div__dvd__iff__mult,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ) ).
% div_dvd_iff_mult
tff(fact_4390_div__dvd__iff__mult,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),C)
<=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B)) ) ) ) ).
% div_dvd_iff_mult
tff(fact_4391_dvd__div__iff__mult,axiom,
! [C: nat,B: nat,A: nat] :
( ( C != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ) ).
% dvd_div_iff_mult
tff(fact_4392_dvd__div__iff__mult,axiom,
! [C: int,B: int,A: int] :
( ( C != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))
<=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ) ).
% dvd_div_iff_mult
tff(fact_4393_dvd__div__iff__mult,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C))
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ) ) ) ).
% dvd_div_iff_mult
tff(fact_4394_dvd__div__iff__mult,axiom,
! [C: code_natural,B: code_natural,A: code_natural] :
( ( C != zero_z2226904508553997617atural )
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),aa_Cod4272041906116450446atural(divide5121882707175180666atural(B),C))
<=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),B) ) ) ) ).
% dvd_div_iff_mult
tff(fact_4395_dvd__div__div__eq__mult,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( ( A != zero_zero_nat )
=> ( ( C != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D2)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D2),C) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D2) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
tff(fact_4396_dvd__div__div__eq__mult,axiom,
! [A: int,C: int,B: int,D2: int] :
( ( A != zero_zero_int )
=> ( ( C != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D2)
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D2),C) )
<=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D2) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
tff(fact_4397_dvd__div__div__eq__mult,axiom,
! [A: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( C != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D2)
=> ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),A) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(D2),C) )
<=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D2) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
tff(fact_4398_dvd__div__div__eq__mult,axiom,
! [A: code_natural,C: code_natural,B: code_natural,D2: code_natural] :
( ( A != zero_z2226904508553997617atural )
=> ( ( C != zero_z2226904508553997617atural )
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),B)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),D2)
=> ( ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(B),A) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(D2),C) )
<=> ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),D2) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
tff(fact_4399_unit__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
<=> ( A = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
tff(fact_4400_unit__div__eq__0__iff,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
<=> ( A = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
tff(fact_4401_unit__div__eq__0__iff,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) = zero_z3403309356797280102nteger )
<=> ( A = zero_z3403309356797280102nteger ) ) ) ).
% unit_div_eq_0_iff
tff(fact_4402_unit__div__eq__0__iff,axiom,
! [B: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
=> ( ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B) = zero_z2226904508553997617atural )
<=> ( A = zero_z2226904508553997617atural ) ) ) ).
% unit_div_eq_0_iff
tff(fact_4403_unit__eq__div1,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = C )
<=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).
% unit_eq_div1
tff(fact_4404_unit__eq__div1,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = C )
<=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ).
% unit_eq_div1
tff(fact_4405_unit__eq__div1,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) = C )
<=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) ) ) ) ).
% unit_eq_div1
tff(fact_4406_unit__eq__div1,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
=> ( ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B) = C )
<=> ( A = aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B) ) ) ) ).
% unit_eq_div1
tff(fact_4407_unit__eq__div2,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C ) ) ) ).
% unit_eq_div2
tff(fact_4408_unit__eq__div2,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( ( A = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B) )
<=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = C ) ) ) ).
% unit_eq_div2
tff(fact_4409_unit__eq__div2,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( ( A = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),B) )
<=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = C ) ) ) ).
% unit_eq_div2
tff(fact_4410_unit__eq__div2,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
=> ( ( A = aa_Cod4272041906116450446atural(divide5121882707175180666atural(C),B) )
<=> ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B) = C ) ) ) ).
% unit_eq_div2
tff(fact_4411_div__mult__unit2,axiom,
! [C: nat,B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).
% div_mult_unit2
tff(fact_4412_div__mult__unit2,axiom,
! [C: int,B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).
% div_mult_unit2
tff(fact_4413_div__mult__unit2,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C) ) ) ) ).
% div_mult_unit2
tff(fact_4414_div__mult__unit2,axiom,
! [C: code_natural,B: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),one_one_Code_natural)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),A)
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),C) ) ) ) ).
% div_mult_unit2
tff(fact_4415_unit__div__commute,axiom,
! [B: nat,A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ).
% unit_div_commute
tff(fact_4416_unit__div__commute,axiom,
! [B: int,A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ).
% unit_div_commute
tff(fact_4417_unit__div__commute,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ) ) ).
% unit_div_commute
tff(fact_4418_unit__div__commute,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
=> ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),C) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),C)),B) ) ) ).
% unit_div_commute
tff(fact_4419_unit__div__mult__swap,axiom,
! [C: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).
% unit_div_mult_swap
tff(fact_4420_unit__div__mult__swap,axiom,
! [C: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).
% unit_div_mult_swap
tff(fact_4421_unit__div__mult__swap,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) ) ) ).
% unit_div_mult_swap
tff(fact_4422_unit__div__mult__swap,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),one_one_Code_natural)
=> ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),aa_Cod4272041906116450446atural(divide5121882707175180666atural(B),C)) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)),C) ) ) ).
% unit_div_mult_swap
tff(fact_4423_is__unit__div__mult2__eq,axiom,
! [B: nat,C: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).
% is_unit_div_mult2_eq
tff(fact_4424_is__unit__div__mult2__eq,axiom,
! [B: int,C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).
% is_unit_div_mult2_eq
tff(fact_4425_is__unit__div__mult2__eq,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),C) ) ) ) ).
% is_unit_div_mult2_eq
tff(fact_4426_is__unit__div__mult2__eq,axiom,
! [B: code_natural,C: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,C),one_one_Code_natural)
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),C)) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),C) ) ) ) ).
% is_unit_div_mult2_eq
tff(fact_4427_unit__imp__mod__eq__0,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat ) ) ).
% unit_imp_mod_eq_0
tff(fact_4428_unit__imp__mod__eq__0,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int ) ) ).
% unit_imp_mod_eq_0
tff(fact_4429_unit__imp__mod__eq__0,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) = zero_z3403309356797280102nteger ) ) ).
% unit_imp_mod_eq_0
tff(fact_4430_unit__imp__mod__eq__0,axiom,
! [B: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
=> ( aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B) = zero_z2226904508553997617atural ) ) ).
% unit_imp_mod_eq_0
tff(fact_4431_is__unit__power__iff,axiom,
! [A: code_integer,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),one_one_Code_integer)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
tff(fact_4432_is__unit__power__iff,axiom,
! [A: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat)
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
tff(fact_4433_is__unit__power__iff,axiom,
! [A: int,N: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),one_one_int)
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
tff(fact_4434_is__unit__div__mult__cancel__right,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).
% is_unit_div_mult_cancel_right
tff(fact_4435_is__unit__div__mult__cancel__right,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).
% is_unit_div_mult_cancel_right
tff(fact_4436_is__unit__div__mult__cancel__right,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),B) ) ) ) ).
% is_unit_div_mult_cancel_right
tff(fact_4437_is__unit__div__mult__cancel__right,axiom,
! [A: code_natural,B: code_natural] :
( ( A != zero_z2226904508553997617atural )
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(B),A)) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(one_one_Code_natural),B) ) ) ) ).
% is_unit_div_mult_cancel_right
tff(fact_4438_is__unit__div__mult__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).
% is_unit_div_mult_cancel_left
tff(fact_4439_is__unit__div__mult__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).
% is_unit_div_mult_cancel_left
tff(fact_4440_is__unit__div__mult__cancel__left,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),B) ) ) ) ).
% is_unit_div_mult_cancel_left
tff(fact_4441_is__unit__div__mult__cancel__left,axiom,
! [A: code_natural,B: code_natural] :
( ( A != zero_z2226904508553997617atural )
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B)) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(one_one_Code_natural),B) ) ) ) ).
% is_unit_div_mult_cancel_left
tff(fact_4442_is__unitE,axiom,
! [A: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ~ ( ( A != zero_zero_nat )
=> ! [B3: nat] :
( ( B3 != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B3),one_one_nat)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B3 )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B3) = A )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B3) = one_one_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3) ) ) ) ) ) ) ) ) ).
% is_unitE
tff(fact_4443_is__unitE,axiom,
! [A: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ~ ( ( A != zero_zero_int )
=> ! [B3: int] :
( ( B3 != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B3),one_one_int)
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A) = B3 )
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B3) = A )
=> ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B3) = one_one_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B3) ) ) ) ) ) ) ) ) ).
% is_unitE
tff(fact_4444_is__unitE,axiom,
! [A: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ~ ( ( A != zero_z3403309356797280102nteger )
=> ! [B3: code_integer] :
( ( B3 != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B3),one_one_Code_integer)
=> ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),A) = B3 )
=> ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),B3) = A )
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B3) = one_one_Code_integer )
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B3) ) ) ) ) ) ) ) ) ).
% is_unitE
tff(fact_4445_is__unitE,axiom,
! [A: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A),one_one_Code_natural)
=> ~ ( ( A != zero_z2226904508553997617atural )
=> ! [B3: code_natural] :
( ( B3 != zero_z2226904508553997617atural )
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B3),one_one_Code_natural)
=> ( ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(one_one_Code_natural),A) = B3 )
=> ( ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(one_one_Code_natural),B3) = A )
=> ( ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),B3) = one_one_Code_natural )
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(C),A) != aa_Cod4272041906116450446atural(times_2397367101498566445atural(C),B3) ) ) ) ) ) ) ) ) ).
% is_unitE
tff(fact_4446_evenE,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
=> ~ ! [B3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B3) ) ).
% evenE
tff(fact_4447_evenE,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A)
=> ~ ! [B3: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),B3) ) ).
% evenE
tff(fact_4448_evenE,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
=> ~ ! [B3: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B3) ) ).
% evenE
tff(fact_4449_evenE,axiom,
! [A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A)
=> ~ ! [B3: code_natural] : A != aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),B3) ) ).
% evenE
tff(fact_4450_odd__one,axiom,
~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),one_one_nat) ).
% odd_one
tff(fact_4451_odd__one,axiom,
~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),one_one_int) ).
% odd_one
tff(fact_4452_odd__one,axiom,
~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer) ).
% odd_one
tff(fact_4453_odd__one,axiom,
~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),one_one_Code_natural) ).
% odd_one
tff(fact_4454_dvd__power__iff,axiom,
! [X: code_integer,M2: nat,N: nat] :
( ( X != zero_z3403309356797280102nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),M2)),aa_nat_Code_integer(power_8256067586552552935nteger(X),N))
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),one_one_Code_integer)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ) ).
% dvd_power_iff
tff(fact_4455_dvd__power__iff,axiom,
! [X: nat,M2: nat,N: nat] :
( ( X != zero_zero_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N))
<=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),one_one_nat)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ) ).
% dvd_power_iff
tff(fact_4456_dvd__power__iff,axiom,
! [X: int,M2: nat,N: nat] :
( ( X != zero_zero_int )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(X),M2)),aa_nat_int(power_power_int(X),N))
<=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),one_one_int)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ) ).
% dvd_power_iff
tff(fact_4457_dvd__power,axiom,
! [N: nat,X: code_integer] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
| ( X = one_one_Code_integer ) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),aa_nat_Code_integer(power_8256067586552552935nteger(X),N)) ) ).
% dvd_power
tff(fact_4458_dvd__power,axiom,
! [N: nat,X: rat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
| ( X = one_one_rat ) )
=> aa_rat_o(dvd_dvd_rat(X),aa_nat_rat(power_power_rat(X),N)) ) ).
% dvd_power
tff(fact_4459_dvd__power,axiom,
! [N: nat,X: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
| ( X = one_one_nat ) )
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)) ) ).
% dvd_power
tff(fact_4460_dvd__power,axiom,
! [N: nat,X: int] :
( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
| ( X = one_one_int ) )
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),aa_nat_int(power_power_int(X),N)) ) ).
% dvd_power
tff(fact_4461_even__two__times__div__two,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))) = A ) ) ).
% even_two_times_div_two
tff(fact_4462_even__two__times__div__two,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(aa_num_num(bit0,one)))) = A ) ) ).
% even_two_times_div_two
tff(fact_4463_even__two__times__div__two,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one)))) = A ) ) ).
% even_two_times_div_two
tff(fact_4464_even__two__times__div__two,axiom,
! [A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A)
=> ( aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),numera5444537566228673987atural(aa_num_num(bit0,one)))) = A ) ) ).
% even_two_times_div_two
tff(fact_4465_odd__iff__mod__2__eq__one,axiom,
! [A: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_nat ) ) ).
% odd_iff_mod_2_eq_one
tff(fact_4466_odd__iff__mod__2__eq__one,axiom,
! [A: int] :
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A)
<=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one))) = one_one_int ) ) ).
% odd_iff_mod_2_eq_one
tff(fact_4467_odd__iff__mod__2__eq__one,axiom,
! [A: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
<=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).
% odd_iff_mod_2_eq_one
tff(fact_4468_odd__iff__mod__2__eq__one,axiom,
! [A: code_natural] :
( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A)
<=> ( aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))) = one_one_Code_natural ) ) ).
% odd_iff_mod_2_eq_one
tff(fact_4469_oddE,axiom,
! [A: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
=> ~ ! [B3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B3)),one_one_nat) ) ).
% oddE
tff(fact_4470_oddE,axiom,
! [A: int] :
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A)
=> ~ ! [B3: int] : A != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),B3)),one_one_int) ) ).
% oddE
tff(fact_4471_oddE,axiom,
! [A: code_integer] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
=> ~ ! [B3: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B3)),one_one_Code_integer) ) ).
% oddE
tff(fact_4472_oddE,axiom,
! [A: code_natural] :
( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A)
=> ~ ! [B3: code_natural] : A != aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),B3)),one_one_Code_natural) ) ).
% oddE
tff(fact_4473_mod2__eq__if,axiom,
! [A: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A),zero_zero_nat,one_one_nat) ).
% mod2_eq_if
tff(fact_4474_mod2__eq__if,axiom,
! [A: int] :
aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A),zero_zero_int,one_one_int) ).
% mod2_eq_if
tff(fact_4475_mod2__eq__if,axiom,
! [A: code_integer] :
aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,one_one_Code_integer) ).
% mod2_eq_if
tff(fact_4476_mod2__eq__if,axiom,
! [A: code_natural] :
aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))) = $ite(aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A),zero_z2226904508553997617atural,one_one_Code_natural) ).
% mod2_eq_if
tff(fact_4477_parity__cases,axiom,
! [A: nat] :
( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) != zero_zero_nat ) )
=> ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) != one_one_nat ) ) ) ).
% parity_cases
tff(fact_4478_parity__cases,axiom,
! [A: int] :
( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A)
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one))) != zero_zero_int ) )
=> ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A)
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one))) != one_one_int ) ) ) ).
% parity_cases
tff(fact_4479_parity__cases,axiom,
! [A: code_integer] :
( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
=> ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).
% parity_cases
tff(fact_4480_parity__cases,axiom,
! [A: code_natural] :
( ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A)
=> ( aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))) != zero_z2226904508553997617atural ) )
=> ~ ( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A)
=> ( aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))) != one_one_Code_natural ) ) ) ).
% parity_cases
tff(fact_4481_minus__one__power__iff,axiom,
! [N: nat] :
aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).
% minus_one_power_iff
tff(fact_4482_minus__one__power__iff,axiom,
! [N: nat] :
aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).
% minus_one_power_iff
tff(fact_4483_minus__one__power__iff,axiom,
! [N: nat] :
aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% minus_one_power_iff
tff(fact_4484_even__mask__div__iff_H,axiom,
! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ).
% even_mask_div_iff'
tff(fact_4485_even__mask__div__iff_H,axiom,
! [M2: nat,N: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),M2)),one_one_int)),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ).
% even_mask_div_iff'
tff(fact_4486_even__mask__div__iff_H,axiom,
! [M2: nat,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M2)),one_one_Code_integer)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ).
% even_mask_div_iff'
tff(fact_4487_even__mask__div__iff_H,axiom,
! [M2: nat,N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M2)),one_one_Code_natural)),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ).
% even_mask_div_iff'
tff(fact_4488_even__mask__div__iff,axiom,
! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))
<=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N) = zero_zero_nat )
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).
% even_mask_div_iff
tff(fact_4489_even__mask__div__iff,axiom,
! [M2: nat,N: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),M2)),one_one_int)),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)))
<=> ( ( aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N) = zero_zero_int )
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).
% even_mask_div_iff
tff(fact_4490_even__mask__div__iff,axiom,
! [M2: nat,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M2)),one_one_Code_integer)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)))
<=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) = zero_z3403309356797280102nteger )
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).
% even_mask_div_iff
tff(fact_4491_even__mask__div__iff,axiom,
! [M2: nat,N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M2)),one_one_Code_natural)),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)))
<=> ( ( aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N) = zero_z2226904508553997617atural )
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).
% even_mask_div_iff
tff(fact_4492_even__mult__exp__div__exp__iff,axiom,
! [A: nat,M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2)
| ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N) = zero_zero_nat )
| ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
& aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ) ).
% even_mult_exp_div_exp_iff
tff(fact_4493_even__mult__exp__div__exp__iff,axiom,
! [A: int,M2: nat,N: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),M2))),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2)
| ( aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N) = zero_zero_int )
| ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
& aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ) ).
% even_mult_exp_div_exp_iff
tff(fact_4494_even__mult__exp__div__exp__iff,axiom,
! [A: code_integer,M2: nat,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M2))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2)
| ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) = zero_z3403309356797280102nteger )
| ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ) ).
% even_mult_exp_div_exp_iff
tff(fact_4495_even__mult__exp__div__exp__iff,axiom,
! [A: code_natural,M2: nat,N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M2))),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2)
| ( aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N) = zero_z2226904508553997617atural )
| ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
& aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ) ).
% even_mult_exp_div_exp_iff
tff(fact_4496_take__bit__rec,axiom,
! [N: nat,A: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = $ite(N = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))))),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))))) ).
% take_bit_rec
tff(fact_4497_take__bit__rec,axiom,
! [N: nat,A: int] :
aa_int_int(bit_se2923211474154528505it_int(N),A) = $ite(N = zero_zero_nat,zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(aa_num_num(bit0,one))))),numeral_numeral_int(aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one))))) ).
% take_bit_rec
tff(fact_4498_take__bit__rec,axiom,
! [N: nat,A: code_integer] :
aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),A) = $ite(N = zero_zero_nat,zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))))),numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).
% take_bit_rec
tff(fact_4499_take__bit__rec,axiom,
! [N: nat,A: code_natural] :
aa_Cod4272041906116450446atural(bit_se569199155075624693atural(N),A) = $ite(N = zero_zero_nat,zero_z2226904508553997617atural,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(bit_se569199155075624693atural(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))))),numera5444537566228673987atural(aa_num_num(bit0,one)))),aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))))) ).
% take_bit_rec
tff(fact_4500_divmod__algorithm__code_I7_J,axiom,
! [M2: num,N: num] :
unique5052692396658037445od_int(aa_num_num(bit0,M2),aa_num_num(bit1,N)) = $ite(ord_less_eq_num(M2,N),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),numeral_numeral_int(aa_num_num(bit0,M2))),unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit0,M2),aa_num_num(bit0,aa_num_num(bit1,N))))) ).
% divmod_algorithm_code(7)
tff(fact_4501_divmod__algorithm__code_I7_J,axiom,
! [M2: num,N: num] :
unique5055182867167087721od_nat(aa_num_num(bit0,M2),aa_num_num(bit1,N)) = $ite(ord_less_eq_num(M2,N),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),numeral_numeral_nat(aa_num_num(bit0,M2))),unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit0,M2),aa_num_num(bit0,aa_num_num(bit1,N))))) ).
% divmod_algorithm_code(7)
tff(fact_4502_divmod__algorithm__code_I7_J,axiom,
! [M2: num,N: num] :
unique3479559517661332726nteger(aa_num_num(bit0,M2),aa_num_num(bit1,N)) = $ite(ord_less_eq_num(M2,N),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,M2))),unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit0,M2),aa_num_num(bit0,aa_num_num(bit1,N))))) ).
% divmod_algorithm_code(7)
tff(fact_4503_divmod__algorithm__code_I8_J,axiom,
! [M2: num,N: num] :
unique5052692396658037445od_int(aa_num_num(bit1,M2),aa_num_num(bit1,N)) = $ite(ord_less_num(M2,N),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),numeral_numeral_int(aa_num_num(bit1,M2))),unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit1,M2),aa_num_num(bit0,aa_num_num(bit1,N))))) ).
% divmod_algorithm_code(8)
tff(fact_4504_divmod__algorithm__code_I8_J,axiom,
! [M2: num,N: num] :
unique5055182867167087721od_nat(aa_num_num(bit1,M2),aa_num_num(bit1,N)) = $ite(ord_less_num(M2,N),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),numeral_numeral_nat(aa_num_num(bit1,M2))),unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit1,M2),aa_num_num(bit0,aa_num_num(bit1,N))))) ).
% divmod_algorithm_code(8)
tff(fact_4505_divmod__algorithm__code_I8_J,axiom,
! [M2: num,N: num] :
unique3479559517661332726nteger(aa_num_num(bit1,M2),aa_num_num(bit1,N)) = $ite(ord_less_num(M2,N),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,M2))),unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit1,M2),aa_num_num(bit0,aa_num_num(bit1,N))))) ).
% divmod_algorithm_code(8)
tff(fact_4506_one__mod__2__pow__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).
% one_mod_2_pow_eq
tff(fact_4507_one__mod__2__pow__eq,axiom,
! [N: nat] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).
% one_mod_2_pow_eq
tff(fact_4508_one__mod__2__pow__eq,axiom,
! [N: nat] : aa_Cod4272041906116450446atural(modulo8411746178871703098atural(one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = zero_n8403883297036319079atural(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).
% one_mod_2_pow_eq
tff(fact_4509_one__mod__2__pow__eq,axiom,
! [N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).
% one_mod_2_pow_eq
tff(fact_4510_push__bit__numeral__minus__1,axiom,
! [N: num] : aa_int_int(bit_se545348938243370406it_int(numeral_numeral_nat(N)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),numeral_numeral_nat(N))) ).
% push_bit_numeral_minus_1
tff(fact_4511_push__bit__numeral__minus__1,axiom,
! [N: num] : aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(numeral_numeral_nat(N)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),numeral_numeral_nat(N))) ).
% push_bit_numeral_minus_1
tff(fact_4512_exp__div__exp__eq,axiom,
! [M2: nat,N: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
aa_o_nat(zero_n2687167440665602831ol_nat,
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M2) != zero_zero_nat )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N))) ).
% exp_div_exp_eq
tff(fact_4513_exp__div__exp__eq,axiom,
! [M2: nat,N: nat] :
aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M2)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
aa_o_Code_integer(zero_n356916108424825756nteger,
( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M2) != zero_z3403309356797280102nteger )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N))) ).
% exp_div_exp_eq
tff(fact_4514_exp__div__exp__eq,axiom,
! [M2: nat,N: nat] :
aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M2)),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(zero_n8403883297036319079atural(( ( aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M2) != zero_z2226904508553997617atural )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ))),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N))) ).
% exp_div_exp_eq
tff(fact_4515_exp__div__exp__eq,axiom,
! [M2: nat,N: nat] :
aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),M2)),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ( aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),M2) != zero_zero_int )
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ))),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N))) ).
% exp_div_exp_eq
tff(fact_4516_divmod__nat__def,axiom,
! [M2: nat,N: nat] : divmod_nat(M2,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)) ).
% divmod_nat_def
tff(fact_4517_set__bit__Suc,axiom,
! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))))) ).
% set_bit_Suc
tff(fact_4518_set__bit__Suc,axiom,
! [N: nat,A: int] : aa_int_int(bit_se7879613467334960850it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(aa_num_num(bit0,one)))))) ).
% set_bit_Suc
tff(fact_4519_set__bit__Suc,axiom,
! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(aa_nat_nat(suc,N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(N),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).
% set_bit_Suc
tff(fact_4520_set__bit__Suc,axiom,
! [N: nat,A: code_natural] : aa_Cod4272041906116450446atural(bit_se1617098188084679374atural(aa_nat_nat(suc,N)),A) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),numera5444537566228673987atural(aa_num_num(bit0,one)))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(bit_se1617098188084679374atural(N),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),numera5444537566228673987atural(aa_num_num(bit0,one)))))) ).
% set_bit_Suc
tff(fact_4521_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( aa_o_Code_integer(zero_n356916108424825756nteger,(P)) = one_one_Code_integer )
<=> (P) ) ).
% of_bool_eq_1_iff
tff(fact_4522_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( aa_o_rat(zero_n2052037380579107095ol_rat,(P)) = one_one_rat )
<=> (P) ) ).
% of_bool_eq_1_iff
tff(fact_4523_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P)) = one_one_nat )
<=> (P) ) ).
% of_bool_eq_1_iff
tff(fact_4524_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( aa_o_int(zero_n2684676970156552555ol_int,(P)) = one_one_int )
<=> (P) ) ).
% of_bool_eq_1_iff
tff(fact_4525_of__bool__eq_I2_J,axiom,
aa_o_Code_integer(zero_n356916108424825756nteger,$true) = one_one_Code_integer ).
% of_bool_eq(2)
tff(fact_4526_of__bool__eq_I2_J,axiom,
aa_o_rat(zero_n2052037380579107095ol_rat,$true) = one_one_rat ).
% of_bool_eq(2)
tff(fact_4527_of__bool__eq_I2_J,axiom,
aa_o_nat(zero_n2687167440665602831ol_nat,$true) = one_one_nat ).
% of_bool_eq(2)
tff(fact_4528_of__bool__eq_I2_J,axiom,
aa_o_int(zero_n2684676970156552555ol_int,$true) = one_one_int ).
% of_bool_eq(2)
tff(fact_4529_take__bit__Suc__1,axiom,
! [N: nat] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N)),one_one_Code_integer) = one_one_Code_integer ).
% take_bit_Suc_1
tff(fact_4530_take__bit__Suc__1,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),one_one_nat) = one_one_nat ).
% take_bit_Suc_1
tff(fact_4531_take__bit__Suc__1,axiom,
! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),one_one_int) = one_one_int ).
% take_bit_Suc_1
tff(fact_4532_of__bool__less__one__iff,axiom,
! [P: $o] :
( aa_rat_o(ord_less_rat(aa_o_rat(zero_n2052037380579107095ol_rat,(P))),one_one_rat)
<=> ~ (P) ) ).
% of_bool_less_one_iff
tff(fact_4533_of__bool__less__one__iff,axiom,
! [P: $o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat)
<=> ~ (P) ) ).
% of_bool_less_one_iff
tff(fact_4534_of__bool__less__one__iff,axiom,
! [P: $o] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(P))),one_one_Code_integer)
<=> ~ (P) ) ).
% of_bool_less_one_iff
tff(fact_4535_of__bool__less__one__iff,axiom,
! [P: $o] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_n8403883297036319079atural((P))),one_one_Code_natural)
<=> ~ (P) ) ).
% of_bool_less_one_iff
tff(fact_4536_of__bool__less__one__iff,axiom,
! [P: $o] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int)
<=> ~ (P) ) ).
% of_bool_less_one_iff
tff(fact_4537_of__bool__not__iff,axiom,
! [P: $o] : aa_o_Code_integer(zero_n356916108424825756nteger,~ (P)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),aa_o_Code_integer(zero_n356916108424825756nteger,(P))) ).
% of_bool_not_iff
tff(fact_4538_of__bool__not__iff,axiom,
! [P: $o] : aa_o_rat(zero_n2052037380579107095ol_rat,~ (P)) = minus_minus_rat(one_one_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(P))) ).
% of_bool_not_iff
tff(fact_4539_of__bool__not__iff,axiom,
! [P: $o] : aa_o_int(zero_n2684676970156552555ol_int,~ (P)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_o_int(zero_n2684676970156552555ol_int,(P))) ).
% of_bool_not_iff
tff(fact_4540_take__bit__numeral__1,axiom,
! [L: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(numeral_numeral_nat(L)),one_one_Code_integer) = one_one_Code_integer ).
% take_bit_numeral_1
tff(fact_4541_take__bit__numeral__1,axiom,
! [L: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,numeral_numeral_nat(L)),one_one_nat) = one_one_nat ).
% take_bit_numeral_1
tff(fact_4542_take__bit__numeral__1,axiom,
! [L: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(L)),one_one_int) = one_one_int ).
% take_bit_numeral_1
tff(fact_4543_signed__take__bit__Suc__1,axiom,
! [N: nat] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(aa_nat_nat(suc,N)),one_one_Code_integer) = one_one_Code_integer ).
% signed_take_bit_Suc_1
tff(fact_4544_signed__take__bit__Suc__1,axiom,
! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),one_one_int) = one_one_int ).
% signed_take_bit_Suc_1
tff(fact_4545_take__bit__of__1__eq__0__iff,axiom,
! [N: nat] :
( ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),one_one_Code_integer) = zero_z3403309356797280102nteger )
<=> ( N = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
tff(fact_4546_take__bit__of__1__eq__0__iff,axiom,
! [N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = zero_zero_nat )
<=> ( N = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
tff(fact_4547_take__bit__of__1__eq__0__iff,axiom,
! [N: nat] :
( ( aa_int_int(bit_se2923211474154528505it_int(N),one_one_int) = zero_zero_int )
<=> ( N = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
tff(fact_4548_sgn__mult__self__eq,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = aa_o_Code_integer(zero_n356916108424825756nteger,A != zero_z3403309356797280102nteger) ).
% sgn_mult_self_eq
tff(fact_4549_sgn__mult__self__eq,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,sgn_sgn_rat(A)),sgn_sgn_rat(A)) = aa_o_rat(zero_n2052037380579107095ol_rat,A != zero_zero_rat) ).
% sgn_mult_self_eq
tff(fact_4550_sgn__mult__self__eq,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ).
% sgn_mult_self_eq
tff(fact_4551_take__bit__of__1,axiom,
! [N: nat] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),one_one_Code_integer) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).
% take_bit_of_1
tff(fact_4552_take__bit__of__1,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).
% take_bit_of_1
tff(fact_4553_take__bit__of__1,axiom,
! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(N),one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).
% take_bit_of_1
tff(fact_4554_dbl__inc__simps_I3_J,axiom,
aa_int_int(neg_nu5851722552734809277nc_int,one_one_int) = numeral_numeral_int(aa_num_num(bit1,one)) ).
% dbl_inc_simps(3)
tff(fact_4555_dbl__inc__simps_I3_J,axiom,
neg_nu5219082963157363817nc_rat(one_one_rat) = numeral_numeral_rat(aa_num_num(bit1,one)) ).
% dbl_inc_simps(3)
tff(fact_4556_dbl__inc__simps_I3_J,axiom,
aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit1,one)) ).
% dbl_inc_simps(3)
tff(fact_4557_push__bit__Suc,axiom,
! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))) ).
% push_bit_Suc
tff(fact_4558_push__bit__Suc,axiom,
! [N: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,N)),A) = aa_int_int(bit_se545348938243370406it_int(N),aa_int_int(aa_int_fun_int_int(times_times_int,A),numeral_numeral_int(aa_num_num(bit0,one)))) ).
% push_bit_Suc
tff(fact_4559_push__bit__Suc,axiom,
! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(aa_nat_nat(suc,N)),A) = aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).
% push_bit_Suc
tff(fact_4560_push__bit__Suc,axiom,
! [N: nat,A: code_natural] : aa_Cod4272041906116450446atural(bit_se6611745700429515170atural(aa_nat_nat(suc,N)),A) = aa_Cod4272041906116450446atural(bit_se6611745700429515170atural(N),aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),numera5444537566228673987atural(aa_num_num(bit0,one)))) ).
% push_bit_Suc
tff(fact_4561_push__bit__of__1,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N) ).
% push_bit_of_1
tff(fact_4562_push__bit__of__1,axiom,
! [N: nat] : aa_int_int(bit_se545348938243370406it_int(N),one_one_int) = aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N) ).
% push_bit_of_1
tff(fact_4563_push__bit__of__1,axiom,
! [N: nat] : aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),one_one_Code_integer) = aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) ).
% push_bit_of_1
tff(fact_4564_push__bit__of__1,axiom,
! [N: nat] : aa_Cod4272041906116450446atural(bit_se6611745700429515170atural(N),one_one_Code_natural) = aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N) ).
% push_bit_of_1
tff(fact_4565_divmod__algorithm__code_I4_J,axiom,
! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),numeral_numeral_nat(one)) ).
% divmod_algorithm_code(4)
tff(fact_4566_divmod__algorithm__code_I4_J,axiom,
! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,N)) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),numeral_numeral_int(one)) ).
% divmod_algorithm_code(4)
tff(fact_4567_divmod__algorithm__code_I4_J,axiom,
! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).
% divmod_algorithm_code(4)
tff(fact_4568_dbl__dec__simps_I4_J,axiom,
aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,numeral_numeral_int(aa_num_num(bit1,one))) ).
% dbl_dec_simps(4)
tff(fact_4569_dbl__dec__simps_I4_J,axiom,
neg_nu3179335615603231917ec_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(aa_num_num(bit1,one))) ).
% dbl_dec_simps(4)
tff(fact_4570_dbl__dec__simps_I4_J,axiom,
aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit1,one))) ).
% dbl_dec_simps(4)
tff(fact_4571_bits__1__div__exp,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_o_nat(zero_n2687167440665602831ol_nat,N = zero_zero_nat) ).
% bits_1_div_exp
tff(fact_4572_bits__1__div__exp,axiom,
! [N: nat] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_o_Code_integer(zero_n356916108424825756nteger,N = zero_zero_nat) ).
% bits_1_div_exp
tff(fact_4573_bits__1__div__exp,axiom,
! [N: nat] : aa_Cod4272041906116450446atural(divide5121882707175180666atural(one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = zero_n8403883297036319079atural(N = zero_zero_nat) ).
% bits_1_div_exp
tff(fact_4574_bits__1__div__exp,axiom,
! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)) = aa_o_int(zero_n2684676970156552555ol_int,N = zero_zero_nat) ).
% bits_1_div_exp
tff(fact_4575_one__div__2__pow__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_o_nat(zero_n2687167440665602831ol_nat,N = zero_zero_nat) ).
% one_div_2_pow_eq
tff(fact_4576_one__div__2__pow__eq,axiom,
! [N: nat] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_o_Code_integer(zero_n356916108424825756nteger,N = zero_zero_nat) ).
% one_div_2_pow_eq
tff(fact_4577_one__div__2__pow__eq,axiom,
! [N: nat] : aa_Cod4272041906116450446atural(divide5121882707175180666atural(one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = zero_n8403883297036319079atural(N = zero_zero_nat) ).
% one_div_2_pow_eq
tff(fact_4578_one__div__2__pow__eq,axiom,
! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)) = aa_o_int(zero_n2684676970156552555ol_int,N = zero_zero_nat) ).
% one_div_2_pow_eq
tff(fact_4579_take__bit__of__exp,axiom,
! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ).
% take_bit_of_exp
tff(fact_4580_take__bit__of__exp,axiom,
! [M2: nat,N: nat] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(M2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ).
% take_bit_of_exp
tff(fact_4581_take__bit__of__exp,axiom,
! [M2: nat,N: nat] : aa_Cod4272041906116450446atural(bit_se569199155075624693atural(M2),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(zero_n8403883297036319079atural(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2))),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) ).
% take_bit_of_exp
tff(fact_4582_take__bit__of__exp,axiom,
! [M2: nat,N: nat] : aa_int_int(bit_se2923211474154528505it_int(M2),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2))),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)) ).
% take_bit_of_exp
tff(fact_4583_take__bit__of__2,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),numeral_numeral_nat(aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))),numeral_numeral_nat(aa_num_num(bit0,one))) ).
% take_bit_of_2
tff(fact_4584_take__bit__of__2,axiom,
! [N: nat] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% take_bit_of_2
tff(fact_4585_take__bit__of__2,axiom,
! [N: nat] : aa_Cod4272041906116450446atural(bit_se569199155075624693atural(N),numera5444537566228673987atural(aa_num_num(bit0,one))) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(zero_n8403883297036319079atural(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))),numera5444537566228673987atural(aa_num_num(bit0,one))) ).
% take_bit_of_2
tff(fact_4586_take__bit__of__2,axiom,
! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(N),numeral_numeral_int(aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))),numeral_numeral_int(aa_num_num(bit0,one))) ).
% take_bit_of_2
tff(fact_4587_of__bool__conj,axiom,
! [P: $o,Q: $o] :
aa_o_Code_integer(zero_n356916108424825756nteger,
( (P)
& (Q) )) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(P))),aa_o_Code_integer(zero_n356916108424825756nteger,(Q))) ).
% of_bool_conj
tff(fact_4588_of__bool__conj,axiom,
! [P: $o,Q: $o] :
aa_o_rat(zero_n2052037380579107095ol_rat,
( (P)
& (Q) )) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(P))),aa_o_rat(zero_n2052037380579107095ol_rat,(Q))) ).
% of_bool_conj
tff(fact_4589_of__bool__conj,axiom,
! [P: $o,Q: $o] :
aa_o_nat(zero_n2687167440665602831ol_nat,
( (P)
& (Q) )) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q))) ).
% of_bool_conj
tff(fact_4590_of__bool__conj,axiom,
! [P: $o,Q: $o] :
aa_o_int(zero_n2684676970156552555ol_int,
( (P)
& (Q) )) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q))) ).
% of_bool_conj
tff(fact_4591_take__bit__Suc__bit1,axiom,
! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),numeral_numeral_nat(aa_num_num(bit1,K))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),numeral_numeral_nat(K))),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_nat) ).
% take_bit_Suc_bit1
tff(fact_4592_take__bit__Suc__bit1,axiom,
! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),numeral_numeral_int(aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),numeral_numeral_int(K))),numeral_numeral_int(aa_num_num(bit0,one)))),one_one_int) ).
% take_bit_Suc_bit1
tff(fact_4593_take__bit__Suc__bit1,axiom,
! [N: nat,K: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N)),numera6620942414471956472nteger(aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),numera6620942414471956472nteger(K))),numera6620942414471956472nteger(aa_num_num(bit0,one)))),one_one_Code_integer) ).
% take_bit_Suc_bit1
tff(fact_4594_take__bit__Suc__bit1,axiom,
! [N: nat,K: num] : aa_Cod4272041906116450446atural(bit_se569199155075624693atural(aa_nat_nat(suc,N)),numera5444537566228673987atural(aa_num_num(bit1,K))) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(bit_se569199155075624693atural(N),numera5444537566228673987atural(K))),numera5444537566228673987atural(aa_num_num(bit0,one)))),one_one_Code_natural) ).
% take_bit_Suc_bit1
tff(fact_4595_take__bit__Suc__bit0,axiom,
! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),numeral_numeral_nat(aa_num_num(bit0,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),numeral_numeral_nat(K))),numeral_numeral_nat(aa_num_num(bit0,one))) ).
% take_bit_Suc_bit0
tff(fact_4596_take__bit__Suc__bit0,axiom,
! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),numeral_numeral_int(aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),numeral_numeral_int(K))),numeral_numeral_int(aa_num_num(bit0,one))) ).
% take_bit_Suc_bit0
tff(fact_4597_take__bit__Suc__bit0,axiom,
! [N: nat,K: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N)),numera6620942414471956472nteger(aa_num_num(bit0,K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),numera6620942414471956472nteger(K))),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% take_bit_Suc_bit0
tff(fact_4598_take__bit__Suc__bit0,axiom,
! [N: nat,K: num] : aa_Cod4272041906116450446atural(bit_se569199155075624693atural(aa_nat_nat(suc,N)),numera5444537566228673987atural(aa_num_num(bit0,K))) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(bit_se569199155075624693atural(N),numera5444537566228673987atural(K))),numera5444537566228673987atural(aa_num_num(bit0,one))) ).
% take_bit_Suc_bit0
tff(fact_4599_of__bool__less__eq__one,axiom,
! [P: $o] : aa_rat_o(ord_less_eq_rat(aa_o_rat(zero_n2052037380579107095ol_rat,(P))),one_one_rat) ).
% of_bool_less_eq_one
tff(fact_4600_of__bool__less__eq__one,axiom,
! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(P))),one_one_Code_integer) ).
% of_bool_less_eq_one
tff(fact_4601_of__bool__less__eq__one,axiom,
! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat) ).
% of_bool_less_eq_one
tff(fact_4602_of__bool__less__eq__one,axiom,
! [P: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int) ).
% of_bool_less_eq_one
tff(fact_4603_split__of__bool__asm,axiom,
! [P: fun_Code_integer_o,P5: $o] :
( aa_Code_integer_o(P,aa_o_Code_integer(zero_n356916108424825756nteger,(P5)))
<=> ~ ( ( (P5)
& ~ aa_Code_integer_o(P,one_one_Code_integer) )
| ( ~ (P5)
& ~ aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).
% split_of_bool_asm
tff(fact_4604_split__of__bool__asm,axiom,
! [P: fun_rat_o,P5: $o] :
( aa_rat_o(P,aa_o_rat(zero_n2052037380579107095ol_rat,(P5)))
<=> ~ ( ( (P5)
& ~ aa_rat_o(P,one_one_rat) )
| ( ~ (P5)
& ~ aa_rat_o(P,zero_zero_rat) ) ) ) ).
% split_of_bool_asm
tff(fact_4605_split__of__bool__asm,axiom,
! [P: fun_nat_o,P5: $o] :
( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P5)))
<=> ~ ( ( (P5)
& ~ aa_nat_o(P,one_one_nat) )
| ( ~ (P5)
& ~ aa_nat_o(P,zero_zero_nat) ) ) ) ).
% split_of_bool_asm
tff(fact_4606_split__of__bool__asm,axiom,
! [P: fun_int_o,P5: $o] :
( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P5)))
<=> ~ ( ( (P5)
& ~ aa_int_o(P,one_one_int) )
| ( ~ (P5)
& ~ aa_int_o(P,zero_zero_int) ) ) ) ).
% split_of_bool_asm
tff(fact_4607_split__of__bool,axiom,
! [P: fun_Code_integer_o,P5: $o] :
( aa_Code_integer_o(P,aa_o_Code_integer(zero_n356916108424825756nteger,(P5)))
<=> ( ( (P5)
=> aa_Code_integer_o(P,one_one_Code_integer) )
& ( ~ (P5)
=> aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).
% split_of_bool
tff(fact_4608_split__of__bool,axiom,
! [P: fun_rat_o,P5: $o] :
( aa_rat_o(P,aa_o_rat(zero_n2052037380579107095ol_rat,(P5)))
<=> ( ( (P5)
=> aa_rat_o(P,one_one_rat) )
& ( ~ (P5)
=> aa_rat_o(P,zero_zero_rat) ) ) ) ).
% split_of_bool
tff(fact_4609_split__of__bool,axiom,
! [P: fun_nat_o,P5: $o] :
( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P5)))
<=> ( ( (P5)
=> aa_nat_o(P,one_one_nat) )
& ( ~ (P5)
=> aa_nat_o(P,zero_zero_nat) ) ) ) ).
% split_of_bool
tff(fact_4610_split__of__bool,axiom,
! [P: fun_int_o,P5: $o] :
( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P5)))
<=> ( ( (P5)
=> aa_int_o(P,one_one_int) )
& ( ~ (P5)
=> aa_int_o(P,zero_zero_int) ) ) ) ).
% split_of_bool
tff(fact_4611_of__bool__def,axiom,
! [P5: $o] :
aa_o_Code_integer(zero_n356916108424825756nteger,(P5)) = $ite((P5),one_one_Code_integer,zero_z3403309356797280102nteger) ).
% of_bool_def
tff(fact_4612_of__bool__def,axiom,
! [P5: $o] :
aa_o_rat(zero_n2052037380579107095ol_rat,(P5)) = $ite((P5),one_one_rat,zero_zero_rat) ).
% of_bool_def
tff(fact_4613_of__bool__def,axiom,
! [P5: $o] :
aa_o_nat(zero_n2687167440665602831ol_nat,(P5)) = $ite((P5),one_one_nat,zero_zero_nat) ).
% of_bool_def
tff(fact_4614_of__bool__def,axiom,
! [P5: $o] :
aa_o_int(zero_n2684676970156552555ol_int,(P5)) = $ite((P5),one_one_int,zero_zero_int) ).
% of_bool_def
tff(fact_4615_xor__num_Ocases,axiom,
! [X: product_prod_num_num] :
( ( X != product_Pair_num_num(one,one) )
=> ( ! [N4: num] : X != product_Pair_num_num(one,aa_num_num(bit0,N4))
=> ( ! [N4: num] : X != product_Pair_num_num(one,aa_num_num(bit1,N4))
=> ( ! [M3: num] : X != product_Pair_num_num(aa_num_num(bit0,M3),one)
=> ( ! [M3: num,N4: num] : X != product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N4))
=> ( ! [M3: num,N4: num] : X != product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N4))
=> ( ! [M3: num] : X != product_Pair_num_num(aa_num_num(bit1,M3),one)
=> ( ! [M3: num,N4: num] : X != product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N4))
=> ~ ! [M3: num,N4: num] : X != product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N4)) ) ) ) ) ) ) ) ) ).
% xor_num.cases
tff(fact_4616_power__Suc,axiom,
! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).
% power_Suc
tff(fact_4617_power__Suc,axiom,
! [A: assn,N: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_nat_assn(power_power_assn(A),N)) ).
% power_Suc
tff(fact_4618_power__Suc,axiom,
! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N)) ).
% power_Suc
tff(fact_4619_power__Suc,axiom,
! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).
% power_Suc
tff(fact_4620_power__Suc,axiom,
! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)) ).
% power_Suc
tff(fact_4621_power__Suc2,axiom,
! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),A) ).
% power_Suc2
tff(fact_4622_power__Suc2,axiom,
! [A: assn,N: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),N)),A) ).
% power_Suc2
tff(fact_4623_power__Suc2,axiom,
! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),N)),A) ).
% power_Suc2
tff(fact_4624_power__Suc2,axiom,
! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),A) ).
% power_Suc2
tff(fact_4625_power__Suc2,axiom,
! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),A) ).
% power_Suc2
tff(fact_4626_numeral__Bit1,axiom,
! [N: num] : numeral_numeral_nat(aa_num_num(bit1,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(N)),numeral_numeral_nat(N))),one_one_nat) ).
% numeral_Bit1
tff(fact_4627_numeral__Bit1,axiom,
! [N: num] : numeral_numeral_int(aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(N)),numeral_numeral_int(N))),one_one_int) ).
% numeral_Bit1
tff(fact_4628_numeral__Bit1,axiom,
! [N: num] : numeral_numeral_rat(aa_num_num(bit1,N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(N)),numeral_numeral_rat(N))),one_one_rat) ).
% numeral_Bit1
tff(fact_4629_numeral__Bit1,axiom,
! [N: num] : numera6620942414471956472nteger(aa_num_num(bit1,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,numera6620942414471956472nteger(N)),numera6620942414471956472nteger(N))),one_one_Code_integer) ).
% numeral_Bit1
tff(fact_4630_numeral__Bit1,axiom,
! [N: num] : numera5444537566228673987atural(aa_num_num(bit1,N)) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(numera5444537566228673987atural(N)),numera5444537566228673987atural(N))),one_one_Code_natural) ).
% numeral_Bit1
tff(fact_4631_power__gt1,axiom,
! [A: rat,N: nat] :
( aa_rat_o(ord_less_rat(one_one_rat),A)
=> aa_rat_o(ord_less_rat(one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))) ) ).
% power_gt1
tff(fact_4632_power__gt1,axiom,
! [A: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))) ) ).
% power_gt1
tff(fact_4633_power__gt1,axiom,
! [A: int,N: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))) ) ).
% power_gt1
tff(fact_4634_power__gt1,axiom,
! [A: code_integer,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))) ) ).
% power_gt1
tff(fact_4635_power__gt1,axiom,
! [A: code_natural,N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(suc,N))) ) ).
% power_gt1
tff(fact_4636_take__bit__Suc__minus__1__eq,axiom,
! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),aa_nat_nat(suc,N))),one_one_int) ).
% take_bit_Suc_minus_1_eq
tff(fact_4637_take__bit__Suc__minus__1__eq,axiom,
! [N: nat] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(suc,N))),one_one_Code_integer) ).
% take_bit_Suc_minus_1_eq
tff(fact_4638_take__bit__Suc,axiom,
! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))))),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))) ).
% take_bit_Suc
tff(fact_4639_take__bit__Suc,axiom,
! [N: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(aa_num_num(bit0,one))))),numeral_numeral_int(aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one)))) ).
% take_bit_Suc
tff(fact_4640_take__bit__Suc,axiom,
! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))))),numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).
% take_bit_Suc
tff(fact_4641_take__bit__Suc,axiom,
! [N: nat,A: code_natural] : aa_Cod4272041906116450446atural(bit_se569199155075624693atural(aa_nat_nat(suc,N)),A) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(bit_se569199155075624693atural(N),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))))),numera5444537566228673987atural(aa_num_num(bit0,one)))),aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),numera5444537566228673987atural(aa_num_num(bit0,one)))) ).
% take_bit_Suc
tff(fact_4642_power__Suc__le__self,axiom,
! [A: rat,N: nat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(A),one_one_rat)
=> aa_rat_o(ord_less_eq_rat(aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),A) ) ) ).
% power_Suc_le_self
tff(fact_4643_power__Suc__le__self,axiom,
! [A: code_integer,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))),A) ) ) ).
% power_Suc_le_self
tff(fact_4644_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),A) ) ) ).
% power_Suc_le_self
tff(fact_4645_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),A) ) ) ).
% power_Suc_le_self
tff(fact_4646_power3__eq__cube,axiom,
! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(aa_num_num(bit1,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A)),A) ).
% power3_eq_cube
tff(fact_4647_power3__eq__cube,axiom,
! [A: assn] : aa_nat_assn(power_power_assn(A),numeral_numeral_nat(aa_num_num(bit1,one))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),A)),A) ).
% power3_eq_cube
tff(fact_4648_power3__eq__cube,axiom,
! [A: rat] : aa_nat_rat(power_power_rat(A),numeral_numeral_nat(aa_num_num(bit1,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),A) ).
% power3_eq_cube
tff(fact_4649_power3__eq__cube,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(aa_num_num(bit1,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A)),A) ).
% power3_eq_cube
tff(fact_4650_power3__eq__cube,axiom,
! [A: int] : aa_nat_int(power_power_int(A),numeral_numeral_nat(aa_num_num(bit1,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),A) ).
% power3_eq_cube
tff(fact_4651_power__Suc__less__one,axiom,
! [A: rat,N: nat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_rat(A),one_one_rat)
=> aa_rat_o(ord_less_rat(aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),one_one_rat) ) ) ).
% power_Suc_less_one
tff(fact_4652_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),one_one_nat) ) ) ).
% power_Suc_less_one
tff(fact_4653_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),one_one_int) ) ) ).
% power_Suc_less_one
tff(fact_4654_power__Suc__less__one,axiom,
! [A: code_integer,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))),one_one_Code_integer) ) ) ).
% power_Suc_less_one
tff(fact_4655_power__Suc__less__one,axiom,
! [A: code_natural,N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A)
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),one_one_Code_natural)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(suc,N))),one_one_Code_natural) ) ) ).
% power_Suc_less_one
tff(fact_4656_push__bit__double,axiom,
! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),A)),numeral_numeral_nat(aa_num_num(bit0,one))) ).
% push_bit_double
tff(fact_4657_push__bit__double,axiom,
! [N: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(N),aa_int_int(aa_int_fun_int_int(times_times_int,A),numeral_numeral_int(aa_num_num(bit0,one)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se545348938243370406it_int(N),A)),numeral_numeral_int(aa_num_num(bit0,one))) ).
% push_bit_double
tff(fact_4658_push__bit__double,axiom,
! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),numera6620942414471956472nteger(aa_num_num(bit0,one)))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),A)),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% push_bit_double
tff(fact_4659_push__bit__double,axiom,
! [N: nat,A: code_natural] : aa_Cod4272041906116450446atural(bit_se6611745700429515170atural(N),aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),numera5444537566228673987atural(aa_num_num(bit0,one)))) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(bit_se6611745700429515170atural(N),A)),numera5444537566228673987atural(aa_num_num(bit0,one))) ).
% push_bit_double
tff(fact_4660_push__bit__eq__mult,axiom,
! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) ).
% push_bit_eq_mult
tff(fact_4661_push__bit__eq__mult,axiom,
! [N: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(N),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)) ).
% push_bit_eq_mult
tff(fact_4662_push__bit__eq__mult,axiom,
! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ).
% push_bit_eq_mult
tff(fact_4663_push__bit__eq__mult,axiom,
! [N: nat,A: code_natural] : aa_Cod4272041906116450446atural(bit_se6611745700429515170atural(N),A) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(A),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) ).
% push_bit_eq_mult
tff(fact_4664_bits__induct,axiom,
! [P: fun_nat_o,A: nat] :
( ! [A3: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),numeral_numeral_nat(aa_num_num(bit0,one))) = A3 )
=> aa_nat_o(P,A3) )
=> ( ! [A3: nat,B3: $o] :
( aa_nat_o(P,A3)
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B3))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A3))),numeral_numeral_nat(aa_num_num(bit0,one))) = A3 )
=> aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B3))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A3))) ) )
=> aa_nat_o(P,A) ) ) ).
% bits_induct
tff(fact_4665_bits__induct,axiom,
! [P: fun_Code_integer_o,A: code_integer] :
( ! [A3: code_integer] :
( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A3),numera6620942414471956472nteger(aa_num_num(bit0,one))) = A3 )
=> aa_Code_integer_o(P,A3) )
=> ( ! [A3: code_integer,B3: $o] :
( aa_Code_integer_o(P,A3)
=> ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(B3))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A3))),numera6620942414471956472nteger(aa_num_num(bit0,one))) = A3 )
=> aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(B3))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A3))) ) )
=> aa_Code_integer_o(P,A) ) ) ).
% bits_induct
tff(fact_4666_bits__induct,axiom,
! [P: fun_Code_natural_o,A: code_natural] :
( ! [A3: code_natural] :
( ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(A3),numera5444537566228673987atural(aa_num_num(bit0,one))) = A3 )
=> aa_Code_natural_o(P,A3) )
=> ( ! [A3: code_natural,B3: $o] :
( aa_Code_natural_o(P,A3)
=> ( ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(zero_n8403883297036319079atural((B3))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),A3))),numera5444537566228673987atural(aa_num_num(bit0,one))) = A3 )
=> aa_Code_natural_o(P,aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(zero_n8403883297036319079atural((B3))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),A3))) ) )
=> aa_Code_natural_o(P,A) ) ) ).
% bits_induct
tff(fact_4667_bits__induct,axiom,
! [P: fun_int_o,A: int] :
( ! [A3: int] :
( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),numeral_numeral_int(aa_num_num(bit0,one))) = A3 )
=> aa_int_o(P,A3) )
=> ( ! [A3: int,B3: $o] :
( aa_int_o(P,A3)
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B3))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),A3))),numeral_numeral_int(aa_num_num(bit0,one))) = A3 )
=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B3))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),A3))) ) )
=> aa_int_o(P,A) ) ) ).
% bits_induct
tff(fact_4668_power__odd__eq,axiom,
! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),numeral_numeral_nat(aa_num_num(bit0,one)))) ).
% power_odd_eq
tff(fact_4669_power__odd__eq,axiom,
! [A: assn,N: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_nat_assn(power_power_assn(aa_nat_assn(power_power_assn(A),N)),numeral_numeral_nat(aa_num_num(bit0,one)))) ).
% power_odd_eq
tff(fact_4670_power__odd__eq,axiom,
! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),N)),numeral_numeral_nat(aa_num_num(bit0,one)))) ).
% power_odd_eq
tff(fact_4671_power__odd__eq,axiom,
! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),numeral_numeral_nat(aa_num_num(bit0,one)))) ).
% power_odd_eq
tff(fact_4672_power__odd__eq,axiom,
! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),N)),numeral_numeral_nat(aa_num_num(bit0,one)))) ).
% power_odd_eq
tff(fact_4673_exp__mod__exp,axiom,
! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M2)) ).
% exp_mod_exp
tff(fact_4674_exp__mod__exp,axiom,
! [M2: nat,N: nat] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M2)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M2)) ).
% exp_mod_exp
tff(fact_4675_exp__mod__exp,axiom,
! [M2: nat,N: nat] : aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M2)),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(zero_n8403883297036319079atural(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N))),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M2)) ).
% exp_mod_exp
tff(fact_4676_exp__mod__exp,axiom,
! [M2: nat,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),M2)),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N))),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),M2)) ).
% exp_mod_exp
tff(fact_4677_power__minus1__odd,axiom,
! [N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) = aa_int_int(uminus_uminus_int,one_one_int) ).
% power_minus1_odd
tff(fact_4678_power__minus1__odd,axiom,
! [N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% power_minus1_odd
tff(fact_4679_power__minus1__odd,axiom,
! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% power_minus1_odd
tff(fact_4680_take__bit__numeral__minus__1__eq,axiom,
! [K: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(K)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),numeral_numeral_nat(K))),one_one_int) ).
% take_bit_numeral_minus_1_eq
tff(fact_4681_take__bit__numeral__minus__1__eq,axiom,
! [K: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(numeral_numeral_nat(K)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),numeral_numeral_nat(K))),one_one_Code_integer) ).
% take_bit_numeral_minus_1_eq
tff(fact_4682_stable__imp__take__bit__eq,axiom,
! [A: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) = A )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),one_one_nat)) ) ) ).
% stable_imp_take_bit_eq
tff(fact_4683_stable__imp__take__bit__eq,axiom,
! [A: int,N: nat] :
( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(aa_num_num(bit0,one))) = A )
=> ( aa_int_int(bit_se2923211474154528505it_int(N),A) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)),one_one_int)) ) ) ).
% stable_imp_take_bit_eq
tff(fact_4684_stable__imp__take__bit__eq,axiom,
! [A: code_integer,N: nat] :
( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
=> ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)),one_one_Code_integer)) ) ) ).
% stable_imp_take_bit_eq
tff(fact_4685_stable__imp__take__bit__eq,axiom,
! [A: code_natural,N: nat] :
( ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))) = A )
=> ( aa_Cod4272041906116450446atural(bit_se569199155075624693atural(N),A) = $ite(aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A),zero_z2226904508553997617atural,aa_Cod4272041906116450446atural(minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)),one_one_Code_natural)) ) ) ).
% stable_imp_take_bit_eq
tff(fact_4686_signed__take__bit__Suc,axiom,
! [N: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(aa_num_num(bit0,one)))))) ).
% signed_take_bit_Suc
tff(fact_4687_signed__take__bit__Suc,axiom,
! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(aa_nat_nat(suc,N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(N),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).
% signed_take_bit_Suc
tff(fact_4688_unset__bit__Suc,axiom,
! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))))) ).
% unset_bit_Suc
tff(fact_4689_unset__bit__Suc,axiom,
! [N: nat,A: int] : aa_int_int(bit_se4203085406695923979it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(aa_num_num(bit0,one)))))) ).
% unset_bit_Suc
tff(fact_4690_unset__bit__Suc,axiom,
! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(aa_nat_nat(suc,N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(N),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).
% unset_bit_Suc
tff(fact_4691_unset__bit__Suc,axiom,
! [N: nat,A: code_natural] : aa_Cod4272041906116450446atural(bit_se7083795435491715335atural(aa_nat_nat(suc,N)),A) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),numera5444537566228673987atural(aa_num_num(bit0,one)))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(bit_se7083795435491715335atural(N),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),numera5444537566228673987atural(aa_num_num(bit0,one)))))) ).
% unset_bit_Suc
tff(fact_4692_flip__bit__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))))) ).
% flip_bit_0
tff(fact_4693_flip__bit__0,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(zero_zero_nat),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).
% flip_bit_0
tff(fact_4694_flip__bit__0,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(bit_se168947363167071951atural(zero_zero_nat),A) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(zero_n8403883297036319079atural(aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))))) ).
% flip_bit_0
tff(fact_4695_flip__bit__0,axiom,
! [A: int] : aa_int_int(bit_se2159334234014336723it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(aa_num_num(bit0,one))))) ).
% flip_bit_0
tff(fact_4696_flip__bit__Suc,axiom,
! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(bit0,one)))))) ).
% flip_bit_Suc
tff(fact_4697_flip__bit__Suc,axiom,
! [N: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(aa_num_num(bit0,one)))))) ).
% flip_bit_Suc
tff(fact_4698_flip__bit__Suc,axiom,
! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(aa_nat_nat(suc,N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(N),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).
% flip_bit_Suc
tff(fact_4699_flip__bit__Suc,axiom,
! [N: nat,A: code_natural] : aa_Cod4272041906116450446atural(bit_se168947363167071951atural(aa_nat_nat(suc,N)),A) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),numera5444537566228673987atural(aa_num_num(bit0,one)))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(bit_se168947363167071951atural(N),aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),numera5444537566228673987atural(aa_num_num(bit0,one)))))) ).
% flip_bit_Suc
tff(fact_4700_take__bit__numeral__bit1,axiom,
! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,numeral_numeral_nat(L)),numeral_numeral_nat(aa_num_num(bit1,K))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,pred_numeral(L)),numeral_numeral_nat(K))),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_nat) ).
% take_bit_numeral_bit1
tff(fact_4701_take__bit__numeral__bit1,axiom,
! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(L)),numeral_numeral_int(aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(pred_numeral(L)),numeral_numeral_int(K))),numeral_numeral_int(aa_num_num(bit0,one)))),one_one_int) ).
% take_bit_numeral_bit1
tff(fact_4702_take__bit__numeral__bit1,axiom,
! [L: num,K: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(numeral_numeral_nat(L)),numera6620942414471956472nteger(aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(pred_numeral(L)),numera6620942414471956472nteger(K))),numera6620942414471956472nteger(aa_num_num(bit0,one)))),one_one_Code_integer) ).
% take_bit_numeral_bit1
tff(fact_4703_take__bit__numeral__bit1,axiom,
! [L: num,K: num] : aa_Cod4272041906116450446atural(bit_se569199155075624693atural(numeral_numeral_nat(L)),numera5444537566228673987atural(aa_num_num(bit1,K))) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(bit_se569199155075624693atural(pred_numeral(L)),numera5444537566228673987atural(K))),numera5444537566228673987atural(aa_num_num(bit0,one)))),one_one_Code_natural) ).
% take_bit_numeral_bit1
tff(fact_4704_xor__numerals_I4_J,axiom,
! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(aa_num_num(bit0,X))),numeral_numeral_nat(aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y)))) ).
% xor_numerals(4)
tff(fact_4705_xor__numerals_I4_J,axiom,
! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(aa_num_num(bit0,X))),numeral_numeral_int(aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(X)),numeral_numeral_int(Y)))) ).
% xor_numerals(4)
tff(fact_4706_xor__numerals_I4_J,axiom,
! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(aa_num_num(bit0,X))),numera6620942414471956472nteger(aa_num_num(bit1,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y)))) ).
% xor_numerals(4)
tff(fact_4707_xor__numerals_I4_J,axiom,
! [X: num,Y: num] : aa_Cod4272041906116450446atural(bit_se2046307713759805098atural(numera5444537566228673987atural(aa_num_num(bit0,X))),numera5444537566228673987atural(aa_num_num(bit1,Y))) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(bit_se2046307713759805098atural(numera5444537566228673987atural(X)),numera5444537566228673987atural(Y)))) ).
% xor_numerals(4)
tff(fact_4708_xor__numerals_I6_J,axiom,
! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(aa_num_num(bit1,X))),numeral_numeral_nat(aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y)))) ).
% xor_numerals(6)
tff(fact_4709_xor__numerals_I6_J,axiom,
! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(aa_num_num(bit1,X))),numeral_numeral_int(aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(X)),numeral_numeral_int(Y)))) ).
% xor_numerals(6)
tff(fact_4710_xor__numerals_I6_J,axiom,
! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(aa_num_num(bit1,X))),numera6620942414471956472nteger(aa_num_num(bit0,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y)))) ).
% xor_numerals(6)
tff(fact_4711_xor__numerals_I6_J,axiom,
! [X: num,Y: num] : aa_Cod4272041906116450446atural(bit_se2046307713759805098atural(numera5444537566228673987atural(aa_num_num(bit1,X))),numera5444537566228673987atural(aa_num_num(bit0,Y))) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(bit_se2046307713759805098atural(numera5444537566228673987atural(X)),numera5444537566228673987atural(Y)))) ).
% xor_numerals(6)
tff(fact_4712_one__xor__eq,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A))) ).
% one_xor_eq
tff(fact_4713_one__xor__eq,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,one_one_Code_integer),A) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)))),aa_o_Code_integer(zero_n356916108424825756nteger,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).
% one_xor_eq
tff(fact_4714_one__xor__eq,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(bit_se2046307713759805098atural(one_one_Code_natural),A) = aa_Cod4272041906116450446atural(minus_7197305767214868737atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),zero_n8403883297036319079atural(aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A)))),zero_n8403883297036319079atural(~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A))) ).
% one_xor_eq
tff(fact_4715_one__xor__eq,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,one_one_int),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A))) ).
% one_xor_eq
tff(fact_4716_xor__one__eq,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A))) ).
% xor_one_eq
tff(fact_4717_xor__one__eq,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,A),one_one_Code_integer) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)))),aa_o_Code_integer(zero_n356916108424825756nteger,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).
% xor_one_eq
tff(fact_4718_xor__one__eq,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(bit_se2046307713759805098atural(A),one_one_Code_natural) = aa_Cod4272041906116450446atural(minus_7197305767214868737atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),zero_n8403883297036319079atural(aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A)))),zero_n8403883297036319079atural(~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A))) ).
% xor_one_eq
tff(fact_4719_xor__one__eq,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),one_one_int) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A))) ).
% xor_one_eq
tff(fact_4720_or__numerals_I4_J,axiom,
! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(aa_num_num(bit0,X))),numeral_numeral_nat(aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y)))) ).
% or_numerals(4)
tff(fact_4721_or__numerals_I4_J,axiom,
! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(aa_num_num(bit0,X))),numeral_numeral_int(aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(X)),numeral_numeral_int(Y)))) ).
% or_numerals(4)
tff(fact_4722_or__numerals_I4_J,axiom,
! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(aa_num_num(bit0,X))),numera6620942414471956472nteger(aa_num_num(bit1,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y)))) ).
% or_numerals(4)
tff(fact_4723_or__numerals_I4_J,axiom,
! [X: num,Y: num] : aa_Cod4272041906116450446atural(bit_se9127793120404214118atural(numera5444537566228673987atural(aa_num_num(bit0,X))),numera5444537566228673987atural(aa_num_num(bit1,Y))) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(bit_se9127793120404214118atural(numera5444537566228673987atural(X)),numera5444537566228673987atural(Y)))) ).
% or_numerals(4)
tff(fact_4724_bit_Odisj__one__left,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,one_one_int)),X) = aa_int_int(uminus_uminus_int,one_one_int) ).
% bit.disj_one_left
tff(fact_4725_bit_Odisj__one__left,axiom,
! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% bit.disj_one_left
tff(fact_4726_bit_Odisj__one__right,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).
% bit.disj_one_right
tff(fact_4727_bit_Odisj__one__right,axiom,
! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% bit.disj_one_right
tff(fact_4728_or__numerals_I2_J,axiom,
! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit1,Y))) = numeral_numeral_nat(aa_num_num(bit1,Y)) ).
% or_numerals(2)
tff(fact_4729_or__numerals_I2_J,axiom,
! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),numeral_numeral_int(aa_num_num(bit1,Y))) = numeral_numeral_int(aa_num_num(bit1,Y)) ).
% or_numerals(2)
tff(fact_4730_or__numerals_I2_J,axiom,
! [Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit1,Y))) = numera6620942414471956472nteger(aa_num_num(bit1,Y)) ).
% or_numerals(2)
tff(fact_4731_or__numerals_I2_J,axiom,
! [Y: num] : aa_Cod4272041906116450446atural(bit_se9127793120404214118atural(one_one_Code_natural),numera5444537566228673987atural(aa_num_num(bit1,Y))) = numera5444537566228673987atural(aa_num_num(bit1,Y)) ).
% or_numerals(2)
tff(fact_4732_or__numerals_I8_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(aa_num_num(bit1,X))),one_one_nat) = numeral_numeral_nat(aa_num_num(bit1,X)) ).
% or_numerals(8)
tff(fact_4733_or__numerals_I8_J,axiom,
! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(aa_num_num(bit1,X))),one_one_int) = numeral_numeral_int(aa_num_num(bit1,X)) ).
% or_numerals(8)
tff(fact_4734_or__numerals_I8_J,axiom,
! [X: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(aa_num_num(bit1,X))),one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit1,X)) ).
% or_numerals(8)
tff(fact_4735_or__numerals_I8_J,axiom,
! [X: num] : aa_Cod4272041906116450446atural(bit_se9127793120404214118atural(numera5444537566228673987atural(aa_num_num(bit1,X))),one_one_Code_natural) = numera5444537566228673987atural(aa_num_num(bit1,X)) ).
% or_numerals(8)
tff(fact_4736_or__numerals_I3_J,axiom,
! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(aa_num_num(bit0,X))),numeral_numeral_nat(aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y))) ).
% or_numerals(3)
tff(fact_4737_or__numerals_I3_J,axiom,
! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(aa_num_num(bit0,X))),numeral_numeral_int(aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(X)),numeral_numeral_int(Y))) ).
% or_numerals(3)
tff(fact_4738_or__numerals_I3_J,axiom,
! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(aa_num_num(bit0,X))),numera6620942414471956472nteger(aa_num_num(bit0,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y))) ).
% or_numerals(3)
tff(fact_4739_or__numerals_I3_J,axiom,
! [X: num,Y: num] : aa_Cod4272041906116450446atural(bit_se9127793120404214118atural(numera5444537566228673987atural(aa_num_num(bit0,X))),numera5444537566228673987atural(aa_num_num(bit0,Y))) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(bit_se9127793120404214118atural(numera5444537566228673987atural(X)),numera5444537566228673987atural(Y))) ).
% or_numerals(3)
tff(fact_4740_xor__numerals_I3_J,axiom,
! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(aa_num_num(bit0,X))),numeral_numeral_nat(aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y))) ).
% xor_numerals(3)
tff(fact_4741_xor__numerals_I3_J,axiom,
! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(aa_num_num(bit0,X))),numeral_numeral_int(aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(X)),numeral_numeral_int(Y))) ).
% xor_numerals(3)
tff(fact_4742_xor__numerals_I3_J,axiom,
! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(aa_num_num(bit0,X))),numera6620942414471956472nteger(aa_num_num(bit0,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y))) ).
% xor_numerals(3)
tff(fact_4743_xor__numerals_I3_J,axiom,
! [X: num,Y: num] : aa_Cod4272041906116450446atural(bit_se2046307713759805098atural(numera5444537566228673987atural(aa_num_num(bit0,X))),numera5444537566228673987atural(aa_num_num(bit0,Y))) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(bit_se2046307713759805098atural(numera5444537566228673987atural(X)),numera5444537566228673987atural(Y))) ).
% xor_numerals(3)
tff(fact_4744_or__numerals_I5_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(aa_num_num(bit0,X))),one_one_nat) = numeral_numeral_nat(aa_num_num(bit1,X)) ).
% or_numerals(5)
tff(fact_4745_or__numerals_I5_J,axiom,
! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(aa_num_num(bit0,X))),one_one_int) = numeral_numeral_int(aa_num_num(bit1,X)) ).
% or_numerals(5)
tff(fact_4746_or__numerals_I5_J,axiom,
! [X: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(aa_num_num(bit0,X))),one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit1,X)) ).
% or_numerals(5)
tff(fact_4747_or__numerals_I5_J,axiom,
! [X: num] : aa_Cod4272041906116450446atural(bit_se9127793120404214118atural(numera5444537566228673987atural(aa_num_num(bit0,X))),one_one_Code_natural) = numera5444537566228673987atural(aa_num_num(bit1,X)) ).
% or_numerals(5)
tff(fact_4748_or__numerals_I1_J,axiom,
! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit0,Y))) = numeral_numeral_nat(aa_num_num(bit1,Y)) ).
% or_numerals(1)
tff(fact_4749_or__numerals_I1_J,axiom,
! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),numeral_numeral_int(aa_num_num(bit0,Y))) = numeral_numeral_int(aa_num_num(bit1,Y)) ).
% or_numerals(1)
tff(fact_4750_or__numerals_I1_J,axiom,
! [Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,Y))) = numera6620942414471956472nteger(aa_num_num(bit1,Y)) ).
% or_numerals(1)
tff(fact_4751_or__numerals_I1_J,axiom,
! [Y: num] : aa_Cod4272041906116450446atural(bit_se9127793120404214118atural(one_one_Code_natural),numera5444537566228673987atural(aa_num_num(bit0,Y))) = numera5444537566228673987atural(aa_num_num(bit1,Y)) ).
% or_numerals(1)
tff(fact_4752_xor__numerals_I8_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(aa_num_num(bit1,X))),one_one_nat) = numeral_numeral_nat(aa_num_num(bit0,X)) ).
% xor_numerals(8)
tff(fact_4753_xor__numerals_I8_J,axiom,
! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(aa_num_num(bit1,X))),one_one_int) = numeral_numeral_int(aa_num_num(bit0,X)) ).
% xor_numerals(8)
tff(fact_4754_xor__numerals_I8_J,axiom,
! [X: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(aa_num_num(bit1,X))),one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit0,X)) ).
% xor_numerals(8)
tff(fact_4755_xor__numerals_I8_J,axiom,
! [X: num] : aa_Cod4272041906116450446atural(bit_se2046307713759805098atural(numera5444537566228673987atural(aa_num_num(bit1,X))),one_one_Code_natural) = numera5444537566228673987atural(aa_num_num(bit0,X)) ).
% xor_numerals(8)
tff(fact_4756_xor__numerals_I5_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(aa_num_num(bit0,X))),one_one_nat) = numeral_numeral_nat(aa_num_num(bit1,X)) ).
% xor_numerals(5)
tff(fact_4757_xor__numerals_I5_J,axiom,
! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(aa_num_num(bit0,X))),one_one_int) = numeral_numeral_int(aa_num_num(bit1,X)) ).
% xor_numerals(5)
tff(fact_4758_xor__numerals_I5_J,axiom,
! [X: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(aa_num_num(bit0,X))),one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit1,X)) ).
% xor_numerals(5)
tff(fact_4759_xor__numerals_I5_J,axiom,
! [X: num] : aa_Cod4272041906116450446atural(bit_se2046307713759805098atural(numera5444537566228673987atural(aa_num_num(bit0,X))),one_one_Code_natural) = numera5444537566228673987atural(aa_num_num(bit1,X)) ).
% xor_numerals(5)
tff(fact_4760_xor__numerals_I2_J,axiom,
! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit1,Y))) = numeral_numeral_nat(aa_num_num(bit0,Y)) ).
% xor_numerals(2)
tff(fact_4761_xor__numerals_I2_J,axiom,
! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,one_one_int),numeral_numeral_int(aa_num_num(bit1,Y))) = numeral_numeral_int(aa_num_num(bit0,Y)) ).
% xor_numerals(2)
tff(fact_4762_xor__numerals_I2_J,axiom,
! [Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit1,Y))) = numera6620942414471956472nteger(aa_num_num(bit0,Y)) ).
% xor_numerals(2)
tff(fact_4763_xor__numerals_I2_J,axiom,
! [Y: num] : aa_Cod4272041906116450446atural(bit_se2046307713759805098atural(one_one_Code_natural),numera5444537566228673987atural(aa_num_num(bit1,Y))) = numera5444537566228673987atural(aa_num_num(bit0,Y)) ).
% xor_numerals(2)
tff(fact_4764_xor__numerals_I1_J,axiom,
! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit0,Y))) = numeral_numeral_nat(aa_num_num(bit1,Y)) ).
% xor_numerals(1)
tff(fact_4765_xor__numerals_I1_J,axiom,
! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,one_one_int),numeral_numeral_int(aa_num_num(bit0,Y))) = numeral_numeral_int(aa_num_num(bit1,Y)) ).
% xor_numerals(1)
tff(fact_4766_xor__numerals_I1_J,axiom,
! [Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,Y))) = numera6620942414471956472nteger(aa_num_num(bit1,Y)) ).
% xor_numerals(1)
tff(fact_4767_xor__numerals_I1_J,axiom,
! [Y: num] : aa_Cod4272041906116450446atural(bit_se2046307713759805098atural(one_one_Code_natural),numera5444537566228673987atural(aa_num_num(bit0,Y))) = numera5444537566228673987atural(aa_num_num(bit1,Y)) ).
% xor_numerals(1)
tff(fact_4768_xor__numerals_I7_J,axiom,
! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(aa_num_num(bit1,X))),numeral_numeral_nat(aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y))) ).
% xor_numerals(7)
tff(fact_4769_xor__numerals_I7_J,axiom,
! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(aa_num_num(bit1,X))),numeral_numeral_int(aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(X)),numeral_numeral_int(Y))) ).
% xor_numerals(7)
tff(fact_4770_xor__numerals_I7_J,axiom,
! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(aa_num_num(bit1,X))),numera6620942414471956472nteger(aa_num_num(bit1,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y))) ).
% xor_numerals(7)
tff(fact_4771_xor__numerals_I7_J,axiom,
! [X: num,Y: num] : aa_Cod4272041906116450446atural(bit_se2046307713759805098atural(numera5444537566228673987atural(aa_num_num(bit1,X))),numera5444537566228673987atural(aa_num_num(bit1,Y))) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(bit_se2046307713759805098atural(numera5444537566228673987atural(X)),numera5444537566228673987atural(Y))) ).
% xor_numerals(7)
tff(fact_4772_or__numerals_I7_J,axiom,
! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(aa_num_num(bit1,X))),numeral_numeral_nat(aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y)))) ).
% or_numerals(7)
tff(fact_4773_or__numerals_I7_J,axiom,
! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(aa_num_num(bit1,X))),numeral_numeral_int(aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(X)),numeral_numeral_int(Y)))) ).
% or_numerals(7)
tff(fact_4774_or__numerals_I7_J,axiom,
! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(aa_num_num(bit1,X))),numera6620942414471956472nteger(aa_num_num(bit1,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y)))) ).
% or_numerals(7)
tff(fact_4775_or__numerals_I7_J,axiom,
! [X: num,Y: num] : aa_Cod4272041906116450446atural(bit_se9127793120404214118atural(numera5444537566228673987atural(aa_num_num(bit1,X))),numera5444537566228673987atural(aa_num_num(bit1,Y))) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(bit_se9127793120404214118atural(numera5444537566228673987atural(X)),numera5444537566228673987atural(Y)))) ).
% or_numerals(7)
tff(fact_4776_or__numerals_I6_J,axiom,
! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(aa_num_num(bit1,X))),numeral_numeral_nat(aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y)))) ).
% or_numerals(6)
tff(fact_4777_or__numerals_I6_J,axiom,
! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(aa_num_num(bit1,X))),numeral_numeral_int(aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(X)),numeral_numeral_int(Y)))) ).
% or_numerals(6)
tff(fact_4778_or__numerals_I6_J,axiom,
! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(aa_num_num(bit1,X))),numera6620942414471956472nteger(aa_num_num(bit0,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y)))) ).
% or_numerals(6)
tff(fact_4779_or__numerals_I6_J,axiom,
! [X: num,Y: num] : aa_Cod4272041906116450446atural(bit_se9127793120404214118atural(numera5444537566228673987atural(aa_num_num(bit1,X))),numera5444537566228673987atural(aa_num_num(bit0,Y))) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(bit_se9127793120404214118atural(numera5444537566228673987atural(X)),numera5444537566228673987atural(Y)))) ).
% or_numerals(6)
tff(fact_4780_flip__bit__eq__xor,axiom,
! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,A),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),one_one_Code_integer)) ).
% flip_bit_eq_xor
tff(fact_4781_flip__bit__eq__xor,axiom,
! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat)) ).
% flip_bit_eq_xor
tff(fact_4782_flip__bit__eq__xor,axiom,
! [N: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(N),A) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),aa_int_int(bit_se545348938243370406it_int(N),one_one_int)) ).
% flip_bit_eq_xor
tff(fact_4783_or_Ocomm__monoid__axioms,axiom,
aa_Code_integer_o(comm_m1654649897431166824nteger(bit_se1080825931792720795nteger),zero_z3403309356797280102nteger) ).
% or.comm_monoid_axioms
tff(fact_4784_or_Ocomm__monoid__axioms,axiom,
aa_nat_o(comm_monoid_nat(bit_se1412395901928357646or_nat),zero_zero_nat) ).
% or.comm_monoid_axioms
tff(fact_4785_or_Ocomm__monoid__axioms,axiom,
aa_int_o(comm_monoid_int(bit_se1409905431419307370or_int),zero_zero_int) ).
% or.comm_monoid_axioms
tff(fact_4786_or_Omonoid__axioms,axiom,
aa_Code_integer_o(monoid_Code_integer(bit_se1080825931792720795nteger),zero_z3403309356797280102nteger) ).
% or.monoid_axioms
tff(fact_4787_or_Omonoid__axioms,axiom,
aa_nat_o(monoid_nat(bit_se1412395901928357646or_nat),zero_zero_nat) ).
% or.monoid_axioms
tff(fact_4788_or_Omonoid__axioms,axiom,
aa_int_o(monoid_int(bit_se1409905431419307370or_int),zero_zero_int) ).
% or.monoid_axioms
tff(fact_4789_or_Osemilattice__neutr__axioms,axiom,
aa_Code_integer_o(semila6106414119144251899nteger(bit_se1080825931792720795nteger),zero_z3403309356797280102nteger) ).
% or.semilattice_neutr_axioms
tff(fact_4790_or_Osemilattice__neutr__axioms,axiom,
aa_int_o(semila9079005292280841162tr_int(bit_se1409905431419307370or_int),zero_zero_int) ).
% or.semilattice_neutr_axioms
tff(fact_4791_or_Osemilattice__neutr__axioms,axiom,
aa_nat_o(semila9081495762789891438tr_nat(bit_se1412395901928357646or_nat),zero_zero_nat) ).
% or.semilattice_neutr_axioms
tff(fact_4792_xor_Ocomm__monoid__axioms,axiom,
aa_Code_integer_o(comm_m1654649897431166824nteger(bit_se3222712562003087583nteger),zero_z3403309356797280102nteger) ).
% xor.comm_monoid_axioms
tff(fact_4793_xor_Ocomm__monoid__axioms,axiom,
aa_nat_o(comm_monoid_nat(bit_se6528837805403552850or_nat),zero_zero_nat) ).
% xor.comm_monoid_axioms
tff(fact_4794_xor_Ocomm__monoid__axioms,axiom,
aa_int_o(comm_monoid_int(bit_se6526347334894502574or_int),zero_zero_int) ).
% xor.comm_monoid_axioms
tff(fact_4795_xor_Omonoid__axioms,axiom,
aa_Code_integer_o(monoid_Code_integer(bit_se3222712562003087583nteger),zero_z3403309356797280102nteger) ).
% xor.monoid_axioms
tff(fact_4796_xor_Omonoid__axioms,axiom,
aa_nat_o(monoid_nat(bit_se6528837805403552850or_nat),zero_zero_nat) ).
% xor.monoid_axioms
tff(fact_4797_xor_Omonoid__axioms,axiom,
aa_int_o(monoid_int(bit_se6526347334894502574or_int),zero_zero_int) ).
% xor.monoid_axioms
tff(fact_4798_set__bit__eq__or,axiom,
! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,A),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),one_one_Code_integer)) ).
% set_bit_eq_or
tff(fact_4799_set__bit__eq__or,axiom,
! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat)) ).
% set_bit_eq_or
tff(fact_4800_set__bit__eq__or,axiom,
! [N: nat,A: int] : aa_int_int(bit_se7879613467334960850it_int(N),A) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),aa_int_int(bit_se545348938243370406it_int(N),one_one_int)) ).
% set_bit_eq_or
tff(fact_4801_take__bit__numeral__bit0,axiom,
! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,numeral_numeral_nat(L)),numeral_numeral_nat(aa_num_num(bit0,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,pred_numeral(L)),numeral_numeral_nat(K))),numeral_numeral_nat(aa_num_num(bit0,one))) ).
% take_bit_numeral_bit0
tff(fact_4802_take__bit__numeral__bit0,axiom,
! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(L)),numeral_numeral_int(aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(pred_numeral(L)),numeral_numeral_int(K))),numeral_numeral_int(aa_num_num(bit0,one))) ).
% take_bit_numeral_bit0
tff(fact_4803_take__bit__numeral__bit0,axiom,
! [L: num,K: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(numeral_numeral_nat(L)),numera6620942414471956472nteger(aa_num_num(bit0,K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(pred_numeral(L)),numera6620942414471956472nteger(K))),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% take_bit_numeral_bit0
tff(fact_4804_take__bit__numeral__bit0,axiom,
! [L: num,K: num] : aa_Cod4272041906116450446atural(bit_se569199155075624693atural(numeral_numeral_nat(L)),numera5444537566228673987atural(aa_num_num(bit0,K))) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(bit_se569199155075624693atural(pred_numeral(L)),numera5444537566228673987atural(K))),numera5444537566228673987atural(aa_num_num(bit0,one))) ).
% take_bit_numeral_bit0
tff(fact_4805_or__one__eq,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A))) ).
% or_one_eq
tff(fact_4806_or__one__eq,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,A),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).
% or_one_eq
tff(fact_4807_or__one__eq,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(bit_se9127793120404214118atural(A),one_one_Code_natural) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),zero_n8403883297036319079atural(aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A))) ).
% or_one_eq
tff(fact_4808_or__one__eq,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),one_one_int) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A))) ).
% or_one_eq
tff(fact_4809_one__or__eq,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A))) ).
% one_or_eq
tff(fact_4810_one__or__eq,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,one_one_Code_integer),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).
% one_or_eq
tff(fact_4811_one__or__eq,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(bit_se9127793120404214118atural(one_one_Code_natural),A) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),zero_n8403883297036319079atural(aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A))) ).
% one_or_eq
tff(fact_4812_one__or__eq,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A))) ).
% one_or_eq
tff(fact_4813_mask__numeral,axiom,
! [N: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,numeral_numeral_nat(N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,pred_numeral(N)))) ).
% mask_numeral
tff(fact_4814_mask__numeral,axiom,
! [N: num] : aa_nat_int(bit_se2000444600071755411sk_int,numeral_numeral_nat(N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,pred_numeral(N)))) ).
% mask_numeral
tff(fact_4815_mask__numeral,axiom,
! [N: num] : aa_nat_Code_integer(bit_se2119862282449309892nteger,numeral_numeral_nat(N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(bit_se2119862282449309892nteger,pred_numeral(N)))) ).
% mask_numeral
tff(fact_4816_mask__numeral,axiom,
! [N: num] : aa_nat_Code_natural(bit_se943457434206027407atural,numeral_numeral_nat(N)) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_nat_Code_natural(bit_se943457434206027407atural,pred_numeral(N)))) ).
% mask_numeral
tff(fact_4817_mask__Suc__double,axiom,
! [N: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ).
% mask_Suc_double
tff(fact_4818_mask__Suc__double,axiom,
! [N: nat] : aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,N))) ).
% mask_Suc_double
tff(fact_4819_mask__Suc__double,axiom,
! [N: nat] : aa_nat_Code_integer(bit_se2119862282449309892nteger,aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(bit_se2119862282449309892nteger,N))) ).
% mask_Suc_double
tff(fact_4820_mask__Suc__double,axiom,
! [N: nat] : aa_nat_Code_natural(bit_se943457434206027407atural,aa_nat_nat(suc,N)) = aa_Cod4272041906116450446atural(bit_se9127793120404214118atural(one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_nat_Code_natural(bit_se943457434206027407atural,N))) ).
% mask_Suc_double
tff(fact_4821_and__numerals_I7_J,axiom,
! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(aa_num_num(bit1,X))),numeral_numeral_nat(aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y)))) ).
% and_numerals(7)
tff(fact_4822_and__numerals_I7_J,axiom,
! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(aa_num_num(bit1,X))),numera6620942414471956472nteger(aa_num_num(bit1,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y)))) ).
% and_numerals(7)
tff(fact_4823_and__numerals_I7_J,axiom,
! [X: num,Y: num] : aa_Cod4272041906116450446atural(bit_se2773287842338716102atural(numera5444537566228673987atural(aa_num_num(bit1,X))),numera5444537566228673987atural(aa_num_num(bit1,Y))) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(bit_se2773287842338716102atural(numera5444537566228673987atural(X)),numera5444537566228673987atural(Y)))) ).
% and_numerals(7)
tff(fact_4824_and__numerals_I7_J,axiom,
! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(aa_num_num(bit1,X))),numeral_numeral_int(aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(X)),numeral_numeral_int(Y)))) ).
% and_numerals(7)
tff(fact_4825_mask__eq__exp__minus__1,axiom,
! [N: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),one_one_nat) ).
% mask_eq_exp_minus_1
tff(fact_4826_mask__eq__exp__minus__1,axiom,
! [N: nat] : aa_nat_int(bit_se2000444600071755411sk_int,N) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)),one_one_int) ).
% mask_eq_exp_minus_1
tff(fact_4827_mask__eq__exp__minus__1,axiom,
! [N: nat] : aa_nat_Code_integer(bit_se2119862282449309892nteger,N) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)),one_one_Code_integer) ).
% mask_eq_exp_minus_1
tff(fact_4828_mask__eq__exp__minus__1,axiom,
! [N: nat] : aa_nat_Code_natural(bit_se943457434206027407atural,N) = aa_Cod4272041906116450446atural(minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)),one_one_Code_natural) ).
% mask_eq_exp_minus_1
tff(fact_4829_bit__sum__mult__2__cases,axiom,
! [A: nat,B: nat,N: nat] :
( ! [J2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,J2))
=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B))),N)
<=> $ite(N = zero_zero_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A),aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B)),N)) ) ) ).
% bit_sum_mult_2_cases
tff(fact_4830_bit__sum__mult__2__cases,axiom,
! [A: int,B: int,N: nat] :
( ! [J2: nat] : ~ aa_nat_o(bit_se1146084159140164899it_int(A),aa_nat_nat(suc,J2))
=> ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),B))),N)
<=> $ite(N = zero_zero_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A),aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),B)),N)) ) ) ).
% bit_sum_mult_2_cases
tff(fact_4831_bit__sum__mult__2__cases,axiom,
! [A: code_integer,B: code_integer,N: nat] :
( ! [J2: nat] : ~ aa_nat_o(bit_se9216721137139052372nteger(A),aa_nat_nat(suc,J2))
=> ( aa_nat_o(bit_se9216721137139052372nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),N)
<=> $ite(N = zero_zero_nat,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),aa_nat_o(bit_se9216721137139052372nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)),N)) ) ) ).
% bit_sum_mult_2_cases
tff(fact_4832_bit__sum__mult__2__cases,axiom,
! [A: code_natural,B: code_natural,N: nat] :
( ! [J2: nat] : ~ aa_nat_o(bit_se8040316288895769887atural(A),aa_nat_nat(suc,J2))
=> ( aa_nat_o(bit_se8040316288895769887atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),B))),N)
<=> $ite(N = zero_zero_nat,~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A),aa_nat_o(bit_se8040316288895769887atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),B)),N)) ) ) ).
% bit_sum_mult_2_cases
tff(fact_4833_diff__numeral__special_I8_J,axiom,
! [M2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_int,one_one_int)) = neg_numeral_sub_int(one,M2) ).
% diff_numeral_special(8)
tff(fact_4834_diff__numeral__special_I8_J,axiom,
! [M2: num] : minus_minus_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M2)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = neg_numeral_sub_rat(one,M2) ).
% diff_numeral_special(8)
tff(fact_4835_diff__numeral__special_I8_J,axiom,
! [M2: num] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = neg_nu5755505904847501662nteger(one,M2) ).
% diff_numeral_special(8)
tff(fact_4836_diff__numeral__special_I7_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = neg_numeral_sub_int(N,one) ).
% diff_numeral_special(7)
tff(fact_4837_diff__numeral__special_I7_J,axiom,
! [N: num] : minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(N))) = neg_numeral_sub_rat(N,one) ).
% diff_numeral_special(7)
tff(fact_4838_diff__numeral__special_I7_J,axiom,
! [N: num] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = neg_nu5755505904847501662nteger(N,one) ).
% diff_numeral_special(7)
tff(fact_4839_minus__sub__one__diff__one,axiom,
! [M2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,neg_numeral_sub_int(M2,one))),one_one_int) = aa_int_int(uminus_uminus_int,numeral_numeral_int(M2)) ).
% minus_sub_one_diff_one
tff(fact_4840_minus__sub__one__diff__one,axiom,
! [M2: num] : minus_minus_rat(aa_rat_rat(uminus_uminus_rat,neg_numeral_sub_rat(M2,one)),one_one_rat) = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M2)) ).
% minus_sub_one_diff_one
tff(fact_4841_minus__sub__one__diff__one,axiom,
! [M2: num] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5755505904847501662nteger(M2,one))),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)) ).
% minus_sub_one_diff_one
tff(fact_4842_bit__0__eq,axiom,
bit_se9216721137139052372nteger(zero_z3403309356797280102nteger) = bot_bot_fun_nat_o ).
% bit_0_eq
tff(fact_4843_bit__0__eq,axiom,
aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,zero_zero_nat) = bot_bot_fun_nat_o ).
% bit_0_eq
tff(fact_4844_bit__0__eq,axiom,
bit_se1146084159140164899it_int(zero_zero_int) = bot_bot_fun_nat_o ).
% bit_0_eq
tff(fact_4845_bit_Oconj__one__right,axiom,
! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = X ).
% bit.conj_one_right
tff(fact_4846_bit_Oconj__one__right,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),aa_int_int(uminus_uminus_int,one_one_int)) = X ).
% bit.conj_one_right
tff(fact_4847_and_Oright__neutral,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = A ).
% and.right_neutral
tff(fact_4848_and_Oright__neutral,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = A ).
% and.right_neutral
tff(fact_4849_and_Oleft__neutral,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),A) = A ).
% and.left_neutral
tff(fact_4850_and_Oleft__neutral,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,one_one_int)),A) = A ).
% and.left_neutral
tff(fact_4851_and__numerals_I8_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(aa_num_num(bit1,X))),one_one_nat) = one_one_nat ).
% and_numerals(8)
tff(fact_4852_and__numerals_I8_J,axiom,
! [X: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(aa_num_num(bit1,X))),one_one_Code_integer) = one_one_Code_integer ).
% and_numerals(8)
tff(fact_4853_and__numerals_I8_J,axiom,
! [X: num] : aa_Cod4272041906116450446atural(bit_se2773287842338716102atural(numera5444537566228673987atural(aa_num_num(bit1,X))),one_one_Code_natural) = one_one_Code_natural ).
% and_numerals(8)
tff(fact_4854_and__numerals_I8_J,axiom,
! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(aa_num_num(bit1,X))),one_one_int) = one_one_int ).
% and_numerals(8)
tff(fact_4855_and__numerals_I2_J,axiom,
! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit1,Y))) = one_one_nat ).
% and_numerals(2)
tff(fact_4856_and__numerals_I2_J,axiom,
! [Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit1,Y))) = one_one_Code_integer ).
% and_numerals(2)
tff(fact_4857_and__numerals_I2_J,axiom,
! [Y: num] : aa_Cod4272041906116450446atural(bit_se2773287842338716102atural(one_one_Code_natural),numera5444537566228673987atural(aa_num_num(bit1,Y))) = one_one_Code_natural ).
% and_numerals(2)
tff(fact_4858_and__numerals_I2_J,axiom,
! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),numeral_numeral_int(aa_num_num(bit1,Y))) = one_one_int ).
% and_numerals(2)
tff(fact_4859_mask__Suc__0,axiom,
aa_nat_Code_integer(bit_se2119862282449309892nteger,aa_nat_nat(suc,zero_zero_nat)) = one_one_Code_integer ).
% mask_Suc_0
tff(fact_4860_mask__Suc__0,axiom,
aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).
% mask_Suc_0
tff(fact_4861_mask__Suc__0,axiom,
aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).
% mask_Suc_0
tff(fact_4862_take__bit__minus__one__eq__mask,axiom,
! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_nat_int(bit_se2000444600071755411sk_int,N) ).
% take_bit_minus_one_eq_mask
tff(fact_4863_take__bit__minus__one__eq__mask,axiom,
! [N: nat] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_nat_Code_integer(bit_se2119862282449309892nteger,N) ).
% take_bit_minus_one_eq_mask
tff(fact_4864_and__numerals_I1_J,axiom,
! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit0,Y))) = zero_zero_nat ).
% and_numerals(1)
tff(fact_4865_and__numerals_I1_J,axiom,
! [Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,Y))) = zero_z3403309356797280102nteger ).
% and_numerals(1)
tff(fact_4866_and__numerals_I1_J,axiom,
! [Y: num] : aa_Cod4272041906116450446atural(bit_se2773287842338716102atural(one_one_Code_natural),numera5444537566228673987atural(aa_num_num(bit0,Y))) = zero_z2226904508553997617atural ).
% and_numerals(1)
tff(fact_4867_and__numerals_I1_J,axiom,
! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),numeral_numeral_int(aa_num_num(bit0,Y))) = zero_zero_int ).
% and_numerals(1)
tff(fact_4868_and__numerals_I5_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(aa_num_num(bit0,X))),one_one_nat) = zero_zero_nat ).
% and_numerals(5)
tff(fact_4869_and__numerals_I5_J,axiom,
! [X: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(aa_num_num(bit0,X))),one_one_Code_integer) = zero_z3403309356797280102nteger ).
% and_numerals(5)
tff(fact_4870_and__numerals_I5_J,axiom,
! [X: num] : aa_Cod4272041906116450446atural(bit_se2773287842338716102atural(numera5444537566228673987atural(aa_num_num(bit0,X))),one_one_Code_natural) = zero_z2226904508553997617atural ).
% and_numerals(5)
tff(fact_4871_and__numerals_I5_J,axiom,
! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(aa_num_num(bit0,X))),one_one_int) = zero_zero_int ).
% and_numerals(5)
tff(fact_4872_and__numerals_I3_J,axiom,
! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(aa_num_num(bit0,X))),numeral_numeral_nat(aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y))) ).
% and_numerals(3)
tff(fact_4873_and__numerals_I3_J,axiom,
! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(aa_num_num(bit0,X))),numera6620942414471956472nteger(aa_num_num(bit0,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y))) ).
% and_numerals(3)
tff(fact_4874_and__numerals_I3_J,axiom,
! [X: num,Y: num] : aa_Cod4272041906116450446atural(bit_se2773287842338716102atural(numera5444537566228673987atural(aa_num_num(bit0,X))),numera5444537566228673987atural(aa_num_num(bit0,Y))) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(bit_se2773287842338716102atural(numera5444537566228673987atural(X)),numera5444537566228673987atural(Y))) ).
% and_numerals(3)
tff(fact_4875_and__numerals_I3_J,axiom,
! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(aa_num_num(bit0,X))),numeral_numeral_int(aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(X)),numeral_numeral_int(Y))) ).
% and_numerals(3)
tff(fact_4876_diff__numeral__special_I2_J,axiom,
! [M2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,numeral_numeral_int(M2)),one_one_int) = neg_numeral_sub_int(M2,one) ).
% diff_numeral_special(2)
tff(fact_4877_diff__numeral__special_I2_J,axiom,
! [M2: num] : minus_minus_rat(numeral_numeral_rat(M2),one_one_rat) = neg_numeral_sub_rat(M2,one) ).
% diff_numeral_special(2)
tff(fact_4878_diff__numeral__special_I2_J,axiom,
! [M2: num] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(numera6620942414471956472nteger(M2)),one_one_Code_integer) = neg_nu5755505904847501662nteger(M2,one) ).
% diff_numeral_special(2)
tff(fact_4879_diff__numeral__special_I1_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),numeral_numeral_int(N)) = neg_numeral_sub_int(one,N) ).
% diff_numeral_special(1)
tff(fact_4880_diff__numeral__special_I1_J,axiom,
! [N: num] : minus_minus_rat(one_one_rat,numeral_numeral_rat(N)) = neg_numeral_sub_rat(one,N) ).
% diff_numeral_special(1)
tff(fact_4881_diff__numeral__special_I1_J,axiom,
! [N: num] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),numera6620942414471956472nteger(N)) = neg_nu5755505904847501662nteger(one,N) ).
% diff_numeral_special(1)
tff(fact_4882_and__numerals_I4_J,axiom,
! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(aa_num_num(bit0,X))),numeral_numeral_nat(aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y))) ).
% and_numerals(4)
tff(fact_4883_and__numerals_I4_J,axiom,
! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(aa_num_num(bit0,X))),numera6620942414471956472nteger(aa_num_num(bit1,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y))) ).
% and_numerals(4)
tff(fact_4884_and__numerals_I4_J,axiom,
! [X: num,Y: num] : aa_Cod4272041906116450446atural(bit_se2773287842338716102atural(numera5444537566228673987atural(aa_num_num(bit0,X))),numera5444537566228673987atural(aa_num_num(bit1,Y))) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(bit_se2773287842338716102atural(numera5444537566228673987atural(X)),numera5444537566228673987atural(Y))) ).
% and_numerals(4)
tff(fact_4885_and__numerals_I4_J,axiom,
! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(aa_num_num(bit0,X))),numeral_numeral_int(aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(X)),numeral_numeral_int(Y))) ).
% and_numerals(4)
tff(fact_4886_and__numerals_I6_J,axiom,
! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(aa_num_num(bit1,X))),numeral_numeral_nat(aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y))) ).
% and_numerals(6)
tff(fact_4887_and__numerals_I6_J,axiom,
! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(aa_num_num(bit1,X))),numera6620942414471956472nteger(aa_num_num(bit0,Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y))) ).
% and_numerals(6)
tff(fact_4888_and__numerals_I6_J,axiom,
! [X: num,Y: num] : aa_Cod4272041906116450446atural(bit_se2773287842338716102atural(numera5444537566228673987atural(aa_num_num(bit1,X))),numera5444537566228673987atural(aa_num_num(bit0,Y))) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(bit_se2773287842338716102atural(numera5444537566228673987atural(X)),numera5444537566228673987atural(Y))) ).
% and_numerals(6)
tff(fact_4889_and__numerals_I6_J,axiom,
! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(aa_num_num(bit1,X))),numeral_numeral_int(aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(X)),numeral_numeral_int(Y))) ).
% and_numerals(6)
tff(fact_4890_add__neg__numeral__special_I1_J,axiom,
! [M2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))) = neg_numeral_sub_int(one,M2) ).
% add_neg_numeral_special(1)
tff(fact_4891_add__neg__numeral__special_I1_J,axiom,
! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M2))) = neg_numeral_sub_rat(one,M2) ).
% add_neg_numeral_special(1)
tff(fact_4892_add__neg__numeral__special_I1_J,axiom,
! [M2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))) = neg_nu5755505904847501662nteger(one,M2) ).
% add_neg_numeral_special(1)
tff(fact_4893_add__neg__numeral__special_I2_J,axiom,
! [M2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),one_one_int) = neg_numeral_sub_int(one,M2) ).
% add_neg_numeral_special(2)
tff(fact_4894_add__neg__numeral__special_I2_J,axiom,
! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(M2))),one_one_rat) = neg_numeral_sub_rat(one,M2) ).
% add_neg_numeral_special(2)
tff(fact_4895_add__neg__numeral__special_I2_J,axiom,
! [M2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),one_one_Code_integer) = neg_nu5755505904847501662nteger(one,M2) ).
% add_neg_numeral_special(2)
tff(fact_4896_add__neg__numeral__special_I3_J,axiom,
! [M2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,one_one_int)) = neg_numeral_sub_int(M2,one) ).
% add_neg_numeral_special(3)
tff(fact_4897_add__neg__numeral__special_I3_J,axiom,
! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(M2)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = neg_numeral_sub_rat(M2,one) ).
% add_neg_numeral_special(3)
tff(fact_4898_add__neg__numeral__special_I3_J,axiom,
! [M2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = neg_nu5755505904847501662nteger(M2,one) ).
% add_neg_numeral_special(3)
tff(fact_4899_add__neg__numeral__special_I4_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),numeral_numeral_int(N)) = neg_numeral_sub_int(N,one) ).
% add_neg_numeral_special(4)
tff(fact_4900_add__neg__numeral__special_I4_J,axiom,
! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),numeral_numeral_rat(N)) = neg_numeral_sub_rat(N,one) ).
% add_neg_numeral_special(4)
tff(fact_4901_add__neg__numeral__special_I4_J,axiom,
! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(N)) = neg_nu5755505904847501662nteger(N,one) ).
% add_neg_numeral_special(4)
tff(fact_4902_and_Osemigroup__axioms,axiom,
semigroup_int(bit_se725231765392027082nd_int) ).
% and.semigroup_axioms
tff(fact_4903_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: code_integer,N: nat] :
( aa_nat_o(bit_se9216721137139052372nteger(A),N)
<=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),one_one_Code_integer)) != zero_z3403309356797280102nteger ) ) ).
% bit_iff_and_push_bit_not_eq_0
tff(fact_4904_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat)) != zero_zero_nat ) ) ).
% bit_iff_and_push_bit_not_eq_0
tff(fact_4905_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: int,N: nat] :
( aa_nat_o(bit_se1146084159140164899it_int(A),N)
<=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(bit_se545348938243370406it_int(N),one_one_int)) != zero_zero_int ) ) ).
% bit_iff_and_push_bit_not_eq_0
tff(fact_4906_and__eq__minus__1__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
<=> ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
& ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).
% and_eq_minus_1_iff
tff(fact_4907_and__eq__minus__1__iff,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B) = aa_int_int(uminus_uminus_int,one_one_int) )
<=> ( ( A = aa_int_int(uminus_uminus_int,one_one_int) )
& ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).
% and_eq_minus_1_iff
tff(fact_4908_bit__1__iff,axiom,
! [N: nat] :
( aa_nat_o(bit_se9216721137139052372nteger(one_one_Code_integer),N)
<=> ( N = zero_zero_nat ) ) ).
% bit_1_iff
tff(fact_4909_bit__1__iff,axiom,
! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),N)
<=> ( N = zero_zero_nat ) ) ).
% bit_1_iff
tff(fact_4910_bit__1__iff,axiom,
! [N: nat] :
( aa_nat_o(bit_se1146084159140164899it_int(one_one_int),N)
<=> ( N = zero_zero_nat ) ) ).
% bit_1_iff
tff(fact_4911_not__bit__1__Suc,axiom,
! [N: nat] : ~ aa_nat_o(bit_se9216721137139052372nteger(one_one_Code_integer),aa_nat_nat(suc,N)) ).
% not_bit_1_Suc
tff(fact_4912_not__bit__1__Suc,axiom,
! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),aa_nat_nat(suc,N)) ).
% not_bit_1_Suc
tff(fact_4913_not__bit__1__Suc,axiom,
! [N: nat] : ~ aa_nat_o(bit_se1146084159140164899it_int(one_one_int),aa_nat_nat(suc,N)) ).
% not_bit_1_Suc
tff(fact_4914_bit__numeral__simps_I1_J,axiom,
! [N: num] : ~ aa_nat_o(bit_se9216721137139052372nteger(one_one_Code_integer),numeral_numeral_nat(N)) ).
% bit_numeral_simps(1)
tff(fact_4915_bit__numeral__simps_I1_J,axiom,
! [N: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),numeral_numeral_nat(N)) ).
% bit_numeral_simps(1)
tff(fact_4916_bit__numeral__simps_I1_J,axiom,
! [N: num] : ~ aa_nat_o(bit_se1146084159140164899it_int(one_one_int),numeral_numeral_nat(N)) ).
% bit_numeral_simps(1)
tff(fact_4917_and_Ocomm__monoid__axioms,axiom,
aa_Code_integer_o(comm_m1654649897431166824nteger(bit_se3949692690581998587nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% and.comm_monoid_axioms
tff(fact_4918_and_Ocomm__monoid__axioms,axiom,
aa_int_o(comm_monoid_int(bit_se725231765392027082nd_int),aa_int_int(uminus_uminus_int,one_one_int)) ).
% and.comm_monoid_axioms
tff(fact_4919_and_Omonoid__axioms,axiom,
aa_Code_integer_o(monoid_Code_integer(bit_se3949692690581998587nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% and.monoid_axioms
tff(fact_4920_and_Omonoid__axioms,axiom,
aa_int_o(monoid_int(bit_se725231765392027082nd_int),aa_int_int(uminus_uminus_int,one_one_int)) ).
% and.monoid_axioms
tff(fact_4921_and_Osemilattice__neutr__axioms,axiom,
aa_Code_integer_o(semila6106414119144251899nteger(bit_se3949692690581998587nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% and.semilattice_neutr_axioms
tff(fact_4922_and_Osemilattice__neutr__axioms,axiom,
aa_int_o(semila9079005292280841162tr_int(bit_se725231765392027082nd_int),aa_int_int(uminus_uminus_int,one_one_int)) ).
% and.semilattice_neutr_axioms
tff(fact_4923_bit_Ocomplement__unique,axiom,
! [A: code_integer,X: code_integer,Y: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),X) = zero_z3403309356797280102nteger )
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,A),X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),Y) = zero_z3403309356797280102nteger )
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,A),Y) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
=> ( X = Y ) ) ) ) ) ).
% bit.complement_unique
tff(fact_4924_bit_Ocomplement__unique,axiom,
! [A: int,X: int,Y: int] :
( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),X) = zero_zero_int )
=> ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),X) = aa_int_int(uminus_uminus_int,one_one_int) )
=> ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),Y) = zero_zero_int )
=> ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),Y) = aa_int_int(uminus_uminus_int,one_one_int) )
=> ( X = Y ) ) ) ) ) ).
% bit.complement_unique
tff(fact_4925_one__and__eq,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) ).
% one_and_eq
tff(fact_4926_one__and__eq,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,one_one_Code_integer),A) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% one_and_eq
tff(fact_4927_one__and__eq,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(bit_se2773287842338716102atural(one_one_Code_natural),A) = aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))) ).
% one_and_eq
tff(fact_4928_one__and__eq,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one))) ).
% one_and_eq
tff(fact_4929_and__one__eq,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(aa_num_num(bit0,one))) ).
% and_one_eq
tff(fact_4930_and__one__eq,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),one_one_Code_integer) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% and_one_eq
tff(fact_4931_and__one__eq,axiom,
! [A: code_natural] : aa_Cod4272041906116450446atural(bit_se2773287842338716102atural(A),one_one_Code_natural) = aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),numera5444537566228673987atural(aa_num_num(bit0,one))) ).
% and_one_eq
tff(fact_4932_and__one__eq,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),one_one_int) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(aa_num_num(bit0,one))) ).
% and_one_eq
tff(fact_4933_even__bit__succ__iff,axiom,
! [A: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),N)
<=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
| ( N = zero_zero_nat ) ) ) ) ).
% even_bit_succ_iff
tff(fact_4934_even__bit__succ__iff,axiom,
! [A: int,N: nat] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),N)
<=> ( aa_nat_o(bit_se1146084159140164899it_int(A),N)
| ( N = zero_zero_nat ) ) ) ) ).
% even_bit_succ_iff
tff(fact_4935_even__bit__succ__iff,axiom,
! [A: code_integer,N: nat] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(bit_se9216721137139052372nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),N)
<=> ( aa_nat_o(bit_se9216721137139052372nteger(A),N)
| ( N = zero_zero_nat ) ) ) ) ).
% even_bit_succ_iff
tff(fact_4936_even__bit__succ__iff,axiom,
! [A: code_natural,N: nat] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(bit_se8040316288895769887atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),A)),N)
<=> ( aa_nat_o(bit_se8040316288895769887atural(A),N)
| ( N = zero_zero_nat ) ) ) ) ).
% even_bit_succ_iff
tff(fact_4937_odd__bit__iff__bit__pred,axiom,
! [A: nat,N: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
<=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),one_one_nat)),N)
| ( N = zero_zero_nat ) ) ) ) ).
% odd_bit_iff_bit_pred
tff(fact_4938_odd__bit__iff__bit__pred,axiom,
! [A: int,N: nat] :
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(bit_se1146084159140164899it_int(A),N)
<=> ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)),N)
| ( N = zero_zero_nat ) ) ) ) ).
% odd_bit_iff_bit_pred
tff(fact_4939_odd__bit__iff__bit__pred,axiom,
! [A: code_integer,N: nat] :
( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(bit_se9216721137139052372nteger(A),N)
<=> ( aa_nat_o(bit_se9216721137139052372nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),one_one_Code_integer)),N)
| ( N = zero_zero_nat ) ) ) ) ).
% odd_bit_iff_bit_pred
tff(fact_4940_odd__bit__iff__bit__pred,axiom,
! [A: code_natural,N: nat] :
( ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,numera5444537566228673987atural(aa_num_num(bit0,one))),A)
=> ( aa_nat_o(bit_se8040316288895769887atural(A),N)
<=> ( aa_nat_o(bit_se8040316288895769887atural(aa_Cod4272041906116450446atural(minus_7197305767214868737atural(A),one_one_Code_natural)),N)
| ( N = zero_zero_nat ) ) ) ) ).
% odd_bit_iff_bit_pred
tff(fact_4941_signed__take__bit__def,axiom,
! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(bit_se9216721137139052372nteger(A),N))),aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_nat_Code_integer(bit_se2119862282449309892nteger,N)))) ).
% signed_take_bit_def
tff(fact_4942_signed__take__bit__def,axiom,
! [N: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(N),A) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(bit_se2923211474154528505it_int(N),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(A),N))),aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,N)))) ).
% signed_take_bit_def
tff(fact_4943_eq__numeral__iff__iszero_I7_J,axiom,
! [X: num] :
( ( aa_int_int(uminus_uminus_int,numeral_numeral_int(X)) = one_one_int )
<=> aa_int_o(ring_1_iszero_int,numeral_numeral_int(plus_plus_num(X,one))) ) ).
% eq_numeral_iff_iszero(7)
tff(fact_4944_eq__numeral__iff__iszero_I7_J,axiom,
! [X: num] :
( ( aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(X)) = one_one_rat )
<=> aa_rat_o(ring_1_iszero_rat,numeral_numeral_rat(plus_plus_num(X,one))) ) ).
% eq_numeral_iff_iszero(7)
tff(fact_4945_eq__numeral__iff__iszero_I7_J,axiom,
! [X: num] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X)) = one_one_Code_integer )
<=> aa_Code_integer_o(ring_16219924574208605041nteger,numera6620942414471956472nteger(plus_plus_num(X,one))) ) ).
% eq_numeral_iff_iszero(7)
tff(fact_4946_eq__numeral__iff__iszero_I8_J,axiom,
! [Y: num] :
( ( one_one_int = aa_int_int(uminus_uminus_int,numeral_numeral_int(Y)) )
<=> aa_int_o(ring_1_iszero_int,numeral_numeral_int(plus_plus_num(one,Y))) ) ).
% eq_numeral_iff_iszero(8)
tff(fact_4947_eq__numeral__iff__iszero_I8_J,axiom,
! [Y: num] :
( ( one_one_rat = aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(Y)) )
<=> aa_rat_o(ring_1_iszero_rat,numeral_numeral_rat(plus_plus_num(one,Y))) ) ).
% eq_numeral_iff_iszero(8)
tff(fact_4948_eq__numeral__iff__iszero_I8_J,axiom,
! [Y: num] :
( ( one_one_Code_integer = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Y)) )
<=> aa_Code_integer_o(ring_16219924574208605041nteger,numera6620942414471956472nteger(plus_plus_num(one,Y))) ) ).
% eq_numeral_iff_iszero(8)
tff(fact_4949_dvd__productE,axiom,
! [P5: nat,A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,P5),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
=> ~ ! [X2: nat,Y2: nat] :
( ( P5 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X2),Y2) )
=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X2),A)
=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y2),B) ) ) ) ).
% dvd_productE
tff(fact_4950_dvd__productE,axiom,
! [P5: int,A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,P5),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
=> ~ ! [X2: int,Y2: int] :
( ( P5 = aa_int_int(aa_int_fun_int_int(times_times_int,X2),Y2) )
=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X2),A)
=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y2),B) ) ) ) ).
% dvd_productE
tff(fact_4951_division__decomp,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
=> ? [B5: nat,C4: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B5),C4) )
& aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B5),B)
& aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C4),C) ) ) ).
% division_decomp
tff(fact_4952_division__decomp,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
=> ? [B5: int,C4: int] :
( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B5),C4) )
& aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B5),B)
& aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C4),C) ) ) ).
% division_decomp
tff(fact_4953_mask__mod__exp,axiom,
! [N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N))),one_one_nat) ).
% mask_mod_exp
tff(fact_4954_mask__mod__exp,axiom,
! [N: nat,M2: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),N)),one_one_int)),aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),M2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N))),one_one_int) ).
% mask_mod_exp
tff(fact_4955_mask__mod__exp,axiom,
! [N: nat,M2: nat] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)),one_one_Code_integer)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M2)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N))),one_one_Code_integer) ).
% mask_mod_exp
tff(fact_4956_mask__mod__exp,axiom,
! [N: nat,M2: nat] : aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),N)),one_one_Code_natural)),aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),M2)) = aa_Cod4272041906116450446atural(minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(numera5444537566228673987atural(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N))),one_one_Code_natural) ).
% mask_mod_exp
tff(fact_4957_min_Obounded__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(ord_min_Code_integer(B),C))
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),C) ) ) ).
% min.bounded_iff
tff(fact_4958_min_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),C))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).
% min.bounded_iff
tff(fact_4959_min_Obounded__iff,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_min_int,B),C))
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).
% min.bounded_iff
tff(fact_4960_min_Oabsorb2,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
=> ( aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B) = B ) ) ).
% min.absorb2
tff(fact_4961_min_Oabsorb2,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) = B ) ) ).
% min.absorb2
tff(fact_4962_min_Oabsorb2,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) = B ) ) ).
% min.absorb2
tff(fact_4963_min_Oabsorb1,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B) = A ) ) ).
% min.absorb1
tff(fact_4964_min_Oabsorb1,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) = A ) ) ).
% min.absorb1
tff(fact_4965_min_Oabsorb1,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) = A ) ) ).
% min.absorb1
tff(fact_4966_min__less__iff__conj,axiom,
! [Z: nat,X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y))
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Y) ) ) ).
% min_less_iff_conj
tff(fact_4967_min__less__iff__conj,axiom,
! [Z: int,X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y))
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X)
& aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Y) ) ) ).
% min_less_iff_conj
tff(fact_4968_min__less__iff__conj,axiom,
! [Z: code_integer,X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),aa_Cod1783974652533396728nteger(ord_min_Code_integer(X),Y))
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),X)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),Y) ) ) ).
% min_less_iff_conj
tff(fact_4969_min__less__iff__conj,axiom,
! [Z: code_natural,X: code_natural,Y: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Z),aa_Cod4272041906116450446atural(ord_min_Code_natural(X),Y))
<=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Z),X)
& aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Z),Y) ) ) ).
% min_less_iff_conj
tff(fact_4970_min_Oabsorb4,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) = B ) ) ).
% min.absorb4
tff(fact_4971_min_Oabsorb4,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) = B ) ) ).
% min.absorb4
tff(fact_4972_min_Oabsorb4,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
=> ( aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B) = B ) ) ).
% min.absorb4
tff(fact_4973_min_Oabsorb4,axiom,
! [B: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),A)
=> ( aa_Cod4272041906116450446atural(ord_min_Code_natural(A),B) = B ) ) ).
% min.absorb4
tff(fact_4974_min_Oabsorb3,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) = A ) ) ).
% min.absorb3
tff(fact_4975_min_Oabsorb3,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) = A ) ) ).
% min.absorb3
tff(fact_4976_min_Oabsorb3,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ( aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B) = A ) ) ).
% min.absorb3
tff(fact_4977_min_Oabsorb3,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ( aa_Cod4272041906116450446atural(ord_min_Code_natural(A),B) = A ) ) ).
% min.absorb3
tff(fact_4978_min__bot,axiom,
! [X: set_Pr1261947904930325089at_nat] : ord_mi4194197829855655752at_nat(bot_bo2099793752762293965at_nat,X) = bot_bo2099793752762293965at_nat ).
% min_bot
tff(fact_4979_min__bot,axiom,
! [X: set_o] : ord_min_set_o(bot_bot_set_o,X) = bot_bot_set_o ).
% min_bot
tff(fact_4980_min__bot,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,bot_bot_set_nat),X) = bot_bot_set_nat ).
% min_bot
tff(fact_4981_min__bot,axiom,
! [X: set_int] : ord_min_set_int(bot_bot_set_int,X) = bot_bot_set_int ).
% min_bot
tff(fact_4982_min__bot,axiom,
! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,bot_bot_nat),X) = bot_bot_nat ).
% min_bot
tff(fact_4983_min__bot2,axiom,
! [X: set_Pr1261947904930325089at_nat] : ord_mi4194197829855655752at_nat(X,bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).
% min_bot2
tff(fact_4984_min__bot2,axiom,
! [X: set_o] : ord_min_set_o(X,bot_bot_set_o) = bot_bot_set_o ).
% min_bot2
tff(fact_4985_min__bot2,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,X),bot_bot_set_nat) = bot_bot_set_nat ).
% min_bot2
tff(fact_4986_min__bot2,axiom,
! [X: set_int] : ord_min_set_int(X,bot_bot_set_int) = bot_bot_set_int ).
% min_bot2
tff(fact_4987_min__bot2,axiom,
! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),bot_bot_nat) = bot_bot_nat ).
% min_bot2
tff(fact_4988_min__top,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,top_top_assn),X) = X ).
% min_top
tff(fact_4989_min__top,axiom,
! [X: set_o] : ord_min_set_o(top_top_set_o,X) = X ).
% min_top
tff(fact_4990_min__top,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,top_top_set_nat),X) = X ).
% min_top
tff(fact_4991_min__top,axiom,
! [X: set_Product_unit] : ord_mi7693666415311043668t_unit(top_to1996260823553986621t_unit,X) = X ).
% min_top
tff(fact_4992_min__top2,axiom,
! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,X),top_top_assn) = X ).
% min_top2
tff(fact_4993_min__top2,axiom,
! [X: set_o] : ord_min_set_o(X,top_top_set_o) = X ).
% min_top2
tff(fact_4994_min__top2,axiom,
! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,X),top_top_set_nat) = X ).
% min_top2
tff(fact_4995_min__top2,axiom,
! [X: set_Product_unit] : ord_mi7693666415311043668t_unit(X,top_to1996260823553986621t_unit) = X ).
% min_top2
tff(fact_4996_min__0__1_I2_J,axiom,
aa_Cod1783974652533396728nteger(ord_min_Code_integer(one_one_Code_integer),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% min_0_1(2)
tff(fact_4997_min__0__1_I2_J,axiom,
ord_min_rat(one_one_rat,zero_zero_rat) = zero_zero_rat ).
% min_0_1(2)
tff(fact_4998_min__0__1_I2_J,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,one_one_nat),zero_zero_nat) = zero_zero_nat ).
% min_0_1(2)
tff(fact_4999_min__0__1_I2_J,axiom,
aa_int_int(aa_int_fun_int_int(ord_min_int,one_one_int),zero_zero_int) = zero_zero_int ).
% min_0_1(2)
tff(fact_5000_min__0__1_I1_J,axiom,
aa_Cod1783974652533396728nteger(ord_min_Code_integer(zero_z3403309356797280102nteger),one_one_Code_integer) = zero_z3403309356797280102nteger ).
% min_0_1(1)
tff(fact_5001_min__0__1_I1_J,axiom,
ord_min_rat(zero_zero_rat,one_one_rat) = zero_zero_rat ).
% min_0_1(1)
tff(fact_5002_min__0__1_I1_J,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),one_one_nat) = zero_zero_nat ).
% min_0_1(1)
tff(fact_5003_min__0__1_I1_J,axiom,
aa_int_int(aa_int_fun_int_int(ord_min_int,zero_zero_int),one_one_int) = zero_zero_int ).
% min_0_1(1)
tff(fact_5004_min__0__1_I5_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,one_one_nat),numeral_numeral_nat(X)) = one_one_nat ).
% min_0_1(5)
tff(fact_5005_min__0__1_I5_J,axiom,
! [X: num] : aa_int_int(aa_int_fun_int_int(ord_min_int,one_one_int),numeral_numeral_int(X)) = one_one_int ).
% min_0_1(5)
tff(fact_5006_min__0__1_I5_J,axiom,
! [X: num] : ord_min_rat(one_one_rat,numeral_numeral_rat(X)) = one_one_rat ).
% min_0_1(5)
tff(fact_5007_min__0__1_I5_J,axiom,
! [X: num] : aa_Cod1783974652533396728nteger(ord_min_Code_integer(one_one_Code_integer),numera6620942414471956472nteger(X)) = one_one_Code_integer ).
% min_0_1(5)
tff(fact_5008_min__0__1_I5_J,axiom,
! [X: num] : aa_Cod4272041906116450446atural(ord_min_Code_natural(one_one_Code_natural),numera5444537566228673987atural(X)) = one_one_Code_natural ).
% min_0_1(5)
tff(fact_5009_min__0__1_I6_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,numeral_numeral_nat(X)),one_one_nat) = one_one_nat ).
% min_0_1(6)
tff(fact_5010_min__0__1_I6_J,axiom,
! [X: num] : aa_int_int(aa_int_fun_int_int(ord_min_int,numeral_numeral_int(X)),one_one_int) = one_one_int ).
% min_0_1(6)
tff(fact_5011_min__0__1_I6_J,axiom,
! [X: num] : ord_min_rat(numeral_numeral_rat(X),one_one_rat) = one_one_rat ).
% min_0_1(6)
tff(fact_5012_min__0__1_I6_J,axiom,
! [X: num] : aa_Cod1783974652533396728nteger(ord_min_Code_integer(numera6620942414471956472nteger(X)),one_one_Code_integer) = one_one_Code_integer ).
% min_0_1(6)
tff(fact_5013_min__0__1_I6_J,axiom,
! [X: num] : aa_Cod4272041906116450446atural(ord_min_Code_natural(numera5444537566228673987atural(X)),one_one_Code_natural) = one_one_Code_natural ).
% min_0_1(6)
tff(fact_5014_bit_Ocompl__one,axiom,
aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).
% bit.compl_one
tff(fact_5015_bit_Ocompl__one,axiom,
aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).
% bit.compl_one
tff(fact_5016_bit_Ocompl__zero,axiom,
aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).
% bit.compl_zero
tff(fact_5017_bit_Ocompl__zero,axiom,
aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% bit.compl_zero
tff(fact_5018_bit_Odisj__cancel__left,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(bit_ri7919022796975470100ot_int,X)),X) = aa_int_int(uminus_uminus_int,one_one_int) ).
% bit.disj_cancel_left
tff(fact_5019_bit_Odisj__cancel__left,axiom,
! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X)),X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% bit.disj_cancel_left
tff(fact_5020_bit_Odisj__cancel__right,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),aa_int_int(bit_ri7919022796975470100ot_int,X)) = aa_int_int(uminus_uminus_int,one_one_int) ).
% bit.disj_cancel_right
tff(fact_5021_bit_Odisj__cancel__right,axiom,
! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,X),aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% bit.disj_cancel_right
tff(fact_5022_bit_Oxor__one__left,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(uminus_uminus_int,one_one_int)),X) = aa_int_int(bit_ri7919022796975470100ot_int,X) ).
% bit.xor_one_left
tff(fact_5023_bit_Oxor__one__left,axiom,
! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X) = aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X) ).
% bit.xor_one_left
tff(fact_5024_bit_Oxor__one__right,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,X) ).
% bit.xor_one_right
tff(fact_5025_bit_Oxor__one__right,axiom,
! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X) ).
% bit.xor_one_right
tff(fact_5026_bit_Oxor__cancel__left,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(bit_ri7919022796975470100ot_int,X)),X) = aa_int_int(uminus_uminus_int,one_one_int) ).
% bit.xor_cancel_left
tff(fact_5027_bit_Oxor__cancel__left,axiom,
! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X)),X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% bit.xor_cancel_left
tff(fact_5028_bit_Oxor__cancel__right,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),aa_int_int(bit_ri7919022796975470100ot_int,X)) = aa_int_int(uminus_uminus_int,one_one_int) ).
% bit.xor_cancel_right
tff(fact_5029_bit_Oxor__cancel__right,axiom,
! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,X),aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% bit.xor_cancel_right
tff(fact_5030_push__bit__minus__one__eq__not__mask,axiom,
! [N: nat] : aa_int_int(bit_se545348938243370406it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,N)) ).
% push_bit_minus_one_eq_not_mask
tff(fact_5031_push__bit__minus__one__eq__not__mask,axiom,
! [N: nat] : aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_nat_Code_integer(bit_se2119862282449309892nteger,N)) ).
% push_bit_minus_one_eq_not_mask
tff(fact_5032_not__one__eq,axiom,
aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) = aa_int_int(uminus_uminus_int,numeral_numeral_int(aa_num_num(bit0,one))) ).
% not_one_eq
tff(fact_5033_not__one__eq,axiom,
aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% not_one_eq
tff(fact_5034_min__absorb2,axiom,
! [Y: code_integer,X: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X)
=> ( aa_Cod1783974652533396728nteger(ord_min_Code_integer(X),Y) = Y ) ) ).
% min_absorb2
tff(fact_5035_min__absorb2,axiom,
! [Y: assn,X: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,X),Y) = Y ) ) ).
% min_absorb2
tff(fact_5036_min__absorb2,axiom,
! [Y: set_nat,X: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,X),Y) = Y ) ) ).
% min_absorb2
tff(fact_5037_min__absorb2,axiom,
! [Y: nat,X: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y) = Y ) ) ).
% min_absorb2
tff(fact_5038_min__absorb2,axiom,
! [Y: int,X: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y) = Y ) ) ).
% min_absorb2
tff(fact_5039_min__absorb1,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
=> ( aa_Cod1783974652533396728nteger(ord_min_Code_integer(X),Y) = X ) ) ).
% min_absorb1
tff(fact_5040_min__absorb1,axiom,
! [X: assn,Y: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,X),Y) = X ) ) ).
% min_absorb1
tff(fact_5041_min__absorb1,axiom,
! [X: set_nat,Y: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,X),Y) = X ) ) ).
% min_absorb1
tff(fact_5042_min__absorb1,axiom,
! [X: nat,Y: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y) = X ) ) ).
% min_absorb1
tff(fact_5043_min__absorb1,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y) = X ) ) ).
% min_absorb1
tff(fact_5044_min__def,axiom,
! [A: code_integer,B: code_integer] :
aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B),A,B) ).
% min_def
tff(fact_5045_min__def,axiom,
! [A: assn,B: assn] :
aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,A),B) = $ite(aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B),A,B) ).
% min_def
tff(fact_5046_min__def,axiom,
! [A: set_nat,B: set_nat] :
aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,A),B) = $ite(aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B),A,B) ).
% min_def
tff(fact_5047_min__def,axiom,
! [A: nat,B: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),A,B) ).
% min_def
tff(fact_5048_min__def,axiom,
! [A: int,B: int] :
aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B),A,B) ).
% min_def
tff(fact_5049_min__le__iff__disj,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(ord_min_Code_integer(X),Y)),Z)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Z)
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),Z) ) ) ).
% min_le_iff_disj
tff(fact_5050_min__le__iff__disj,axiom,
! [X: nat,Y: nat,Z: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y)),Z)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z) ) ) ).
% min_le_iff_disj
tff(fact_5051_min__le__iff__disj,axiom,
! [X: int,Y: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y)),Z)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z)
| aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z) ) ) ).
% min_le_iff_disj
tff(fact_5052_min_OcoboundedI2,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B)),C) ) ).
% min.coboundedI2
tff(fact_5053_min_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),C) ) ).
% min.coboundedI2
tff(fact_5054_min_OcoboundedI2,axiom,
! [B: int,C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A),B)),C) ) ).
% min.coboundedI2
tff(fact_5055_min_OcoboundedI1,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B)),C) ) ).
% min.coboundedI1
tff(fact_5056_min_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),C) ) ).
% min.coboundedI1
tff(fact_5057_min_OcoboundedI1,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A),B)),C) ) ).
% min.coboundedI1
tff(fact_5058_min_Oabsorb__iff2,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
<=> ( aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B) = B ) ) ).
% min.absorb_iff2
tff(fact_5059_min_Oabsorb__iff2,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) = B ) ) ).
% min.absorb_iff2
tff(fact_5060_min_Oabsorb__iff2,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
<=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) = B ) ) ).
% min.absorb_iff2
tff(fact_5061_min_Oabsorb__iff1,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
<=> ( aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B) = A ) ) ).
% min.absorb_iff1
tff(fact_5062_min_Oabsorb__iff1,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) = A ) ) ).
% min.absorb_iff1
tff(fact_5063_min_Oabsorb__iff1,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
<=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) = A ) ) ).
% min.absorb_iff1
tff(fact_5064_min_Ocobounded2,axiom,
! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B)),B) ).
% min.cobounded2
tff(fact_5065_min_Ocobounded2,axiom,
! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),B) ).
% min.cobounded2
tff(fact_5066_min_Ocobounded2,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A),B)),B) ).
% min.cobounded2
tff(fact_5067_min_Ocobounded1,axiom,
! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B)),A) ).
% min.cobounded1
tff(fact_5068_min_Ocobounded1,axiom,
! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),A) ).
% min.cobounded1
tff(fact_5069_min_Ocobounded1,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A),B)),A) ).
% min.cobounded1
tff(fact_5070_min_Oorder__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
<=> ( A = aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B) ) ) ).
% min.order_iff
tff(fact_5071_min_Oorder__iff,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
<=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) ) ) ).
% min.order_iff
tff(fact_5072_min_Oorder__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
<=> ( A = aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) ) ) ).
% min.order_iff
tff(fact_5073_min_OboundedI,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(ord_min_Code_integer(B),C)) ) ) ).
% min.boundedI
tff(fact_5074_min_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),C)) ) ) ).
% min.boundedI
tff(fact_5075_min_OboundedI,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_min_int,B),C)) ) ) ).
% min.boundedI
tff(fact_5076_min_OboundedE,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(ord_min_Code_integer(B),C))
=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),C) ) ) ).
% min.boundedE
tff(fact_5077_min_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),C))
=> ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).
% min.boundedE
tff(fact_5078_min_OboundedE,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_min_int,B),C))
=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).
% min.boundedE
tff(fact_5079_min_OorderI,axiom,
! [A: code_integer,B: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B) )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).
% min.orderI
tff(fact_5080_min_OorderI,axiom,
! [A: nat,B: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).
% min.orderI
tff(fact_5081_min_OorderI,axiom,
! [A: int,B: int] :
( ( A = aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% min.orderI
tff(fact_5082_min_OorderE,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( A = aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B) ) ) ).
% min.orderE
tff(fact_5083_min_OorderE,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) ) ) ).
% min.orderE
tff(fact_5084_min_OorderE,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( A = aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) ) ) ).
% min.orderE
tff(fact_5085_min_Omono,axiom,
! [A: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),C)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),D2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B)),aa_Cod1783974652533396728nteger(ord_min_Code_integer(C),D2)) ) ) ).
% min.mono
tff(fact_5086_min_Omono,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,C),D2)) ) ) ).
% min.mono
tff(fact_5087_min_Omono,axiom,
! [A: int,C: int,B: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D2)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A),B)),aa_int_int(aa_int_fun_int_int(ord_min_int,C),D2)) ) ) ).
% min.mono
tff(fact_5088_min_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),C) ) ).
% min.strict_coboundedI2
tff(fact_5089_min_Ostrict__coboundedI2,axiom,
! [B: int,C: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A),B)),C) ) ).
% min.strict_coboundedI2
tff(fact_5090_min_Ostrict__coboundedI2,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B)),C) ) ).
% min.strict_coboundedI2
tff(fact_5091_min_Ostrict__coboundedI2,axiom,
! [B: code_natural,C: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(ord_min_Code_natural(A),B)),C) ) ).
% min.strict_coboundedI2
tff(fact_5092_min_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),C) ) ).
% min.strict_coboundedI1
tff(fact_5093_min_Ostrict__coboundedI1,axiom,
! [A: int,C: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),C)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A),B)),C) ) ).
% min.strict_coboundedI1
tff(fact_5094_min_Ostrict__coboundedI1,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),C)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B)),C) ) ).
% min.strict_coboundedI1
tff(fact_5095_min_Ostrict__coboundedI1,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),C)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(ord_min_Code_natural(A),B)),C) ) ).
% min.strict_coboundedI1
tff(fact_5096_min_Ostrict__order__iff,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
<=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B) )
& ( A != B ) ) ) ).
% min.strict_order_iff
tff(fact_5097_min_Ostrict__order__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
<=> ( ( A = aa_int_int(aa_int_fun_int_int(ord_min_int,A),B) )
& ( A != B ) ) ) ).
% min.strict_order_iff
tff(fact_5098_min_Ostrict__order__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
<=> ( ( A = aa_Cod1783974652533396728nteger(ord_min_Code_integer(A),B) )
& ( A != B ) ) ) ).
% min.strict_order_iff
tff(fact_5099_min_Ostrict__order__iff,axiom,
! [A: code_natural,B: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
<=> ( ( A = aa_Cod4272041906116450446atural(ord_min_Code_natural(A),B) )
& ( A != B ) ) ) ).
% min.strict_order_iff
tff(fact_5100_min_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),C))
=> ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).
% min.strict_boundedE
tff(fact_5101_min_Ostrict__boundedE,axiom,
! [A: int,B: int,C: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(ord_min_int,B),C))
=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).
% min.strict_boundedE
tff(fact_5102_min_Ostrict__boundedE,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(ord_min_Code_integer(B),C))
=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),C) ) ) ).
% min.strict_boundedE
tff(fact_5103_min_Ostrict__boundedE,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),aa_Cod4272041906116450446atural(ord_min_Code_natural(B),C))
=> ~ ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B)
=> ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),C) ) ) ).
% min.strict_boundedE
tff(fact_5104_min__less__iff__disj,axiom,
! [X: nat,Y: nat,Z: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y)),Z)
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ) ).
% min_less_iff_disj
tff(fact_5105_min__less__iff__disj,axiom,
! [X: int,Y: int,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y)),Z)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z)
| aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z) ) ) ).
% min_less_iff_disj
tff(fact_5106_min__less__iff__disj,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(ord_min_Code_integer(X),Y)),Z)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z)
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Z) ) ) ).
% min_less_iff_disj
tff(fact_5107_min__less__iff__disj,axiom,
! [X: code_natural,Y: code_natural,Z: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(ord_min_Code_natural(X),Y)),Z)
<=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Z)
| aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Y),Z) ) ) ).
% min_less_iff_disj
tff(fact_5108_min__add__distrib__right,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),aa_Cod1783974652533396728nteger(ord_min_Code_integer(Y),Z)) = aa_Cod1783974652533396728nteger(ord_min_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Z)) ).
% min_add_distrib_right
tff(fact_5109_min__add__distrib__right,axiom,
! [X: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),ord_min_rat(Y,Z)) = ord_min_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Z)) ).
% min_add_distrib_right
tff(fact_5110_min__add__distrib__right,axiom,
! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)) ).
% min_add_distrib_right
tff(fact_5111_min__add__distrib__right,axiom,
! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(ord_min_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(ord_min_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)) ).
% min_add_distrib_right
tff(fact_5112_min__add__distrib__left,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(ord_min_Code_integer(X),Y)),Z) = aa_Cod1783974652533396728nteger(ord_min_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Y),Z)) ).
% min_add_distrib_left
tff(fact_5113_min__add__distrib__left,axiom,
! [X: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,ord_min_rat(X,Y)),Z) = ord_min_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Z),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),Z)) ).
% min_add_distrib_left
tff(fact_5114_min__add__distrib__left,axiom,
! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).
% min_add_distrib_left
tff(fact_5115_min__add__distrib__left,axiom,
! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_min_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ).
% min_add_distrib_left
tff(fact_5116_min__diff__distrib__left,axiom,
! [X: rat,Y: rat,Z: rat] : minus_minus_rat(ord_min_rat(X,Y),Z) = ord_min_rat(minus_minus_rat(X,Z),minus_minus_rat(Y,Z)) ).
% min_diff_distrib_left
tff(fact_5117_min__diff__distrib__left,axiom,
! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_min_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ).
% min_diff_distrib_left
tff(fact_5118_not__iszero__1,axiom,
~ aa_Code_integer_o(ring_16219924574208605041nteger,one_one_Code_integer) ).
% not_iszero_1
tff(fact_5119_not__iszero__1,axiom,
~ aa_rat_o(ring_1_iszero_rat,one_one_rat) ).
% not_iszero_1
tff(fact_5120_not__iszero__1,axiom,
~ aa_int_o(ring_1_iszero_int,one_one_int) ).
% not_iszero_1
tff(fact_5121_not__iszero__neg__1,axiom,
~ aa_int_o(ring_1_iszero_int,aa_int_int(uminus_uminus_int,one_one_int)) ).
% not_iszero_neg_1
tff(fact_5122_not__iszero__neg__1,axiom,
~ aa_rat_o(ring_1_iszero_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).
% not_iszero_neg_1
tff(fact_5123_not__iszero__neg__1,axiom,
~ aa_Code_integer_o(ring_16219924574208605041nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% not_iszero_neg_1
tff(fact_5124_minus__eq__not__plus__1,axiom,
! [A: int] : aa_int_int(uminus_uminus_int,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri7919022796975470100ot_int,A)),one_one_int) ).
% minus_eq_not_plus_1
tff(fact_5125_minus__eq__not__plus__1,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,A)),one_one_Code_integer) ).
% minus_eq_not_plus_1
tff(fact_5126_not__eq__complement,axiom,
! [A: int] : aa_int_int(bit_ri7919022796975470100ot_int,A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),one_one_int) ).
% not_eq_complement
tff(fact_5127_not__eq__complement,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,A) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),one_one_Code_integer) ).
% not_eq_complement
tff(fact_5128_minus__eq__not__minus__1,axiom,
! [A: int] : aa_int_int(uminus_uminus_int,A) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)) ).
% minus_eq_not_minus_1
tff(fact_5129_minus__eq__not__minus__1,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),one_one_Code_integer)) ).
% minus_eq_not_minus_1
tff(fact_5130_unset__bit__eq__and__not,axiom,
! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),one_one_Code_integer))) ).
% unset_bit_eq_and_not
tff(fact_5131_unset__bit__eq__and__not,axiom,
! [N: nat,A: int] : aa_int_int(bit_se4203085406695923979it_int(N),A) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(N),one_one_int))) ).
% unset_bit_eq_and_not
tff(fact_5132_bit_Ocompl__unique,axiom,
! [X: code_integer,Y: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,X),Y) = zero_z3403309356797280102nteger )
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,X),Y) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
=> ( aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X) = Y ) ) ) ).
% bit.compl_unique
tff(fact_5133_bit_Ocompl__unique,axiom,
! [X: int,Y: int] :
( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y) = zero_zero_int )
=> ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y) = aa_int_int(uminus_uminus_int,one_one_int) )
=> ( aa_int_int(bit_ri7919022796975470100ot_int,X) = Y ) ) ) ).
% bit.compl_unique
tff(fact_5134_eq__numeral__iff__iszero_I5_J,axiom,
! [X: num] :
( ( numeral_numeral_int(X) = one_one_int )
<=> aa_int_o(ring_1_iszero_int,neg_numeral_sub_int(X,one)) ) ).
% eq_numeral_iff_iszero(5)
tff(fact_5135_eq__numeral__iff__iszero_I5_J,axiom,
! [X: num] :
( ( numeral_numeral_rat(X) = one_one_rat )
<=> aa_rat_o(ring_1_iszero_rat,neg_numeral_sub_rat(X,one)) ) ).
% eq_numeral_iff_iszero(5)
tff(fact_5136_eq__numeral__iff__iszero_I5_J,axiom,
! [X: num] :
( ( numera6620942414471956472nteger(X) = one_one_Code_integer )
<=> aa_Code_integer_o(ring_16219924574208605041nteger,neg_nu5755505904847501662nteger(X,one)) ) ).
% eq_numeral_iff_iszero(5)
tff(fact_5137_eq__numeral__iff__iszero_I6_J,axiom,
! [Y: num] :
( ( one_one_int = numeral_numeral_int(Y) )
<=> aa_int_o(ring_1_iszero_int,neg_numeral_sub_int(one,Y)) ) ).
% eq_numeral_iff_iszero(6)
tff(fact_5138_eq__numeral__iff__iszero_I6_J,axiom,
! [Y: num] :
( ( one_one_rat = numeral_numeral_rat(Y) )
<=> aa_rat_o(ring_1_iszero_rat,neg_numeral_sub_rat(one,Y)) ) ).
% eq_numeral_iff_iszero(6)
tff(fact_5139_eq__numeral__iff__iszero_I6_J,axiom,
! [Y: num] :
( ( one_one_Code_integer = numera6620942414471956472nteger(Y) )
<=> aa_Code_integer_o(ring_16219924574208605041nteger,neg_nu5755505904847501662nteger(one,Y)) ) ).
% eq_numeral_iff_iszero(6)
tff(fact_5140_bit_Oabstract__boolean__algebra__axioms,axiom,
aa_Code_integer_o(aa_Cod326357343816142457eger_o(boolea5643492148547655015nteger(bit_se3949692690581998587nteger,bit_se1080825931792720795nteger,bit_ri7632146776885996613nteger),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).
% bit.abstract_boolean_algebra_axioms
tff(fact_5141_bit_Oabstract__boolean__algebra__axioms,axiom,
aa_int_o(aa_int_fun_int_o(boolea5469654029284665142ra_int(bit_se725231765392027082nd_int,bit_se1409905431419307370or_int,bit_ri7919022796975470100ot_int),zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).
% bit.abstract_boolean_algebra_axioms
tff(fact_5142_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
boolea2445317508997433345nteger(bit_se3949692690581998587nteger,bit_se1080825931792720795nteger,bit_ri7632146776885996613nteger,zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),bit_se3222712562003087583nteger) ).
% bit.abstract_boolean_algebra_sym_diff_axioms
tff(fact_5143_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
boolea8527374999097803216ff_int(bit_se725231765392027082nd_int,bit_se1409905431419307370or_int,bit_ri7919022796975470100ot_int,zero_zero_int,aa_int_int(uminus_uminus_int,one_one_int),bit_se6526347334894502574or_int) ).
% bit.abstract_boolean_algebra_sym_diff_axioms
tff(fact_5144_abs__square__less__1,axiom,
! [X: rat] :
( aa_rat_o(ord_less_rat(aa_nat_rat(power_power_rat(X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_rat)
<=> aa_rat_o(ord_less_rat(abs_abs_rat(X)),one_one_rat) ) ).
% abs_square_less_1
tff(fact_5145_abs__square__less__1,axiom,
! [X: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int) ) ).
% abs_square_less_1
tff(fact_5146_abs__square__less__1,axiom,
! [X: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_Code_integer)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)),one_one_Code_integer) ) ).
% abs_square_less_1
tff(fact_5147_abs__square__le__1,axiom,
! [X: rat] :
( aa_rat_o(ord_less_eq_rat(aa_nat_rat(power_power_rat(X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_rat)
<=> aa_rat_o(ord_less_eq_rat(abs_abs_rat(X)),one_one_rat) ) ).
% abs_square_le_1
tff(fact_5148_abs__square__le__1,axiom,
! [X: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_Code_integer)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)),one_one_Code_integer) ) ).
% abs_square_le_1
tff(fact_5149_abs__square__le__1,axiom,
! [X: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(X),numeral_numeral_nat(aa_num_num(bit0,one)))),one_one_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X)),one_one_int) ) ).
% abs_square_le_1
tff(fact_5150_euclidean__size__times__nonunit,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( B != zero_z3403309356797280102nteger )
=> ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(euclid6377331345833325938nteger,B)),aa_Code_integer_nat(euclid6377331345833325938nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))) ) ) ) ).
% euclidean_size_times_nonunit
tff(fact_5151_euclidean__size__times__nonunit,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(euclid4777050414544973029ze_nat,B)),aa_nat_nat(euclid4777050414544973029ze_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ) ) ) ).
% euclidean_size_times_nonunit
tff(fact_5152_euclidean__size__times__nonunit,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(euclid4774559944035922753ze_int,B)),aa_int_nat(euclid4774559944035922753ze_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ) ).
% euclidean_size_times_nonunit
tff(fact_5153_gbinomial__reduce__nat,axiom,
! [K: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
=> ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K)) ) ) ).
% gbinomial_reduce_nat
tff(fact_5154_odd__card__imp__not__empty,axiom,
! [Aa: set_Pr1261947904930325089at_nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),finite711546835091564841at_nat(Aa))
=> ( Aa != bot_bo2099793752762293965at_nat ) ) ).
% odd_card_imp_not_empty
tff(fact_5155_odd__card__imp__not__empty,axiom,
! [Aa: set_o] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),finite_card_o(Aa))
=> ( Aa != bot_bot_set_o ) ) ).
% odd_card_imp_not_empty
tff(fact_5156_odd__card__imp__not__empty,axiom,
! [Aa: set_nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),finite_card_nat(Aa))
=> ( Aa != bot_bot_set_nat ) ) ).
% odd_card_imp_not_empty
tff(fact_5157_odd__card__imp__not__empty,axiom,
! [Aa: set_int] :
( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),finite_card_int(Aa))
=> ( Aa != bot_bot_set_int ) ) ).
% odd_card_imp_not_empty
tff(fact_5158_abs__square__eq__1,axiom,
! [X: code_integer] :
( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_Code_integer )
<=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X) = one_one_Code_integer ) ) ).
% abs_square_eq_1
tff(fact_5159_abs__square__eq__1,axiom,
! [X: rat] :
( ( aa_nat_rat(power_power_rat(X),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_rat )
<=> ( abs_abs_rat(X) = one_one_rat ) ) ).
% abs_square_eq_1
tff(fact_5160_abs__square__eq__1,axiom,
! [X: int] :
( ( aa_nat_int(power_power_int(X),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_int )
<=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).
% abs_square_eq_1
tff(fact_5161_abs__idempotent,axiom,
! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).
% abs_idempotent
tff(fact_5162_abs__idempotent,axiom,
! [A: rat] : abs_abs_rat(abs_abs_rat(A)) = abs_abs_rat(A) ).
% abs_idempotent
tff(fact_5163_abs__idempotent,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) ).
% abs_idempotent
tff(fact_5164_abs__0__eq,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% abs_0_eq
tff(fact_5165_abs__0__eq,axiom,
! [A: rat] :
( ( zero_zero_rat = abs_abs_rat(A) )
<=> ( A = zero_zero_rat ) ) ).
% abs_0_eq
tff(fact_5166_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int = aa_int_int(abs_abs_int,A) )
<=> ( A = zero_zero_int ) ) ).
% abs_0_eq
tff(fact_5167_abs__eq__0,axiom,
! [A: code_integer] :
( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = zero_z3403309356797280102nteger )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% abs_eq_0
tff(fact_5168_abs__eq__0,axiom,
! [A: rat] :
( ( abs_abs_rat(A) = zero_zero_rat )
<=> ( A = zero_zero_rat ) ) ).
% abs_eq_0
tff(fact_5169_abs__eq__0,axiom,
! [A: int] :
( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
<=> ( A = zero_zero_int ) ) ).
% abs_eq_0
tff(fact_5170_abs__zero,axiom,
aa_Cod1783974652533396728nteger(abs_abs_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% abs_zero
tff(fact_5171_abs__zero,axiom,
abs_abs_rat(zero_zero_rat) = zero_zero_rat ).
% abs_zero
tff(fact_5172_abs__zero,axiom,
aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).
% abs_zero
tff(fact_5173_abs__add__abs,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)) ).
% abs_add_abs
tff(fact_5174_abs__add__abs,axiom,
! [A: rat,B: rat] : abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B)) ).
% abs_add_abs
tff(fact_5175_abs__add__abs,axiom,
! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).
% abs_add_abs
tff(fact_5176_abs__mult__self__eq,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) ).
% abs_mult_self_eq
tff(fact_5177_abs__mult__self__eq,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(A)),abs_abs_rat(A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) ).
% abs_mult_self_eq
tff(fact_5178_abs__mult__self__eq,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,A)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).
% abs_mult_self_eq
tff(fact_5179_abs__1,axiom,
aa_Cod1783974652533396728nteger(abs_abs_Code_integer,one_one_Code_integer) = one_one_Code_integer ).
% abs_1
tff(fact_5180_abs__1,axiom,
abs_abs_rat(one_one_rat) = one_one_rat ).
% abs_1
tff(fact_5181_abs__1,axiom,
aa_int_int(abs_abs_int,one_one_int) = one_one_int ).
% abs_1
tff(fact_5182_abs__minus__cancel,axiom,
! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).
% abs_minus_cancel
tff(fact_5183_abs__minus__cancel,axiom,
! [A: rat] : abs_abs_rat(aa_rat_rat(uminus_uminus_rat,A)) = abs_abs_rat(A) ).
% abs_minus_cancel
tff(fact_5184_abs__minus__cancel,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) ).
% abs_minus_cancel
tff(fact_5185_abs__of__nonneg,axiom,
! [A: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( abs_abs_rat(A) = A ) ) ).
% abs_of_nonneg
tff(fact_5186_abs__of__nonneg,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = A ) ) ).
% abs_of_nonneg
tff(fact_5187_abs__of__nonneg,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
=> ( aa_int_int(abs_abs_int,A) = A ) ) ).
% abs_of_nonneg
tff(fact_5188_abs__le__self__iff,axiom,
! [A: rat] :
( aa_rat_o(ord_less_eq_rat(abs_abs_rat(A)),A)
<=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),A) ) ).
% abs_le_self_iff
tff(fact_5189_abs__le__self__iff,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),A)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).
% abs_le_self_iff
tff(fact_5190_abs__le__self__iff,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).
% abs_le_self_iff
tff(fact_5191_abs__le__zero__iff,axiom,
! [A: rat] :
( aa_rat_o(ord_less_eq_rat(abs_abs_rat(A)),zero_zero_rat)
<=> ( A = zero_zero_rat ) ) ).
% abs_le_zero_iff
tff(fact_5192_abs__le__zero__iff,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),zero_z3403309356797280102nteger)
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% abs_le_zero_iff
tff(fact_5193_abs__le__zero__iff,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int)
<=> ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
tff(fact_5194_zero__less__abs__iff,axiom,
! [A: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),abs_abs_rat(A))
<=> ( A != zero_zero_rat ) ) ).
% zero_less_abs_iff
tff(fact_5195_zero__less__abs__iff,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A))
<=> ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
tff(fact_5196_zero__less__abs__iff,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A))
<=> ( A != zero_z3403309356797280102nteger ) ) ).
% zero_less_abs_iff
tff(fact_5197_abs__neg__one,axiom,
aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,one_one_int)) = one_one_int ).
% abs_neg_one
tff(fact_5198_abs__neg__one,axiom,
abs_abs_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = one_one_rat ).
% abs_neg_one
tff(fact_5199_abs__neg__one,axiom,
aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).
% abs_neg_one
tff(fact_5200_gbinomial__0_I1_J,axiom,
! [A: code_integer] : aa_nat_Code_integer(gbinom8545251970709558553nteger(A),zero_zero_nat) = one_one_Code_integer ).
% gbinomial_0(1)
tff(fact_5201_gbinomial__0_I1_J,axiom,
! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).
% gbinomial_0(1)
tff(fact_5202_gbinomial__0_I1_J,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),zero_zero_nat) = one_one_nat ).
% gbinomial_0(1)
tff(fact_5203_gbinomial__0_I1_J,axiom,
! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).
% gbinomial_0(1)
tff(fact_5204_euclidean__size__1,axiom,
aa_Code_integer_nat(euclid6377331345833325938nteger,one_one_Code_integer) = one_one_nat ).
% euclidean_size_1
tff(fact_5205_euclidean__size__1,axiom,
aa_nat_nat(euclid4777050414544973029ze_nat,one_one_nat) = one_one_nat ).
% euclidean_size_1
tff(fact_5206_euclidean__size__1,axiom,
aa_int_nat(euclid4774559944035922753ze_int,one_one_int) = one_one_nat ).
% euclidean_size_1
tff(fact_5207_abs__of__nonpos,axiom,
! [A: rat] :
( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
=> ( abs_abs_rat(A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).
% abs_of_nonpos
tff(fact_5208_abs__of__nonpos,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).
% abs_of_nonpos
tff(fact_5209_abs__of__nonpos,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
=> ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).
% abs_of_nonpos
tff(fact_5210_abs__sgn__eq__1,axiom,
! [A: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = one_one_Code_integer ) ) ).
% abs_sgn_eq_1
tff(fact_5211_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
tff(fact_5212_abs__sgn__eq__1,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = one_one_int ) ) ).
% abs_sgn_eq_1
tff(fact_5213_abs__le__D1,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(abs_abs_rat(A)),B)
=> aa_rat_o(ord_less_eq_rat(A),B) ) ).
% abs_le_D1
tff(fact_5214_abs__le__D1,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).
% abs_le_D1
tff(fact_5215_abs__le__D1,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% abs_le_D1
tff(fact_5216_abs__ge__self,axiom,
! [A: rat] : aa_rat_o(ord_less_eq_rat(A),abs_abs_rat(A)) ).
% abs_ge_self
tff(fact_5217_abs__ge__self,axiom,
! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) ).
% abs_ge_self
tff(fact_5218_abs__ge__self,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(abs_abs_int,A)) ).
% abs_ge_self
tff(fact_5219_abs__mult,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)) ).
% abs_mult
tff(fact_5220_abs__mult,axiom,
! [A: rat,B: rat] : abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(A)),abs_abs_rat(B)) ).
% abs_mult
tff(fact_5221_abs__mult,axiom,
! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).
% abs_mult
tff(fact_5222_abs__one,axiom,
aa_Cod1783974652533396728nteger(abs_abs_Code_integer,one_one_Code_integer) = one_one_Code_integer ).
% abs_one
tff(fact_5223_abs__one,axiom,
abs_abs_rat(one_one_rat) = one_one_rat ).
% abs_one
tff(fact_5224_abs__one,axiom,
aa_int_int(abs_abs_int,one_one_int) = one_one_int ).
% abs_one
tff(fact_5225_abs__minus__commute,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A)) ).
% abs_minus_commute
tff(fact_5226_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
tff(fact_5227_abs__minus__commute,axiom,
! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)) ).
% abs_minus_commute
tff(fact_5228_abs__ge__zero,axiom,
! [A: rat] : aa_rat_o(ord_less_eq_rat(zero_zero_rat),abs_abs_rat(A)) ).
% abs_ge_zero
tff(fact_5229_abs__ge__zero,axiom,
! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) ).
% abs_ge_zero
tff(fact_5230_abs__ge__zero,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A)) ).
% abs_ge_zero
tff(fact_5231_abs__not__less__zero,axiom,
! [A: rat] : ~ aa_rat_o(ord_less_rat(abs_abs_rat(A)),zero_zero_rat) ).
% abs_not_less_zero
tff(fact_5232_abs__not__less__zero,axiom,
! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int) ).
% abs_not_less_zero
tff(fact_5233_abs__not__less__zero,axiom,
! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),zero_z3403309356797280102nteger) ).
% abs_not_less_zero
tff(fact_5234_abs__of__pos,axiom,
! [A: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),A)
=> ( abs_abs_rat(A) = A ) ) ).
% abs_of_pos
tff(fact_5235_abs__of__pos,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
=> ( aa_int_int(abs_abs_int,A) = A ) ) ).
% abs_of_pos
tff(fact_5236_abs__of__pos,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = A ) ) ).
% abs_of_pos
tff(fact_5237_abs__triangle__ineq,axiom,
! [A: rat,B: rat] : aa_rat_o(ord_less_eq_rat(abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B))) ).
% abs_triangle_ineq
tff(fact_5238_abs__triangle__ineq,axiom,
! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))) ).
% abs_triangle_ineq
tff(fact_5239_abs__triangle__ineq,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).
% abs_triangle_ineq
tff(fact_5240_abs__triangle__ineq2,axiom,
! [A: rat,B: rat] : aa_rat_o(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
tff(fact_5241_abs__triangle__ineq2,axiom,
! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B))) ).
% abs_triangle_ineq2
tff(fact_5242_abs__triangle__ineq2,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ).
% abs_triangle_ineq2
tff(fact_5243_abs__triangle__ineq3,axiom,
! [A: rat,B: rat] : aa_rat_o(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
tff(fact_5244_abs__triangle__ineq3,axiom,
! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B))) ).
% abs_triangle_ineq3
tff(fact_5245_abs__triangle__ineq3,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ).
% abs_triangle_ineq3
tff(fact_5246_abs__triangle__ineq2__sym,axiom,
! [A: rat,B: rat] : aa_rat_o(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
tff(fact_5247_abs__triangle__ineq2__sym,axiom,
! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A))) ).
% abs_triangle_ineq2_sym
tff(fact_5248_abs__triangle__ineq2__sym,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A))) ).
% abs_triangle_ineq2_sym
tff(fact_5249_abs__mult__less,axiom,
! [A: rat,C: rat,B: rat,D2: rat] :
( aa_rat_o(ord_less_rat(abs_abs_rat(A)),C)
=> ( aa_rat_o(ord_less_rat(abs_abs_rat(B)),D2)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(A)),abs_abs_rat(B))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),D2)) ) ) ).
% abs_mult_less
tff(fact_5250_abs__mult__less,axiom,
! [A: int,C: int,B: int,D2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),C)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B)),D2)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D2)) ) ) ).
% abs_mult_less
tff(fact_5251_abs__mult__less,axiom,
! [A: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),C)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)),D2)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D2)) ) ) ).
% abs_mult_less
tff(fact_5252_abs__leI,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(A),B)
=> ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,A)),B)
=> aa_rat_o(ord_less_eq_rat(abs_abs_rat(A)),B) ) ) ).
% abs_leI
tff(fact_5253_abs__leI,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B) ) ) ).
% abs_leI
tff(fact_5254_abs__leI,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B) ) ) ).
% abs_leI
tff(fact_5255_abs__le__D2,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(abs_abs_rat(A)),B)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ) ).
% abs_le_D2
tff(fact_5256_abs__le__D2,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ).
% abs_le_D2
tff(fact_5257_abs__le__D2,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ).
% abs_le_D2
tff(fact_5258_abs__le__iff,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(abs_abs_rat(A)),B)
<=> ( aa_rat_o(ord_less_eq_rat(A),B)
& aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).
% abs_le_iff
tff(fact_5259_abs__le__iff,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B)
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).
% abs_le_iff
tff(fact_5260_abs__le__iff,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).
% abs_le_iff
tff(fact_5261_abs__ge__minus__self,axiom,
! [A: rat] : aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,A)),abs_abs_rat(A)) ).
% abs_ge_minus_self
tff(fact_5262_abs__ge__minus__self,axiom,
! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) ).
% abs_ge_minus_self
tff(fact_5263_abs__ge__minus__self,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A)) ).
% abs_ge_minus_self
tff(fact_5264_gbinomial__Suc__Suc,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) ).
% gbinomial_Suc_Suc
tff(fact_5265_mult__sgn__abs,axiom,
! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)) = X ).
% mult_sgn_abs
tff(fact_5266_mult__sgn__abs,axiom,
! [X: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,sgn_sgn_rat(X)),abs_abs_rat(X)) = X ).
% mult_sgn_abs
tff(fact_5267_mult__sgn__abs,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,X)),aa_int_int(abs_abs_int,X)) = X ).
% mult_sgn_abs
tff(fact_5268_sgn__mult__abs,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) = A ).
% sgn_mult_abs
tff(fact_5269_sgn__mult__abs,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,sgn_sgn_rat(A)),abs_abs_rat(A)) = A ).
% sgn_mult_abs
tff(fact_5270_sgn__mult__abs,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(abs_abs_int,A)) = A ).
% sgn_mult_abs
tff(fact_5271_abs__mult__sgn,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = A ).
% abs_mult_sgn
tff(fact_5272_abs__mult__sgn,axiom,
! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(A)),sgn_sgn_rat(A)) = A ).
% abs_mult_sgn
tff(fact_5273_abs__mult__sgn,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(sgn_sgn_int,A)) = A ).
% abs_mult_sgn
tff(fact_5274_linordered__idom__class_Oabs__sgn,axiom,
! [K: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K)) ).
% linordered_idom_class.abs_sgn
tff(fact_5275_linordered__idom__class_Oabs__sgn,axiom,
! [K: rat] : abs_abs_rat(K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),sgn_sgn_rat(K)) ).
% linordered_idom_class.abs_sgn
tff(fact_5276_linordered__idom__class_Oabs__sgn,axiom,
! [K: int] : aa_int_int(abs_abs_int,K) = aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,K)) ).
% linordered_idom_class.abs_sgn
tff(fact_5277_euclidean__size__unit,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ( aa_Code_integer_nat(euclid6377331345833325938nteger,A) = aa_Code_integer_nat(euclid6377331345833325938nteger,one_one_Code_integer) ) ) ).
% euclidean_size_unit
tff(fact_5278_euclidean__size__unit,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ( aa_nat_nat(euclid4777050414544973029ze_nat,A) = aa_nat_nat(euclid4777050414544973029ze_nat,one_one_nat) ) ) ).
% euclidean_size_unit
tff(fact_5279_euclidean__size__unit,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( aa_int_nat(euclid4774559944035922753ze_int,A) = aa_int_nat(euclid4774559944035922753ze_int,one_one_int) ) ) ).
% euclidean_size_unit
tff(fact_5280_euclidean__size__mult,axiom,
! [A: code_integer,B: code_integer] : aa_Code_integer_nat(euclid6377331345833325938nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Code_integer_nat(euclid6377331345833325938nteger,A)),aa_Code_integer_nat(euclid6377331345833325938nteger,B)) ).
% euclidean_size_mult
tff(fact_5281_euclidean__size__mult,axiom,
! [A: nat,B: nat] : aa_nat_nat(euclid4777050414544973029ze_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(euclid4777050414544973029ze_nat,A)),aa_nat_nat(euclid4777050414544973029ze_nat,B)) ).
% euclidean_size_mult
tff(fact_5282_euclidean__size__mult,axiom,
! [A: int,B: int] : aa_int_nat(euclid4774559944035922753ze_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(euclid4774559944035922753ze_int,A)),aa_int_nat(euclid4774559944035922753ze_int,B)) ).
% euclidean_size_mult
tff(fact_5283_gbinomial__addition__formula,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K)) ).
% gbinomial_addition_formula
tff(fact_5284_dense__eq0__I,axiom,
! [X: rat] :
( ! [E2: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),E2)
=> aa_rat_o(ord_less_eq_rat(abs_abs_rat(X)),E2) )
=> ( X = zero_zero_rat ) ) ).
% dense_eq0_I
tff(fact_5285_abs__eq__mult,axiom,
! [A: rat,B: rat] :
( ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
| aa_rat_o(ord_less_eq_rat(A),zero_zero_rat) )
& ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),B)
| aa_rat_o(ord_less_eq_rat(B),zero_zero_rat) ) )
=> ( abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(A)),abs_abs_rat(B)) ) ) ).
% abs_eq_mult
tff(fact_5286_abs__eq__mult,axiom,
! [A: code_integer,B: code_integer] :
( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
& ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) )
=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)) ) ) ).
% abs_eq_mult
tff(fact_5287_abs__eq__mult,axiom,
! [A: int,B: int] :
( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
| aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
& ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
| aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
=> ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).
% abs_eq_mult
tff(fact_5288_abs__mult__pos,axiom,
! [X: rat,Y: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),X)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(Y)),X) = abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X)) ) ) ).
% abs_mult_pos
tff(fact_5289_abs__mult__pos,axiom,
! [X: code_integer,Y: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Y)),X) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)) ) ) ).
% abs_mult_pos
tff(fact_5290_abs__mult__pos,axiom,
! [X: int,Y: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),X) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)) ) ) ).
% abs_mult_pos
tff(fact_5291_abs__minus__le__zero,axiom,
! [A: rat] : aa_rat_o(ord_less_eq_rat(aa_rat_rat(uminus_uminus_rat,abs_abs_rat(A))),zero_zero_rat) ).
% abs_minus_le_zero
tff(fact_5292_abs__minus__le__zero,axiom,
! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A))),zero_z3403309356797280102nteger) ).
% abs_minus_le_zero
tff(fact_5293_abs__minus__le__zero,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A))),zero_zero_int) ).
% abs_minus_le_zero
tff(fact_5294_abs__triangle__ineq4,axiom,
! [A: rat,B: rat] : aa_rat_o(ord_less_eq_rat(abs_abs_rat(minus_minus_rat(A,B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B))) ).
% abs_triangle_ineq4
tff(fact_5295_abs__triangle__ineq4,axiom,
! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))) ).
% abs_triangle_ineq4
tff(fact_5296_abs__triangle__ineq4,axiom,
! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).
% abs_triangle_ineq4
tff(fact_5297_abs__diff__triangle__ineq,axiom,
! [A: rat,B: rat,C: rat,D2: rat] : aa_rat_o(ord_less_eq_rat(abs_abs_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),D2)))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(minus_minus_rat(A,C))),abs_abs_rat(minus_minus_rat(B,D2)))) ).
% abs_diff_triangle_ineq
tff(fact_5298_abs__diff__triangle__ineq,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),D2)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),C))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),D2)))) ).
% abs_diff_triangle_ineq
tff(fact_5299_abs__diff__triangle__ineq,axiom,
! [A: int,B: int,C: int,D2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),D2)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D2)))) ).
% abs_diff_triangle_ineq
tff(fact_5300_abs__of__neg,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
=> ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).
% abs_of_neg
tff(fact_5301_abs__of__neg,axiom,
! [A: rat] :
( aa_rat_o(ord_less_rat(A),zero_zero_rat)
=> ( abs_abs_rat(A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).
% abs_of_neg
tff(fact_5302_abs__of__neg,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).
% abs_of_neg
tff(fact_5303_abs__sgn__eq,axiom,
! [A: code_integer] :
aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = $ite(A = zero_z3403309356797280102nteger,zero_z3403309356797280102nteger,one_one_Code_integer) ).
% abs_sgn_eq
tff(fact_5304_abs__sgn__eq,axiom,
! [A: rat] :
abs_abs_rat(sgn_sgn_rat(A)) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).
% abs_sgn_eq
tff(fact_5305_abs__sgn__eq,axiom,
! [A: int] :
aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = $ite(A = zero_zero_int,zero_zero_int,one_one_int) ).
% abs_sgn_eq
tff(fact_5306_unit__iff__euclidean__size,axiom,
! [A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
<=> ( ( aa_Code_integer_nat(euclid6377331345833325938nteger,A) = aa_Code_integer_nat(euclid6377331345833325938nteger,one_one_Code_integer) )
& ( A != zero_z3403309356797280102nteger ) ) ) ).
% unit_iff_euclidean_size
tff(fact_5307_unit__iff__euclidean__size,axiom,
! [A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
<=> ( ( aa_nat_nat(euclid4777050414544973029ze_nat,A) = aa_nat_nat(euclid4777050414544973029ze_nat,one_one_nat) )
& ( A != zero_zero_nat ) ) ) ).
% unit_iff_euclidean_size
tff(fact_5308_unit__iff__euclidean__size,axiom,
! [A: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
<=> ( ( aa_int_nat(euclid4774559944035922753ze_int,A) = aa_int_nat(euclid4774559944035922753ze_int,one_one_int) )
& ( A != zero_zero_int ) ) ) ).
% unit_iff_euclidean_size
tff(fact_5309_size__mult__mono,axiom,
! [B: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(euclid6377331345833325938nteger,A)),aa_Code_integer_nat(euclid6377331345833325938nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))) ) ).
% size_mult_mono
tff(fact_5310_size__mult__mono,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(euclid4777050414544973029ze_nat,A)),aa_nat_nat(euclid4777050414544973029ze_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ) ).
% size_mult_mono
tff(fact_5311_size__mult__mono,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(euclid4774559944035922753ze_int,A)),aa_int_nat(euclid4774559944035922753ze_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ).
% size_mult_mono
tff(fact_5312_size__mult__mono_H,axiom,
! [B: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(euclid6377331345833325938nteger,A)),aa_Code_integer_nat(euclid6377331345833325938nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A))) ) ).
% size_mult_mono'
tff(fact_5313_size__mult__mono_H,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(euclid4777050414544973029ze_nat,A)),aa_nat_nat(euclid4777050414544973029ze_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))) ) ).
% size_mult_mono'
tff(fact_5314_size__mult__mono_H,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(euclid4774559944035922753ze_int,A)),aa_int_nat(euclid4774559944035922753ze_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A))) ) ).
% size_mult_mono'
tff(fact_5315_euclidean__size__times__unit,axiom,
! [A: code_integer,B: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
=> ( aa_Code_integer_nat(euclid6377331345833325938nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Code_integer_nat(euclid6377331345833325938nteger,B) ) ) ).
% euclidean_size_times_unit
tff(fact_5316_euclidean__size__times__unit,axiom,
! [A: nat,B: nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ( aa_nat_nat(euclid4777050414544973029ze_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(euclid4777050414544973029ze_nat,B) ) ) ).
% euclidean_size_times_unit
tff(fact_5317_euclidean__size__times__unit,axiom,
! [A: int,B: int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( aa_int_nat(euclid4774559944035922753ze_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_nat(euclid4774559944035922753ze_int,B) ) ) ).
% euclidean_size_times_unit
tff(fact_5318_abs__add__one__gt__zero,axiom,
! [X: rat] : aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),abs_abs_rat(X))) ).
% abs_add_one_gt_zero
tff(fact_5319_abs__add__one__gt__zero,axiom,
! [X: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(abs_abs_int,X))) ).
% abs_add_one_gt_zero
tff(fact_5320_abs__add__one__gt__zero,axiom,
! [X: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X))) ).
% abs_add_one_gt_zero
tff(fact_5321_card_Oempty,axiom,
finite711546835091564841at_nat(bot_bo2099793752762293965at_nat) = zero_zero_nat ).
% card.empty
tff(fact_5322_card_Oempty,axiom,
finite_card_o(bot_bot_set_o) = zero_zero_nat ).
% card.empty
tff(fact_5323_card_Oempty,axiom,
finite_card_nat(bot_bot_set_nat) = zero_zero_nat ).
% card.empty
tff(fact_5324_card_Oempty,axiom,
finite_card_int(bot_bot_set_int) = zero_zero_nat ).
% card.empty
tff(fact_5325_divmod__cases,axiom,
! [B: nat,A: nat] :
( ( ( B != zero_zero_nat )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
=> ( A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B) ) ) )
=> ( ( ( B != zero_zero_nat )
=> ! [Q5: nat,R3: nat] :
( ( aa_nat_nat(euclid3398187327856392827nt_nat,R3) = aa_nat_nat(euclid3398187327856392827nt_nat,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(euclid4777050414544973029ze_nat,R3)),aa_nat_nat(euclid4777050414544973029ze_nat,B))
=> ( ( R3 != zero_zero_nat )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = Q5 )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = R3 )
=> ( A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q5),B)),R3) ) ) ) ) ) ) )
=> ( B = zero_zero_nat ) ) ) ).
% divmod_cases
tff(fact_5326_divmod__cases,axiom,
! [B: int,A: int] :
( ( ( B != zero_zero_int )
=> ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
=> ( A != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B) ) ) )
=> ( ( ( B != zero_zero_int )
=> ! [Q5: int,R3: int] :
( ( aa_int_int(euclid3395696857347342551nt_int,R3) = aa_int_int(euclid3395696857347342551nt_int,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(euclid4774559944035922753ze_int,R3)),aa_int_nat(euclid4774559944035922753ze_int,B))
=> ( ( R3 != zero_zero_int )
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q5 )
=> ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R3 )
=> ( A != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q5),B)),R3) ) ) ) ) ) ) )
=> ( B = zero_zero_int ) ) ) ).
% divmod_cases
tff(fact_5327_divmod__cases,axiom,
! [B: code_integer,A: code_integer] :
( ( ( B != zero_z3403309356797280102nteger )
=> ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) = zero_z3403309356797280102nteger )
=> ( A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B)),B) ) ) )
=> ( ( ( B != zero_z3403309356797280102nteger )
=> ! [Q5: code_integer,R3: code_integer] :
( ( aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,R3) = aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(euclid6377331345833325938nteger,R3)),aa_Code_integer_nat(euclid6377331345833325938nteger,B))
=> ( ( R3 != zero_z3403309356797280102nteger )
=> ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) = Q5 )
=> ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) = R3 )
=> ( A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Q5),B)),R3) ) ) ) ) ) ) )
=> ( B = zero_z3403309356797280102nteger ) ) ) ).
% divmod_cases
tff(fact_5328_divmod__cases,axiom,
! [B: code_natural,A: code_natural] :
( ( ( B != zero_z2226904508553997617atural )
=> ( ( aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B) = zero_z2226904508553997617atural )
=> ( A != aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B)),B) ) ) )
=> ( ( ( B != zero_z2226904508553997617atural )
=> ! [Q5: code_natural,R3: code_natural] :
( ( aa_Cod4272041906116450446atural(euclid5112970336976722131atural,R3) = aa_Cod4272041906116450446atural(euclid5112970336976722131atural,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_natural_nat(euclid5200926497590043453atural,R3)),aa_Code_natural_nat(euclid5200926497590043453atural,B))
=> ( ( R3 != zero_z2226904508553997617atural )
=> ( ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B) = Q5 )
=> ( ( aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B) = R3 )
=> ( A != aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(Q5),B)),R3) ) ) ) ) ) ) )
=> ( B = zero_z2226904508553997617atural ) ) ) ).
% divmod_cases
tff(fact_5329_gbinomial__absorption_H,axiom,
! [K: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
=> ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,divide_divide_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).
% gbinomial_absorption'
tff(fact_5330_mod__eqI,axiom,
! [B: nat,R2: nat,Q4: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( aa_nat_nat(euclid3398187327856392827nt_nat,R2) = aa_nat_nat(euclid3398187327856392827nt_nat,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(euclid4777050414544973029ze_nat,R2)),aa_nat_nat(euclid4777050414544973029ze_nat,B))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q4),B)),R2) = A )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = R2 ) ) ) ) ) ).
% mod_eqI
tff(fact_5331_mod__eqI,axiom,
! [B: int,R2: int,Q4: int,A: int] :
( ( B != zero_zero_int )
=> ( ( aa_int_int(euclid3395696857347342551nt_int,R2) = aa_int_int(euclid3395696857347342551nt_int,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(euclid4774559944035922753ze_int,R2)),aa_int_nat(euclid4774559944035922753ze_int,B))
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),B)),R2) = A )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R2 ) ) ) ) ) ).
% mod_eqI
tff(fact_5332_mod__eqI,axiom,
! [B: code_integer,R2: code_integer,Q4: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( ( aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,R2) = aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(euclid6377331345833325938nteger,R2)),aa_Code_integer_nat(euclid6377331345833325938nteger,B))
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Q4),B)),R2) = A )
=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),B) = R2 ) ) ) ) ) ).
% mod_eqI
tff(fact_5333_mod__eqI,axiom,
! [B: code_natural,R2: code_natural,Q4: code_natural,A: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( ( aa_Cod4272041906116450446atural(euclid5112970336976722131atural,R2) = aa_Cod4272041906116450446atural(euclid5112970336976722131atural,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_natural_nat(euclid5200926497590043453atural,R2)),aa_Code_natural_nat(euclid5200926497590043453atural,B))
=> ( ( aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(Q4),B)),R2) = A )
=> ( aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),B) = R2 ) ) ) ) ) ).
% mod_eqI
tff(fact_5334_div__bounded,axiom,
! [B: nat,R2: nat,Q4: nat] :
( ( B != zero_zero_nat )
=> ( ( aa_nat_nat(euclid3398187327856392827nt_nat,R2) = aa_nat_nat(euclid3398187327856392827nt_nat,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(euclid4777050414544973029ze_nat,R2)),aa_nat_nat(euclid4777050414544973029ze_nat,B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q4),B)),R2)),B) = Q4 ) ) ) ) ).
% div_bounded
tff(fact_5335_div__bounded,axiom,
! [B: int,R2: int,Q4: int] :
( ( B != zero_zero_int )
=> ( ( aa_int_int(euclid3395696857347342551nt_int,R2) = aa_int_int(euclid3395696857347342551nt_int,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(euclid4774559944035922753ze_int,R2)),aa_int_nat(euclid4774559944035922753ze_int,B))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),B)),R2)),B) = Q4 ) ) ) ) ).
% div_bounded
tff(fact_5336_div__bounded,axiom,
! [B: code_integer,R2: code_integer,Q4: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( ( aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,R2) = aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(euclid6377331345833325938nteger,R2)),aa_Code_integer_nat(euclid6377331345833325938nteger,B))
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Q4),B)),R2)),B) = Q4 ) ) ) ) ).
% div_bounded
tff(fact_5337_div__bounded,axiom,
! [B: code_natural,R2: code_natural,Q4: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( ( aa_Cod4272041906116450446atural(euclid5112970336976722131atural,R2) = aa_Cod4272041906116450446atural(euclid5112970336976722131atural,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_natural_nat(euclid5200926497590043453atural,R2)),aa_Code_natural_nat(euclid5200926497590043453atural,B))
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(Q4),B)),R2)),B) = Q4 ) ) ) ) ).
% div_bounded
tff(fact_5338_division__segment__1,axiom,
aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,one_one_Code_integer) = one_one_Code_integer ).
% division_segment_1
tff(fact_5339_division__segment__1,axiom,
aa_nat_nat(euclid3398187327856392827nt_nat,one_one_nat) = one_one_nat ).
% division_segment_1
tff(fact_5340_division__segment__1,axiom,
aa_int_int(euclid3395696857347342551nt_int,one_one_int) = one_one_int ).
% division_segment_1
tff(fact_5341_division__segment__numeral,axiom,
! [K: num] : aa_nat_nat(euclid3398187327856392827nt_nat,numeral_numeral_nat(K)) = one_one_nat ).
% division_segment_numeral
tff(fact_5342_division__segment__numeral,axiom,
! [K: num] : aa_int_int(euclid3395696857347342551nt_int,numeral_numeral_int(K)) = one_one_int ).
% division_segment_numeral
tff(fact_5343_division__segment__numeral,axiom,
! [K: num] : aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,numera6620942414471956472nteger(K)) = one_one_Code_integer ).
% division_segment_numeral
tff(fact_5344_division__segment__numeral,axiom,
! [K: num] : aa_Cod4272041906116450446atural(euclid5112970336976722131atural,numera5444537566228673987atural(K)) = one_one_Code_natural ).
% division_segment_numeral
tff(fact_5345_division__segment__of__nat,axiom,
! [N: nat] : aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = one_one_Code_integer ).
% division_segment_of_nat
tff(fact_5346_division__segment__of__nat,axiom,
! [N: nat] : aa_nat_nat(euclid3398187327856392827nt_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)) = one_one_nat ).
% division_segment_of_nat
tff(fact_5347_division__segment__of__nat,axiom,
! [N: nat] : aa_int_int(euclid3395696857347342551nt_int,aa_nat_int(semiri1314217659103216013at_int,N)) = one_one_int ).
% division_segment_of_nat
tff(fact_5348_division__segment__euclidean__size,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,A)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_Code_integer_nat(euclid6377331345833325938nteger,A))) = A ).
% division_segment_euclidean_size
tff(fact_5349_division__segment__euclidean__size,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(euclid3398187327856392827nt_nat,A)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(euclid4777050414544973029ze_nat,A))) = A ).
% division_segment_euclidean_size
tff(fact_5350_division__segment__euclidean__size,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(euclid3395696857347342551nt_int,A)),aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(euclid4774559944035922753ze_int,A))) = A ).
% division_segment_euclidean_size
tff(fact_5351_div__mult2__eq_H,axiom,
! [A: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M2))),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).
% div_mult2_eq'
tff(fact_5352_div__mult2__eq_H,axiom,
! [A: code_integer,M2: nat,N: nat] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M2)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_nat_Code_integer(semiri4939895301339042750nteger,M2))),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).
% div_mult2_eq'
tff(fact_5353_div__mult2__eq_H,axiom,
! [A: code_natural,M2: nat,N: nat] : aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,M2)),aa_nat_Code_natural(semiri3763490453095760265atural,N))) = aa_Cod4272041906116450446atural(divide5121882707175180666atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),aa_nat_Code_natural(semiri3763490453095760265atural,M2))),aa_nat_Code_natural(semiri3763490453095760265atural,N)) ).
% div_mult2_eq'
tff(fact_5354_div__mult2__eq_H,axiom,
! [A: int,M2: nat,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_nat_int(semiri1314217659103216013at_int,N)) ).
% div_mult2_eq'
tff(fact_5355_mult__inverse__of__nat__commute,axiom,
! [Xa: nat,X: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa))),X) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa))) ).
% mult_inverse_of_nat_commute
tff(fact_5356_division__segment__mult,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( B != zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,A)),aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,B)) ) ) ) ).
% division_segment_mult
tff(fact_5357_division__segment__mult,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( aa_nat_nat(euclid3398187327856392827nt_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(euclid3398187327856392827nt_nat,A)),aa_nat_nat(euclid3398187327856392827nt_nat,B)) ) ) ) ).
% division_segment_mult
tff(fact_5358_division__segment__mult,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( aa_int_int(euclid3395696857347342551nt_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(euclid3395696857347342551nt_int,A)),aa_int_int(euclid3395696857347342551nt_int,B)) ) ) ) ).
% division_segment_mult
tff(fact_5359_is__unit__division__segment,axiom,
! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,A)),one_one_Code_integer) ).
% is_unit_division_segment
tff(fact_5360_is__unit__division__segment,axiom,
! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(euclid3398187327856392827nt_nat,A)),one_one_nat) ).
% is_unit_division_segment
tff(fact_5361_is__unit__division__segment,axiom,
! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(euclid3395696857347342551nt_int,A)),one_one_int) ).
% is_unit_division_segment
tff(fact_5362_mod__mult2__eq_H,axiom,
! [A: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M2))),aa_nat_nat(semiri1316708129612266289at_nat,N)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M2))) ).
% mod_mult2_eq'
tff(fact_5363_mod__mult2__eq_H,axiom,
! [A: code_integer,M2: nat,N: nat] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M2)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M2)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),aa_nat_Code_integer(semiri4939895301339042750nteger,M2))),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A),aa_nat_Code_integer(semiri4939895301339042750nteger,M2))) ).
% mod_mult2_eq'
tff(fact_5364_mod__mult2__eq_H,axiom,
! [A: code_natural,M2: nat,N: nat] : aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,M2)),aa_nat_Code_natural(semiri3763490453095760265atural,N))) = aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,M2)),aa_Cod4272041906116450446atural(modulo8411746178871703098atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),aa_nat_Code_natural(semiri3763490453095760265atural,M2))),aa_nat_Code_natural(semiri3763490453095760265atural,N)))),aa_Cod4272041906116450446atural(modulo8411746178871703098atural(A),aa_nat_Code_natural(semiri3763490453095760265atural,M2))) ).
% mod_mult2_eq'
tff(fact_5365_mod__mult2__eq_H,axiom,
! [A: int,M2: nat,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_nat_int(semiri1314217659103216013at_int,N)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(semiri1314217659103216013at_int,M2))) ).
% mod_mult2_eq'
tff(fact_5366_gbinomial__absorb__comp,axiom,
! [A: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K)) ).
% gbinomial_absorb_comp
tff(fact_5367_gbinomial__mult__1,axiom,
! [A: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).
% gbinomial_mult_1
tff(fact_5368_gbinomial__mult__1_H,axiom,
! [A: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).
% gbinomial_mult_1'
tff(fact_5369_Suc__times__gbinomial,axiom,
! [K: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K)) ).
% Suc_times_gbinomial
tff(fact_5370_gbinomial__absorption,axiom,
! [K: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K)) ).
% gbinomial_absorption
tff(fact_5371_gbinomial__trinomial__revision,axiom,
! [K: nat,M2: nat,A: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M2)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),M2)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,M2)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K))) ) ) ).
% gbinomial_trinomial_revision
tff(fact_5372_gbinomial__rec,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))) ).
% gbinomial_rec
tff(fact_5373_gbinomial__factors,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))),aa_nat_rat(gbinomial_rat(A),K)) ).
% gbinomial_factors
tff(fact_5374_gbinomial__index__swap,axiom,
! [K: nat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(gbinomial_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),N)) ).
% gbinomial_index_swap
tff(fact_5375_gbinomial__negated__upper,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,K),A),one_one_rat)),K)) ).
% gbinomial_negated_upper
tff(fact_5376_gbinomial__minus,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,A)),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),K)) ).
% gbinomial_minus
tff(fact_5377_div__eqI,axiom,
! [B: nat,R2: nat,Q4: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( aa_nat_nat(euclid3398187327856392827nt_nat,R2) = aa_nat_nat(euclid3398187327856392827nt_nat,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(euclid4777050414544973029ze_nat,R2)),aa_nat_nat(euclid4777050414544973029ze_nat,B))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q4),B)),R2) = A )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = Q4 ) ) ) ) ) ).
% div_eqI
tff(fact_5378_div__eqI,axiom,
! [B: int,R2: int,Q4: int,A: int] :
( ( B != zero_zero_int )
=> ( ( aa_int_int(euclid3395696857347342551nt_int,R2) = aa_int_int(euclid3395696857347342551nt_int,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(euclid4774559944035922753ze_int,R2)),aa_int_nat(euclid4774559944035922753ze_int,B))
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),B)),R2) = A )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q4 ) ) ) ) ) ).
% div_eqI
tff(fact_5379_div__eqI,axiom,
! [B: code_integer,R2: code_integer,Q4: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( ( aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,R2) = aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(euclid6377331345833325938nteger,R2)),aa_Code_integer_nat(euclid6377331345833325938nteger,B))
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Q4),B)),R2) = A )
=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A),B) = Q4 ) ) ) ) ) ).
% div_eqI
tff(fact_5380_div__eqI,axiom,
! [B: code_natural,R2: code_natural,Q4: code_natural,A: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( ( aa_Cod4272041906116450446atural(euclid5112970336976722131atural,R2) = aa_Cod4272041906116450446atural(euclid5112970336976722131atural,B) )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_natural_nat(euclid5200926497590043453atural,R2)),aa_Code_natural_nat(euclid5200926497590043453atural,B))
=> ( ( aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(times_2397367101498566445atural(Q4),B)),R2) = A )
=> ( aa_Cod4272041906116450446atural(divide5121882707175180666atural(A),B) = Q4 ) ) ) ) ) ).
% div_eqI
tff(fact_5381_of__nat__Suc,axiom,
! [M2: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,M2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_nat_Code_integer(semiri4939895301339042750nteger,M2)) ).
% of_nat_Suc
tff(fact_5382_of__nat__Suc,axiom,
! [M2: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M2)) ).
% of_nat_Suc
tff(fact_5383_of__nat__Suc,axiom,
! [M2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M2)) ).
% of_nat_Suc
tff(fact_5384_of__nat__Suc,axiom,
! [M2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,M2)) ).
% of_nat_Suc
tff(fact_5385_of__nat__1,axiom,
aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).
% of_nat_1
tff(fact_5386_of__nat__1,axiom,
aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).
% of_nat_1
tff(fact_5387_of__nat__1,axiom,
aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).
% of_nat_1
tff(fact_5388_of__nat__1,axiom,
aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).
% of_nat_1
tff(fact_5389_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_Code_integer = aa_nat_Code_integer(semiri4939895301339042750nteger,N) )
<=> ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_5390_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
<=> ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_5391_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
<=> ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_5392_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,N) )
<=> ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_5393_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( aa_nat_Code_integer(semiri4939895301339042750nteger,N) = one_one_Code_integer )
<=> ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_5394_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( aa_nat_rat(semiri681578069525770553at_rat,N) = one_one_rat )
<=> ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_5395_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( aa_nat_nat(semiri1316708129612266289at_nat,N) = one_one_nat )
<=> ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_5396_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( aa_nat_int(semiri1314217659103216013at_int,N) = one_one_int )
<=> ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_5397_of__nat__mult,axiom,
! [M2: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M2)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).
% of_nat_mult
tff(fact_5398_of__nat__mult,axiom,
! [M2: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).
% of_nat_mult
tff(fact_5399_of__nat__mult,axiom,
! [M2: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).
% of_nat_mult
tff(fact_5400_of__nat__mult,axiom,
! [M2: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N)) ).
% of_nat_mult
tff(fact_5401_nat__approx__posE,axiom,
! [E: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),E)
=> ~ ! [N4: nat] : ~ aa_rat_o(ord_less_rat(divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N4)))),E) ) ).
% nat_approx_posE
tff(fact_5402_ex__less__of__nat__mult,axiom,
! [X: rat,Y: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),X)
=> ? [N4: nat] : aa_rat_o(ord_less_rat(Y),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N4)),X)) ) ).
% ex_less_of_nat_mult
tff(fact_5403_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
tff(fact_5404_mult__of__nat__commute,axiom,
! [X: nat,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_nat_Code_integer(semiri4939895301339042750nteger,X)) ).
% mult_of_nat_commute
tff(fact_5405_mult__of__nat__commute,axiom,
! [X: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(semiri681578069525770553at_rat,X)) ).
% mult_of_nat_commute
tff(fact_5406_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,X)) ).
% mult_of_nat_commute
tff(fact_5407_mult__of__nat__commute,axiom,
! [X: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,X)) ).
% mult_of_nat_commute
tff(fact_5408_pochhammer__double,axiom,
! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),N)) ).
% pochhammer_double
tff(fact_5409_pochhammer__minus_H,axiom,
! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(B,aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K)) ).
% pochhammer_minus'
tff(fact_5410_pochhammer__minus_H,axiom,
! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),aa_nat_Code_integer(semiri4939895301339042750nteger,K))),one_one_Code_integer)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K)) ).
% pochhammer_minus'
tff(fact_5411_pochhammer__minus_H,axiom,
! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K)) ).
% pochhammer_minus'
tff(fact_5412_pochhammer__minus,axiom,
! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(B,aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).
% pochhammer_minus
tff(fact_5413_pochhammer__minus,axiom,
! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),aa_nat_Code_integer(semiri4939895301339042750nteger,K))),one_one_Code_integer)),K)) ).
% pochhammer_minus
tff(fact_5414_pochhammer__minus,axiom,
! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K)) ).
% pochhammer_minus
tff(fact_5415_pochhammer__absorb__comp,axiom,
! [R2: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(R2,aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R2)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,R2)),one_one_rat)),K)) ).
% pochhammer_absorb_comp
tff(fact_5416_pochhammer__absorb__comp,axiom,
! [R2: code_integer,K: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(R2),aa_nat_Code_integer(semiri4939895301339042750nteger,K))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2)),K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2)),one_one_Code_integer)),K)) ).
% pochhammer_absorb_comp
tff(fact_5417_pochhammer__absorb__comp,axiom,
! [R2: int,K: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R2)),K)) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R2)),one_one_int)),K)) ).
% pochhammer_absorb_comp
tff(fact_5418_fact__reduce,axiom,
! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_nat_Code_integer(semiri3624122377584611663nteger,N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_5419_fact__reduce,axiom,
! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_5420_fact__reduce,axiom,
! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_5421_fact__reduce,axiom,
! [N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
=> ( aa_nat_int(semiri1406184849735516958ct_int,N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_5422_fact__num__eq__if,axiom,
! [M2: nat] :
aa_nat_Code_integer(semiri3624122377584611663nteger,M2) = $ite(M2 = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M2)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).
% fact_num_eq_if
tff(fact_5423_fact__num__eq__if,axiom,
! [M2: nat] :
aa_nat_rat(semiri773545260158071498ct_rat,M2) = $ite(M2 = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).
% fact_num_eq_if
tff(fact_5424_fact__num__eq__if,axiom,
! [M2: nat] :
aa_nat_nat(semiri1408675320244567234ct_nat,M2) = $ite(M2 = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).
% fact_num_eq_if
tff(fact_5425_fact__num__eq__if,axiom,
! [M2: nat] :
aa_nat_int(semiri1406184849735516958ct_int,M2) = $ite(M2 = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).
% fact_num_eq_if
tff(fact_5426_fact__0,axiom,
aa_nat_Code_integer(semiri3624122377584611663nteger,zero_zero_nat) = one_one_Code_integer ).
% fact_0
tff(fact_5427_fact__0,axiom,
aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).
% fact_0
tff(fact_5428_fact__0,axiom,
aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).
% fact_0
tff(fact_5429_fact__0,axiom,
aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).
% fact_0
tff(fact_5430_pochhammer__0,axiom,
! [A: code_integer] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),zero_zero_nat) = one_one_Code_integer ).
% pochhammer_0
tff(fact_5431_pochhammer__0,axiom,
! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).
% pochhammer_0
tff(fact_5432_pochhammer__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),zero_zero_nat) = one_one_nat ).
% pochhammer_0
tff(fact_5433_pochhammer__0,axiom,
! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).
% pochhammer_0
tff(fact_5434_fact__1,axiom,
aa_nat_Code_integer(semiri3624122377584611663nteger,one_one_nat) = one_one_Code_integer ).
% fact_1
tff(fact_5435_fact__1,axiom,
aa_nat_rat(semiri773545260158071498ct_rat,one_one_nat) = one_one_rat ).
% fact_1
tff(fact_5436_fact__1,axiom,
aa_nat_nat(semiri1408675320244567234ct_nat,one_one_nat) = one_one_nat ).
% fact_1
tff(fact_5437_fact__1,axiom,
aa_nat_int(semiri1406184849735516958ct_int,one_one_nat) = one_one_int ).
% fact_1
tff(fact_5438_fact__Suc__0,axiom,
aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,zero_zero_nat)) = one_one_Code_integer ).
% fact_Suc_0
tff(fact_5439_fact__Suc__0,axiom,
aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).
% fact_Suc_0
tff(fact_5440_fact__Suc__0,axiom,
aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).
% fact_Suc_0
tff(fact_5441_fact__Suc__0,axiom,
aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).
% fact_Suc_0
tff(fact_5442_fact__Suc,axiom,
! [N: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,N))),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ).
% fact_Suc
tff(fact_5443_fact__Suc,axiom,
! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N))),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).
% fact_Suc
tff(fact_5444_fact__Suc,axiom,
! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N))),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ).
% fact_Suc
tff(fact_5445_fact__Suc,axiom,
! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))),aa_nat_int(semiri1406184849735516958ct_int,N)) ).
% fact_Suc
tff(fact_5446_pochhammer__fact,axiom,
! [N: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,N) = aa_nat_Code_integer(comm_s8582702949713902594nteger(one_one_Code_integer),N) ).
% pochhammer_fact
tff(fact_5447_pochhammer__fact,axiom,
! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_nat_rat(comm_s4028243227959126397er_rat(one_one_rat),N) ).
% pochhammer_fact
tff(fact_5448_pochhammer__fact,axiom,
! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,one_one_nat),N) ).
% pochhammer_fact
tff(fact_5449_pochhammer__fact,axiom,
! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) = aa_nat_int(comm_s4660882817536571857er_int(one_one_int),N) ).
% pochhammer_fact
tff(fact_5450_fact__ge__1,axiom,
! [N: nat] : aa_rat_o(ord_less_eq_rat(one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).
% fact_ge_1
tff(fact_5451_fact__ge__1,axiom,
! [N: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ).
% fact_ge_1
tff(fact_5452_fact__ge__1,axiom,
! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ).
% fact_ge_1
tff(fact_5453_fact__ge__1,axiom,
! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,N)) ).
% fact_ge_1
tff(fact_5454_pochhammer__same,axiom,
! [N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).
% pochhammer_same
tff(fact_5455_pochhammer__same,axiom,
! [N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ).
% pochhammer_same
tff(fact_5456_pochhammer__same,axiom,
! [N: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(semiri1406184849735516958ct_int,N)) ).
% pochhammer_same
tff(fact_5457_gbinomial__pochhammer_H,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = divide_divide_rat(aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).
% gbinomial_pochhammer'
tff(fact_5458_gbinomial__pochhammer,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,A)),K)),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).
% gbinomial_pochhammer
tff(fact_5459_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,N))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N))) ).
% fact_fact_dvd_fact
tff(fact_5460_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : aa_rat_o(dvd_dvd_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,N))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N))) ).
% fact_fact_dvd_fact
tff(fact_5461_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,N))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N))) ).
% fact_fact_dvd_fact
tff(fact_5462_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,N))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N))) ).
% fact_fact_dvd_fact
tff(fact_5463_pochhammer__0__left,axiom,
! [N: nat] :
aa_nat_Code_integer(comm_s8582702949713902594nteger(zero_z3403309356797280102nteger),N) = $ite(N = zero_zero_nat,one_one_Code_integer,zero_z3403309356797280102nteger) ).
% pochhammer_0_left
tff(fact_5464_pochhammer__0__left,axiom,
! [N: nat] :
aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = $ite(N = zero_zero_nat,one_one_rat,zero_zero_rat) ).
% pochhammer_0_left
tff(fact_5465_pochhammer__0__left,axiom,
! [N: nat] :
aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = $ite(N = zero_zero_nat,one_one_nat,zero_zero_nat) ).
% pochhammer_0_left
tff(fact_5466_pochhammer__0__left,axiom,
! [N: nat] :
aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = $ite(N = zero_zero_nat,one_one_int,zero_zero_int) ).
% pochhammer_0_left
tff(fact_5467_choose__dvd,axiom,
! [K: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ) ).
% choose_dvd
tff(fact_5468_choose__dvd,axiom,
! [K: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
=> aa_rat_o(dvd_dvd_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_rat(semiri773545260158071498ct_rat,N)) ) ).
% choose_dvd
tff(fact_5469_choose__dvd,axiom,
! [K: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ) ).
% choose_dvd
tff(fact_5470_choose__dvd,axiom,
! [K: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_int(semiri1406184849735516958ct_int,N)) ) ).
% choose_dvd
tff(fact_5471_pochhammer__rec,axiom,
! [A: code_integer,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),N)) ).
% pochhammer_rec
tff(fact_5472_pochhammer__rec,axiom,
! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),N)) ).
% pochhammer_rec
tff(fact_5473_pochhammer__rec,axiom,
! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),N)) ).
% pochhammer_rec
tff(fact_5474_pochhammer__rec,axiom,
! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),N)) ).
% pochhammer_rec
tff(fact_5475_pochhammer__rec_H,axiom,
! [Z: code_integer,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,N))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N)) ).
% pochhammer_rec'
tff(fact_5476_pochhammer__rec_H,axiom,
! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,N))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)) ).
% pochhammer_rec'
tff(fact_5477_pochhammer__rec_H,axiom,
! [Z: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)) ).
% pochhammer_rec'
tff(fact_5478_pochhammer__rec_H,axiom,
! [Z: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(comm_s4660882817536571857er_int(Z),N)) ).
% pochhammer_rec'
tff(fact_5479_pochhammer__Suc,axiom,
! [A: code_integer,N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) ).
% pochhammer_Suc
tff(fact_5480_pochhammer__Suc,axiom,
! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(A),N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,N))) ).
% pochhammer_Suc
tff(fact_5481_pochhammer__Suc,axiom,
! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).
% pochhammer_Suc
tff(fact_5482_pochhammer__Suc,axiom,
! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,N))) ).
% pochhammer_Suc
tff(fact_5483_pochhammer__product_H,axiom,
! [Z: code_integer,N: nat,M2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,N))),M2)) ).
% pochhammer_product'
tff(fact_5484_pochhammer__product_H,axiom,
! [Z: rat,N: nat,M2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,N))),M2)) ).
% pochhammer_product'
tff(fact_5485_pochhammer__product_H,axiom,
! [Z: nat,N: nat,M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),M2)) ).
% pochhammer_product'
tff(fact_5486_pochhammer__product_H,axiom,
! [Z: int,N: nat,M2: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),N)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),M2)) ).
% pochhammer_product'
tff(fact_5487_fact__numeral,axiom,
! [K: num] : aa_nat_nat(semiri1408675320244567234ct_nat,numeral_numeral_nat(K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(K)),aa_nat_nat(semiri1408675320244567234ct_nat,pred_numeral(K))) ).
% fact_numeral
tff(fact_5488_fact__numeral,axiom,
! [K: num] : aa_nat_int(semiri1406184849735516958ct_int,numeral_numeral_nat(K)) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(K)),aa_nat_int(semiri1406184849735516958ct_int,pred_numeral(K))) ).
% fact_numeral
tff(fact_5489_fact__numeral,axiom,
! [K: num] : aa_nat_rat(semiri773545260158071498ct_rat,numeral_numeral_nat(K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(K)),aa_nat_rat(semiri773545260158071498ct_rat,pred_numeral(K))) ).
% fact_numeral
tff(fact_5490_fact__numeral,axiom,
! [K: num] : aa_nat_Code_integer(semiri3624122377584611663nteger,numeral_numeral_nat(K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(K)),aa_nat_Code_integer(semiri3624122377584611663nteger,pred_numeral(K))) ).
% fact_numeral
tff(fact_5491_fact__numeral,axiom,
! [K: num] : aa_nat_Code_natural(semiri2447717529341329178atural,numeral_numeral_nat(K)) = aa_Cod4272041906116450446atural(times_2397367101498566445atural(numera5444537566228673987atural(K)),aa_nat_Code_natural(semiri2447717529341329178atural,pred_numeral(K))) ).
% fact_numeral
tff(fact_5492_fact__double,axiom,
! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))),aa_nat_rat(comm_s4028243227959126397er_rat(divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))),N))),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).
% fact_double
tff(fact_5493_pochhammer__product,axiom,
! [M2: nat,N: nat,Z: code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),M2)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,M2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).
% pochhammer_product
tff(fact_5494_pochhammer__product,axiom,
! [M2: nat,N: nat,Z: rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),M2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,M2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).
% pochhammer_product
tff(fact_5495_pochhammer__product,axiom,
! [M2: nat,N: nat,Z: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),M2)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,M2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).
% pochhammer_product
tff(fact_5496_pochhammer__product,axiom,
! [M2: nat,N: nat,Z: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_nat_int(comm_s4660882817536571857er_int(Z),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),M2)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,M2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ).
% pochhammer_product
tff(fact_5497_card__doubleton__eq__2__iff,axiom,
! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
( ( finite3771342082235030671at_nat(aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(B),bot_bo228742789529271731at_nat))) = numeral_numeral_nat(aa_num_num(bit0,one)) )
<=> ( A != B ) ) ).
% card_doubleton_eq_2_iff
tff(fact_5498_card__doubleton__eq__2__iff,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( finite711546835091564841at_nat(aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(B),bot_bo2099793752762293965at_nat))) = numeral_numeral_nat(aa_num_num(bit0,one)) )
<=> ( A != B ) ) ).
% card_doubleton_eq_2_iff
tff(fact_5499_card__doubleton__eq__2__iff,axiom,
! [A: $o,B: $o] :
( ( finite_card_o(aa_set_o_set_o(insert_o((A)),aa_set_o_set_o(insert_o((B)),bot_bot_set_o))) = numeral_numeral_nat(aa_num_num(bit0,one)) )
<=> ~ ( (A)
<=> (B) ) ) ).
% card_doubleton_eq_2_iff
tff(fact_5500_card__doubleton__eq__2__iff,axiom,
! [A: nat,B: nat] :
( ( finite_card_nat(aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat))) = numeral_numeral_nat(aa_num_num(bit0,one)) )
<=> ( A != B ) ) ).
% card_doubleton_eq_2_iff
tff(fact_5501_card__doubleton__eq__2__iff,axiom,
! [A: int,B: int] :
( ( finite_card_int(aa_set_int_set_int(insert_int(A),aa_set_int_set_int(insert_int(B),bot_bot_set_int))) = numeral_numeral_nat(aa_num_num(bit0,one)) )
<=> ( A != B ) ) ).
% card_doubleton_eq_2_iff
tff(fact_5502_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y)
=> ( bezw(X,Y) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ).
% bezw_non_0
tff(fact_5503_neg__numeral__le__ceiling,axiom,
! [V: num,X: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(V))),archim2889992004027027881ng_rat(X))
<=> aa_rat_o(ord_less_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V)),one_one_rat)),X) ) ).
% neg_numeral_le_ceiling
tff(fact_5504_ceiling__less__neg__numeral,axiom,
! [X: rat,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(X)),aa_int_int(uminus_uminus_int,numeral_numeral_int(V)))
<=> aa_rat_o(ord_less_eq_rat(X),minus_minus_rat(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V)),one_one_rat)) ) ).
% ceiling_less_neg_numeral
tff(fact_5505_binomial__fact,axiom,
! [K: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
=> ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,N),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).
% binomial_fact
tff(fact_5506_fact__binomial,axiom,
! [K: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) = divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,N),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ) ).
% fact_binomial
tff(fact_5507_floor__le__neg__numeral,axiom,
! [X: rat,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(X)),aa_int_int(uminus_uminus_int,numeral_numeral_int(V)))
<=> aa_rat_o(ord_less_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),one_one_rat)) ) ).
% floor_le_neg_numeral
tff(fact_5508_singletonI,axiom,
! [A: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),bot_bo228742789529271731at_nat)) ).
% singletonI
tff(fact_5509_singletonI,axiom,
! [A: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A),bot_bo8632255151191677115_nat_o)) ).
% singletonI
tff(fact_5510_singletonI,axiom,
! [A: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),bot_bo2099793752762293965at_nat)) ).
% singletonI
tff(fact_5511_singletonI,axiom,
! [A: $o] : aa_set_o_o(member_o((A)),aa_set_o_set_o(insert_o((A)),bot_bot_set_o)) ).
% singletonI
tff(fact_5512_singletonI,axiom,
! [A: nat] : aa_set_nat_o(member_nat(A),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)) ).
% singletonI
tff(fact_5513_singletonI,axiom,
! [A: int] : aa_set_int_o(member_int(A),aa_set_int_set_int(insert_int(A),bot_bot_set_int)) ).
% singletonI
tff(fact_5514_floor__one,axiom,
archim3151403230148437115or_rat(one_one_rat) = one_one_int ).
% floor_one
tff(fact_5515_ceiling__one,axiom,
archim2889992004027027881ng_rat(one_one_rat) = one_one_int ).
% ceiling_one
tff(fact_5516_singleton__insert__inj__eq,axiom,
! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,Aa: set_Pr4329608150637261639at_nat] :
( ( aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(B),bot_bo228742789529271731at_nat) = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),Aa) )
<=> ( ( A = B )
& aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(B),bot_bo228742789529271731at_nat)) ) ) ).
% singleton_insert_inj_eq
tff(fact_5517_singleton__insert__inj__eq,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat,Aa: set_Pr1261947904930325089at_nat] :
( ( aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(B),bot_bo2099793752762293965at_nat) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),Aa) )
<=> ( ( A = B )
& aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Aa),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(B),bot_bo2099793752762293965at_nat)) ) ) ).
% singleton_insert_inj_eq
tff(fact_5518_singleton__insert__inj__eq,axiom,
! [B: $o,A: $o,Aa: set_o] :
( ( aa_set_o_set_o(insert_o((B)),bot_bot_set_o) = aa_set_o_set_o(insert_o((A)),Aa) )
<=> ( ( (A)
<=> (B) )
& aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Aa),aa_set_o_set_o(insert_o((B)),bot_bot_set_o)) ) ) ).
% singleton_insert_inj_eq
tff(fact_5519_singleton__insert__inj__eq,axiom,
! [B: int,A: int,Aa: set_int] :
( ( aa_set_int_set_int(insert_int(B),bot_bot_set_int) = aa_set_int_set_int(insert_int(A),Aa) )
<=> ( ( A = B )
& aa_set_int_o(ord_less_eq_set_int(Aa),aa_set_int_set_int(insert_int(B),bot_bot_set_int)) ) ) ).
% singleton_insert_inj_eq
tff(fact_5520_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,Aa: set_nat] :
( ( aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat) = aa_set_nat_set_nat(insert_nat(A),Aa) )
<=> ( ( A = B )
& aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat)) ) ) ).
% singleton_insert_inj_eq
tff(fact_5521_singleton__insert__inj__eq_H,axiom,
! [A: produc3843707927480180839at_nat,Aa: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
( ( aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),Aa) = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(B),bot_bo228742789529271731at_nat) )
<=> ( ( A = B )
& aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(B),bot_bo228742789529271731at_nat)) ) ) ).
% singleton_insert_inj_eq'
tff(fact_5522_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_nat_nat,Aa: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
( ( aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),Aa) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(B),bot_bo2099793752762293965at_nat) )
<=> ( ( A = B )
& aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Aa),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(B),bot_bo2099793752762293965at_nat)) ) ) ).
% singleton_insert_inj_eq'
tff(fact_5523_singleton__insert__inj__eq_H,axiom,
! [A: $o,Aa: set_o,B: $o] :
( ( aa_set_o_set_o(insert_o((A)),Aa) = aa_set_o_set_o(insert_o((B)),bot_bot_set_o) )
<=> ( ( (A)
<=> (B) )
& aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Aa),aa_set_o_set_o(insert_o((B)),bot_bot_set_o)) ) ) ).
% singleton_insert_inj_eq'
tff(fact_5524_singleton__insert__inj__eq_H,axiom,
! [A: int,Aa: set_int,B: int] :
( ( aa_set_int_set_int(insert_int(A),Aa) = aa_set_int_set_int(insert_int(B),bot_bot_set_int) )
<=> ( ( A = B )
& aa_set_int_o(ord_less_eq_set_int(Aa),aa_set_int_set_int(insert_int(B),bot_bot_set_int)) ) ) ).
% singleton_insert_inj_eq'
tff(fact_5525_singleton__insert__inj__eq_H,axiom,
! [A: nat,Aa: set_nat,B: nat] :
( ( aa_set_nat_set_nat(insert_nat(A),Aa) = aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat) )
<=> ( ( A = B )
& aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat)) ) ) ).
% singleton_insert_inj_eq'
tff(fact_5526_disjoint__insert_I2_J,axiom,
! [Aa: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,Ba: set_Pr4329608150637261639at_nat] :
( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(B),Ba)) )
<=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),Aa)
& ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Aa),Ba) ) ) ) ).
% disjoint_insert(2)
tff(fact_5527_disjoint__insert_I2_J,axiom,
! [Aa: set_fu3555266933746557007_nat_o,B: fun_Pr6460944699146269935_nat_o,Ba: set_fu3555266933746557007_nat_o] :
( ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(Aa,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),Ba)) )
<=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),Aa)
& ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(Aa,Ba) ) ) ) ).
% disjoint_insert(2)
tff(fact_5528_disjoint__insert_I2_J,axiom,
! [Aa: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,Ba: set_Pr1261947904930325089at_nat] :
( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Aa),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(B),Ba)) )
<=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),Aa)
& ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Aa),Ba) ) ) ) ).
% disjoint_insert(2)
tff(fact_5529_disjoint__insert_I2_J,axiom,
! [Aa: set_o,B: $o,Ba: set_o] :
( ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Aa),aa_set_o_set_o(insert_o((B)),Ba)) )
<=> ( ~ aa_set_o_o(member_o((B)),Aa)
& ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Aa),Ba) ) ) ) ).
% disjoint_insert(2)
tff(fact_5530_disjoint__insert_I2_J,axiom,
! [Aa: set_nat,B: nat,Ba: set_nat] :
( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),aa_set_nat_set_nat(insert_nat(B),Ba)) )
<=> ( ~ aa_set_nat_o(member_nat(B),Aa)
& ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),Ba) ) ) ) ).
% disjoint_insert(2)
tff(fact_5531_disjoint__insert_I2_J,axiom,
! [Aa: set_int,B: int,Ba: set_int] :
( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Aa),aa_set_int_set_int(insert_int(B),Ba)) )
<=> ( ~ aa_set_int_o(member_int(B),Aa)
& ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Aa),Ba) ) ) ) ).
% disjoint_insert(2)
tff(fact_5532_disjoint__insert_I1_J,axiom,
! [Ba: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,Aa: set_Pr4329608150637261639at_nat] :
( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Ba),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),Aa)) = bot_bo228742789529271731at_nat )
<=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),Ba)
& ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Ba),Aa) = bot_bo228742789529271731at_nat ) ) ) ).
% disjoint_insert(1)
tff(fact_5533_disjoint__insert_I1_J,axiom,
! [Ba: set_fu3555266933746557007_nat_o,A: fun_Pr6460944699146269935_nat_o,Aa: set_fu3555266933746557007_nat_o] :
( ( inf_in3376754634429691197_nat_o(Ba,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A),Aa)) = bot_bo8632255151191677115_nat_o )
<=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),Ba)
& ( inf_in3376754634429691197_nat_o(Ba,Aa) = bot_bo8632255151191677115_nat_o ) ) ) ).
% disjoint_insert(1)
tff(fact_5534_disjoint__insert_I1_J,axiom,
! [Ba: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,Aa: set_Pr1261947904930325089at_nat] :
( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ba),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),Aa)) = bot_bo2099793752762293965at_nat )
<=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),Ba)
& ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ba),Aa) = bot_bo2099793752762293965at_nat ) ) ) ).
% disjoint_insert(1)
tff(fact_5535_disjoint__insert_I1_J,axiom,
! [Ba: set_o,A: $o,Aa: set_o] :
( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Ba),aa_set_o_set_o(insert_o((A)),Aa)) = bot_bot_set_o )
<=> ( ~ aa_set_o_o(member_o((A)),Ba)
& ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Ba),Aa) = bot_bot_set_o ) ) ) ).
% disjoint_insert(1)
tff(fact_5536_disjoint__insert_I1_J,axiom,
! [Ba: set_nat,A: nat,Aa: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Ba),aa_set_nat_set_nat(insert_nat(A),Aa)) = bot_bot_set_nat )
<=> ( ~ aa_set_nat_o(member_nat(A),Ba)
& ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Ba),Aa) = bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
tff(fact_5537_disjoint__insert_I1_J,axiom,
! [Ba: set_int,A: int,Aa: set_int] :
( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Ba),aa_set_int_set_int(insert_int(A),Aa)) = bot_bot_set_int )
<=> ( ~ aa_set_int_o(member_int(A),Ba)
& ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Ba),Aa) = bot_bot_set_int ) ) ) ).
% disjoint_insert(1)
tff(fact_5538_insert__disjoint_I2_J,axiom,
! [A: produc3843707927480180839at_nat,Aa: set_Pr4329608150637261639at_nat,Ba: set_Pr4329608150637261639at_nat] :
( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),Aa)),Ba) )
<=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),Ba)
& ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Aa),Ba) ) ) ) ).
% insert_disjoint(2)
tff(fact_5539_insert__disjoint_I2_J,axiom,
! [A: fun_Pr6460944699146269935_nat_o,Aa: set_fu3555266933746557007_nat_o,Ba: set_fu3555266933746557007_nat_o] :
( ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A),Aa),Ba) )
<=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),Ba)
& ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(Aa,Ba) ) ) ) ).
% insert_disjoint(2)
tff(fact_5540_insert__disjoint_I2_J,axiom,
! [A: product_prod_nat_nat,Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),Aa)),Ba) )
<=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),Ba)
& ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Aa),Ba) ) ) ) ).
% insert_disjoint(2)
tff(fact_5541_insert__disjoint_I2_J,axiom,
! [A: $o,Aa: set_o,Ba: set_o] :
( ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(insert_o((A)),Aa)),Ba) )
<=> ( ~ aa_set_o_o(member_o((A)),Ba)
& ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Aa),Ba) ) ) ) ).
% insert_disjoint(2)
tff(fact_5542_insert__disjoint_I2_J,axiom,
! [A: nat,Aa: set_nat,Ba: set_nat] :
( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(insert_nat(A),Aa)),Ba) )
<=> ( ~ aa_set_nat_o(member_nat(A),Ba)
& ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),Ba) ) ) ) ).
% insert_disjoint(2)
tff(fact_5543_insert__disjoint_I2_J,axiom,
! [A: int,Aa: set_int,Ba: set_int] :
( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(insert_int(A),Aa)),Ba) )
<=> ( ~ aa_set_int_o(member_int(A),Ba)
& ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Aa),Ba) ) ) ) ).
% insert_disjoint(2)
tff(fact_5544_insert__disjoint_I1_J,axiom,
! [A: produc3843707927480180839at_nat,Aa: set_Pr4329608150637261639at_nat,Ba: set_Pr4329608150637261639at_nat] :
( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),Aa)),Ba) = bot_bo228742789529271731at_nat )
<=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),Ba)
& ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Aa),Ba) = bot_bo228742789529271731at_nat ) ) ) ).
% insert_disjoint(1)
tff(fact_5545_insert__disjoint_I1_J,axiom,
! [A: fun_Pr6460944699146269935_nat_o,Aa: set_fu3555266933746557007_nat_o,Ba: set_fu3555266933746557007_nat_o] :
( ( inf_in3376754634429691197_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A),Aa),Ba) = bot_bo8632255151191677115_nat_o )
<=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),Ba)
& ( inf_in3376754634429691197_nat_o(Aa,Ba) = bot_bo8632255151191677115_nat_o ) ) ) ).
% insert_disjoint(1)
tff(fact_5546_insert__disjoint_I1_J,axiom,
! [A: product_prod_nat_nat,Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),Aa)),Ba) = bot_bo2099793752762293965at_nat )
<=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),Ba)
& ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Aa),Ba) = bot_bo2099793752762293965at_nat ) ) ) ).
% insert_disjoint(1)
tff(fact_5547_insert__disjoint_I1_J,axiom,
! [A: $o,Aa: set_o,Ba: set_o] :
( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,aa_set_o_set_o(insert_o((A)),Aa)),Ba) = bot_bot_set_o )
<=> ( ~ aa_set_o_o(member_o((A)),Ba)
& ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Aa),Ba) = bot_bot_set_o ) ) ) ).
% insert_disjoint(1)
tff(fact_5548_insert__disjoint_I1_J,axiom,
! [A: nat,Aa: set_nat,Ba: set_nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(insert_nat(A),Aa)),Ba) = bot_bot_set_nat )
<=> ( ~ aa_set_nat_o(member_nat(A),Ba)
& ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),Ba) = bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
tff(fact_5549_insert__disjoint_I1_J,axiom,
! [A: int,Aa: set_int,Ba: set_int] :
( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(insert_int(A),Aa)),Ba) = bot_bot_set_int )
<=> ( ~ aa_set_int_o(member_int(A),Ba)
& ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Aa),Ba) = bot_bot_set_int ) ) ) ).
% insert_disjoint(1)
tff(fact_5550_insert__Diff__single,axiom,
! [A: produc3843707927480180839at_nat,Aa: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),bot_bo228742789529271731at_nat))) = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),Aa) ).
% insert_Diff_single
tff(fact_5551_insert__Diff__single,axiom,
! [A: product_prod_nat_nat,Aa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),bot_bo2099793752762293965at_nat))) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),Aa) ).
% insert_Diff_single
tff(fact_5552_insert__Diff__single,axiom,
! [A: $o,Aa: set_o] : aa_set_o_set_o(insert_o((A)),minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((A)),bot_bot_set_o))) = aa_set_o_set_o(insert_o((A)),Aa) ).
% insert_Diff_single
tff(fact_5553_insert__Diff__single,axiom,
! [A: int,Aa: set_int] : aa_set_int_set_int(insert_int(A),minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(A),bot_bot_set_int))) = aa_set_int_set_int(insert_int(A),Aa) ).
% insert_Diff_single
tff(fact_5554_insert__Diff__single,axiom,
! [A: nat,Aa: set_nat] : aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) = aa_set_nat_set_nat(insert_nat(A),Aa) ).
% insert_Diff_single
tff(fact_5555_ceiling__add__one,axiom,
! [X: rat] : archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(X)),one_one_int) ).
% ceiling_add_one
tff(fact_5556_bezw__0,axiom,
! [X: nat] : bezw(X,zero_zero_nat) = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int) ).
% bezw_0
tff(fact_5557_one__le__floor,axiom,
! [X: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),archim3151403230148437115or_rat(X))
<=> aa_rat_o(ord_less_eq_rat(one_one_rat),X) ) ).
% one_le_floor
tff(fact_5558_zero__less__floor,axiom,
! [X: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),archim3151403230148437115or_rat(X))
<=> aa_rat_o(ord_less_eq_rat(one_one_rat),X) ) ).
% zero_less_floor
tff(fact_5559_floor__le__zero,axiom,
! [X: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(X)),zero_zero_int)
<=> aa_rat_o(ord_less_rat(X),one_one_rat) ) ).
% floor_le_zero
tff(fact_5560_floor__less__one,axiom,
! [X: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(X)),one_one_int)
<=> aa_rat_o(ord_less_rat(X),one_one_rat) ) ).
% floor_less_one
tff(fact_5561_subset__Compl__singleton,axiom,
! [Aa: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Aa),aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(B),bot_bo228742789529271731at_nat)))
<=> ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),Aa) ) ).
% subset_Compl_singleton
tff(fact_5562_subset__Compl__singleton,axiom,
! [Aa: set_fu3555266933746557007_nat_o,B: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(Aa),uminus2199602881456817606_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o)))
<=> ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),Aa) ) ).
% subset_Compl_singleton
tff(fact_5563_subset__Compl__singleton,axiom,
! [Aa: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Aa),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(B),bot_bo2099793752762293965at_nat)))
<=> ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),Aa) ) ).
% subset_Compl_singleton
tff(fact_5564_subset__Compl__singleton,axiom,
! [Aa: set_o,B: $o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Aa),aa_set_o_set_o(uminus_uminus_set_o,aa_set_o_set_o(insert_o((B)),bot_bot_set_o)))
<=> ~ aa_set_o_o(member_o((B)),Aa) ) ).
% subset_Compl_singleton
tff(fact_5565_subset__Compl__singleton,axiom,
! [Aa: set_int,B: int] :
( aa_set_int_o(ord_less_eq_set_int(Aa),aa_set_int_set_int(uminus1532241313380277803et_int,aa_set_int_set_int(insert_int(B),bot_bot_set_int)))
<=> ~ aa_set_int_o(member_int(B),Aa) ) ).
% subset_Compl_singleton
tff(fact_5566_subset__Compl__singleton,axiom,
! [Aa: set_nat,B: nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat)))
<=> ~ aa_set_nat_o(member_nat(B),Aa) ) ).
% subset_Compl_singleton
tff(fact_5567_ceiling__le__one,axiom,
! [X: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(X)),one_one_int)
<=> aa_rat_o(ord_less_eq_rat(X),one_one_rat) ) ).
% ceiling_le_one
tff(fact_5568_one__less__ceiling,axiom,
! [X: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),archim2889992004027027881ng_rat(X))
<=> aa_rat_o(ord_less_rat(one_one_rat),X) ) ).
% one_less_ceiling
tff(fact_5569_floor__diff__one,axiom,
! [X: rat] : archim3151403230148437115or_rat(minus_minus_rat(X,one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim3151403230148437115or_rat(X)),one_one_int) ).
% floor_diff_one
tff(fact_5570_ceiling__diff__one,axiom,
! [X: rat] : archim2889992004027027881ng_rat(minus_minus_rat(X,one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(X)),one_one_int) ).
% ceiling_diff_one
tff(fact_5571_ceiling__less__zero,axiom,
! [X: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(X)),zero_zero_int)
<=> aa_rat_o(ord_less_eq_rat(X),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).
% ceiling_less_zero
tff(fact_5572_zero__le__ceiling,axiom,
! [X: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),archim2889992004027027881ng_rat(X))
<=> aa_rat_o(ord_less_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),X) ) ).
% zero_le_ceiling
tff(fact_5573_numeral__less__floor,axiom,
! [V: num,X: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,numeral_numeral_int(V)),archim3151403230148437115or_rat(X))
<=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(V)),one_one_rat)),X) ) ).
% numeral_less_floor
tff(fact_5574_floor__le__numeral,axiom,
! [X: rat,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(X)),numeral_numeral_int(V))
<=> aa_rat_o(ord_less_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(V)),one_one_rat)) ) ).
% floor_le_numeral
tff(fact_5575_ceiling__less__numeral,axiom,
! [X: rat,V: num] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(X)),numeral_numeral_int(V))
<=> aa_rat_o(ord_less_eq_rat(X),minus_minus_rat(numeral_numeral_rat(V),one_one_rat)) ) ).
% ceiling_less_numeral
tff(fact_5576_numeral__le__ceiling,axiom,
! [V: num,X: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(V)),archim2889992004027027881ng_rat(X))
<=> aa_rat_o(ord_less_rat(minus_minus_rat(numeral_numeral_rat(V),one_one_rat)),X) ) ).
% numeral_le_ceiling
tff(fact_5577_neg__numeral__less__floor,axiom,
! [V: num,X: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(V))),archim3151403230148437115or_rat(X))
<=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(V))),one_one_rat)),X) ) ).
% neg_numeral_less_floor
tff(fact_5578_divmod__int__def,axiom,
! [M2: num,N: num] : unique5052692396658037445od_int(M2,N) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,numeral_numeral_int(M2)),numeral_numeral_int(N))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,numeral_numeral_int(M2)),numeral_numeral_int(N))) ).
% divmod_int_def
tff(fact_5579_singleton__inject,axiom,
! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
( ( aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),bot_bo228742789529271731at_nat) = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(B),bot_bo228742789529271731at_nat) )
=> ( A = B ) ) ).
% singleton_inject
tff(fact_5580_singleton__inject,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),bot_bo2099793752762293965at_nat) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(B),bot_bo2099793752762293965at_nat) )
=> ( A = B ) ) ).
% singleton_inject
tff(fact_5581_singleton__inject,axiom,
! [A: $o,B: $o] :
( ( aa_set_o_set_o(insert_o((A)),bot_bot_set_o) = aa_set_o_set_o(insert_o((B)),bot_bot_set_o) )
=> ( (A)
<=> (B) ) ) ).
% singleton_inject
tff(fact_5582_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) = aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat) )
=> ( A = B ) ) ).
% singleton_inject
tff(fact_5583_singleton__inject,axiom,
! [A: int,B: int] :
( ( aa_set_int_set_int(insert_int(A),bot_bot_set_int) = aa_set_int_set_int(insert_int(B),bot_bot_set_int) )
=> ( A = B ) ) ).
% singleton_inject
tff(fact_5584_insert__not__empty,axiom,
! [A: produc3843707927480180839at_nat,Aa: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),Aa) != bot_bo228742789529271731at_nat ).
% insert_not_empty
tff(fact_5585_insert__not__empty,axiom,
! [A: product_prod_nat_nat,Aa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),Aa) != bot_bo2099793752762293965at_nat ).
% insert_not_empty
tff(fact_5586_insert__not__empty,axiom,
! [A: $o,Aa: set_o] : aa_set_o_set_o(insert_o((A)),Aa) != bot_bot_set_o ).
% insert_not_empty
tff(fact_5587_insert__not__empty,axiom,
! [A: nat,Aa: set_nat] : aa_set_nat_set_nat(insert_nat(A),Aa) != bot_bot_set_nat ).
% insert_not_empty
tff(fact_5588_insert__not__empty,axiom,
! [A: int,Aa: set_int] : aa_set_int_set_int(insert_int(A),Aa) != bot_bot_set_int ).
% insert_not_empty
tff(fact_5589_doubleton__eq__iff,axiom,
! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C: produc3843707927480180839at_nat,D2: produc3843707927480180839at_nat] :
( ( aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(B),bot_bo228742789529271731at_nat)) = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(C),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(D2),bot_bo228742789529271731at_nat)) )
<=> ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
tff(fact_5590_doubleton__eq__iff,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D2: product_prod_nat_nat] :
( ( aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(B),bot_bo2099793752762293965at_nat)) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(C),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(D2),bot_bo2099793752762293965at_nat)) )
<=> ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
tff(fact_5591_doubleton__eq__iff,axiom,
! [A: $o,B: $o,C: $o,D2: $o] :
( ( aa_set_o_set_o(insert_o((A)),aa_set_o_set_o(insert_o((B)),bot_bot_set_o)) = aa_set_o_set_o(insert_o((C)),aa_set_o_set_o(insert_o((D2)),bot_bot_set_o)) )
<=> ( ( ( (A)
<=> (C) )
& ( (B)
<=> (D2) ) )
| ( ( (A)
<=> (D2) )
& ( (B)
<=> (C) ) ) ) ) ).
% doubleton_eq_iff
tff(fact_5592_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat)) = aa_set_nat_set_nat(insert_nat(C),aa_set_nat_set_nat(insert_nat(D2),bot_bot_set_nat)) )
<=> ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
tff(fact_5593_doubleton__eq__iff,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( aa_set_int_set_int(insert_int(A),aa_set_int_set_int(insert_int(B),bot_bot_set_int)) = aa_set_int_set_int(insert_int(C),aa_set_int_set_int(insert_int(D2),bot_bot_set_int)) )
<=> ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
tff(fact_5594_singleton__iff,axiom,
! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),bot_bo228742789529271731at_nat))
<=> ( B = A ) ) ).
% singleton_iff
tff(fact_5595_singleton__iff,axiom,
! [B: fun_Pr6460944699146269935_nat_o,A: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A),bot_bo8632255151191677115_nat_o))
<=> ( B = A ) ) ).
% singleton_iff
tff(fact_5596_singleton__iff,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),bot_bo2099793752762293965at_nat))
<=> ( B = A ) ) ).
% singleton_iff
tff(fact_5597_singleton__iff,axiom,
! [B: $o,A: $o] :
( aa_set_o_o(member_o((B)),aa_set_o_set_o(insert_o((A)),bot_bot_set_o))
<=> ( (B)
<=> (A) ) ) ).
% singleton_iff
tff(fact_5598_singleton__iff,axiom,
! [B: nat,A: nat] :
( aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))
<=> ( B = A ) ) ).
% singleton_iff
tff(fact_5599_singleton__iff,axiom,
! [B: int,A: int] :
( aa_set_int_o(member_int(B),aa_set_int_set_int(insert_int(A),bot_bot_set_int))
<=> ( B = A ) ) ).
% singleton_iff
tff(fact_5600_singletonD,axiom,
! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),bot_bo228742789529271731at_nat))
=> ( B = A ) ) ).
% singletonD
tff(fact_5601_singletonD,axiom,
! [B: fun_Pr6460944699146269935_nat_o,A: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A),bot_bo8632255151191677115_nat_o))
=> ( B = A ) ) ).
% singletonD
tff(fact_5602_singletonD,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),bot_bo2099793752762293965at_nat))
=> ( B = A ) ) ).
% singletonD
tff(fact_5603_singletonD,axiom,
! [B: $o,A: $o] :
( aa_set_o_o(member_o((B)),aa_set_o_set_o(insert_o((A)),bot_bot_set_o))
=> ( (B)
<=> (A) ) ) ).
% singletonD
tff(fact_5604_singletonD,axiom,
! [B: nat,A: nat] :
( aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))
=> ( B = A ) ) ).
% singletonD
tff(fact_5605_singletonD,axiom,
! [B: int,A: int] :
( aa_set_int_o(member_int(B),aa_set_int_set_int(insert_int(A),bot_bot_set_int))
=> ( B = A ) ) ).
% singletonD
tff(fact_5606_one__add__floor,axiom,
! [X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(X)),one_one_int) = archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) ).
% one_add_floor
tff(fact_5607_and__int_Oelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa) = Y )
=> ( Y = $ite(
( aa_set_int_o(member_int(X),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))
& aa_set_int_o(member_int(Xa),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int))) ),
aa_int_int(uminus_uminus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),X)
& ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),Xa) ))),
aa_int_int(aa_int_fun_int_int(plus_plus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),X)
& ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),Xa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),numeral_numeral_int(aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),numeral_numeral_int(aa_num_num(bit0,one)))))) ) ) ) ).
% and_int.elims
tff(fact_5608_and__int_Osimps,axiom,
! [K: int,L: int] :
aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = $ite(
( aa_set_int_o(member_int(K),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))
& aa_set_int_o(member_int(L),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int))) ),
aa_int_int(uminus_uminus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),K)
& ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),L) ))),
aa_int_int(aa_int_fun_int_int(plus_plus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),K)
& ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),numeral_numeral_int(aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),numeral_numeral_int(aa_num_num(bit0,one)))))) ) ).
% and_int.simps
tff(fact_5609_subset__singletonD,axiom,
! [Aa: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat))
=> ( ( Aa = bot_bo228742789529271731at_nat )
| ( Aa = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat) ) ) ) ).
% subset_singletonD
tff(fact_5610_subset__singletonD,axiom,
! [Aa: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Aa),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat))
=> ( ( Aa = bot_bo2099793752762293965at_nat )
| ( Aa = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat) ) ) ) ).
% subset_singletonD
tff(fact_5611_subset__singletonD,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Aa),aa_set_o_set_o(insert_o((X)),bot_bot_set_o))
=> ( ( Aa = bot_bot_set_o )
| ( Aa = aa_set_o_set_o(insert_o((X)),bot_bot_set_o) ) ) ) ).
% subset_singletonD
tff(fact_5612_subset__singletonD,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(ord_less_eq_set_int(Aa),aa_set_int_set_int(insert_int(X),bot_bot_set_int))
=> ( ( Aa = bot_bot_set_int )
| ( Aa = aa_set_int_set_int(insert_int(X),bot_bot_set_int) ) ) ) ).
% subset_singletonD
tff(fact_5613_subset__singletonD,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))
=> ( ( Aa = bot_bot_set_nat )
| ( Aa = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) ) ) ) ).
% subset_singletonD
tff(fact_5614_subset__singleton__iff,axiom,
! [X5: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X5),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),bot_bo228742789529271731at_nat))
<=> ( ( X5 = bot_bo228742789529271731at_nat )
| ( X5 = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),bot_bo228742789529271731at_nat) ) ) ) ).
% subset_singleton_iff
tff(fact_5615_subset__singleton__iff,axiom,
! [X5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(X5),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),bot_bo2099793752762293965at_nat))
<=> ( ( X5 = bot_bo2099793752762293965at_nat )
| ( X5 = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),bot_bo2099793752762293965at_nat) ) ) ) ).
% subset_singleton_iff
tff(fact_5616_subset__singleton__iff,axiom,
! [X5: set_o,A: $o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,X5),aa_set_o_set_o(insert_o((A)),bot_bot_set_o))
<=> ( ( X5 = bot_bot_set_o )
| ( X5 = aa_set_o_set_o(insert_o((A)),bot_bot_set_o) ) ) ) ).
% subset_singleton_iff
tff(fact_5617_subset__singleton__iff,axiom,
! [X5: set_int,A: int] :
( aa_set_int_o(ord_less_eq_set_int(X5),aa_set_int_set_int(insert_int(A),bot_bot_set_int))
<=> ( ( X5 = bot_bot_set_int )
| ( X5 = aa_set_int_set_int(insert_int(A),bot_bot_set_int) ) ) ) ).
% subset_singleton_iff
tff(fact_5618_subset__singleton__iff,axiom,
! [X5: set_nat,A: nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X5),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))
<=> ( ( X5 = bot_bot_set_nat )
| ( X5 = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ) ) ) ).
% subset_singleton_iff
tff(fact_5619_insert__is__Un,axiom,
! [A: produc3843707927480180839at_nat,Aa: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),Aa) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),bot_bo228742789529271731at_nat)),Aa) ).
% insert_is_Un
tff(fact_5620_insert__is__Un,axiom,
! [A: product_prod_nat_nat,Aa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),Aa) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),bot_bo2099793752762293965at_nat)),Aa) ).
% insert_is_Un
tff(fact_5621_insert__is__Un,axiom,
! [A: $o,Aa: set_o] : aa_set_o_set_o(insert_o((A)),Aa) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(insert_o((A)),bot_bot_set_o)),Aa) ).
% insert_is_Un
tff(fact_5622_insert__is__Un,axiom,
! [A: nat,Aa: set_nat] : aa_set_nat_set_nat(insert_nat(A),Aa) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)),Aa) ).
% insert_is_Un
tff(fact_5623_insert__is__Un,axiom,
! [A: int,Aa: set_int] : aa_set_int_set_int(insert_int(A),Aa) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(insert_int(A),bot_bot_set_int)),Aa) ).
% insert_is_Un
tff(fact_5624_Un__singleton__iff,axiom,
! [Aa: set_Pr4329608150637261639at_nat,Ba: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Aa),Ba) = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat) )
<=> ( ( ( Aa = bot_bo228742789529271731at_nat )
& ( Ba = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat) ) )
| ( ( Aa = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat) )
& ( Ba = bot_bo228742789529271731at_nat ) )
| ( ( Aa = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat) )
& ( Ba = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat) ) ) ) ) ).
% Un_singleton_iff
tff(fact_5625_Un__singleton__iff,axiom,
! [Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Aa),Ba) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat) )
<=> ( ( ( Aa = bot_bo2099793752762293965at_nat )
& ( Ba = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat) ) )
| ( ( Aa = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat) )
& ( Ba = bot_bo2099793752762293965at_nat ) )
| ( ( Aa = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat) )
& ( Ba = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat) ) ) ) ) ).
% Un_singleton_iff
tff(fact_5626_Un__singleton__iff,axiom,
! [Aa: set_o,Ba: set_o,X: $o] :
( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Aa),Ba) = aa_set_o_set_o(insert_o((X)),bot_bot_set_o) )
<=> ( ( ( Aa = bot_bot_set_o )
& ( Ba = aa_set_o_set_o(insert_o((X)),bot_bot_set_o) ) )
| ( ( Aa = aa_set_o_set_o(insert_o((X)),bot_bot_set_o) )
& ( Ba = bot_bot_set_o ) )
| ( ( Aa = aa_set_o_set_o(insert_o((X)),bot_bot_set_o) )
& ( Ba = aa_set_o_set_o(insert_o((X)),bot_bot_set_o) ) ) ) ) ).
% Un_singleton_iff
tff(fact_5627_Un__singleton__iff,axiom,
! [Aa: set_nat,Ba: set_nat,X: nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Aa),Ba) = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) )
<=> ( ( ( Aa = bot_bot_set_nat )
& ( Ba = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) ) )
| ( ( Aa = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) )
& ( Ba = bot_bot_set_nat ) )
| ( ( Aa = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) )
& ( Ba = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) ) ) ) ) ).
% Un_singleton_iff
tff(fact_5628_Un__singleton__iff,axiom,
! [Aa: set_int,Ba: set_int,X: int] :
( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Aa),Ba) = aa_set_int_set_int(insert_int(X),bot_bot_set_int) )
<=> ( ( ( Aa = bot_bot_set_int )
& ( Ba = aa_set_int_set_int(insert_int(X),bot_bot_set_int) ) )
| ( ( Aa = aa_set_int_set_int(insert_int(X),bot_bot_set_int) )
& ( Ba = bot_bot_set_int ) )
| ( ( Aa = aa_set_int_set_int(insert_int(X),bot_bot_set_int) )
& ( Ba = aa_set_int_set_int(insert_int(X),bot_bot_set_int) ) ) ) ) ).
% Un_singleton_iff
tff(fact_5629_singleton__Un__iff,axiom,
! [X: produc3843707927480180839at_nat,Aa: set_Pr4329608150637261639at_nat,Ba: set_Pr4329608150637261639at_nat] :
( ( aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Aa),Ba) )
<=> ( ( ( Aa = bot_bo228742789529271731at_nat )
& ( Ba = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat) ) )
| ( ( Aa = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat) )
& ( Ba = bot_bo228742789529271731at_nat ) )
| ( ( Aa = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat) )
& ( Ba = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat) ) ) ) ) ).
% singleton_Un_iff
tff(fact_5630_singleton__Un__iff,axiom,
! [X: product_prod_nat_nat,Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
( ( aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Aa),Ba) )
<=> ( ( ( Aa = bot_bo2099793752762293965at_nat )
& ( Ba = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat) ) )
| ( ( Aa = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat) )
& ( Ba = bot_bo2099793752762293965at_nat ) )
| ( ( Aa = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat) )
& ( Ba = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat) ) ) ) ) ).
% singleton_Un_iff
tff(fact_5631_singleton__Un__iff,axiom,
! [X: $o,Aa: set_o,Ba: set_o] :
( ( aa_set_o_set_o(insert_o((X)),bot_bot_set_o) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Aa),Ba) )
<=> ( ( ( Aa = bot_bot_set_o )
& ( Ba = aa_set_o_set_o(insert_o((X)),bot_bot_set_o) ) )
| ( ( Aa = aa_set_o_set_o(insert_o((X)),bot_bot_set_o) )
& ( Ba = bot_bot_set_o ) )
| ( ( Aa = aa_set_o_set_o(insert_o((X)),bot_bot_set_o) )
& ( Ba = aa_set_o_set_o(insert_o((X)),bot_bot_set_o) ) ) ) ) ).
% singleton_Un_iff
tff(fact_5632_singleton__Un__iff,axiom,
! [X: nat,Aa: set_nat,Ba: set_nat] :
( ( aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Aa),Ba) )
<=> ( ( ( Aa = bot_bot_set_nat )
& ( Ba = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) ) )
| ( ( Aa = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) )
& ( Ba = bot_bot_set_nat ) )
| ( ( Aa = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) )
& ( Ba = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) ) ) ) ) ).
% singleton_Un_iff
tff(fact_5633_singleton__Un__iff,axiom,
! [X: int,Aa: set_int,Ba: set_int] :
( ( aa_set_int_set_int(insert_int(X),bot_bot_set_int) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Aa),Ba) )
<=> ( ( ( Aa = bot_bot_set_int )
& ( Ba = aa_set_int_set_int(insert_int(X),bot_bot_set_int) ) )
| ( ( Aa = aa_set_int_set_int(insert_int(X),bot_bot_set_int) )
& ( Ba = bot_bot_set_int ) )
| ( ( Aa = aa_set_int_set_int(insert_int(X),bot_bot_set_int) )
& ( Ba = aa_set_int_set_int(insert_int(X),bot_bot_set_int) ) ) ) ) ).
% singleton_Un_iff
tff(fact_5634_Diff__insert,axiom,
! [Aa: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,Ba: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),Ba)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),Ba)),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),bot_bo228742789529271731at_nat)) ).
% Diff_insert
tff(fact_5635_Diff__insert,axiom,
! [Aa: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,Ba: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),Ba)) = minus_1356011639430497352at_nat(minus_1356011639430497352at_nat(Aa,Ba),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),bot_bo2099793752762293965at_nat)) ).
% Diff_insert
tff(fact_5636_Diff__insert,axiom,
! [Aa: set_o,A: $o,Ba: set_o] : minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((A)),Ba)) = minus_minus_set_o(minus_minus_set_o(Aa,Ba),aa_set_o_set_o(insert_o((A)),bot_bot_set_o)) ).
% Diff_insert
tff(fact_5637_Diff__insert,axiom,
! [Aa: set_int,A: int,Ba: set_int] : minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(A),Ba)) = minus_minus_set_int(minus_minus_set_int(Aa,Ba),aa_set_int_set_int(insert_int(A),bot_bot_set_int)) ).
% Diff_insert
tff(fact_5638_Diff__insert,axiom,
! [Aa: set_nat,A: nat,Ba: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(A),Ba)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),Ba)),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)) ).
% Diff_insert
tff(fact_5639_insert__Diff,axiom,
! [A: produc3843707927480180839at_nat,Aa: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),Aa)
=> ( aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),bot_bo228742789529271731at_nat))) = Aa ) ) ).
% insert_Diff
tff(fact_5640_insert__Diff,axiom,
! [A: fun_Pr6460944699146269935_nat_o,Aa: set_fu3555266933746557007_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),Aa)
=> ( aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A),minus_6456595905716665654_nat_o(Aa,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A),bot_bo8632255151191677115_nat_o))) = Aa ) ) ).
% insert_Diff
tff(fact_5641_insert__Diff,axiom,
! [A: product_prod_nat_nat,Aa: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),Aa)
=> ( aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),bot_bo2099793752762293965at_nat))) = Aa ) ) ).
% insert_Diff
tff(fact_5642_insert__Diff,axiom,
! [A: $o,Aa: set_o] :
( aa_set_o_o(member_o((A)),Aa)
=> ( aa_set_o_set_o(insert_o((A)),minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((A)),bot_bot_set_o))) = Aa ) ) ).
% insert_Diff
tff(fact_5643_insert__Diff,axiom,
! [A: int,Aa: set_int] :
( aa_set_int_o(member_int(A),Aa)
=> ( aa_set_int_set_int(insert_int(A),minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(A),bot_bot_set_int))) = Aa ) ) ).
% insert_Diff
tff(fact_5644_insert__Diff,axiom,
! [A: nat,Aa: set_nat] :
( aa_set_nat_o(member_nat(A),Aa)
=> ( aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) = Aa ) ) ).
% insert_Diff
tff(fact_5645_Diff__insert2,axiom,
! [Aa: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,Ba: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),Ba)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),bot_bo228742789529271731at_nat))),Ba) ).
% Diff_insert2
tff(fact_5646_Diff__insert2,axiom,
! [Aa: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,Ba: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),Ba)) = minus_1356011639430497352at_nat(minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),bot_bo2099793752762293965at_nat)),Ba) ).
% Diff_insert2
tff(fact_5647_Diff__insert2,axiom,
! [Aa: set_o,A: $o,Ba: set_o] : minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((A)),Ba)) = minus_minus_set_o(minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((A)),bot_bot_set_o)),Ba) ).
% Diff_insert2
tff(fact_5648_Diff__insert2,axiom,
! [Aa: set_int,A: int,Ba: set_int] : minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(A),Ba)) = minus_minus_set_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(A),bot_bot_set_int)),Ba) ).
% Diff_insert2
tff(fact_5649_Diff__insert2,axiom,
! [Aa: set_nat,A: nat,Ba: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(A),Ba)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))),Ba) ).
% Diff_insert2
tff(fact_5650_Diff__insert__absorb,axiom,
! [X: produc3843707927480180839at_nat,Aa: set_Pr4329608150637261639at_nat] :
( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),Aa)
=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),Aa)),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat)) = Aa ) ) ).
% Diff_insert_absorb
tff(fact_5651_Diff__insert__absorb,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Aa: set_fu3555266933746557007_nat_o] :
( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Aa)
=> ( minus_6456595905716665654_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),Aa),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = Aa ) ) ).
% Diff_insert_absorb
tff(fact_5652_Diff__insert__absorb,axiom,
! [X: product_prod_nat_nat,Aa: set_Pr1261947904930325089at_nat] :
( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),Aa)
=> ( minus_1356011639430497352at_nat(aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),Aa),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat)) = Aa ) ) ).
% Diff_insert_absorb
tff(fact_5653_Diff__insert__absorb,axiom,
! [X: $o,Aa: set_o] :
( ~ aa_set_o_o(member_o((X)),Aa)
=> ( minus_minus_set_o(aa_set_o_set_o(insert_o((X)),Aa),aa_set_o_set_o(insert_o((X)),bot_bot_set_o)) = Aa ) ) ).
% Diff_insert_absorb
tff(fact_5654_Diff__insert__absorb,axiom,
! [X: int,Aa: set_int] :
( ~ aa_set_int_o(member_int(X),Aa)
=> ( minus_minus_set_int(aa_set_int_set_int(insert_int(X),Aa),aa_set_int_set_int(insert_int(X),bot_bot_set_int)) = Aa ) ) ).
% Diff_insert_absorb
tff(fact_5655_Diff__insert__absorb,axiom,
! [X: nat,Aa: set_nat] :
( ~ aa_set_nat_o(member_nat(X),Aa)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(insert_nat(X),Aa)),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) = Aa ) ) ).
% Diff_insert_absorb
tff(fact_5656_set__minus__singleton__eq,axiom,
! [X: produc3843707927480180839at_nat,X5: set_Pr4329608150637261639at_nat] :
( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),X5)
=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,X5),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat)) = X5 ) ) ).
% set_minus_singleton_eq
tff(fact_5657_set__minus__singleton__eq,axiom,
! [X: fun_Pr6460944699146269935_nat_o,X5: set_fu3555266933746557007_nat_o] :
( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),X5)
=> ( minus_6456595905716665654_nat_o(X5,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = X5 ) ) ).
% set_minus_singleton_eq
tff(fact_5658_set__minus__singleton__eq,axiom,
! [X: product_prod_nat_nat,X5: set_Pr1261947904930325089at_nat] :
( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),X5)
=> ( minus_1356011639430497352at_nat(X5,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat)) = X5 ) ) ).
% set_minus_singleton_eq
tff(fact_5659_set__minus__singleton__eq,axiom,
! [X: $o,X5: set_o] :
( ~ aa_set_o_o(member_o((X)),X5)
=> ( minus_minus_set_o(X5,aa_set_o_set_o(insert_o((X)),bot_bot_set_o)) = X5 ) ) ).
% set_minus_singleton_eq
tff(fact_5660_set__minus__singleton__eq,axiom,
! [X: int,X5: set_int] :
( ~ aa_set_int_o(member_int(X),X5)
=> ( minus_minus_set_int(X5,aa_set_int_set_int(insert_int(X),bot_bot_set_int)) = X5 ) ) ).
% set_minus_singleton_eq
tff(fact_5661_set__minus__singleton__eq,axiom,
! [X: nat,X5: set_nat] :
( ~ aa_set_nat_o(member_nat(X),X5)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X5),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) = X5 ) ) ).
% set_minus_singleton_eq
tff(fact_5662_insert__minus__eq,axiom,
! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,Aa: set_Pr4329608150637261639at_nat] :
( ( X != Y )
=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),Aa)),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(Y),bot_bo228742789529271731at_nat)) = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(Y),bot_bo228742789529271731at_nat))) ) ) ).
% insert_minus_eq
tff(fact_5663_insert__minus__eq,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,Aa: set_Pr1261947904930325089at_nat] :
( ( X != Y )
=> ( minus_1356011639430497352at_nat(aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),Aa),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Y),bot_bo2099793752762293965at_nat)) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Y),bot_bo2099793752762293965at_nat))) ) ) ).
% insert_minus_eq
tff(fact_5664_insert__minus__eq,axiom,
! [X: $o,Y: $o,Aa: set_o] :
( ~ ( (X)
<=> (Y) )
=> ( minus_minus_set_o(aa_set_o_set_o(insert_o((X)),Aa),aa_set_o_set_o(insert_o((Y)),bot_bot_set_o)) = aa_set_o_set_o(insert_o((X)),minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((Y)),bot_bot_set_o))) ) ) ).
% insert_minus_eq
tff(fact_5665_insert__minus__eq,axiom,
! [X: int,Y: int,Aa: set_int] :
( ( X != Y )
=> ( minus_minus_set_int(aa_set_int_set_int(insert_int(X),Aa),aa_set_int_set_int(insert_int(Y),bot_bot_set_int)) = aa_set_int_set_int(insert_int(X),minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(Y),bot_bot_set_int))) ) ) ).
% insert_minus_eq
tff(fact_5666_insert__minus__eq,axiom,
! [X: nat,Y: nat,Aa: set_nat] :
( ( X != Y )
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(insert_nat(X),Aa)),aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat)) = aa_set_nat_set_nat(insert_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat))) ) ) ).
% insert_minus_eq
tff(fact_5667_le__mult__floor,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,archim3151403230148437115or_rat(A)),archim3151403230148437115or_rat(B))),archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).
% le_mult_floor
tff(fact_5668_mult__ceiling__le,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),B)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,archim2889992004027027881ng_rat(A)),archim2889992004027027881ng_rat(B))) ) ) ).
% mult_ceiling_le
tff(fact_5669_subset__insert__iff,axiom,
! [Aa: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,Ba: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),Ba))
<=> $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),Aa),aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat))),Ba),aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Aa),Ba)) ) ).
% subset_insert_iff
tff(fact_5670_subset__insert__iff,axiom,
! [Aa: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,Ba: set_fu3555266933746557007_nat_o] :
( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(Aa),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),Ba))
<=> $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Aa),aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(minus_6456595905716665654_nat_o(Aa,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))),Ba),aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(Aa),Ba)) ) ).
% subset_insert_iff
tff(fact_5671_subset__insert__iff,axiom,
! [Aa: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Ba: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Aa),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),Ba))
<=> $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),Aa),aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat))),Ba),aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Aa),Ba)) ) ).
% subset_insert_iff
tff(fact_5672_subset__insert__iff,axiom,
! [Aa: set_o,X: $o,Ba: set_o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Aa),aa_set_o_set_o(insert_o((X)),Ba))
<=> $ite(aa_set_o_o(member_o((X)),Aa),aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))),Ba),aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Aa),Ba)) ) ).
% subset_insert_iff
tff(fact_5673_subset__insert__iff,axiom,
! [Aa: set_int,X: int,Ba: set_int] :
( aa_set_int_o(ord_less_eq_set_int(Aa),aa_set_int_set_int(insert_int(X),Ba))
<=> $ite(aa_set_int_o(member_int(X),Aa),aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int))),Ba),aa_set_int_o(ord_less_eq_set_int(Aa),Ba)) ) ).
% subset_insert_iff
tff(fact_5674_subset__insert__iff,axiom,
! [Aa: set_nat,X: nat,Ba: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),Ba))
<=> $ite(aa_set_nat_o(member_nat(X),Aa),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))),Ba),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),Ba)) ) ).
% subset_insert_iff
tff(fact_5675_Diff__single__insert,axiom,
! [Aa: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,Ba: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat))),Ba)
=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),Ba)) ) ).
% Diff_single_insert
tff(fact_5676_Diff__single__insert,axiom,
! [Aa: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Ba: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat))),Ba)
=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Aa),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),Ba)) ) ).
% Diff_single_insert
tff(fact_5677_Diff__single__insert,axiom,
! [Aa: set_o,X: $o,Ba: set_o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))),Ba)
=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Aa),aa_set_o_set_o(insert_o((X)),Ba)) ) ).
% Diff_single_insert
tff(fact_5678_Diff__single__insert,axiom,
! [Aa: set_int,X: int,Ba: set_int] :
( aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int))),Ba)
=> aa_set_int_o(ord_less_eq_set_int(Aa),aa_set_int_set_int(insert_int(X),Ba)) ) ).
% Diff_single_insert
tff(fact_5679_Diff__single__insert,axiom,
! [Aa: set_nat,X: nat,Ba: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))),Ba)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),Ba)) ) ).
% Diff_single_insert
tff(fact_5680_card__1__singletonE,axiom,
! [Aa: set_Pr4329608150637261639at_nat] :
( ( finite3771342082235030671at_nat(Aa) = one_one_nat )
=> ~ ! [X2: produc3843707927480180839at_nat] : Aa != aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X2),bot_bo228742789529271731at_nat) ) ).
% card_1_singletonE
tff(fact_5681_card__1__singletonE,axiom,
! [Aa: set_Pr1261947904930325089at_nat] :
( ( finite711546835091564841at_nat(Aa) = one_one_nat )
=> ~ ! [X2: product_prod_nat_nat] : Aa != aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X2),bot_bo2099793752762293965at_nat) ) ).
% card_1_singletonE
tff(fact_5682_card__1__singletonE,axiom,
! [Aa: set_o] :
( ( finite_card_o(Aa) = one_one_nat )
=> ~ ! [X2: $o] : Aa != aa_set_o_set_o(insert_o((X2)),bot_bot_set_o) ) ).
% card_1_singletonE
tff(fact_5683_card__1__singletonE,axiom,
! [Aa: set_nat] :
( ( finite_card_nat(Aa) = one_one_nat )
=> ~ ! [X2: nat] : Aa != aa_set_nat_set_nat(insert_nat(X2),bot_bot_set_nat) ) ).
% card_1_singletonE
tff(fact_5684_card__1__singletonE,axiom,
! [Aa: set_int] :
( ( finite_card_int(Aa) = one_one_nat )
=> ~ ! [X2: int] : Aa != aa_set_int_set_int(insert_int(X2),bot_bot_set_int) ) ).
% card_1_singletonE
tff(fact_5685_remove__subset,axiom,
! [X: produc3843707927480180839at_nat,S: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),S)
=> aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,S),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat))),S) ) ).
% remove_subset
tff(fact_5686_remove__subset,axiom,
! [X: fun_Pr6460944699146269935_nat_o,S: set_fu3555266933746557007_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),S)
=> aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(minus_6456595905716665654_nat_o(S,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))),S) ) ).
% remove_subset
tff(fact_5687_remove__subset,axiom,
! [X: product_prod_nat_nat,S: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),S)
=> aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(minus_1356011639430497352at_nat(S,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat))),S) ) ).
% remove_subset
tff(fact_5688_remove__subset,axiom,
! [X: $o,S: set_o] :
( aa_set_o_o(member_o((X)),S)
=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,minus_minus_set_o(S,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))),S) ) ).
% remove_subset
tff(fact_5689_remove__subset,axiom,
! [X: int,S: set_int] :
( aa_set_int_o(member_int(X),S)
=> aa_set_int_o(ord_less_set_int(minus_minus_set_int(S,aa_set_int_set_int(insert_int(X),bot_bot_set_int))),S) ) ).
% remove_subset
tff(fact_5690_remove__subset,axiom,
! [X: nat,S: set_nat] :
( aa_set_nat_o(member_nat(X),S)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))),S) ) ).
% remove_subset
tff(fact_5691_Compl__insert,axiom,
! [X: produc3843707927480180839at_nat,Aa: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),Aa)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(uminus935396558254630718at_nat,Aa)),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat)) ).
% Compl_insert
tff(fact_5692_Compl__insert,axiom,
! [X: product_prod_nat_nat,Aa: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),Aa)) = minus_1356011639430497352at_nat(aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Aa),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat)) ).
% Compl_insert
tff(fact_5693_Compl__insert,axiom,
! [X: $o,Aa: set_o] : aa_set_o_set_o(uminus_uminus_set_o,aa_set_o_set_o(insert_o((X)),Aa)) = minus_minus_set_o(aa_set_o_set_o(uminus_uminus_set_o,Aa),aa_set_o_set_o(insert_o((X)),bot_bot_set_o)) ).
% Compl_insert
tff(fact_5694_Compl__insert,axiom,
! [X: int,Aa: set_int] : aa_set_int_set_int(uminus1532241313380277803et_int,aa_set_int_set_int(insert_int(X),Aa)) = minus_minus_set_int(aa_set_int_set_int(uminus1532241313380277803et_int,Aa),aa_set_int_set_int(insert_int(X),bot_bot_set_int)) ).
% Compl_insert
tff(fact_5695_Compl__insert,axiom,
! [X: nat,Aa: set_nat] : aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(insert_nat(X),Aa)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,Aa)),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) ).
% Compl_insert
tff(fact_5696_card__Suc__eq,axiom,
! [Aa: set_Pr4329608150637261639at_nat,K: nat] :
( ( finite3771342082235030671at_nat(Aa) = aa_nat_nat(suc,K) )
<=> ? [B4: produc3843707927480180839at_nat,B6: set_Pr4329608150637261639at_nat] :
( ( Aa = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(B4),B6) )
& ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B4),B6)
& ( finite3771342082235030671at_nat(B6) = K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bo228742789529271731at_nat ) ) ) ) ).
% card_Suc_eq
tff(fact_5697_card__Suc__eq,axiom,
! [Aa: set_fu3555266933746557007_nat_o,K: nat] :
( ( finite5795495896937551127_nat_o(Aa) = aa_nat_nat(suc,K) )
<=> ? [B4: fun_Pr6460944699146269935_nat_o,B6: set_fu3555266933746557007_nat_o] :
( ( Aa = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B4),B6) )
& ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B4),B6)
& ( finite5795495896937551127_nat_o(B6) = K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bo8632255151191677115_nat_o ) ) ) ) ).
% card_Suc_eq
tff(fact_5698_card__Suc__eq,axiom,
! [Aa: set_Pr1261947904930325089at_nat,K: nat] :
( ( finite711546835091564841at_nat(Aa) = aa_nat_nat(suc,K) )
<=> ? [B4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
( ( Aa = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(B4),B6) )
& ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B4),B6)
& ( finite711546835091564841at_nat(B6) = K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bo2099793752762293965at_nat ) ) ) ) ).
% card_Suc_eq
tff(fact_5699_card__Suc__eq,axiom,
! [Aa: set_o,K: nat] :
( ( finite_card_o(Aa) = aa_nat_nat(suc,K) )
<=> ? [B4: $o,B6: set_o] :
( ( Aa = aa_set_o_set_o(insert_o((B4)),B6) )
& ~ aa_set_o_o(member_o((B4)),B6)
& ( finite_card_o(B6) = K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_o ) ) ) ) ).
% card_Suc_eq
tff(fact_5700_card__Suc__eq,axiom,
! [Aa: set_nat,K: nat] :
( ( finite_card_nat(Aa) = aa_nat_nat(suc,K) )
<=> ? [B4: nat,B6: set_nat] :
( ( Aa = aa_set_nat_set_nat(insert_nat(B4),B6) )
& ~ aa_set_nat_o(member_nat(B4),B6)
& ( finite_card_nat(B6) = K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_nat ) ) ) ) ).
% card_Suc_eq
tff(fact_5701_card__Suc__eq,axiom,
! [Aa: set_int,K: nat] :
( ( finite_card_int(Aa) = aa_nat_nat(suc,K) )
<=> ? [B4: int,B6: set_int] :
( ( Aa = aa_set_int_set_int(insert_int(B4),B6) )
& ~ aa_set_int_o(member_int(B4),B6)
& ( finite_card_int(B6) = K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_int ) ) ) ) ).
% card_Suc_eq
tff(fact_5702_card__eq__SucD,axiom,
! [Aa: set_Pr4329608150637261639at_nat,K: nat] :
( ( finite3771342082235030671at_nat(Aa) = aa_nat_nat(suc,K) )
=> ? [B3: produc3843707927480180839at_nat,B7: set_Pr4329608150637261639at_nat] :
( ( Aa = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(B3),B7) )
& ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B3),B7)
& ( finite3771342082235030671at_nat(B7) = K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bo228742789529271731at_nat ) ) ) ) ).
% card_eq_SucD
tff(fact_5703_card__eq__SucD,axiom,
! [Aa: set_fu3555266933746557007_nat_o,K: nat] :
( ( finite5795495896937551127_nat_o(Aa) = aa_nat_nat(suc,K) )
=> ? [B3: fun_Pr6460944699146269935_nat_o,B7: set_fu3555266933746557007_nat_o] :
( ( Aa = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B3),B7) )
& ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B3),B7)
& ( finite5795495896937551127_nat_o(B7) = K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bo8632255151191677115_nat_o ) ) ) ) ).
% card_eq_SucD
tff(fact_5704_card__eq__SucD,axiom,
! [Aa: set_Pr1261947904930325089at_nat,K: nat] :
( ( finite711546835091564841at_nat(Aa) = aa_nat_nat(suc,K) )
=> ? [B3: product_prod_nat_nat,B7: set_Pr1261947904930325089at_nat] :
( ( Aa = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(B3),B7) )
& ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B3),B7)
& ( finite711546835091564841at_nat(B7) = K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bo2099793752762293965at_nat ) ) ) ) ).
% card_eq_SucD
tff(fact_5705_card__eq__SucD,axiom,
! [Aa: set_o,K: nat] :
( ( finite_card_o(Aa) = aa_nat_nat(suc,K) )
=> ? [B3: $o,B7: set_o] :
( ( Aa = aa_set_o_set_o(insert_o((B3)),B7) )
& ~ aa_set_o_o(member_o((B3)),B7)
& ( finite_card_o(B7) = K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_o ) ) ) ) ).
% card_eq_SucD
tff(fact_5706_card__eq__SucD,axiom,
! [Aa: set_nat,K: nat] :
( ( finite_card_nat(Aa) = aa_nat_nat(suc,K) )
=> ? [B3: nat,B7: set_nat] :
( ( Aa = aa_set_nat_set_nat(insert_nat(B3),B7) )
& ~ aa_set_nat_o(member_nat(B3),B7)
& ( finite_card_nat(B7) = K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_nat ) ) ) ) ).
% card_eq_SucD
tff(fact_5707_card__eq__SucD,axiom,
! [Aa: set_int,K: nat] :
( ( finite_card_int(Aa) = aa_nat_nat(suc,K) )
=> ? [B3: int,B7: set_int] :
( ( Aa = aa_set_int_set_int(insert_int(B3),B7) )
& ~ aa_set_int_o(member_int(B3),B7)
& ( finite_card_int(B7) = K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_int ) ) ) ) ).
% card_eq_SucD
tff(fact_5708_card__1__singleton__iff,axiom,
! [Aa: set_Pr4329608150637261639at_nat] :
( ( finite3771342082235030671at_nat(Aa) = aa_nat_nat(suc,zero_zero_nat) )
<=> ? [X3: produc3843707927480180839at_nat] : Aa = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X3),bot_bo228742789529271731at_nat) ) ).
% card_1_singleton_iff
tff(fact_5709_card__1__singleton__iff,axiom,
! [Aa: set_Pr1261947904930325089at_nat] :
( ( finite711546835091564841at_nat(Aa) = aa_nat_nat(suc,zero_zero_nat) )
<=> ? [X3: product_prod_nat_nat] : Aa = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X3),bot_bo2099793752762293965at_nat) ) ).
% card_1_singleton_iff
tff(fact_5710_card__1__singleton__iff,axiom,
! [Aa: set_o] :
( ( finite_card_o(Aa) = aa_nat_nat(suc,zero_zero_nat) )
<=> ? [X3: $o] : Aa = aa_set_o_set_o(insert_o((X3)),bot_bot_set_o) ) ).
% card_1_singleton_iff
tff(fact_5711_card__1__singleton__iff,axiom,
! [Aa: set_nat] :
( ( finite_card_nat(Aa) = aa_nat_nat(suc,zero_zero_nat) )
<=> ? [X3: nat] : Aa = aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat) ) ).
% card_1_singleton_iff
tff(fact_5712_card__1__singleton__iff,axiom,
! [Aa: set_int] :
( ( finite_card_int(Aa) = aa_nat_nat(suc,zero_zero_nat) )
<=> ? [X3: int] : Aa = aa_set_int_set_int(insert_int(X3),bot_bot_set_int) ) ).
% card_1_singleton_iff
tff(fact_5713_card__Diff1__le,axiom,
! [Aa: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite3771342082235030671at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat)))),finite3771342082235030671at_nat(Aa)) ).
% card_Diff1_le
tff(fact_5714_card__Diff1__le,axiom,
! [Aa: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite711546835091564841at_nat(minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat)))),finite711546835091564841at_nat(Aa)) ).
% card_Diff1_le
tff(fact_5715_card__Diff1__le,axiom,
! [Aa: set_o,X: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o)))),finite_card_o(Aa)) ).
% card_Diff1_le
tff(fact_5716_card__Diff1__le,axiom,
! [Aa: set_int,X: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int)))),finite_card_int(Aa)) ).
% card_Diff1_le
tff(fact_5717_card__Diff1__le,axiom,
! [Aa: set_nat,X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))),finite_card_nat(Aa)) ).
% card_Diff1_le
tff(fact_5718_psubset__insert__iff,axiom,
! [Aa: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,Ba: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),Ba))
<=> $ite(
aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),Ba),
aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(Aa),Ba),
$ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),Aa),aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat))),Ba),aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Aa),Ba)) ) ) ).
% psubset_insert_iff
tff(fact_5719_psubset__insert__iff,axiom,
! [Aa: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,Ba: set_fu3555266933746557007_nat_o] :
( aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(Aa),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),Ba))
<=> $ite(
aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Ba),
aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(Aa),Ba),
$ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Aa),aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(minus_6456595905716665654_nat_o(Aa,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))),Ba),aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(Aa),Ba)) ) ) ).
% psubset_insert_iff
tff(fact_5720_psubset__insert__iff,axiom,
! [Aa: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Ba: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(Aa),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),Ba))
<=> $ite(
aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),Ba),
aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(Aa),Ba),
$ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),Aa),aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat))),Ba),aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Aa),Ba)) ) ) ).
% psubset_insert_iff
tff(fact_5721_psubset__insert__iff,axiom,
! [Aa: set_o,X: $o,Ba: set_o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,Aa),aa_set_o_set_o(insert_o((X)),Ba))
<=> $ite(
aa_set_o_o(member_o((X)),Ba),
aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,Aa),Ba),
$ite(aa_set_o_o(member_o((X)),Aa),aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))),Ba),aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Aa),Ba)) ) ) ).
% psubset_insert_iff
tff(fact_5722_psubset__insert__iff,axiom,
! [Aa: set_int,X: int,Ba: set_int] :
( aa_set_int_o(ord_less_set_int(Aa),aa_set_int_set_int(insert_int(X),Ba))
<=> $ite(
aa_set_int_o(member_int(X),Ba),
aa_set_int_o(ord_less_set_int(Aa),Ba),
$ite(aa_set_int_o(member_int(X),Aa),aa_set_int_o(ord_less_set_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int))),Ba),aa_set_int_o(ord_less_eq_set_int(Aa),Ba)) ) ) ).
% psubset_insert_iff
tff(fact_5723_psubset__insert__iff,axiom,
! [Aa: set_nat,X: nat,Ba: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),Ba))
<=> $ite(
aa_set_nat_o(member_nat(X),Ba),
aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Aa),Ba),
$ite(aa_set_nat_o(member_nat(X),Aa),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))),Ba),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),Ba)) ) ) ).
% psubset_insert_iff
tff(fact_5724_bezw_Osimps,axiom,
! [X: nat,Y: nat] :
bezw(X,Y) = $ite(Y = zero_zero_nat,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)))))) ).
% bezw.simps
tff(fact_5725_bezw_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_int_int] :
( ( bezw(X,Xa) = Y )
=> ( Y = $ite(Xa = zero_zero_nat,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa)))))) ) ) ).
% bezw.elims
tff(fact_5726_card__2__iff,axiom,
! [S: set_Pr4329608150637261639at_nat] :
( ( finite3771342082235030671at_nat(S) = numeral_numeral_nat(aa_num_num(bit0,one)) )
<=> ? [X3: produc3843707927480180839at_nat,Y4: produc3843707927480180839at_nat] :
( ( S = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X3),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(Y4),bot_bo228742789529271731at_nat)) )
& ( X3 != Y4 ) ) ) ).
% card_2_iff
tff(fact_5727_card__2__iff,axiom,
! [S: set_Pr1261947904930325089at_nat] :
( ( finite711546835091564841at_nat(S) = numeral_numeral_nat(aa_num_num(bit0,one)) )
<=> ? [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] :
( ( S = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X3),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Y4),bot_bo2099793752762293965at_nat)) )
& ( X3 != Y4 ) ) ) ).
% card_2_iff
tff(fact_5728_card__2__iff,axiom,
! [S: set_o] :
( ( finite_card_o(S) = numeral_numeral_nat(aa_num_num(bit0,one)) )
<=> ? [X3: $o,Y4: $o] :
( ( S = aa_set_o_set_o(insert_o((X3)),aa_set_o_set_o(insert_o((Y4)),bot_bot_set_o)) )
& ~ ( (X3)
<=> (Y4) ) ) ) ).
% card_2_iff
tff(fact_5729_card__2__iff,axiom,
! [S: set_nat] :
( ( finite_card_nat(S) = numeral_numeral_nat(aa_num_num(bit0,one)) )
<=> ? [X3: nat,Y4: nat] :
( ( S = aa_set_nat_set_nat(insert_nat(X3),aa_set_nat_set_nat(insert_nat(Y4),bot_bot_set_nat)) )
& ( X3 != Y4 ) ) ) ).
% card_2_iff
tff(fact_5730_card__2__iff,axiom,
! [S: set_int] :
( ( finite_card_int(S) = numeral_numeral_nat(aa_num_num(bit0,one)) )
<=> ? [X3: int,Y4: int] :
( ( S = aa_set_int_set_int(insert_int(X3),aa_set_int_set_int(insert_int(Y4),bot_bot_set_int)) )
& ( X3 != Y4 ) ) ) ).
% card_2_iff
tff(fact_5731_card__3__iff,axiom,
! [S: set_Pr4329608150637261639at_nat] :
( ( finite3771342082235030671at_nat(S) = numeral_numeral_nat(aa_num_num(bit1,one)) )
<=> ? [X3: produc3843707927480180839at_nat,Y4: produc3843707927480180839at_nat,Z3: produc3843707927480180839at_nat] :
( ( S = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X3),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(Y4),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(Z3),bot_bo228742789529271731at_nat))) )
& ( X3 != Y4 )
& ( Y4 != Z3 )
& ( X3 != Z3 ) ) ) ).
% card_3_iff
tff(fact_5732_card__3__iff,axiom,
! [S: set_Pr1261947904930325089at_nat] :
( ( finite711546835091564841at_nat(S) = numeral_numeral_nat(aa_num_num(bit1,one)) )
<=> ? [X3: product_prod_nat_nat,Y4: product_prod_nat_nat,Z3: product_prod_nat_nat] :
( ( S = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X3),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Y4),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Z3),bot_bo2099793752762293965at_nat))) )
& ( X3 != Y4 )
& ( Y4 != Z3 )
& ( X3 != Z3 ) ) ) ).
% card_3_iff
tff(fact_5733_card__3__iff,axiom,
! [S: set_o] :
( ( finite_card_o(S) = numeral_numeral_nat(aa_num_num(bit1,one)) )
<=> ? [X3: $o,Y4: $o,Z3: $o] :
( ( S = aa_set_o_set_o(insert_o((X3)),aa_set_o_set_o(insert_o((Y4)),aa_set_o_set_o(insert_o((Z3)),bot_bot_set_o))) )
& ~ ( (X3)
<=> (Y4) )
& ~ ( (Y4)
<=> (Z3) )
& ~ ( (X3)
<=> (Z3) ) ) ) ).
% card_3_iff
tff(fact_5734_card__3__iff,axiom,
! [S: set_nat] :
( ( finite_card_nat(S) = numeral_numeral_nat(aa_num_num(bit1,one)) )
<=> ? [X3: nat,Y4: nat,Z3: nat] :
( ( S = aa_set_nat_set_nat(insert_nat(X3),aa_set_nat_set_nat(insert_nat(Y4),aa_set_nat_set_nat(insert_nat(Z3),bot_bot_set_nat))) )
& ( X3 != Y4 )
& ( Y4 != Z3 )
& ( X3 != Z3 ) ) ) ).
% card_3_iff
tff(fact_5735_card__3__iff,axiom,
! [S: set_int] :
( ( finite_card_int(S) = numeral_numeral_nat(aa_num_num(bit1,one)) )
<=> ? [X3: int,Y4: int,Z3: int] :
( ( S = aa_set_int_set_int(insert_int(X3),aa_set_int_set_int(insert_int(Y4),aa_set_int_set_int(insert_int(Z3),bot_bot_set_int))) )
& ( X3 != Y4 )
& ( Y4 != Z3 )
& ( X3 != Z3 ) ) ) ).
% card_3_iff
tff(fact_5736_card__Diff__singleton__if,axiom,
! [Aa: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
finite3771342082235030671at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat))) = $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite3771342082235030671at_nat(Aa)),one_one_nat),finite3771342082235030671at_nat(Aa)) ).
% card_Diff_singleton_if
tff(fact_5737_card__Diff__singleton__if,axiom,
! [Aa: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(Aa,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))) = $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite5795495896937551127_nat_o(Aa)),one_one_nat),finite5795495896937551127_nat_o(Aa)) ).
% card_Diff_singleton_if
tff(fact_5738_card__Diff__singleton__if,axiom,
! [Aa: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
finite711546835091564841at_nat(minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat))) = $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite711546835091564841at_nat(Aa)),one_one_nat),finite711546835091564841at_nat(Aa)) ).
% card_Diff_singleton_if
tff(fact_5739_card__Diff__singleton__if,axiom,
! [Aa: set_o,X: $o] :
finite_card_o(minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((X)),Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_o(Aa)),one_one_nat),finite_card_o(Aa)) ).
% card_Diff_singleton_if
tff(fact_5740_card__Diff__singleton__if,axiom,
! [Aa: set_int,X: int] :
finite_card_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int))) = $ite(aa_set_int_o(member_int(X),Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(Aa)),one_one_nat),finite_card_int(Aa)) ).
% card_Diff_singleton_if
tff(fact_5741_card__Diff__singleton__if,axiom,
! [Aa: set_nat,X: nat] :
finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(X),Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_nat(Aa)),one_one_nat),finite_card_nat(Aa)) ).
% card_Diff_singleton_if
tff(fact_5742_card__Diff__singleton,axiom,
! [X: produc3843707927480180839at_nat,Aa: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),Aa)
=> ( finite3771342082235030671at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite3771342082235030671at_nat(Aa)),one_one_nat) ) ) ).
% card_Diff_singleton
tff(fact_5743_card__Diff__singleton,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Aa: set_fu3555266933746557007_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Aa)
=> ( finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(Aa,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite5795495896937551127_nat_o(Aa)),one_one_nat) ) ) ).
% card_Diff_singleton
tff(fact_5744_card__Diff__singleton,axiom,
! [X: product_prod_nat_nat,Aa: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),Aa)
=> ( finite711546835091564841at_nat(minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite711546835091564841at_nat(Aa)),one_one_nat) ) ) ).
% card_Diff_singleton
tff(fact_5745_card__Diff__singleton,axiom,
! [X: $o,Aa: set_o] :
( aa_set_o_o(member_o((X)),Aa)
=> ( finite_card_o(minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_o(Aa)),one_one_nat) ) ) ).
% card_Diff_singleton
tff(fact_5746_card__Diff__singleton,axiom,
! [X: int,Aa: set_int] :
( aa_set_int_o(member_int(X),Aa)
=> ( finite_card_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(Aa)),one_one_nat) ) ) ).
% card_Diff_singleton
tff(fact_5747_card__Diff__singleton,axiom,
! [X: nat,Aa: set_nat] :
( aa_set_nat_o(member_nat(X),Aa)
=> ( finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_nat(Aa)),one_one_nat) ) ) ).
% card_Diff_singleton
tff(fact_5748_bezw_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_int_int] :
( ( bezw(X,Xa) = Y )
=> ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa))
=> ~ ( ( Y = $ite(Xa = zero_zero_nat,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa)))))) )
=> ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)) ) ) ) ).
% bezw.pelims
tff(fact_5749_neg__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q4: int,R2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
=> ( aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2))
=> aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),R2)),one_one_int))) ) ) ).
% neg_eucl_rel_int_mult_2
tff(fact_5750_round__def,axiom,
! [X: rat] : archim7778729529865785530nd_rat(X) = archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))) ).
% round_def
tff(fact_5751_the__elem__eq,axiom,
! [X: produc3843707927480180839at_nat] : the_el221668144340439132at_nat(aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat)) = X ).
% the_elem_eq
tff(fact_5752_the__elem__eq,axiom,
! [X: product_prod_nat_nat] : the_el2281957884133575798at_nat(aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat)) = X ).
% the_elem_eq
tff(fact_5753_the__elem__eq,axiom,
! [X: $o] :
( aa_set_o_o(the_elem_o,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))
<=> (X) ) ).
% the_elem_eq
tff(fact_5754_the__elem__eq,axiom,
! [X: nat] : the_elem_nat(aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) = X ).
% the_elem_eq
tff(fact_5755_the__elem__eq,axiom,
! [X: int] : the_elem_int(aa_set_int_set_int(insert_int(X),bot_bot_set_int)) = X ).
% the_elem_eq
tff(fact_5756_pos__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q4: int,R2: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
=> ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2))
=> aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),R2)))) ) ) ).
% pos_eucl_rel_int_mult_2
tff(fact_5757_is__singletonI,axiom,
! [X: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(is_sin2937591304547752795at_nat,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat)) ).
% is_singletonI
tff(fact_5758_is__singletonI,axiom,
! [X: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat)) ).
% is_singletonI
tff(fact_5759_is__singletonI,axiom,
! [X: $o] : aa_set_o_o(is_singleton_o,aa_set_o_set_o(insert_o((X)),bot_bot_set_o)) ).
% is_singletonI
tff(fact_5760_is__singletonI,axiom,
! [X: nat] : aa_set_nat_o(is_singleton_nat,aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) ).
% is_singletonI
tff(fact_5761_is__singletonI,axiom,
! [X: int] : aa_set_int_o(is_singleton_int,aa_set_int_set_int(insert_int(X),bot_bot_set_int)) ).
% is_singletonI
tff(fact_5762_divmod__BitM__2__eq,axiom,
! [M2: num] : unique5052692396658037445od_int(bitM(M2),aa_num_num(bit0,one)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,numeral_numeral_int(M2)),one_one_int)),one_one_int) ).
% divmod_BitM_2_eq
tff(fact_5763_round__1,axiom,
archim7778729529865785530nd_rat(one_one_rat) = one_one_int ).
% round_1
tff(fact_5764_mod__int__unique,axiom,
! [K: int,L: int,Q4: int,R2: int] :
( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = R2 ) ) ).
% mod_int_unique
tff(fact_5765_eucl__rel__int,axiom,
! [K: int,L: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ).
% eucl_rel_int
tff(fact_5766_div__int__unique,axiom,
! [K: int,L: int,Q4: int,R2: int] :
( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = Q4 ) ) ).
% div_int_unique
tff(fact_5767_unique__quotient,axiom,
! [A: int,B: int,Q4: int,R2: int,Q6: int,R4: int] :
( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2))
=> ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(product_Pair_int_int(Q6),R4))
=> ( Q4 = Q6 ) ) ) ).
% unique_quotient
tff(fact_5768_unique__remainder,axiom,
! [A: int,B: int,Q4: int,R2: int,Q6: int,R4: int] :
( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2))
=> ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(product_Pair_int_int(Q6),R4))
=> ( R2 = R4 ) ) ) ).
% unique_remainder
tff(fact_5769_eucl__rel__int__by0,axiom,
! [K: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K,zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),K)) ).
% eucl_rel_int_by0
tff(fact_5770_is__singleton__the__elem,axiom,
! [Aa: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(is_sin2937591304547752795at_nat,Aa)
<=> ( Aa = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(the_el221668144340439132at_nat(Aa)),bot_bo228742789529271731at_nat) ) ) ).
% is_singleton_the_elem
tff(fact_5771_is__singleton__the__elem,axiom,
! [Aa: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,Aa)
<=> ( Aa = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(the_el2281957884133575798at_nat(Aa)),bot_bo2099793752762293965at_nat) ) ) ).
% is_singleton_the_elem
tff(fact_5772_is__singleton__the__elem,axiom,
! [Aa: set_o] :
( aa_set_o_o(is_singleton_o,Aa)
<=> ( Aa = aa_set_o_set_o(insert_o(aa_set_o_o(the_elem_o,Aa)),bot_bot_set_o) ) ) ).
% is_singleton_the_elem
tff(fact_5773_is__singleton__the__elem,axiom,
! [Aa: set_nat] :
( aa_set_nat_o(is_singleton_nat,Aa)
<=> ( Aa = aa_set_nat_set_nat(insert_nat(the_elem_nat(Aa)),bot_bot_set_nat) ) ) ).
% is_singleton_the_elem
tff(fact_5774_is__singleton__the__elem,axiom,
! [Aa: set_int] :
( aa_set_int_o(is_singleton_int,Aa)
<=> ( Aa = aa_set_int_set_int(insert_int(the_elem_int(Aa)),bot_bot_set_int) ) ) ).
% is_singleton_the_elem
tff(fact_5775_is__singletonI_H,axiom,
! [Aa: set_fu3555266933746557007_nat_o] :
( ( Aa != bot_bo8632255151191677115_nat_o )
=> ( ! [X2: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),Aa)
=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),Aa)
=> ( X2 = Y2 ) ) )
=> aa_set5091322549519198405at_o_o(is_sin624665226112233955_nat_o,Aa) ) ) ).
% is_singletonI'
tff(fact_5776_is__singletonI_H,axiom,
! [Aa: set_Pr1261947904930325089at_nat] :
( ( Aa != bot_bo2099793752762293965at_nat )
=> ( ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),Aa)
=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y2),Aa)
=> ( X2 = Y2 ) ) )
=> aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,Aa) ) ) ).
% is_singletonI'
tff(fact_5777_is__singletonI_H,axiom,
! [Aa: set_o] :
( ( Aa != bot_bot_set_o )
=> ( ! [X2: $o,Y2: $o] :
( aa_set_o_o(member_o((X2)),Aa)
=> ( aa_set_o_o(member_o((Y2)),Aa)
=> ( (X2)
<=> (Y2) ) ) )
=> aa_set_o_o(is_singleton_o,Aa) ) ) ).
% is_singletonI'
tff(fact_5778_is__singletonI_H,axiom,
! [Aa: set_nat] :
( ( Aa != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] :
( aa_set_nat_o(member_nat(X2),Aa)
=> ( aa_set_nat_o(member_nat(Y2),Aa)
=> ( X2 = Y2 ) ) )
=> aa_set_nat_o(is_singleton_nat,Aa) ) ) ).
% is_singletonI'
tff(fact_5779_is__singletonI_H,axiom,
! [Aa: set_int] :
( ( Aa != bot_bot_set_int )
=> ( ! [X2: int,Y2: int] :
( aa_set_int_o(member_int(X2),Aa)
=> ( aa_set_int_o(member_int(Y2),Aa)
=> ( X2 = Y2 ) ) )
=> aa_set_int_o(is_singleton_int,Aa) ) ) ).
% is_singletonI'
tff(fact_5780_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q4: int] :
( ( L != zero_zero_int )
=> ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L) )
=> aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),zero_zero_int)) ) ) ).
% eucl_rel_int_dividesI
tff(fact_5781_eucl__rel__int__remainderI,axiom,
! [R2: int,L: int,K: int,Q4: int] :
( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L))
=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L)),R2) )
=> aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2)) ) ) ) ).
% eucl_rel_int_remainderI
tff(fact_5782_eucl__rel__int_Osimps,axiom,
! [A1: int,A22: int,A32: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
<=> ( ? [K4: int] :
( ( A1 = K4 )
& ( A22 = zero_zero_int )
& ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),K4) ) )
| ? [L2: int,K4: int,Q7: int] :
( ( A1 = K4 )
& ( A22 = L2 )
& ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q7),zero_zero_int) )
& ( L2 != zero_zero_int )
& ( K4 = aa_int_int(aa_int_fun_int_int(times_times_int,Q7),L2) ) )
| ? [R5: int,L2: int,K4: int,Q7: int] :
( ( A1 = K4 )
& ( A22 = L2 )
& ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q7),R5) )
& ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L2) )
& aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L2))
& ( K4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q7),L2)),R5) ) ) ) ) ).
% eucl_rel_int.simps
tff(fact_5783_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A32: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
=> ( ( ( A22 = zero_zero_int )
=> ( A32 != aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),A1) ) )
=> ( ! [Q5: int] :
( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),zero_zero_int) )
=> ( ( A22 != zero_zero_int )
=> ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q5),A22) ) ) )
=> ~ ! [R3: int,Q5: int] :
( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),R3) )
=> ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A22) )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,A22))
=> ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q5),A22)),R3) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
tff(fact_5784_zminus1__lemma,axiom,
! [A: int,B: int,Q4: int,R2: int] :
( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2))
=> ( ( B != zero_zero_int )
=> aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(uminus_uminus_int,A),B),
aa_int7462565939943104719nt_int(product_Pair_int_int($ite(R2 = zero_zero_int,aa_int_int(uminus_uminus_int,Q4),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q4)),one_one_int))),
$ite(R2 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R2)))) ) ) ).
% zminus1_lemma
tff(fact_5785_numeral__BitM,axiom,
! [N: num] : numeral_numeral_int(bitM(N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,numeral_numeral_int(aa_num_num(bit0,N))),one_one_int) ).
% numeral_BitM
tff(fact_5786_numeral__BitM,axiom,
! [N: num] : numeral_numeral_rat(bitM(N)) = minus_minus_rat(numeral_numeral_rat(aa_num_num(bit0,N)),one_one_rat) ).
% numeral_BitM
tff(fact_5787_numeral__BitM,axiom,
! [N: num] : numera6620942414471956472nteger(bitM(N)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(numera6620942414471956472nteger(aa_num_num(bit0,N))),one_one_Code_integer) ).
% numeral_BitM
tff(fact_5788_is__singleton__def,axiom,
! [Aa: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(is_sin2937591304547752795at_nat,Aa)
<=> ? [X3: produc3843707927480180839at_nat] : Aa = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X3),bot_bo228742789529271731at_nat) ) ).
% is_singleton_def
tff(fact_5789_is__singleton__def,axiom,
! [Aa: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,Aa)
<=> ? [X3: product_prod_nat_nat] : Aa = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X3),bot_bo2099793752762293965at_nat) ) ).
% is_singleton_def
tff(fact_5790_is__singleton__def,axiom,
! [Aa: set_o] :
( aa_set_o_o(is_singleton_o,Aa)
<=> ? [X3: $o] : Aa = aa_set_o_set_o(insert_o((X3)),bot_bot_set_o) ) ).
% is_singleton_def
tff(fact_5791_is__singleton__def,axiom,
! [Aa: set_nat] :
( aa_set_nat_o(is_singleton_nat,Aa)
<=> ? [X3: nat] : Aa = aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat) ) ).
% is_singleton_def
tff(fact_5792_is__singleton__def,axiom,
! [Aa: set_int] :
( aa_set_int_o(is_singleton_int,Aa)
<=> ? [X3: int] : Aa = aa_set_int_set_int(insert_int(X3),bot_bot_set_int) ) ).
% is_singleton_def
tff(fact_5793_is__singletonE,axiom,
! [Aa: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(is_sin2937591304547752795at_nat,Aa)
=> ~ ! [X2: produc3843707927480180839at_nat] : Aa != aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X2),bot_bo228742789529271731at_nat) ) ).
% is_singletonE
tff(fact_5794_is__singletonE,axiom,
! [Aa: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,Aa)
=> ~ ! [X2: product_prod_nat_nat] : Aa != aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X2),bot_bo2099793752762293965at_nat) ) ).
% is_singletonE
tff(fact_5795_is__singletonE,axiom,
! [Aa: set_o] :
( aa_set_o_o(is_singleton_o,Aa)
=> ~ ! [X2: $o] : Aa != aa_set_o_set_o(insert_o((X2)),bot_bot_set_o) ) ).
% is_singletonE
tff(fact_5796_is__singletonE,axiom,
! [Aa: set_nat] :
( aa_set_nat_o(is_singleton_nat,Aa)
=> ~ ! [X2: nat] : Aa != aa_set_nat_set_nat(insert_nat(X2),bot_bot_set_nat) ) ).
% is_singletonE
tff(fact_5797_is__singletonE,axiom,
! [Aa: set_int] :
( aa_set_int_o(is_singleton_int,Aa)
=> ~ ! [X2: int] : Aa != aa_set_int_set_int(insert_int(X2),bot_bot_set_int) ) ).
% is_singletonE
tff(fact_5798_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q4: int,R2: int] :
( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2))
<=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q4)),R2) )
& $ite(
aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L),
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
& aa_int_o(aa_int_fun_int_o(ord_less_int,R2),L) ),
$ite(
aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int),
( aa_int_o(aa_int_fun_int_o(ord_less_int,L),R2)
& aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int) ),
Q4 = zero_zero_int ) ) ) ) ).
% eucl_rel_int_iff
tff(fact_5799_Divides_Oadjust__div__eq,axiom,
! [Q4: int,R2: int] : adjust_div(aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q4),aa_o_int(zero_n2684676970156552555ol_int,R2 != zero_zero_int)) ).
% Divides.adjust_div_eq
tff(fact_5800_round__altdef,axiom,
! [X: rat] :
archim7778729529865785530nd_rat(X) = $ite(aa_rat_o(ord_less_eq_rat(divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))),archimedean_frac_rat(X)),archim2889992004027027881ng_rat(X),archim3151403230148437115or_rat(X)) ).
% round_altdef
tff(fact_5801_and__int_Opelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa) = Y )
=> ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa))
=> ~ ( ( Y = $ite(
( aa_set_int_o(member_int(X),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))
& aa_set_int_o(member_int(Xa),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int))) ),
aa_int_int(uminus_uminus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),X)
& ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),Xa) ))),
aa_int_int(aa_int_fun_int_int(plus_plus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),X)
& ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),Xa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),numeral_numeral_int(aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),numeral_numeral_int(aa_num_num(bit0,one)))))) ) )
=> ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa)) ) ) ) ).
% and_int.pelims
tff(fact_5802_and__int_Opsimps,axiom,
! [K: int,L: int] :
( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(K),L))
=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = $ite(
( aa_set_int_o(member_int(K),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))
& aa_set_int_o(member_int(L),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int))) ),
aa_int_int(uminus_uminus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),K)
& ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),L) ))),
aa_int_int(aa_int_fun_int_int(plus_plus_int,
aa_o_int(zero_n2684676970156552555ol_int,
( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),K)
& ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),numeral_numeral_int(aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),numeral_numeral_int(aa_num_num(bit0,one)))))) ) ) ) ).
% and_int.psimps
tff(fact_5803_round__unique,axiom,
! [X: rat,Y: int] :
( aa_rat_o(ord_less_rat(minus_minus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y))
=> ( aa_rat_o(ord_less_eq_rat(aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))))
=> ( archim7778729529865785530nd_rat(X) = Y ) ) ) ).
% round_unique
tff(fact_5804_normalize__negative,axiom,
! [Q4: int,P5: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Q4),zero_zero_int)
=> ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P5),Q4)) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,P5)),aa_int_int(uminus_uminus_int,Q4))) ) ) ).
% normalize_negative
tff(fact_5805_round__unique_H,axiom,
! [X: rat,N: int] :
( aa_rat_o(ord_less_rat(abs_abs_rat(minus_minus_rat(X,aa_int_rat(ring_1_of_int_rat,N)))),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))
=> ( archim7778729529865785530nd_rat(X) = N ) ) ).
% round_unique'
tff(fact_5806_of__int__1,axiom,
aa_int_Code_integer(ring_18347121197199848620nteger,one_one_int) = one_one_Code_integer ).
% of_int_1
tff(fact_5807_of__int__1,axiom,
aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).
% of_int_1
tff(fact_5808_of__int__1,axiom,
aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).
% of_int_1
tff(fact_5809_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( aa_int_Code_integer(ring_18347121197199848620nteger,Z) = one_one_Code_integer )
<=> ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
tff(fact_5810_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( aa_int_int(ring_1_of_int_int,Z) = one_one_int )
<=> ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
tff(fact_5811_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( aa_int_rat(ring_1_of_int_rat,Z) = one_one_rat )
<=> ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
tff(fact_5812_of__int__mult,axiom,
! [W: int,Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).
% of_int_mult
tff(fact_5813_of__int__mult,axiom,
! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).
% of_int_mult
tff(fact_5814_of__int__mult,axiom,
! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).
% of_int_mult
tff(fact_5815_normalize__denom__zero,axiom,
! [P5: int] : normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P5),zero_zero_int)) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int) ).
% normalize_denom_zero
tff(fact_5816_of__int__le__1__iff,axiom,
! [Z: int] :
( aa_rat_o(ord_less_eq_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).
% of_int_le_1_iff
tff(fact_5817_of__int__le__1__iff,axiom,
! [Z: int] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),one_one_Code_integer)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).
% of_int_le_1_iff
tff(fact_5818_of__int__le__1__iff,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).
% of_int_le_1_iff
tff(fact_5819_of__int__1__le__iff,axiom,
! [Z: int] :
( aa_rat_o(ord_less_eq_rat(one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).
% of_int_1_le_iff
tff(fact_5820_of__int__1__le__iff,axiom,
! [Z: int] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).
% of_int_1_le_iff
tff(fact_5821_of__int__1__le__iff,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).
% of_int_1_le_iff
tff(fact_5822_of__int__less__1__iff,axiom,
! [Z: int] :
( aa_rat_o(ord_less_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).
% of_int_less_1_iff
tff(fact_5823_of__int__less__1__iff,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).
% of_int_less_1_iff
tff(fact_5824_of__int__less__1__iff,axiom,
! [Z: int] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),one_one_Code_integer)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).
% of_int_less_1_iff
tff(fact_5825_of__int__1__less__iff,axiom,
! [Z: int] :
( aa_rat_o(ord_less_rat(one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).
% of_int_1_less_iff
tff(fact_5826_of__int__1__less__iff,axiom,
! [Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).
% of_int_1_less_iff
tff(fact_5827_of__int__1__less__iff,axiom,
! [Z: int] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).
% of_int_1_less_iff
tff(fact_5828_mult__of__int__commute,axiom,
! [X: int,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(ring_18347121197199848620nteger,X)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_int_Code_integer(ring_18347121197199848620nteger,X)) ).
% mult_of_int_commute
tff(fact_5829_mult__of__int__commute,axiom,
! [X: int,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,X)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_int_rat(ring_1_of_int_rat,X)) ).
% mult_of_int_commute
tff(fact_5830_mult__of__int__commute,axiom,
! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_int_int(ring_1_of_int_int,X)) ).
% mult_of_int_commute
tff(fact_5831_mult__inverse__of__int__commute,axiom,
! [Xa: int,X: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xa))),X) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xa))) ).
% mult_inverse_of_int_commute
tff(fact_5832_frac__lt__1,axiom,
! [X: rat] : aa_rat_o(ord_less_rat(archimedean_frac_rat(X)),one_one_rat) ).
% frac_lt_1
tff(fact_5833_frac__1__eq,axiom,
! [X: rat] : archimedean_frac_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) = archimedean_frac_rat(X) ).
% frac_1_eq
tff(fact_5834_of__int__leD,axiom,
! [N: int,X: rat] :
( aa_rat_o(ord_less_eq_rat(abs_abs_rat(aa_int_rat(ring_1_of_int_rat,N))),X)
=> ( ( N = zero_zero_int )
| aa_rat_o(ord_less_eq_rat(one_one_rat),X) ) ) ).
% of_int_leD
tff(fact_5835_of__int__leD,axiom,
! [N: int,X: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(ring_18347121197199848620nteger,N))),X)
=> ( ( N = zero_zero_int )
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),X) ) ) ).
% of_int_leD
tff(fact_5836_of__int__leD,axiom,
! [N: int,X: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X)
=> ( ( N = zero_zero_int )
| aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),X) ) ) ).
% of_int_leD
tff(fact_5837_of__int__lessD,axiom,
! [N: int,X: rat] :
( aa_rat_o(ord_less_rat(abs_abs_rat(aa_int_rat(ring_1_of_int_rat,N))),X)
=> ( ( N = zero_zero_int )
| aa_rat_o(ord_less_rat(one_one_rat),X) ) ) ).
% of_int_lessD
tff(fact_5838_of__int__lessD,axiom,
! [N: int,X: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X)
=> ( ( N = zero_zero_int )
| aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),X) ) ) ).
% of_int_lessD
tff(fact_5839_of__int__lessD,axiom,
! [N: int,X: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(ring_18347121197199848620nteger,N))),X)
=> ( ( N = zero_zero_int )
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),X) ) ) ).
% of_int_lessD
tff(fact_5840_floor__split,axiom,
! [P: fun_int_o,T: rat] :
( aa_int_o(P,archim3151403230148437115or_rat(T))
<=> ! [I2: int] :
( ( aa_rat_o(ord_less_eq_rat(aa_int_rat(ring_1_of_int_rat,I2)),T)
& aa_rat_o(ord_less_rat(T),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,I2)),one_one_rat)) )
=> aa_int_o(P,I2) ) ) ).
% floor_split
tff(fact_5841_floor__eq__iff,axiom,
! [X: rat,A: int] :
( ( archim3151403230148437115or_rat(X) = A )
<=> ( aa_rat_o(ord_less_eq_rat(aa_int_rat(ring_1_of_int_rat,A)),X)
& aa_rat_o(ord_less_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)) ) ) ).
% floor_eq_iff
tff(fact_5842_floor__unique,axiom,
! [Z: int,X: rat] :
( aa_rat_o(ord_less_eq_rat(aa_int_rat(ring_1_of_int_rat,Z)),X)
=> ( aa_rat_o(ord_less_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
=> ( archim3151403230148437115or_rat(X) = Z ) ) ) ).
% floor_unique
tff(fact_5843_ceiling__correct,axiom,
! [X: rat] :
( aa_rat_o(ord_less_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(X)),one_one_rat)),X)
& aa_rat_o(ord_less_eq_rat(X),aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(X))) ) ).
% ceiling_correct
tff(fact_5844_ceiling__unique,axiom,
! [Z: int,X: rat] :
( aa_rat_o(ord_less_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),X)
=> ( aa_rat_o(ord_less_eq_rat(X),aa_int_rat(ring_1_of_int_rat,Z))
=> ( archim2889992004027027881ng_rat(X) = Z ) ) ) ).
% ceiling_unique
tff(fact_5845_ceiling__eq__iff,axiom,
! [X: rat,A: int] :
( ( archim2889992004027027881ng_rat(X) = A )
<=> ( aa_rat_o(ord_less_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A),one_one_rat)),X)
& aa_rat_o(ord_less_eq_rat(X),aa_int_rat(ring_1_of_int_rat,A)) ) ) ).
% ceiling_eq_iff
tff(fact_5846_ceiling__split,axiom,
! [P: fun_int_o,T: rat] :
( aa_int_o(P,archim2889992004027027881ng_rat(T))
<=> ! [I2: int] :
( ( aa_rat_o(ord_less_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I2),one_one_rat)),T)
& aa_rat_o(ord_less_eq_rat(T),aa_int_rat(ring_1_of_int_rat,I2)) )
=> aa_int_o(P,I2) ) ) ).
% ceiling_split
tff(fact_5847_less__floor__iff,axiom,
! [Z: int,X: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),archim3151403230148437115or_rat(X))
<=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),X) ) ).
% less_floor_iff
tff(fact_5848_floor__le__iff,axiom,
! [X: rat,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(X)),Z)
<=> aa_rat_o(ord_less_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)) ) ).
% floor_le_iff
tff(fact_5849_ceiling__less__iff,axiom,
! [X: rat,Z: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(X)),Z)
<=> aa_rat_o(ord_less_eq_rat(X),minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)) ) ).
% ceiling_less_iff
tff(fact_5850_le__ceiling__iff,axiom,
! [Z: int,X: rat] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),archim2889992004027027881ng_rat(X))
<=> aa_rat_o(ord_less_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),X) ) ).
% le_ceiling_iff
tff(fact_5851_normalize__denom__pos,axiom,
! [R2: product_prod_int_int,P5: int,Q4: int] :
( ( normalize(R2) = aa_int7462565939943104719nt_int(product_Pair_int_int(P5),Q4) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q4) ) ).
% normalize_denom_pos
tff(fact_5852_normalize__crossproduct,axiom,
! [Q4: int,S2: int,P5: int,R2: int] :
( ( Q4 != zero_zero_int )
=> ( ( S2 != zero_zero_int )
=> ( ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P5),Q4)) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(R2),S2)) )
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,P5),S2) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q4) ) ) ) ) ).
% normalize_crossproduct
tff(fact_5853_frac__eq,axiom,
! [X: rat] :
( ( archimedean_frac_rat(X) = X )
<=> ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),X)
& aa_rat_o(ord_less_rat(X),one_one_rat) ) ) ).
% frac_eq
tff(fact_5854_frac__add,axiom,
! [X: rat,Y: rat] :
archimedean_frac_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = $ite(aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,archimedean_frac_rat(X)),archimedean_frac_rat(Y))),one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,archimedean_frac_rat(X)),archimedean_frac_rat(Y)),minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,archimedean_frac_rat(X)),archimedean_frac_rat(Y)),one_one_rat)) ).
% frac_add
tff(fact_5855_floor__divide__lower,axiom,
! [Q4: rat,P5: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),Q4)
=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(divide_divide_rat(P5,Q4)))),Q4)),P5) ) ).
% floor_divide_lower
tff(fact_5856_ceiling__divide__upper,axiom,
! [Q4: rat,P5: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),Q4)
=> aa_rat_o(ord_less_eq_rat(P5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P5,Q4)))),Q4)) ) ).
% ceiling_divide_upper
tff(fact_5857_floor__divide__upper,axiom,
! [Q4: rat,P5: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),Q4)
=> aa_rat_o(ord_less_rat(P5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(divide_divide_rat(P5,Q4)))),one_one_rat)),Q4)) ) ).
% floor_divide_upper
tff(fact_5858_ceiling__divide__lower,axiom,
! [Q4: rat,P5: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),Q4)
=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P5,Q4))),one_one_rat)),Q4)),P5) ) ).
% ceiling_divide_lower
tff(fact_5859_of__int__round__le,axiom,
! [X: rat] : aa_rat_o(ord_less_eq_rat(aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))) ).
% of_int_round_le
tff(fact_5860_of__int__round__ge,axiom,
! [X: rat] : aa_rat_o(ord_less_eq_rat(minus_minus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X))) ).
% of_int_round_ge
tff(fact_5861_of__int__round__gt,axiom,
! [X: rat] : aa_rat_o(ord_less_rat(minus_minus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X))) ).
% of_int_round_gt
tff(fact_5862_floor__add,axiom,
! [X: rat,Y: rat] :
archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = $ite(aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,archimedean_frac_rat(X)),archimedean_frac_rat(Y))),one_one_rat),aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(X)),archim3151403230148437115or_rat(Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(X)),archim3151403230148437115or_rat(Y))),one_one_int)) ).
% floor_add
tff(fact_5863_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: fun_int_fun_int_o] :
( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(A0),A1))
=> ( ! [K2: int,L3: int] :
( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(K2),L3))
=> ( ( ~ ( aa_set_int_o(member_int(K2),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))
& aa_set_int_o(member_int(L3),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int))) )
=> aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),numeral_numeral_int(aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L3),numeral_numeral_int(aa_num_num(bit0,one)))) )
=> aa_int_o(aa_int_fun_int_o(P,K2),L3) ) )
=> aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).
% and_int.pinduct
tff(fact_5864_of__int__round__abs__le,axiom,
! [X: rat] : aa_rat_o(ord_less_eq_rat(abs_abs_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X)),X))),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))) ).
% of_int_round_abs_le
tff(fact_5865_upto_Opinduct,axiom,
! [A0: int,A1: int,P: fun_int_fun_int_o] :
( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(A0),A1))
=> ( ! [I3: int,J2: int] :
( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(I3),J2))
=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J2)
=> aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)),J2) )
=> aa_int_o(aa_int_fun_int_o(P,I3),J2) ) )
=> aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).
% upto.pinduct
tff(fact_5866_mult__ceiling__le__Ints,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( member_rat(A,ring_1_Ints_rat)
=> aa_rat_o(ord_less_eq_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,archim2889992004027027881ng_rat(A)),archim2889992004027027881ng_rat(B)))) ) ) ).
% mult_ceiling_le_Ints
tff(fact_5867_mult__ceiling__le__Ints,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( member_rat(A,ring_1_Ints_rat)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(times_times_int,archim2889992004027027881ng_rat(A)),archim2889992004027027881ng_rat(B)))) ) ) ).
% mult_ceiling_le_Ints
tff(fact_5868_mult__ceiling__le__Ints,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( member_rat(A,ring_1_Ints_rat)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,archim2889992004027027881ng_rat(A)),archim2889992004027027881ng_rat(B)))) ) ) ).
% mult_ceiling_le_Ints
tff(fact_5869_le__mult__floor__Ints,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( member_rat(A,ring_1_Ints_rat)
=> aa_rat_o(ord_less_eq_rat(aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,archim3151403230148437115or_rat(A)),archim3151403230148437115or_rat(B)))),aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))) ) ) ).
% le_mult_floor_Ints
tff(fact_5870_le__mult__floor__Ints,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( member_rat(A,ring_1_Ints_rat)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(times_times_int,archim3151403230148437115or_rat(A)),archim3151403230148437115or_rat(B)))),aa_int_Code_integer(ring_18347121197199848620nteger,archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))) ) ) ).
% le_mult_floor_Ints
tff(fact_5871_le__mult__floor__Ints,axiom,
! [A: rat,B: rat] :
( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
=> ( member_rat(A,ring_1_Ints_rat)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,archim3151403230148437115or_rat(A)),archim3151403230148437115or_rat(B)))),aa_int_int(ring_1_of_int_int,archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))) ) ) ).
% le_mult_floor_Ints
tff(fact_5872_frac__unique__iff,axiom,
! [X: rat,A: rat] :
( ( archimedean_frac_rat(X) = A )
<=> ( member_rat(minus_minus_rat(X,A),ring_1_Ints_rat)
& aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
& aa_rat_o(ord_less_rat(A),one_one_rat) ) ) ).
% frac_unique_iff
tff(fact_5873_Gcd__0__iff,axiom,
! [Aa: set_int] :
( ( gcd_Gcd_int(Aa) = zero_zero_int )
<=> aa_set_int_o(ord_less_eq_set_int(Aa),aa_set_int_set_int(insert_int(zero_zero_int),bot_bot_set_int)) ) ).
% Gcd_0_iff
tff(fact_5874_Gcd__0__iff,axiom,
! [Aa: set_nat] :
( ( gcd_Gcd_nat(Aa) = zero_zero_nat )
<=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat)) ) ).
% Gcd_0_iff
tff(fact_5875_remove__def,axiom,
! [X: produc3843707927480180839at_nat,Aa: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(remove5300788332298244428at_nat(X),Aa) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat)) ).
% remove_def
tff(fact_5876_remove__def,axiom,
! [X: product_prod_nat_nat,Aa: set_Pr1261947904930325089at_nat] : remove6466555014256735590at_nat(X,Aa) = minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat)) ).
% remove_def
tff(fact_5877_remove__def,axiom,
! [X: $o,Aa: set_o] : remove_o((X),Aa) = minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o)) ).
% remove_def
tff(fact_5878_remove__def,axiom,
! [X: int,Aa: set_int] : remove_int(X,Aa) = minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int)) ).
% remove_def
tff(fact_5879_remove__def,axiom,
! [X: nat,Aa: set_nat] : aa_set_nat_set_nat(remove_nat(X),Aa) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) ).
% remove_def
tff(fact_5880_quotient__of__number_I5_J,axiom,
! [K: num] : quotient_of(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(K))) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(K))),one_one_int) ).
% quotient_of_number(5)
tff(fact_5881_Gcd__empty,axiom,
gcd_Gcd_nat(bot_bot_set_nat) = zero_zero_nat ).
% Gcd_empty
tff(fact_5882_Gcd__empty,axiom,
gcd_Gcd_int(bot_bot_set_int) = zero_zero_int ).
% Gcd_empty
tff(fact_5883_Gcd__UNIV,axiom,
gcd_Gcd_int(top_top_set_int) = one_one_int ).
% Gcd_UNIV
tff(fact_5884_Gcd__UNIV,axiom,
gcd_Gcd_nat(top_top_set_nat) = one_one_nat ).
% Gcd_UNIV
tff(fact_5885_rat__one__code,axiom,
quotient_of(one_one_rat) = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int) ).
% rat_one_code
tff(fact_5886_rat__zero__code,axiom,
quotient_of(zero_zero_rat) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int) ).
% rat_zero_code
tff(fact_5887_quotient__of__number_I4_J,axiom,
quotient_of(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).
% quotient_of_number(4)
tff(fact_5888_quotient__of__number_I3_J,axiom,
! [K: num] : quotient_of(numeral_numeral_rat(K)) = aa_int7462565939943104719nt_int(product_Pair_int_int(numeral_numeral_int(K)),one_one_int) ).
% quotient_of_number(3)
tff(fact_5889_Ints__mult,axiom,
! [A: code_integer,B: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(A),ring_11222124179247155820nteger)
=> ( aa_set5643387332196575111eger_o(member_Code_integer(B),ring_11222124179247155820nteger)
=> aa_set5643387332196575111eger_o(member_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),ring_11222124179247155820nteger) ) ) ).
% Ints_mult
tff(fact_5890_Ints__mult,axiom,
! [A: rat,B: rat] :
( member_rat(A,ring_1_Ints_rat)
=> ( member_rat(B,ring_1_Ints_rat)
=> member_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),ring_1_Ints_rat) ) ) ).
% Ints_mult
tff(fact_5891_Ints__mult,axiom,
! [A: int,B: int] :
( aa_set_int_o(member_int(A),ring_1_Ints_int)
=> ( aa_set_int_o(member_int(B),ring_1_Ints_int)
=> aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),ring_1_Ints_int) ) ) ).
% Ints_mult
tff(fact_5892_Ints__1,axiom,
aa_set5643387332196575111eger_o(member_Code_integer(one_one_Code_integer),ring_11222124179247155820nteger) ).
% Ints_1
tff(fact_5893_Ints__1,axiom,
member_rat(one_one_rat,ring_1_Ints_rat) ).
% Ints_1
tff(fact_5894_Ints__1,axiom,
aa_set_int_o(member_int(one_one_int),ring_1_Ints_int) ).
% Ints_1
tff(fact_5895_quotient__of__div,axiom,
! [R2: rat,N: int,D2: int] :
( ( quotient_of(R2) = aa_int7462565939943104719nt_int(product_Pair_int_int(N),D2) )
=> ( R2 = divide_divide_rat(aa_int_rat(ring_1_of_int_rat,N),aa_int_rat(ring_1_of_int_rat,D2)) ) ) ).
% quotient_of_div
tff(fact_5896_Gcd__1,axiom,
! [Aa: set_nat] :
( aa_set_nat_o(member_nat(one_one_nat),Aa)
=> ( gcd_Gcd_nat(Aa) = one_one_nat ) ) ).
% Gcd_1
tff(fact_5897_Gcd__1,axiom,
! [Aa: set_int] :
( aa_set_int_o(member_int(one_one_int),Aa)
=> ( gcd_Gcd_int(Aa) = one_one_int ) ) ).
% Gcd_1
tff(fact_5898_Gcd__eq__1__I,axiom,
! [A: nat,Aa: set_nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
=> ( aa_set_nat_o(member_nat(A),Aa)
=> ( gcd_Gcd_nat(Aa) = one_one_nat ) ) ) ).
% Gcd_eq_1_I
tff(fact_5899_Gcd__eq__1__I,axiom,
! [A: int,Aa: set_int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
=> ( aa_set_int_o(member_int(A),Aa)
=> ( gcd_Gcd_int(Aa) = one_one_int ) ) ) ).
% Gcd_eq_1_I
tff(fact_5900_Ints__odd__nonzero,axiom,
! [A: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(A),ring_11222124179247155820nteger)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),A) != zero_z3403309356797280102nteger ) ) ).
% Ints_odd_nonzero
tff(fact_5901_Ints__odd__nonzero,axiom,
! [A: rat] :
( member_rat(A,ring_1_Ints_rat)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),A)),A) != zero_zero_rat ) ) ).
% Ints_odd_nonzero
tff(fact_5902_Ints__odd__nonzero,axiom,
! [A: int] :
( aa_set_int_o(member_int(A),ring_1_Ints_int)
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),A) != zero_zero_int ) ) ).
% Ints_odd_nonzero
tff(fact_5903_quotient__of__denom__pos,axiom,
! [R2: rat,P5: int,Q4: int] :
( ( quotient_of(R2) = aa_int7462565939943104719nt_int(product_Pair_int_int(P5),Q4) )
=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q4) ) ).
% quotient_of_denom_pos
tff(fact_5904_Ints__odd__less__0,axiom,
! [A: rat] :
( member_rat(A,ring_1_Ints_rat)
=> ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),A)),A)),zero_zero_rat)
<=> aa_rat_o(ord_less_rat(A),zero_zero_rat) ) ) ).
% Ints_odd_less_0
tff(fact_5905_Ints__odd__less__0,axiom,
! [A: int] :
( aa_set_int_o(member_int(A),ring_1_Ints_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),A)),zero_zero_int)
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).
% Ints_odd_less_0
tff(fact_5906_Ints__odd__less__0,axiom,
! [A: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(A),ring_11222124179247155820nteger)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),A)),zero_z3403309356797280102nteger)
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ) ).
% Ints_odd_less_0
tff(fact_5907_Ints__nonzero__abs__ge1,axiom,
! [X: rat] :
( member_rat(X,ring_1_Ints_rat)
=> ( ( X != zero_zero_rat )
=> aa_rat_o(ord_less_eq_rat(one_one_rat),abs_abs_rat(X)) ) ) ).
% Ints_nonzero_abs_ge1
tff(fact_5908_Ints__nonzero__abs__ge1,axiom,
! [X: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(X),ring_11222124179247155820nteger)
=> ( ( X != zero_z3403309356797280102nteger )
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)) ) ) ).
% Ints_nonzero_abs_ge1
tff(fact_5909_Ints__nonzero__abs__ge1,axiom,
! [X: int] :
( aa_set_int_o(member_int(X),ring_1_Ints_int)
=> ( ( X != zero_zero_int )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(abs_abs_int,X)) ) ) ).
% Ints_nonzero_abs_ge1
tff(fact_5910_Ints__nonzero__abs__less1,axiom,
! [X: rat] :
( member_rat(X,ring_1_Ints_rat)
=> ( aa_rat_o(ord_less_rat(abs_abs_rat(X)),one_one_rat)
=> ( X = zero_zero_rat ) ) ) ).
% Ints_nonzero_abs_less1
tff(fact_5911_Ints__nonzero__abs__less1,axiom,
! [X: int] :
( aa_set_int_o(member_int(X),ring_1_Ints_int)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int)
=> ( X = zero_zero_int ) ) ) ).
% Ints_nonzero_abs_less1
tff(fact_5912_Ints__nonzero__abs__less1,axiom,
! [X: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(X),ring_11222124179247155820nteger)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)),one_one_Code_integer)
=> ( X = zero_z3403309356797280102nteger ) ) ) ).
% Ints_nonzero_abs_less1
tff(fact_5913_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 )
<=> aa_rat_o(ord_less_rat(abs_abs_rat(minus_minus_rat(X,Y))),one_one_rat) ) ) ) ).
% Ints_eq_abs_less1
tff(fact_5914_Ints__eq__abs__less1,axiom,
! [X: int,Y: int] :
( aa_set_int_o(member_int(X),ring_1_Ints_int)
=> ( aa_set_int_o(member_int(Y),ring_1_Ints_int)
=> ( ( X = Y )
<=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y))),one_one_int) ) ) ) ).
% Ints_eq_abs_less1
tff(fact_5915_Ints__eq__abs__less1,axiom,
! [X: code_integer,Y: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(X),ring_11222124179247155820nteger)
=> ( aa_set5643387332196575111eger_o(member_Code_integer(Y),ring_11222124179247155820nteger)
=> ( ( X = Y )
<=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),Y))),one_one_Code_integer) ) ) ) ).
% Ints_eq_abs_less1
tff(fact_5916_rat__sgn__code,axiom,
! [P5: rat] : quotient_of(sgn_sgn_rat(P5)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(sgn_sgn_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,quotient_of(P5)))),one_one_int) ).
% rat_sgn_code
tff(fact_5917_frac__neg,axiom,
! [X: rat] :
archimedean_frac_rat(aa_rat_rat(uminus_uminus_rat,X)) = $ite(member_rat(X,ring_1_Ints_rat),zero_zero_rat,minus_minus_rat(one_one_rat,archimedean_frac_rat(X))) ).
% frac_neg
tff(fact_5918_quotient__of__int,axiom,
! [A: int] : quotient_of(aa_int_rat(of_int,A)) = aa_int7462565939943104719nt_int(product_Pair_int_int(A),one_one_int) ).
% quotient_of_int
tff(fact_5919_Frct__code__post_I4_J,axiom,
! [K: num] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(numeral_numeral_int(K)),one_one_int)) = numeral_numeral_rat(K) ).
% Frct_code_post(4)
tff(fact_5920_Frct__code__post_I5_J,axiom,
! [K: num] : frct(aa_int7462565939943104719nt_int(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)
tff(fact_5921_Frct__code__post_I6_J,axiom,
! [K: num,L: num] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(numeral_numeral_int(K)),numeral_numeral_int(L))) = divide_divide_rat(numeral_numeral_rat(K),numeral_numeral_rat(L)) ).
% Frct_code_post(6)
tff(fact_5922_Frct__code__post_I8_J,axiom,
! [A: int,B: int] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(A),aa_int_int(uminus_uminus_int,B))) = aa_rat_rat(uminus_uminus_rat,frct(aa_int7462565939943104719nt_int(product_Pair_int_int(A),B))) ).
% Frct_code_post(8)
tff(fact_5923_Frct__code__post_I7_J,axiom,
! [A: int,B: int] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,A)),B)) = aa_rat_rat(uminus_uminus_rat,frct(aa_int7462565939943104719nt_int(product_Pair_int_int(A),B))) ).
% Frct_code_post(7)
tff(fact_5924_numeral__num__of__nat__unfold,axiom,
! [N: nat] :
numeral_numeral_nat(num_of_nat(N)) = $ite(N = zero_zero_nat,one_one_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)) ).
% numeral_num_of_nat_unfold
tff(fact_5925_numeral__num__of__nat__unfold,axiom,
! [N: nat] :
numeral_numeral_rat(num_of_nat(N)) = $ite(N = zero_zero_nat,one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) ).
% numeral_num_of_nat_unfold
tff(fact_5926_numeral__num__of__nat__unfold,axiom,
! [N: nat] :
numera6620942414471956472nteger(num_of_nat(N)) = $ite(N = zero_zero_nat,one_one_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).
% numeral_num_of_nat_unfold
tff(fact_5927_numeral__num__of__nat__unfold,axiom,
! [N: nat] :
numera5444537566228673987atural(num_of_nat(N)) = $ite(N = zero_zero_nat,one_one_Code_natural,aa_nat_Code_natural(semiri3763490453095760265atural,N)) ).
% numeral_num_of_nat_unfold
tff(fact_5928_numeral__num__of__nat__unfold,axiom,
! [N: nat] :
numeral_numeral_int(num_of_nat(N)) = $ite(N = zero_zero_nat,one_one_int,aa_nat_int(semiri1314217659103216013at_int,N)) ).
% numeral_num_of_nat_unfold
tff(fact_5929_Frct__code__post_I1_J,axiom,
! [A: int] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),A)) = zero_zero_rat ).
% Frct_code_post(1)
tff(fact_5930_Frct__code__post_I2_J,axiom,
! [A: int] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(A),zero_zero_int)) = zero_zero_rat ).
% Frct_code_post(2)
tff(fact_5931_Frct__code__post_I3_J,axiom,
frct(aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)) = one_one_rat ).
% Frct_code_post(3)
tff(fact_5932_zero__rat__def,axiom,
zero_zero_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int)) ).
% zero_rat_def
tff(fact_5933_zero__rat_Otransfer,axiom,
aa_rat_o(aa_Pro2298929076809959388_rat_o(pcr_rat,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int)),zero_zero_rat) ).
% zero_rat.transfer
tff(fact_5934_of__rat__neg__one,axiom,
field_2639924705303425560at_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).
% of_rat_neg_one
tff(fact_5935_simp__from__to,axiom,
! [I: int,J: int] :
set_or1266510415728281911st_int(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),bot_bot_set_int,aa_set_int_set_int(insert_int(I),set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J))) ).
% simp_from_to
tff(fact_5936_one__int_Otransfer,axiom,
aa_int_o(aa_Pro2912763917461432128_int_o(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),one_one_int) ).
% one_int.transfer
tff(fact_5937_card__Un__disjoint,axiom,
! [Aa: set_Pr4329608150637261639at_nat,Ba: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,Aa)
=> ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,Ba)
=> ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Aa),Ba) = bot_bo228742789529271731at_nat )
=> ( finite3771342082235030671at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Aa),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite3771342082235030671at_nat(Aa)),finite3771342082235030671at_nat(Ba)) ) ) ) ) ).
% card_Un_disjoint
tff(fact_5938_card__Un__disjoint,axiom,
! [Aa: set_Pr1261947904930325089at_nat,Ba: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Aa)
=> ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Ba)
=> ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Aa),Ba) = bot_bo2099793752762293965at_nat )
=> ( finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Aa),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite711546835091564841at_nat(Aa)),finite711546835091564841at_nat(Ba)) ) ) ) ) ).
% card_Un_disjoint
tff(fact_5939_card__Un__disjoint,axiom,
! [Aa: set_o,Ba: set_o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( aa_set_o_o(finite_finite_o,Ba)
=> ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,Aa),Ba) = bot_bot_set_o )
=> ( finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Aa),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_o(Aa)),finite_card_o(Ba)) ) ) ) ) ).
% card_Un_disjoint
tff(fact_5940_card__Un__disjoint,axiom,
! [Aa: set_nat,Ba: set_nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( aa_set_nat_o(finite_finite_nat,Ba)
=> ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Aa),Ba) = bot_bot_set_nat )
=> ( finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Aa),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_nat(Aa)),finite_card_nat(Ba)) ) ) ) ) ).
% card_Un_disjoint
tff(fact_5941_card__Un__disjoint,axiom,
! [Aa: set_int,Ba: set_int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( aa_set_int_o(finite_finite_int,Ba)
=> ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Aa),Ba) = bot_bot_set_int )
=> ( finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Aa),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_int(Aa)),finite_card_int(Ba)) ) ) ) ) ).
% card_Un_disjoint
tff(fact_5942_card__Diff1__less,axiom,
! [Aa: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,Aa)
=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite3771342082235030671at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat)))),finite3771342082235030671at_nat(Aa)) ) ) ).
% card_Diff1_less
tff(fact_5943_card__Diff1__less,axiom,
! [Aa: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(Aa,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)))),finite5795495896937551127_nat_o(Aa)) ) ) ).
% card_Diff1_less
tff(fact_5944_card__Diff1__less,axiom,
! [Aa: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Aa)
=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite711546835091564841at_nat(minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat)))),finite711546835091564841at_nat(Aa)) ) ) ).
% card_Diff1_less
tff(fact_5945_card__Diff1__less,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( aa_set_o_o(member_o((X)),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_o(minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o)))),finite_card_o(Aa)) ) ) ).
% card_Diff1_less
tff(fact_5946_card__Diff1__less,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( aa_set_int_o(member_int(X),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int)))),finite_card_int(Aa)) ) ) ).
% card_Diff1_less
tff(fact_5947_card__Diff1__less,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( aa_set_nat_o(member_nat(X),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))),finite_card_nat(Aa)) ) ) ).
% card_Diff1_less
tff(fact_5948_atLeastatMost__empty__iff2,axiom,
! [A: $o,B: $o] :
( ( bot_bot_set_o = set_or8904488021354931149Most_o((A),(B)) )
<=> ~ aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(B)) ) ).
% atLeastatMost_empty_iff2
tff(fact_5949_atLeastatMost__empty__iff2,axiom,
! [A: code_integer,B: code_integer] :
( ( bot_bo3990330152332043303nteger = set_or189985376899183464nteger(A,B) )
<=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).
% atLeastatMost_empty_iff2
tff(fact_5950_atLeastatMost__empty__iff2,axiom,
! [A: assn,B: assn] :
( ( bot_bot_set_assn = set_or7959216805967363635t_assn(A,B) )
<=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B) ) ).
% atLeastatMost_empty_iff2
tff(fact_5951_atLeastatMost__empty__iff2,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A,B) )
<=> ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ).
% atLeastatMost_empty_iff2
tff(fact_5952_atLeastatMost__empty__iff2,axiom,
! [A: int,B: int] :
( ( bot_bot_set_int = set_or1266510415728281911st_int(A,B) )
<=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% atLeastatMost_empty_iff2
tff(fact_5953_atLeastatMost__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat = aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) )
<=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).
% atLeastatMost_empty_iff2
tff(fact_5954_atLeastatMost__empty__iff,axiom,
! [A: $o,B: $o] :
( ( set_or8904488021354931149Most_o((A),(B)) = bot_bot_set_o )
<=> ~ aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(B)) ) ).
% atLeastatMost_empty_iff
tff(fact_5955_atLeastatMost__empty__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( set_or189985376899183464nteger(A,B) = bot_bo3990330152332043303nteger )
<=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).
% atLeastatMost_empty_iff
tff(fact_5956_atLeastatMost__empty__iff,axiom,
! [A: assn,B: assn] :
( ( set_or7959216805967363635t_assn(A,B) = bot_bot_set_assn )
<=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B) ) ).
% atLeastatMost_empty_iff
tff(fact_5957_atLeastatMost__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( set_or4548717258645045905et_nat(A,B) = bot_bot_set_set_nat )
<=> ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ).
% atLeastatMost_empty_iff
tff(fact_5958_atLeastatMost__empty__iff,axiom,
! [A: int,B: int] :
( ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int )
<=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).
% atLeastatMost_empty_iff
tff(fact_5959_atLeastatMost__empty__iff,axiom,
! [A: nat,B: nat] :
( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat )
<=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).
% atLeastatMost_empty_iff
tff(fact_5960_atLeastatMost__empty,axiom,
! [B: $o,A: $o] :
( aa_o_o(aa_o_fun_o_o(ord_less_o,(B)),(A))
=> ( set_or8904488021354931149Most_o((A),(B)) = bot_bot_set_o ) ) ).
% atLeastatMost_empty
tff(fact_5961_atLeastatMost__empty,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
=> ( set_or7959216805967363635t_assn(A,B) = bot_bot_set_assn ) ) ).
% atLeastatMost_empty
tff(fact_5962_atLeastatMost__empty,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
=> ( set_or189985376899183464nteger(A,B) = bot_bo3990330152332043303nteger ) ) ).
% atLeastatMost_empty
tff(fact_5963_atLeastatMost__empty,axiom,
! [B: code_natural,A: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),A)
=> ( set_or8236952565510676787atural(A,B) = bot_bo8448779972843232370atural ) ) ).
% atLeastatMost_empty
tff(fact_5964_atLeastatMost__empty,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
=> ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int ) ) ).
% atLeastatMost_empty
tff(fact_5965_atLeastatMost__empty,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
=> ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat ) ) ).
% atLeastatMost_empty
tff(fact_5966_atLeastAtMost__singleton,axiom,
! [A: $o] : set_or8904488021354931149Most_o((A),(A)) = aa_set_o_set_o(insert_o((A)),bot_bot_set_o) ).
% atLeastAtMost_singleton
tff(fact_5967_atLeastAtMost__singleton,axiom,
! [A: int] : set_or1266510415728281911st_int(A,A) = aa_set_int_set_int(insert_int(A),bot_bot_set_int) ).
% atLeastAtMost_singleton
tff(fact_5968_atLeastAtMost__singleton,axiom,
! [A: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(A),A) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ).
% atLeastAtMost_singleton
tff(fact_5969_atLeastAtMost__singleton__iff,axiom,
! [A: $o,B: $o,C: $o] :
( ( set_or8904488021354931149Most_o((A),(B)) = aa_set_o_set_o(insert_o((C)),bot_bot_set_o) )
<=> ( ( (A)
<=> (B) )
& ( (B)
<=> (C) ) ) ) ).
% atLeastAtMost_singleton_iff
tff(fact_5970_atLeastAtMost__singleton__iff,axiom,
! [A: int,B: int,C: int] :
( ( set_or1266510415728281911st_int(A,B) = aa_set_int_set_int(insert_int(C),bot_bot_set_int) )
<=> ( ( A = B )
& ( B = C ) ) ) ).
% atLeastAtMost_singleton_iff
tff(fact_5971_atLeastAtMost__singleton__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = aa_set_nat_set_nat(insert_nat(C),bot_bot_set_nat) )
<=> ( ( A = B )
& ( B = C ) ) ) ).
% atLeastAtMost_singleton_iff
tff(fact_5972_of__rat__1,axiom,
field_2639924705303425560at_rat(one_one_rat) = one_one_rat ).
% of_rat_1
tff(fact_5973_of__rat__eq__1__iff,axiom,
! [A: rat] :
( ( field_2639924705303425560at_rat(A) = one_one_rat )
<=> ( A = one_one_rat ) ) ).
% of_rat_eq_1_iff
tff(fact_5974_one__eq__of__rat__iff,axiom,
! [A: rat] :
( ( one_one_rat = field_2639924705303425560at_rat(A) )
<=> ( one_one_rat = A ) ) ).
% one_eq_of_rat_iff
tff(fact_5975_card__0__eq,axiom,
! [Aa: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Aa)
=> ( ( finite711546835091564841at_nat(Aa) = zero_zero_nat )
<=> ( Aa = bot_bo2099793752762293965at_nat ) ) ) ).
% card_0_eq
tff(fact_5976_card__0__eq,axiom,
! [Aa: set_o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( finite_card_o(Aa) = zero_zero_nat )
<=> ( Aa = bot_bot_set_o ) ) ) ).
% card_0_eq
tff(fact_5977_card__0__eq,axiom,
! [Aa: set_nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( finite_card_nat(Aa) = zero_zero_nat )
<=> ( Aa = bot_bot_set_nat ) ) ) ).
% card_0_eq
tff(fact_5978_card__0__eq,axiom,
! [Aa: set_int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( finite_card_int(Aa) = zero_zero_nat )
<=> ( Aa = bot_bot_set_int ) ) ) ).
% card_0_eq
tff(fact_5979_of__rat__le__1__iff,axiom,
! [R2: rat] :
( aa_rat_o(ord_less_eq_rat(field_2639924705303425560at_rat(R2)),one_one_rat)
<=> aa_rat_o(ord_less_eq_rat(R2),one_one_rat) ) ).
% of_rat_le_1_iff
tff(fact_5980_one__le__of__rat__iff,axiom,
! [R2: rat] :
( aa_rat_o(ord_less_eq_rat(one_one_rat),field_2639924705303425560at_rat(R2))
<=> aa_rat_o(ord_less_eq_rat(one_one_rat),R2) ) ).
% one_le_of_rat_iff
tff(fact_5981_of__rat__less__1__iff,axiom,
! [R2: rat] :
( aa_rat_o(ord_less_rat(field_2639924705303425560at_rat(R2)),one_one_rat)
<=> aa_rat_o(ord_less_rat(R2),one_one_rat) ) ).
% of_rat_less_1_iff
tff(fact_5982_one__less__of__rat__iff,axiom,
! [R2: rat] :
( aa_rat_o(ord_less_rat(one_one_rat),field_2639924705303425560at_rat(R2))
<=> aa_rat_o(ord_less_rat(one_one_rat),R2) ) ).
% one_less_of_rat_iff
tff(fact_5983_of__rat__mult,axiom,
! [A: rat,B: rat] : field_2639924705303425560at_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,field_2639924705303425560at_rat(A)),field_2639924705303425560at_rat(B)) ).
% of_rat_mult
tff(fact_5984_finite_OemptyI,axiom,
aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,bot_bo2099793752762293965at_nat) ).
% finite.emptyI
tff(fact_5985_finite_OemptyI,axiom,
aa_set_o_o(finite_finite_o,bot_bot_set_o) ).
% finite.emptyI
tff(fact_5986_finite_OemptyI,axiom,
aa_set_nat_o(finite_finite_nat,bot_bot_set_nat) ).
% finite.emptyI
tff(fact_5987_finite_OemptyI,axiom,
aa_set_int_o(finite_finite_int,bot_bot_set_int) ).
% finite.emptyI
tff(fact_5988_infinite__imp__nonempty,axiom,
! [S: set_Pr1261947904930325089at_nat] :
( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
=> ( S != bot_bo2099793752762293965at_nat ) ) ).
% infinite_imp_nonempty
tff(fact_5989_infinite__imp__nonempty,axiom,
! [S: set_o] :
( ~ aa_set_o_o(finite_finite_o,S)
=> ( S != bot_bot_set_o ) ) ).
% infinite_imp_nonempty
tff(fact_5990_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ aa_set_nat_o(finite_finite_nat,S)
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
tff(fact_5991_infinite__imp__nonempty,axiom,
! [S: set_int] :
( ~ aa_set_int_o(finite_finite_int,S)
=> ( S != bot_bot_set_int ) ) ).
% infinite_imp_nonempty
tff(fact_5992_atLeastAtMost__singleton_H,axiom,
! [A: $o,B: $o] :
( ( (A)
<=> (B) )
=> ( set_or8904488021354931149Most_o((A),(B)) = aa_set_o_set_o(insert_o((A)),bot_bot_set_o) ) ) ).
% atLeastAtMost_singleton'
tff(fact_5993_atLeastAtMost__singleton_H,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( set_or1266510415728281911st_int(A,B) = aa_set_int_set_int(insert_int(A),bot_bot_set_int) ) ) ).
% atLeastAtMost_singleton'
tff(fact_5994_atLeastAtMost__singleton_H,axiom,
! [A: nat,B: nat] :
( ( A = B )
=> ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ) ) ).
% atLeastAtMost_singleton'
tff(fact_5995_finite__has__minimal,axiom,
! [Aa: set_o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> ? [X2: $o] :
( aa_set_o_o(member_o((X2)),Aa)
& ! [Xa3: $o] :
( aa_set_o_o(member_o((Xa3)),Aa)
=> ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa3)),(X2))
=> ( (X2)
<=> (Xa3) ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_5996_finite__has__minimal,axiom,
! [Aa: set_Code_integer] :
( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Aa)
=> ( ( Aa != bot_bo3990330152332043303nteger )
=> ? [X2: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(X2),Aa)
& ! [Xa3: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),Aa)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa3),X2)
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_5997_finite__has__minimal,axiom,
! [Aa: set_assn] :
( finite_finite_assn(Aa)
=> ( ( Aa != bot_bot_set_assn )
=> ? [X2: assn] :
( member_assn(X2,Aa)
& ! [Xa3: assn] :
( member_assn(Xa3,Aa)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Xa3),X2)
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_5998_finite__has__minimal,axiom,
! [Aa: set_set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ( Aa != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( member_set_nat(X2,Aa)
& ! [Xa3: set_nat] :
( member_set_nat(Xa3,Aa)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa3),X2)
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_5999_finite__has__minimal,axiom,
! [Aa: set_nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ? [X2: nat] :
( aa_set_nat_o(member_nat(X2),Aa)
& ! [Xa3: nat] :
( aa_set_nat_o(member_nat(Xa3),Aa)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa3),X2)
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_6000_finite__has__minimal,axiom,
! [Aa: set_int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> ? [X2: int] :
( aa_set_int_o(member_int(X2),Aa)
& ! [Xa3: int] :
( aa_set_int_o(member_int(Xa3),Aa)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa3),X2)
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_6001_finite__has__maximal,axiom,
! [Aa: set_o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> ? [X2: $o] :
( aa_set_o_o(member_o((X2)),Aa)
& ! [Xa3: $o] :
( aa_set_o_o(member_o((Xa3)),Aa)
=> ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X2)),(Xa3))
=> ( (X2)
<=> (Xa3) ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_6002_finite__has__maximal,axiom,
! [Aa: set_Code_integer] :
( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Aa)
=> ( ( Aa != bot_bo3990330152332043303nteger )
=> ? [X2: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(X2),Aa)
& ! [Xa3: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),Aa)
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Xa3)
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_6003_finite__has__maximal,axiom,
! [Aa: set_assn] :
( finite_finite_assn(Aa)
=> ( ( Aa != bot_bot_set_assn )
=> ? [X2: assn] :
( member_assn(X2,Aa)
& ! [Xa3: assn] :
( member_assn(Xa3,Aa)
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Xa3)
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_6004_finite__has__maximal,axiom,
! [Aa: set_set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ( Aa != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( member_set_nat(X2,Aa)
& ! [Xa3: set_nat] :
( member_set_nat(Xa3,Aa)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),Xa3)
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_6005_finite__has__maximal,axiom,
! [Aa: set_nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ? [X2: nat] :
( aa_set_nat_o(member_nat(X2),Aa)
& ! [Xa3: nat] :
( aa_set_nat_o(member_nat(Xa3),Aa)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Xa3)
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_6006_finite__has__maximal,axiom,
! [Aa: set_int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> ? [X2: int] :
( aa_set_int_o(member_int(X2),Aa)
& ! [Xa3: int] :
( aa_set_int_o(member_int(Xa3),Aa)
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Xa3)
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_6007_finite_Ocases,axiom,
! [A: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A)
=> ( ( A != bot_bo228742789529271731at_nat )
=> ~ ! [A5: set_Pr4329608150637261639at_nat] :
( ? [A3: produc3843707927480180839at_nat] : A = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A3),A5)
=> ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A5) ) ) ) ).
% finite.cases
tff(fact_6008_finite_Ocases,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A)
=> ( ( A != bot_bo2099793752762293965at_nat )
=> ~ ! [A5: set_Pr1261947904930325089at_nat] :
( ? [A3: product_prod_nat_nat] : A = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A3),A5)
=> ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A5) ) ) ) ).
% finite.cases
tff(fact_6009_finite_Ocases,axiom,
! [A: set_o] :
( aa_set_o_o(finite_finite_o,A)
=> ( ( A != bot_bot_set_o )
=> ~ ! [A5: set_o] :
( ? [A3: $o] : A = aa_set_o_set_o(insert_o((A3)),A5)
=> ~ aa_set_o_o(finite_finite_o,A5) ) ) ) ).
% finite.cases
tff(fact_6010_finite_Ocases,axiom,
! [A: set_nat] :
( aa_set_nat_o(finite_finite_nat,A)
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A5: set_nat] :
( ? [A3: nat] : A = aa_set_nat_set_nat(insert_nat(A3),A5)
=> ~ aa_set_nat_o(finite_finite_nat,A5) ) ) ) ).
% finite.cases
tff(fact_6011_finite_Ocases,axiom,
! [A: set_int] :
( aa_set_int_o(finite_finite_int,A)
=> ( ( A != bot_bot_set_int )
=> ~ ! [A5: set_int] :
( ? [A3: int] : A = aa_set_int_set_int(insert_int(A3),A5)
=> ~ aa_set_int_o(finite_finite_int,A5) ) ) ) ).
% finite.cases
tff(fact_6012_finite_Osimps,axiom,
! [A: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A)
<=> ( ( A = bot_bo228742789529271731at_nat )
| ? [A6: set_Pr4329608150637261639at_nat,A4: produc3843707927480180839at_nat] :
( ( A = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A4),A6) )
& aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A6) ) ) ) ).
% finite.simps
tff(fact_6013_finite_Osimps,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A)
<=> ( ( A = bot_bo2099793752762293965at_nat )
| ? [A6: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat] :
( ( A = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A4),A6) )
& aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A6) ) ) ) ).
% finite.simps
tff(fact_6014_finite_Osimps,axiom,
! [A: set_o] :
( aa_set_o_o(finite_finite_o,A)
<=> ( ( A = bot_bot_set_o )
| ? [A6: set_o,A4: $o] :
( ( A = aa_set_o_set_o(insert_o((A4)),A6) )
& aa_set_o_o(finite_finite_o,A6) ) ) ) ).
% finite.simps
tff(fact_6015_finite_Osimps,axiom,
! [A: set_nat] :
( aa_set_nat_o(finite_finite_nat,A)
<=> ( ( A = bot_bot_set_nat )
| ? [A6: set_nat,A4: nat] :
( ( A = aa_set_nat_set_nat(insert_nat(A4),A6) )
& aa_set_nat_o(finite_finite_nat,A6) ) ) ) ).
% finite.simps
tff(fact_6016_finite_Osimps,axiom,
! [A: set_int] :
( aa_set_int_o(finite_finite_int,A)
<=> ( ( A = bot_bot_set_int )
| ? [A6: set_int,A4: int] :
( ( A = aa_set_int_set_int(insert_int(A4),A6) )
& aa_set_int_o(finite_finite_int,A6) ) ) ) ).
% finite.simps
tff(fact_6017_finite__induct,axiom,
! [F3: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
=> ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
=> ( ! [X2: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X2),F4)
=> ( aa_set5844364881195198285_nat_o(P,F4)
=> aa_set5844364881195198285_nat_o(P,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X2),F4)) ) ) )
=> aa_set5844364881195198285_nat_o(P,F3) ) ) ) ).
% finite_induct
tff(fact_6018_finite__induct,axiom,
! [F3: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
=> ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
=> ( ! [X2: fun_Pr6460944699146269935_nat_o,F4: set_fu3555266933746557007_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F4)
=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),F4)
=> ( aa_set5091322549519198405at_o_o(P,F4)
=> aa_set5091322549519198405at_o_o(P,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X2),F4)) ) ) )
=> aa_set5091322549519198405at_o_o(P,F3) ) ) ) ).
% finite_induct
tff(fact_6019_finite__induct,axiom,
! [F3: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
=> ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
=> ( ! [X2: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),F4)
=> ( aa_set3702741264012733747_nat_o(P,F4)
=> aa_set3702741264012733747_nat_o(P,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X2),F4)) ) ) )
=> aa_set3702741264012733747_nat_o(P,F3) ) ) ) ).
% finite_induct
tff(fact_6020_finite__induct,axiom,
! [F3: set_o,P: fun_set_o_o] :
( aa_set_o_o(finite_finite_o,F3)
=> ( aa_set_o_o(P,bot_bot_set_o)
=> ( ! [X2: $o,F4: set_o] :
( aa_set_o_o(finite_finite_o,F4)
=> ( ~ aa_set_o_o(member_o((X2)),F4)
=> ( aa_set_o_o(P,F4)
=> aa_set_o_o(P,aa_set_o_set_o(insert_o((X2)),F4)) ) ) )
=> aa_set_o_o(P,F3) ) ) ) ).
% finite_induct
tff(fact_6021_finite__induct,axiom,
! [F3: set_nat,P: fun_set_nat_o] :
( aa_set_nat_o(finite_finite_nat,F3)
=> ( aa_set_nat_o(P,bot_bot_set_nat)
=> ( ! [X2: nat,F4: set_nat] :
( aa_set_nat_o(finite_finite_nat,F4)
=> ( ~ aa_set_nat_o(member_nat(X2),F4)
=> ( aa_set_nat_o(P,F4)
=> aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(X2),F4)) ) ) )
=> aa_set_nat_o(P,F3) ) ) ) ).
% finite_induct
tff(fact_6022_finite__induct,axiom,
! [F3: set_int,P: fun_set_int_o] :
( aa_set_int_o(finite_finite_int,F3)
=> ( aa_set_int_o(P,bot_bot_set_int)
=> ( ! [X2: int,F4: set_int] :
( aa_set_int_o(finite_finite_int,F4)
=> ( ~ aa_set_int_o(member_int(X2),F4)
=> ( aa_set_int_o(P,F4)
=> aa_set_int_o(P,aa_set_int_set_int(insert_int(X2),F4)) ) ) )
=> aa_set_int_o(P,F3) ) ) ) ).
% finite_induct
tff(fact_6023_finite__ne__induct,axiom,
! [F3: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
=> ( ( F3 != bot_bo228742789529271731at_nat )
=> ( ! [X2: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(P,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X2),bot_bo228742789529271731at_nat))
=> ( ! [X2: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
=> ( ( F4 != bot_bo228742789529271731at_nat )
=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X2),F4)
=> ( aa_set5844364881195198285_nat_o(P,F4)
=> aa_set5844364881195198285_nat_o(P,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X2),F4)) ) ) ) )
=> aa_set5844364881195198285_nat_o(P,F3) ) ) ) ) ).
% finite_ne_induct
tff(fact_6024_finite__ne__induct,axiom,
! [F3: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
=> ( ( F3 != bot_bo8632255151191677115_nat_o )
=> ( ! [X2: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(P,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X2),bot_bo8632255151191677115_nat_o))
=> ( ! [X2: fun_Pr6460944699146269935_nat_o,F4: set_fu3555266933746557007_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F4)
=> ( ( F4 != bot_bo8632255151191677115_nat_o )
=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),F4)
=> ( aa_set5091322549519198405at_o_o(P,F4)
=> aa_set5091322549519198405at_o_o(P,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X2),F4)) ) ) ) )
=> aa_set5091322549519198405at_o_o(P,F3) ) ) ) ) ).
% finite_ne_induct
tff(fact_6025_finite__ne__induct,axiom,
! [F3: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
=> ( ( F3 != bot_bo2099793752762293965at_nat )
=> ( ! [X2: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(P,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X2),bot_bo2099793752762293965at_nat))
=> ( ! [X2: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
=> ( ( F4 != bot_bo2099793752762293965at_nat )
=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),F4)
=> ( aa_set3702741264012733747_nat_o(P,F4)
=> aa_set3702741264012733747_nat_o(P,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X2),F4)) ) ) ) )
=> aa_set3702741264012733747_nat_o(P,F3) ) ) ) ) ).
% finite_ne_induct
tff(fact_6026_finite__ne__induct,axiom,
! [F3: set_o,P: fun_set_o_o] :
( aa_set_o_o(finite_finite_o,F3)
=> ( ( F3 != bot_bot_set_o )
=> ( ! [X2: $o] : aa_set_o_o(P,aa_set_o_set_o(insert_o((X2)),bot_bot_set_o))
=> ( ! [X2: $o,F4: set_o] :
( aa_set_o_o(finite_finite_o,F4)
=> ( ( F4 != bot_bot_set_o )
=> ( ~ aa_set_o_o(member_o((X2)),F4)
=> ( aa_set_o_o(P,F4)
=> aa_set_o_o(P,aa_set_o_set_o(insert_o((X2)),F4)) ) ) ) )
=> aa_set_o_o(P,F3) ) ) ) ) ).
% finite_ne_induct
tff(fact_6027_finite__ne__induct,axiom,
! [F3: set_nat,P: fun_set_nat_o] :
( aa_set_nat_o(finite_finite_nat,F3)
=> ( ( F3 != bot_bot_set_nat )
=> ( ! [X2: nat] : aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(X2),bot_bot_set_nat))
=> ( ! [X2: nat,F4: set_nat] :
( aa_set_nat_o(finite_finite_nat,F4)
=> ( ( F4 != bot_bot_set_nat )
=> ( ~ aa_set_nat_o(member_nat(X2),F4)
=> ( aa_set_nat_o(P,F4)
=> aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(X2),F4)) ) ) ) )
=> aa_set_nat_o(P,F3) ) ) ) ) ).
% finite_ne_induct
tff(fact_6028_finite__ne__induct,axiom,
! [F3: set_int,P: fun_set_int_o] :
( aa_set_int_o(finite_finite_int,F3)
=> ( ( F3 != bot_bot_set_int )
=> ( ! [X2: int] : aa_set_int_o(P,aa_set_int_set_int(insert_int(X2),bot_bot_set_int))
=> ( ! [X2: int,F4: set_int] :
( aa_set_int_o(finite_finite_int,F4)
=> ( ( F4 != bot_bot_set_int )
=> ( ~ aa_set_int_o(member_int(X2),F4)
=> ( aa_set_int_o(P,F4)
=> aa_set_int_o(P,aa_set_int_set_int(insert_int(X2),F4)) ) ) ) )
=> aa_set_int_o(P,F3) ) ) ) ) ).
% finite_ne_induct
tff(fact_6029_infinite__finite__induct,axiom,
! [P: fun_se7788542033685991124_nat_o,Aa: set_Pr4329608150637261639at_nat] :
( ! [A5: set_Pr4329608150637261639at_nat] :
( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A5)
=> aa_set5844364881195198285_nat_o(P,A5) )
=> ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
=> ( ! [X2: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X2),F4)
=> ( aa_set5844364881195198285_nat_o(P,F4)
=> aa_set5844364881195198285_nat_o(P,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X2),F4)) ) ) )
=> aa_set5844364881195198285_nat_o(P,Aa) ) ) ) ).
% infinite_finite_induct
tff(fact_6030_infinite__finite__induct,axiom,
! [P: fun_se7459991829846211660at_o_o,Aa: set_fu3555266933746557007_nat_o] :
( ! [A5: set_fu3555266933746557007_nat_o] :
( ~ aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A5)
=> aa_set5091322549519198405at_o_o(P,A5) )
=> ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
=> ( ! [X2: fun_Pr6460944699146269935_nat_o,F4: set_fu3555266933746557007_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F4)
=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),F4)
=> ( aa_set5091322549519198405at_o_o(P,F4)
=> aa_set5091322549519198405at_o_o(P,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X2),F4)) ) ) )
=> aa_set5091322549519198405at_o_o(P,Aa) ) ) ) ).
% infinite_finite_induct
tff(fact_6031_infinite__finite__induct,axiom,
! [P: fun_se4782206882984618170_nat_o,Aa: set_Pr1261947904930325089at_nat] :
( ! [A5: set_Pr1261947904930325089at_nat] :
( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A5)
=> aa_set3702741264012733747_nat_o(P,A5) )
=> ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
=> ( ! [X2: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),F4)
=> ( aa_set3702741264012733747_nat_o(P,F4)
=> aa_set3702741264012733747_nat_o(P,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X2),F4)) ) ) )
=> aa_set3702741264012733747_nat_o(P,Aa) ) ) ) ).
% infinite_finite_induct
tff(fact_6032_infinite__finite__induct,axiom,
! [P: fun_set_o_o,Aa: set_o] :
( ! [A5: set_o] :
( ~ aa_set_o_o(finite_finite_o,A5)
=> aa_set_o_o(P,A5) )
=> ( aa_set_o_o(P,bot_bot_set_o)
=> ( ! [X2: $o,F4: set_o] :
( aa_set_o_o(finite_finite_o,F4)
=> ( ~ aa_set_o_o(member_o((X2)),F4)
=> ( aa_set_o_o(P,F4)
=> aa_set_o_o(P,aa_set_o_set_o(insert_o((X2)),F4)) ) ) )
=> aa_set_o_o(P,Aa) ) ) ) ).
% infinite_finite_induct
tff(fact_6033_infinite__finite__induct,axiom,
! [P: fun_set_nat_o,Aa: set_nat] :
( ! [A5: set_nat] :
( ~ aa_set_nat_o(finite_finite_nat,A5)
=> aa_set_nat_o(P,A5) )
=> ( aa_set_nat_o(P,bot_bot_set_nat)
=> ( ! [X2: nat,F4: set_nat] :
( aa_set_nat_o(finite_finite_nat,F4)
=> ( ~ aa_set_nat_o(member_nat(X2),F4)
=> ( aa_set_nat_o(P,F4)
=> aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(X2),F4)) ) ) )
=> aa_set_nat_o(P,Aa) ) ) ) ).
% infinite_finite_induct
tff(fact_6034_infinite__finite__induct,axiom,
! [P: fun_set_int_o,Aa: set_int] :
( ! [A5: set_int] :
( ~ aa_set_int_o(finite_finite_int,A5)
=> aa_set_int_o(P,A5) )
=> ( aa_set_int_o(P,bot_bot_set_int)
=> ( ! [X2: int,F4: set_int] :
( aa_set_int_o(finite_finite_int,F4)
=> ( ~ aa_set_int_o(member_int(X2),F4)
=> ( aa_set_int_o(P,F4)
=> aa_set_int_o(P,aa_set_int_set_int(insert_int(X2),F4)) ) ) )
=> aa_set_int_o(P,Aa) ) ) ) ).
% infinite_finite_induct
tff(fact_6035_Gcd__remove0__nat,axiom,
! [M4: set_nat] :
( aa_set_nat_o(finite_finite_nat,M4)
=> ( gcd_Gcd_nat(M4) = gcd_Gcd_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,M4),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat))) ) ) ).
% Gcd_remove0_nat
tff(fact_6036_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( set_or2784603332384507286at_nat(X,Y) = top_to7629004291339433233at_nat )
<=> ( ( X = bot_bo2099793752762293965at_nat )
& ( Y = top_to4669805908274784177at_nat ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
tff(fact_6037_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: set_int,Y: set_int] :
( ( set_or370866239135849197et_int(X,Y) = top_top_set_set_int )
<=> ( ( X = bot_bot_set_int )
& ( Y = top_top_set_int ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
tff(fact_6038_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: assn,Y: assn] :
( ( set_or7959216805967363635t_assn(X,Y) = top_top_set_assn )
<=> ( ( X = bot_bot_assn )
& ( Y = top_top_assn ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
tff(fact_6039_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: set_o,Y: set_o] :
( ( set_or266616469461849773_set_o(X,Y) = top_top_set_set_o )
<=> ( ( X = bot_bot_set_o )
& ( Y = top_top_set_o ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
tff(fact_6040_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( set_or4548717258645045905et_nat(X,Y) = top_top_set_set_nat )
<=> ( ( X = bot_bot_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
tff(fact_6041_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( set_or269330187037242274t_unit(X,Y) = top_to1767297665138865437t_unit )
<=> ( ( X = bot_bo3957492148770167129t_unit )
& ( Y = top_to1996260823553986621t_unit ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
tff(fact_6042_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: $o,Y: $o] :
( ( set_or8904488021354931149Most_o((X),(Y)) = top_top_set_o )
<=> ( ( (X)
<=> bot_bot_o )
& ( (Y)
<=> top_top_o ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
tff(fact_6043_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: product_unit,Y: product_unit] :
( ( set_or5692540242118139586t_unit(X,Y) = top_to1996260823553986621t_unit )
<=> ( ( X = bot_bot_Product_unit )
& ( Y = top_top_Product_unit ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
tff(fact_6044_finite__subset__induct_H,axiom,
! [F3: set_Pr4329608150637261639at_nat,Aa: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(F3),Aa)
=> ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
=> ( ! [A3: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),Aa)
=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(F4),Aa)
=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),F4)
=> ( aa_set5844364881195198285_nat_o(P,F4)
=> aa_set5844364881195198285_nat_o(P,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A3),F4)) ) ) ) ) )
=> aa_set5844364881195198285_nat_o(P,F3) ) ) ) ) ).
% finite_subset_induct'
tff(fact_6045_finite__subset__induct_H,axiom,
! [F3: set_fu3555266933746557007_nat_o,Aa: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
=> ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(F3),Aa)
=> ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
=> ( ! [A3: fun_Pr6460944699146269935_nat_o,F4: set_fu3555266933746557007_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F4)
=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),Aa)
=> ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(F4),Aa)
=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),F4)
=> ( aa_set5091322549519198405at_o_o(P,F4)
=> aa_set5091322549519198405at_o_o(P,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A3),F4)) ) ) ) ) )
=> aa_set5091322549519198405at_o_o(P,F3) ) ) ) ) ).
% finite_subset_induct'
tff(fact_6046_finite__subset__induct_H,axiom,
! [F3: set_Pr1261947904930325089at_nat,Aa: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
=> ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(F3),Aa)
=> ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
=> ( ! [A3: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),Aa)
=> ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(F4),Aa)
=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),F4)
=> ( aa_set3702741264012733747_nat_o(P,F4)
=> aa_set3702741264012733747_nat_o(P,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A3),F4)) ) ) ) ) )
=> aa_set3702741264012733747_nat_o(P,F3) ) ) ) ) ).
% finite_subset_induct'
tff(fact_6047_finite__subset__induct_H,axiom,
! [F3: set_o,Aa: set_o,P: fun_set_o_o] :
( aa_set_o_o(finite_finite_o,F3)
=> ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,F3),Aa)
=> ( aa_set_o_o(P,bot_bot_set_o)
=> ( ! [A3: $o,F4: set_o] :
( aa_set_o_o(finite_finite_o,F4)
=> ( aa_set_o_o(member_o((A3)),Aa)
=> ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,F4),Aa)
=> ( ~ aa_set_o_o(member_o((A3)),F4)
=> ( aa_set_o_o(P,F4)
=> aa_set_o_o(P,aa_set_o_set_o(insert_o((A3)),F4)) ) ) ) ) )
=> aa_set_o_o(P,F3) ) ) ) ) ).
% finite_subset_induct'
tff(fact_6048_finite__subset__induct_H,axiom,
! [F3: set_int,Aa: set_int,P: fun_set_int_o] :
( aa_set_int_o(finite_finite_int,F3)
=> ( aa_set_int_o(ord_less_eq_set_int(F3),Aa)
=> ( aa_set_int_o(P,bot_bot_set_int)
=> ( ! [A3: int,F4: set_int] :
( aa_set_int_o(finite_finite_int,F4)
=> ( aa_set_int_o(member_int(A3),Aa)
=> ( aa_set_int_o(ord_less_eq_set_int(F4),Aa)
=> ( ~ aa_set_int_o(member_int(A3),F4)
=> ( aa_set_int_o(P,F4)
=> aa_set_int_o(P,aa_set_int_set_int(insert_int(A3),F4)) ) ) ) ) )
=> aa_set_int_o(P,F3) ) ) ) ) ).
% finite_subset_induct'
tff(fact_6049_finite__subset__induct_H,axiom,
! [F3: set_nat,Aa: set_nat,P: fun_set_nat_o] :
( aa_set_nat_o(finite_finite_nat,F3)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F3),Aa)
=> ( aa_set_nat_o(P,bot_bot_set_nat)
=> ( ! [A3: nat,F4: set_nat] :
( aa_set_nat_o(finite_finite_nat,F4)
=> ( aa_set_nat_o(member_nat(A3),Aa)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F4),Aa)
=> ( ~ aa_set_nat_o(member_nat(A3),F4)
=> ( aa_set_nat_o(P,F4)
=> aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(A3),F4)) ) ) ) ) )
=> aa_set_nat_o(P,F3) ) ) ) ) ).
% finite_subset_induct'
tff(fact_6050_finite__subset__induct,axiom,
! [F3: set_Pr4329608150637261639at_nat,Aa: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F3)
=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(F3),Aa)
=> ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
=> ( ! [A3: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),Aa)
=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),F4)
=> ( aa_set5844364881195198285_nat_o(P,F4)
=> aa_set5844364881195198285_nat_o(P,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A3),F4)) ) ) ) )
=> aa_set5844364881195198285_nat_o(P,F3) ) ) ) ) ).
% finite_subset_induct
tff(fact_6051_finite__subset__induct,axiom,
! [F3: set_fu3555266933746557007_nat_o,Aa: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
=> ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(F3),Aa)
=> ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
=> ( ! [A3: fun_Pr6460944699146269935_nat_o,F4: set_fu3555266933746557007_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F4)
=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),Aa)
=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),F4)
=> ( aa_set5091322549519198405at_o_o(P,F4)
=> aa_set5091322549519198405at_o_o(P,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A3),F4)) ) ) ) )
=> aa_set5091322549519198405at_o_o(P,F3) ) ) ) ) ).
% finite_subset_induct
tff(fact_6052_finite__subset__induct,axiom,
! [F3: set_Pr1261947904930325089at_nat,Aa: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
=> ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(F3),Aa)
=> ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
=> ( ! [A3: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),Aa)
=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),F4)
=> ( aa_set3702741264012733747_nat_o(P,F4)
=> aa_set3702741264012733747_nat_o(P,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A3),F4)) ) ) ) )
=> aa_set3702741264012733747_nat_o(P,F3) ) ) ) ) ).
% finite_subset_induct
tff(fact_6053_finite__subset__induct,axiom,
! [F3: set_o,Aa: set_o,P: fun_set_o_o] :
( aa_set_o_o(finite_finite_o,F3)
=> ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,F3),Aa)
=> ( aa_set_o_o(P,bot_bot_set_o)
=> ( ! [A3: $o,F4: set_o] :
( aa_set_o_o(finite_finite_o,F4)
=> ( aa_set_o_o(member_o((A3)),Aa)
=> ( ~ aa_set_o_o(member_o((A3)),F4)
=> ( aa_set_o_o(P,F4)
=> aa_set_o_o(P,aa_set_o_set_o(insert_o((A3)),F4)) ) ) ) )
=> aa_set_o_o(P,F3) ) ) ) ) ).
% finite_subset_induct
tff(fact_6054_finite__subset__induct,axiom,
! [F3: set_int,Aa: set_int,P: fun_set_int_o] :
( aa_set_int_o(finite_finite_int,F3)
=> ( aa_set_int_o(ord_less_eq_set_int(F3),Aa)
=> ( aa_set_int_o(P,bot_bot_set_int)
=> ( ! [A3: int,F4: set_int] :
( aa_set_int_o(finite_finite_int,F4)
=> ( aa_set_int_o(member_int(A3),Aa)
=> ( ~ aa_set_int_o(member_int(A3),F4)
=> ( aa_set_int_o(P,F4)
=> aa_set_int_o(P,aa_set_int_set_int(insert_int(A3),F4)) ) ) ) )
=> aa_set_int_o(P,F3) ) ) ) ) ).
% finite_subset_induct
tff(fact_6055_finite__subset__induct,axiom,
! [F3: set_nat,Aa: set_nat,P: fun_set_nat_o] :
( aa_set_nat_o(finite_finite_nat,F3)
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F3),Aa)
=> ( aa_set_nat_o(P,bot_bot_set_nat)
=> ( ! [A3: nat,F4: set_nat] :
( aa_set_nat_o(finite_finite_nat,F4)
=> ( aa_set_nat_o(member_nat(A3),Aa)
=> ( ~ aa_set_nat_o(member_nat(A3),F4)
=> ( aa_set_nat_o(P,F4)
=> aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(A3),F4)) ) ) ) )
=> aa_set_nat_o(P,F3) ) ) ) ) ).
% finite_subset_induct
tff(fact_6056_finite__empty__induct,axiom,
! [Aa: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,Aa)
=> ( aa_set5844364881195198285_nat_o(P,Aa)
=> ( ! [A3: produc3843707927480180839at_nat,A5: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A5)
=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),A5)
=> ( aa_set5844364881195198285_nat_o(P,A5)
=> aa_set5844364881195198285_nat_o(P,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A5),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A3),bot_bo228742789529271731at_nat))) ) ) )
=> aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat) ) ) ) ).
% finite_empty_induct
tff(fact_6057_finite__empty__induct,axiom,
! [Aa: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( aa_set5091322549519198405at_o_o(P,Aa)
=> ( ! [A3: fun_Pr6460944699146269935_nat_o,A5: set_fu3555266933746557007_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A5)
=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),A5)
=> ( aa_set5091322549519198405at_o_o(P,A5)
=> aa_set5091322549519198405at_o_o(P,minus_6456595905716665654_nat_o(A5,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A3),bot_bo8632255151191677115_nat_o))) ) ) )
=> aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o) ) ) ) ).
% finite_empty_induct
tff(fact_6058_finite__empty__induct,axiom,
! [Aa: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Aa)
=> ( aa_set3702741264012733747_nat_o(P,Aa)
=> ( ! [A3: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A5)
=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),A5)
=> ( aa_set3702741264012733747_nat_o(P,A5)
=> aa_set3702741264012733747_nat_o(P,minus_1356011639430497352at_nat(A5,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A3),bot_bo2099793752762293965at_nat))) ) ) )
=> aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat) ) ) ) ).
% finite_empty_induct
tff(fact_6059_finite__empty__induct,axiom,
! [Aa: set_o,P: fun_set_o_o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( aa_set_o_o(P,Aa)
=> ( ! [A3: $o,A5: set_o] :
( aa_set_o_o(finite_finite_o,A5)
=> ( aa_set_o_o(member_o((A3)),A5)
=> ( aa_set_o_o(P,A5)
=> aa_set_o_o(P,minus_minus_set_o(A5,aa_set_o_set_o(insert_o((A3)),bot_bot_set_o))) ) ) )
=> aa_set_o_o(P,bot_bot_set_o) ) ) ) ).
% finite_empty_induct
tff(fact_6060_finite__empty__induct,axiom,
! [Aa: set_int,P: fun_set_int_o] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( aa_set_int_o(P,Aa)
=> ( ! [A3: int,A5: set_int] :
( aa_set_int_o(finite_finite_int,A5)
=> ( aa_set_int_o(member_int(A3),A5)
=> ( aa_set_int_o(P,A5)
=> aa_set_int_o(P,minus_minus_set_int(A5,aa_set_int_set_int(insert_int(A3),bot_bot_set_int))) ) ) )
=> aa_set_int_o(P,bot_bot_set_int) ) ) ) ).
% finite_empty_induct
tff(fact_6061_finite__empty__induct,axiom,
! [Aa: set_nat,P: fun_set_nat_o] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( aa_set_nat_o(P,Aa)
=> ( ! [A3: nat,A5: set_nat] :
( aa_set_nat_o(finite_finite_nat,A5)
=> ( aa_set_nat_o(member_nat(A3),A5)
=> ( aa_set_nat_o(P,A5)
=> aa_set_nat_o(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A5),aa_set_nat_set_nat(insert_nat(A3),bot_bot_set_nat))) ) ) )
=> aa_set_nat_o(P,bot_bot_set_nat) ) ) ) ).
% finite_empty_induct
tff(fact_6062_infinite__coinduct,axiom,
! [X5: fun_se7788542033685991124_nat_o,Aa: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(X5,Aa)
=> ( ! [A5: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(X5,A5)
=> ? [X4: produc3843707927480180839at_nat] :
( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),A5)
& ( aa_set5844364881195198285_nat_o(X5,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A5),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X4),bot_bo228742789529271731at_nat)))
| ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A5),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X4),bot_bo228742789529271731at_nat))) ) ) )
=> ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,Aa) ) ) ).
% infinite_coinduct
tff(fact_6063_infinite__coinduct,axiom,
! [X5: fun_se4782206882984618170_nat_o,Aa: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(X5,Aa)
=> ( ! [A5: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(X5,A5)
=> ? [X4: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A5)
& ( aa_set3702741264012733747_nat_o(X5,minus_1356011639430497352at_nat(A5,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X4),bot_bo2099793752762293965at_nat)))
| ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A5,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X4),bot_bo2099793752762293965at_nat))) ) ) )
=> ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Aa) ) ) ).
% infinite_coinduct
tff(fact_6064_infinite__coinduct,axiom,
! [X5: fun_set_o_o,Aa: set_o] :
( aa_set_o_o(X5,Aa)
=> ( ! [A5: set_o] :
( aa_set_o_o(X5,A5)
=> ? [X4: $o] :
( aa_set_o_o(member_o((X4)),A5)
& ( aa_set_o_o(X5,minus_minus_set_o(A5,aa_set_o_set_o(insert_o((X4)),bot_bot_set_o)))
| ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(A5,aa_set_o_set_o(insert_o((X4)),bot_bot_set_o))) ) ) )
=> ~ aa_set_o_o(finite_finite_o,Aa) ) ) ).
% infinite_coinduct
tff(fact_6065_infinite__coinduct,axiom,
! [X5: fun_set_int_o,Aa: set_int] :
( aa_set_int_o(X5,Aa)
=> ( ! [A5: set_int] :
( aa_set_int_o(X5,A5)
=> ? [X4: int] :
( aa_set_int_o(member_int(X4),A5)
& ( aa_set_int_o(X5,minus_minus_set_int(A5,aa_set_int_set_int(insert_int(X4),bot_bot_set_int)))
| ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(A5,aa_set_int_set_int(insert_int(X4),bot_bot_set_int))) ) ) )
=> ~ aa_set_int_o(finite_finite_int,Aa) ) ) ).
% infinite_coinduct
tff(fact_6066_infinite__coinduct,axiom,
! [X5: fun_set_nat_o,Aa: set_nat] :
( aa_set_nat_o(X5,Aa)
=> ( ! [A5: set_nat] :
( aa_set_nat_o(X5,A5)
=> ? [X4: nat] :
( aa_set_nat_o(member_nat(X4),A5)
& ( aa_set_nat_o(X5,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A5),aa_set_nat_set_nat(insert_nat(X4),bot_bot_set_nat)))
| ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A5),aa_set_nat_set_nat(insert_nat(X4),bot_bot_set_nat))) ) ) )
=> ~ aa_set_nat_o(finite_finite_nat,Aa) ) ) ).
% infinite_coinduct
tff(fact_6067_infinite__remove,axiom,
! [S: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,S)
=> ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,S),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),bot_bo228742789529271731at_nat))) ) ).
% infinite_remove
tff(fact_6068_infinite__remove,axiom,
! [S: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
=> ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(S,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),bot_bo2099793752762293965at_nat))) ) ).
% infinite_remove
tff(fact_6069_infinite__remove,axiom,
! [S: set_o,A: $o] :
( ~ aa_set_o_o(finite_finite_o,S)
=> ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(S,aa_set_o_set_o(insert_o((A)),bot_bot_set_o))) ) ).
% infinite_remove
tff(fact_6070_infinite__remove,axiom,
! [S: set_int,A: int] :
( ~ aa_set_int_o(finite_finite_int,S)
=> ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(S,aa_set_int_set_int(insert_int(A),bot_bot_set_int))) ) ).
% infinite_remove
tff(fact_6071_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ aa_set_nat_o(finite_finite_nat,S)
=> ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) ) ).
% infinite_remove
tff(fact_6072_card__eq__0__iff,axiom,
! [Aa: set_Pr1261947904930325089at_nat] :
( ( finite711546835091564841at_nat(Aa) = zero_zero_nat )
<=> ( ( Aa = bot_bo2099793752762293965at_nat )
| ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Aa) ) ) ).
% card_eq_0_iff
tff(fact_6073_card__eq__0__iff,axiom,
! [Aa: set_o] :
( ( finite_card_o(Aa) = zero_zero_nat )
<=> ( ( Aa = bot_bot_set_o )
| ~ aa_set_o_o(finite_finite_o,Aa) ) ) ).
% card_eq_0_iff
tff(fact_6074_card__eq__0__iff,axiom,
! [Aa: set_nat] :
( ( finite_card_nat(Aa) = zero_zero_nat )
<=> ( ( Aa = bot_bot_set_nat )
| ~ aa_set_nat_o(finite_finite_nat,Aa) ) ) ).
% card_eq_0_iff
tff(fact_6075_card__eq__0__iff,axiom,
! [Aa: set_int] :
( ( finite_card_int(Aa) = zero_zero_nat )
<=> ( ( Aa = bot_bot_set_int )
| ~ aa_set_int_o(finite_finite_int,Aa) ) ) ).
% card_eq_0_iff
tff(fact_6076_zero__int_Otransfer,axiom,
aa_int_o(aa_Pro2912763917461432128_int_o(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),zero_zero_int) ).
% zero_int.transfer
tff(fact_6077_remove__induct,axiom,
! [P: fun_se7788542033685991124_nat_o,Ba: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
=> ( ( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,Ba)
=> aa_set5844364881195198285_nat_o(P,Ba) )
=> ( ! [A5: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A5)
=> ( ( A5 != bot_bo228742789529271731at_nat )
=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A5),Ba)
=> ( ! [X4: produc3843707927480180839at_nat] :
( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),A5)
=> aa_set5844364881195198285_nat_o(P,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A5),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X4),bot_bo228742789529271731at_nat))) )
=> aa_set5844364881195198285_nat_o(P,A5) ) ) ) )
=> aa_set5844364881195198285_nat_o(P,Ba) ) ) ) ).
% remove_induct
tff(fact_6078_remove__induct,axiom,
! [P: fun_se7459991829846211660at_o_o,Ba: set_fu3555266933746557007_nat_o] :
( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
=> ( ( ~ aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Ba)
=> aa_set5091322549519198405at_o_o(P,Ba) )
=> ( ! [A5: set_fu3555266933746557007_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A5)
=> ( ( A5 != bot_bo8632255151191677115_nat_o )
=> ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A5),Ba)
=> ( ! [X4: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),A5)
=> aa_set5091322549519198405at_o_o(P,minus_6456595905716665654_nat_o(A5,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X4),bot_bo8632255151191677115_nat_o))) )
=> aa_set5091322549519198405at_o_o(P,A5) ) ) ) )
=> aa_set5091322549519198405at_o_o(P,Ba) ) ) ) ).
% remove_induct
tff(fact_6079_remove__induct,axiom,
! [P: fun_se4782206882984618170_nat_o,Ba: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
=> ( ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Ba)
=> aa_set3702741264012733747_nat_o(P,Ba) )
=> ( ! [A5: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A5)
=> ( ( A5 != bot_bo2099793752762293965at_nat )
=> ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A5),Ba)
=> ( ! [X4: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A5)
=> aa_set3702741264012733747_nat_o(P,minus_1356011639430497352at_nat(A5,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X4),bot_bo2099793752762293965at_nat))) )
=> aa_set3702741264012733747_nat_o(P,A5) ) ) ) )
=> aa_set3702741264012733747_nat_o(P,Ba) ) ) ) ).
% remove_induct
tff(fact_6080_remove__induct,axiom,
! [P: fun_set_o_o,Ba: set_o] :
( aa_set_o_o(P,bot_bot_set_o)
=> ( ( ~ aa_set_o_o(finite_finite_o,Ba)
=> aa_set_o_o(P,Ba) )
=> ( ! [A5: set_o] :
( aa_set_o_o(finite_finite_o,A5)
=> ( ( A5 != bot_bot_set_o )
=> ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A5),Ba)
=> ( ! [X4: $o] :
( aa_set_o_o(member_o((X4)),A5)
=> aa_set_o_o(P,minus_minus_set_o(A5,aa_set_o_set_o(insert_o((X4)),bot_bot_set_o))) )
=> aa_set_o_o(P,A5) ) ) ) )
=> aa_set_o_o(P,Ba) ) ) ) ).
% remove_induct
tff(fact_6081_remove__induct,axiom,
! [P: fun_set_int_o,Ba: set_int] :
( aa_set_int_o(P,bot_bot_set_int)
=> ( ( ~ aa_set_int_o(finite_finite_int,Ba)
=> aa_set_int_o(P,Ba) )
=> ( ! [A5: set_int] :
( aa_set_int_o(finite_finite_int,A5)
=> ( ( A5 != bot_bot_set_int )
=> ( aa_set_int_o(ord_less_eq_set_int(A5),Ba)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),A5)
=> aa_set_int_o(P,minus_minus_set_int(A5,aa_set_int_set_int(insert_int(X4),bot_bot_set_int))) )
=> aa_set_int_o(P,A5) ) ) ) )
=> aa_set_int_o(P,Ba) ) ) ) ).
% remove_induct
tff(fact_6082_remove__induct,axiom,
! [P: fun_set_nat_o,Ba: set_nat] :
( aa_set_nat_o(P,bot_bot_set_nat)
=> ( ( ~ aa_set_nat_o(finite_finite_nat,Ba)
=> aa_set_nat_o(P,Ba) )
=> ( ! [A5: set_nat] :
( aa_set_nat_o(finite_finite_nat,A5)
=> ( ( A5 != bot_bot_set_nat )
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A5),Ba)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A5)
=> aa_set_nat_o(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A5),aa_set_nat_set_nat(insert_nat(X4),bot_bot_set_nat))) )
=> aa_set_nat_o(P,A5) ) ) ) )
=> aa_set_nat_o(P,Ba) ) ) ) ).
% remove_induct
tff(fact_6083_finite__remove__induct,axiom,
! [Ba: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,Ba)
=> ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
=> ( ! [A5: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A5)
=> ( ( A5 != bot_bo228742789529271731at_nat )
=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A5),Ba)
=> ( ! [X4: produc3843707927480180839at_nat] :
( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),A5)
=> aa_set5844364881195198285_nat_o(P,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A5),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X4),bot_bo228742789529271731at_nat))) )
=> aa_set5844364881195198285_nat_o(P,A5) ) ) ) )
=> aa_set5844364881195198285_nat_o(P,Ba) ) ) ) ).
% finite_remove_induct
tff(fact_6084_finite__remove__induct,axiom,
! [Ba: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Ba)
=> ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
=> ( ! [A5: set_fu3555266933746557007_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A5)
=> ( ( A5 != bot_bo8632255151191677115_nat_o )
=> ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A5),Ba)
=> ( ! [X4: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),A5)
=> aa_set5091322549519198405at_o_o(P,minus_6456595905716665654_nat_o(A5,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X4),bot_bo8632255151191677115_nat_o))) )
=> aa_set5091322549519198405at_o_o(P,A5) ) ) ) )
=> aa_set5091322549519198405at_o_o(P,Ba) ) ) ) ).
% finite_remove_induct
tff(fact_6085_finite__remove__induct,axiom,
! [Ba: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Ba)
=> ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
=> ( ! [A5: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A5)
=> ( ( A5 != bot_bo2099793752762293965at_nat )
=> ( aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(A5),Ba)
=> ( ! [X4: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A5)
=> aa_set3702741264012733747_nat_o(P,minus_1356011639430497352at_nat(A5,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X4),bot_bo2099793752762293965at_nat))) )
=> aa_set3702741264012733747_nat_o(P,A5) ) ) ) )
=> aa_set3702741264012733747_nat_o(P,Ba) ) ) ) ).
% finite_remove_induct
tff(fact_6086_finite__remove__induct,axiom,
! [Ba: set_o,P: fun_set_o_o] :
( aa_set_o_o(finite_finite_o,Ba)
=> ( aa_set_o_o(P,bot_bot_set_o)
=> ( ! [A5: set_o] :
( aa_set_o_o(finite_finite_o,A5)
=> ( ( A5 != bot_bot_set_o )
=> ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A5),Ba)
=> ( ! [X4: $o] :
( aa_set_o_o(member_o((X4)),A5)
=> aa_set_o_o(P,minus_minus_set_o(A5,aa_set_o_set_o(insert_o((X4)),bot_bot_set_o))) )
=> aa_set_o_o(P,A5) ) ) ) )
=> aa_set_o_o(P,Ba) ) ) ) ).
% finite_remove_induct
tff(fact_6087_finite__remove__induct,axiom,
! [Ba: set_int,P: fun_set_int_o] :
( aa_set_int_o(finite_finite_int,Ba)
=> ( aa_set_int_o(P,bot_bot_set_int)
=> ( ! [A5: set_int] :
( aa_set_int_o(finite_finite_int,A5)
=> ( ( A5 != bot_bot_set_int )
=> ( aa_set_int_o(ord_less_eq_set_int(A5),Ba)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),A5)
=> aa_set_int_o(P,minus_minus_set_int(A5,aa_set_int_set_int(insert_int(X4),bot_bot_set_int))) )
=> aa_set_int_o(P,A5) ) ) ) )
=> aa_set_int_o(P,Ba) ) ) ) ).
% finite_remove_induct
tff(fact_6088_finite__remove__induct,axiom,
! [Ba: set_nat,P: fun_set_nat_o] :
( aa_set_nat_o(finite_finite_nat,Ba)
=> ( aa_set_nat_o(P,bot_bot_set_nat)
=> ( ! [A5: set_nat] :
( aa_set_nat_o(finite_finite_nat,A5)
=> ( ( A5 != bot_bot_set_nat )
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A5),Ba)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A5)
=> aa_set_nat_o(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A5),aa_set_nat_set_nat(insert_nat(X4),bot_bot_set_nat))) )
=> aa_set_nat_o(P,A5) ) ) ) )
=> aa_set_nat_o(P,Ba) ) ) ) ).
% finite_remove_induct
tff(fact_6089_card__gt__0__iff,axiom,
! [Aa: set_Pr1261947904930325089at_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite711546835091564841at_nat(Aa))
<=> ( ( Aa != bot_bo2099793752762293965at_nat )
& aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Aa) ) ) ).
% card_gt_0_iff
tff(fact_6090_card__gt__0__iff,axiom,
! [Aa: set_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_o(Aa))
<=> ( ( Aa != bot_bot_set_o )
& aa_set_o_o(finite_finite_o,Aa) ) ) ).
% card_gt_0_iff
tff(fact_6091_card__gt__0__iff,axiom,
! [Aa: set_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_nat(Aa))
<=> ( ( Aa != bot_bot_set_nat )
& aa_set_nat_o(finite_finite_nat,Aa) ) ) ).
% card_gt_0_iff
tff(fact_6092_card__gt__0__iff,axiom,
! [Aa: set_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_int(Aa))
<=> ( ( Aa != bot_bot_set_int )
& aa_set_int_o(finite_finite_int,Aa) ) ) ).
% card_gt_0_iff
tff(fact_6093_card__1__singletonI,axiom,
! [S: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,S)
=> ( ( finite3771342082235030671at_nat(S) = one_one_nat )
=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),S)
=> ( S = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat) ) ) ) ) ).
% card_1_singletonI
tff(fact_6094_card__1__singletonI,axiom,
! [S: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,S)
=> ( ( finite5795495896937551127_nat_o(S) = one_one_nat )
=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),S)
=> ( S = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o) ) ) ) ) ).
% card_1_singletonI
tff(fact_6095_card__1__singletonI,axiom,
! [S: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
=> ( ( finite711546835091564841at_nat(S) = one_one_nat )
=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),S)
=> ( S = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat) ) ) ) ) ).
% card_1_singletonI
tff(fact_6096_card__1__singletonI,axiom,
! [S: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,S)
=> ( ( finite_card_o(S) = one_one_nat )
=> ( aa_set_o_o(member_o((X)),S)
=> ( S = aa_set_o_set_o(insert_o((X)),bot_bot_set_o) ) ) ) ) ).
% card_1_singletonI
tff(fact_6097_card__1__singletonI,axiom,
! [S: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,S)
=> ( ( finite_card_nat(S) = one_one_nat )
=> ( aa_set_nat_o(member_nat(X),S)
=> ( S = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) ) ) ) ) ).
% card_1_singletonI
tff(fact_6098_card__1__singletonI,axiom,
! [S: set_int,X: int] :
( aa_set_int_o(finite_finite_int,S)
=> ( ( finite_card_int(S) = one_one_nat )
=> ( aa_set_int_o(member_int(X),S)
=> ( S = aa_set_int_set_int(insert_int(X),bot_bot_set_int) ) ) ) ) ).
% card_1_singletonI
tff(fact_6099_finite__induct__select,axiom,
! [S: set_Pr4329608150637261639at_nat,P: fun_se7788542033685991124_nat_o] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,S)
=> ( aa_set5844364881195198285_nat_o(P,bot_bo228742789529271731at_nat)
=> ( ! [T2: set_Pr4329608150637261639at_nat] :
( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(T2),S)
=> ( aa_set5844364881195198285_nat_o(P,T2)
=> ? [X4: produc3843707927480180839at_nat] :
( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,S),T2))
& aa_set5844364881195198285_nat_o(P,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X4),T2)) ) ) )
=> aa_set5844364881195198285_nat_o(P,S) ) ) ) ).
% finite_induct_select
tff(fact_6100_finite__induct__select,axiom,
! [S: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
=> ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
=> ( ! [T2: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(T2),S)
=> ( aa_set3702741264012733747_nat_o(P,T2)
=> ? [X4: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),minus_1356011639430497352at_nat(S,T2))
& aa_set3702741264012733747_nat_o(P,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X4),T2)) ) ) )
=> aa_set3702741264012733747_nat_o(P,S) ) ) ) ).
% finite_induct_select
tff(fact_6101_finite__induct__select,axiom,
! [S: set_o,P: fun_set_o_o] :
( aa_set_o_o(finite_finite_o,S)
=> ( aa_set_o_o(P,bot_bot_set_o)
=> ( ! [T2: set_o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,T2),S)
=> ( aa_set_o_o(P,T2)
=> ? [X4: $o] :
( aa_set_o_o(member_o((X4)),minus_minus_set_o(S,T2))
& aa_set_o_o(P,aa_set_o_set_o(insert_o((X4)),T2)) ) ) )
=> aa_set_o_o(P,S) ) ) ) ).
% finite_induct_select
tff(fact_6102_finite__induct__select,axiom,
! [S: set_int,P: fun_set_int_o] :
( aa_set_int_o(finite_finite_int,S)
=> ( aa_set_int_o(P,bot_bot_set_int)
=> ( ! [T2: set_int] :
( aa_set_int_o(ord_less_set_int(T2),S)
=> ( aa_set_int_o(P,T2)
=> ? [X4: int] :
( aa_set_int_o(member_int(X4),minus_minus_set_int(S,T2))
& aa_set_int_o(P,aa_set_int_set_int(insert_int(X4),T2)) ) ) )
=> aa_set_int_o(P,S) ) ) ) ).
% finite_induct_select
tff(fact_6103_finite__induct__select,axiom,
! [S: set_nat,P: fun_set_nat_o] :
( aa_set_nat_o(finite_finite_nat,S)
=> ( aa_set_nat_o(P,bot_bot_set_nat)
=> ( ! [T2: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,T2),S)
=> ( aa_set_nat_o(P,T2)
=> ? [X4: nat] :
( aa_set_nat_o(member_nat(X4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),T2))
& aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(X4),T2)) ) ) )
=> aa_set_nat_o(P,S) ) ) ) ).
% finite_induct_select
tff(fact_6104_one__rat_Otransfer,axiom,
aa_rat_o(aa_Pro2298929076809959388_rat_o(pcr_rat,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)),one_one_rat) ).
% one_rat.transfer
tff(fact_6105_one__rat__def,axiom,
one_one_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)) ).
% one_rat_def
tff(fact_6106_card__Suc__Diff1,axiom,
! [Aa: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,Aa)
=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),Aa)
=> ( aa_nat_nat(suc,finite3771342082235030671at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat)))) = finite3771342082235030671at_nat(Aa) ) ) ) ).
% card_Suc_Diff1
tff(fact_6107_card__Suc__Diff1,axiom,
! [Aa: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Aa)
=> ( aa_nat_nat(suc,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(Aa,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)))) = finite5795495896937551127_nat_o(Aa) ) ) ) ).
% card_Suc_Diff1
tff(fact_6108_card__Suc__Diff1,axiom,
! [Aa: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Aa)
=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),Aa)
=> ( aa_nat_nat(suc,finite711546835091564841at_nat(minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat)))) = finite711546835091564841at_nat(Aa) ) ) ) ).
% card_Suc_Diff1
tff(fact_6109_card__Suc__Diff1,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( aa_set_o_o(member_o((X)),Aa)
=> ( aa_nat_nat(suc,finite_card_o(minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o)))) = finite_card_o(Aa) ) ) ) ).
% card_Suc_Diff1
tff(fact_6110_card__Suc__Diff1,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( aa_set_int_o(member_int(X),Aa)
=> ( aa_nat_nat(suc,finite_card_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int)))) = finite_card_int(Aa) ) ) ) ).
% card_Suc_Diff1
tff(fact_6111_card__Suc__Diff1,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( aa_set_nat_o(member_nat(X),Aa)
=> ( aa_nat_nat(suc,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))) = finite_card_nat(Aa) ) ) ) ).
% card_Suc_Diff1
tff(fact_6112_card_Oinsert__remove,axiom,
! [Aa: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,Aa)
=> ( finite3771342082235030671at_nat(aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),Aa)) = aa_nat_nat(suc,finite3771342082235030671at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat)))) ) ) ).
% card.insert_remove
tff(fact_6113_card_Oinsert__remove,axiom,
! [Aa: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Aa)
=> ( finite711546835091564841at_nat(aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),Aa)) = aa_nat_nat(suc,finite711546835091564841at_nat(minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat)))) ) ) ).
% card.insert_remove
tff(fact_6114_card_Oinsert__remove,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( finite_card_o(aa_set_o_set_o(insert_o((X)),Aa)) = aa_nat_nat(suc,finite_card_o(minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o)))) ) ) ).
% card.insert_remove
tff(fact_6115_card_Oinsert__remove,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( finite_card_int(aa_set_int_set_int(insert_int(X),Aa)) = aa_nat_nat(suc,finite_card_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int)))) ) ) ).
% card.insert_remove
tff(fact_6116_card_Oinsert__remove,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( finite_card_nat(aa_set_nat_set_nat(insert_nat(X),Aa)) = aa_nat_nat(suc,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))) ) ) ).
% card.insert_remove
tff(fact_6117_card_Oremove,axiom,
! [Aa: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,Aa)
=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),Aa)
=> ( finite3771342082235030671at_nat(Aa) = aa_nat_nat(suc,finite3771342082235030671at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat)))) ) ) ) ).
% card.remove
tff(fact_6118_card_Oremove,axiom,
! [Aa: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Aa)
=> ( finite5795495896937551127_nat_o(Aa) = aa_nat_nat(suc,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(Aa,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)))) ) ) ) ).
% card.remove
tff(fact_6119_card_Oremove,axiom,
! [Aa: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Aa)
=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),Aa)
=> ( finite711546835091564841at_nat(Aa) = aa_nat_nat(suc,finite711546835091564841at_nat(minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat)))) ) ) ) ).
% card.remove
tff(fact_6120_card_Oremove,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( aa_set_o_o(member_o((X)),Aa)
=> ( finite_card_o(Aa) = aa_nat_nat(suc,finite_card_o(minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o)))) ) ) ) ).
% card.remove
tff(fact_6121_card_Oremove,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( aa_set_int_o(member_int(X),Aa)
=> ( finite_card_int(Aa) = aa_nat_nat(suc,finite_card_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int)))) ) ) ) ).
% card.remove
tff(fact_6122_card_Oremove,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( aa_set_nat_o(member_nat(X),Aa)
=> ( finite_card_nat(Aa) = aa_nat_nat(suc,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))) ) ) ) ).
% card.remove
tff(fact_6123_card__Diff1__less__iff,axiom,
! [Aa: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite3771342082235030671at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat)))),finite3771342082235030671at_nat(Aa))
<=> ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,Aa)
& aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),Aa) ) ) ).
% card_Diff1_less_iff
tff(fact_6124_card__Diff1__less__iff,axiom,
! [Aa: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(Aa,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)))),finite5795495896937551127_nat_o(Aa))
<=> ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
& aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Aa) ) ) ).
% card_Diff1_less_iff
tff(fact_6125_card__Diff1__less__iff,axiom,
! [Aa: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite711546835091564841at_nat(minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat)))),finite711546835091564841at_nat(Aa))
<=> ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Aa)
& aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),Aa) ) ) ).
% card_Diff1_less_iff
tff(fact_6126_card__Diff1__less__iff,axiom,
! [Aa: set_o,X: $o] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_o(minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o)))),finite_card_o(Aa))
<=> ( aa_set_o_o(finite_finite_o,Aa)
& aa_set_o_o(member_o((X)),Aa) ) ) ).
% card_Diff1_less_iff
tff(fact_6127_card__Diff1__less__iff,axiom,
! [Aa: set_int,X: int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int)))),finite_card_int(Aa))
<=> ( aa_set_int_o(finite_finite_int,Aa)
& aa_set_int_o(member_int(X),Aa) ) ) ).
% card_Diff1_less_iff
tff(fact_6128_card__Diff1__less__iff,axiom,
! [Aa: set_nat,X: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))),finite_card_nat(Aa))
<=> ( aa_set_nat_o(finite_finite_nat,Aa)
& aa_set_nat_o(member_nat(X),Aa) ) ) ).
% card_Diff1_less_iff
tff(fact_6129_card__Diff2__less,axiom,
! [Aa: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,Aa)
=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),Aa)
=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Y),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite3771342082235030671at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat))),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(Y),bot_bo228742789529271731at_nat)))),finite3771342082235030671at_nat(Aa)) ) ) ) ).
% card_Diff2_less
tff(fact_6130_card__Diff2__less,axiom,
! [Aa: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Aa)
=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(minus_6456595905716665654_nat_o(Aa,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(Y),bot_bo8632255151191677115_nat_o)))),finite5795495896937551127_nat_o(Aa)) ) ) ) ).
% card_Diff2_less
tff(fact_6131_card__Diff2__less,axiom,
! [Aa: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Aa)
=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),Aa)
=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite711546835091564841at_nat(minus_1356011639430497352at_nat(minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat)),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Y),bot_bo2099793752762293965at_nat)))),finite711546835091564841at_nat(Aa)) ) ) ) ).
% card_Diff2_less
tff(fact_6132_card__Diff2__less,axiom,
! [Aa: set_o,X: $o,Y: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( aa_set_o_o(member_o((X)),Aa)
=> ( aa_set_o_o(member_o((Y)),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_o(minus_minus_set_o(minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o)),aa_set_o_set_o(insert_o((Y)),bot_bot_set_o)))),finite_card_o(Aa)) ) ) ) ).
% card_Diff2_less
tff(fact_6133_card__Diff2__less,axiom,
! [Aa: set_int,X: int,Y: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( aa_set_int_o(member_int(X),Aa)
=> ( aa_set_int_o(member_int(Y),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(minus_minus_set_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int)),aa_set_int_set_int(insert_int(Y),bot_bot_set_int)))),finite_card_int(Aa)) ) ) ) ).
% card_Diff2_less
tff(fact_6134_card__Diff2__less,axiom,
! [Aa: set_nat,X: nat,Y: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( aa_set_nat_o(member_nat(X),Aa)
=> ( aa_set_nat_o(member_nat(Y),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))),aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat)))),finite_card_nat(Aa)) ) ) ) ).
% card_Diff2_less
tff(fact_6135_finite__linorder__max__induct,axiom,
! [Aa: set_o,P: fun_set_o_o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( aa_set_o_o(P,bot_bot_set_o)
=> ( ! [B3: $o,A5: set_o] :
( aa_set_o_o(finite_finite_o,A5)
=> ( ! [X4: $o] :
( aa_set_o_o(member_o((X4)),A5)
=> aa_o_o(aa_o_fun_o_o(ord_less_o,(X4)),(B3)) )
=> ( aa_set_o_o(P,A5)
=> aa_set_o_o(P,aa_set_o_set_o(insert_o((B3)),A5)) ) ) )
=> aa_set_o_o(P,Aa) ) ) ) ).
% finite_linorder_max_induct
tff(fact_6136_finite__linorder__max__induct,axiom,
! [Aa: set_nat,P: fun_set_nat_o] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( aa_set_nat_o(P,bot_bot_set_nat)
=> ( ! [B3: nat,A5: set_nat] :
( aa_set_nat_o(finite_finite_nat,A5)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A5)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),B3) )
=> ( aa_set_nat_o(P,A5)
=> aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(B3),A5)) ) ) )
=> aa_set_nat_o(P,Aa) ) ) ) ).
% finite_linorder_max_induct
tff(fact_6137_finite__linorder__max__induct,axiom,
! [Aa: set_int,P: fun_set_int_o] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( aa_set_int_o(P,bot_bot_set_int)
=> ( ! [B3: int,A5: set_int] :
( aa_set_int_o(finite_finite_int,A5)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),A5)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,X4),B3) )
=> ( aa_set_int_o(P,A5)
=> aa_set_int_o(P,aa_set_int_set_int(insert_int(B3),A5)) ) ) )
=> aa_set_int_o(P,Aa) ) ) ) ).
% finite_linorder_max_induct
tff(fact_6138_finite__linorder__max__induct,axiom,
! [Aa: set_Code_integer,P: fun_se7289222521248229150eger_o] :
( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Aa)
=> ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
=> ( ! [B3: code_integer,A5: set_Code_integer] :
( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A5)
=> ( ! [X4: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(X4),A5)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X4),B3) )
=> ( aa_set5643387332196575111eger_o(P,A5)
=> aa_set5643387332196575111eger_o(P,insert_Code_integer(B3,A5)) ) ) )
=> aa_set5643387332196575111eger_o(P,Aa) ) ) ) ).
% finite_linorder_max_induct
tff(fact_6139_finite__linorder__max__induct,axiom,
! [Aa: set_Code_natural,P: fun_se3210475987521176083ural_o] :
( aa_set1564640798469522044ural_o(finite4840673202314680255atural,Aa)
=> ( aa_set1564640798469522044ural_o(P,bot_bo8448779972843232370atural)
=> ( ! [B3: code_natural,A5: set_Code_natural] :
( aa_set1564640798469522044ural_o(finite4840673202314680255atural,A5)
=> ( ! [X4: code_natural] :
( aa_set1564640798469522044ural_o(member_Code_natural(X4),A5)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X4),B3) )
=> ( aa_set1564640798469522044ural_o(P,A5)
=> aa_set1564640798469522044ural_o(P,insert_Code_natural(B3,A5)) ) ) )
=> aa_set1564640798469522044ural_o(P,Aa) ) ) ) ).
% finite_linorder_max_induct
tff(fact_6140_finite__linorder__min__induct,axiom,
! [Aa: set_o,P: fun_set_o_o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( aa_set_o_o(P,bot_bot_set_o)
=> ( ! [B3: $o,A5: set_o] :
( aa_set_o_o(finite_finite_o,A5)
=> ( ! [X4: $o] :
( aa_set_o_o(member_o((X4)),A5)
=> aa_o_o(aa_o_fun_o_o(ord_less_o,(B3)),(X4)) )
=> ( aa_set_o_o(P,A5)
=> aa_set_o_o(P,aa_set_o_set_o(insert_o((B3)),A5)) ) ) )
=> aa_set_o_o(P,Aa) ) ) ) ).
% finite_linorder_min_induct
tff(fact_6141_finite__linorder__min__induct,axiom,
! [Aa: set_nat,P: fun_set_nat_o] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( aa_set_nat_o(P,bot_bot_set_nat)
=> ( ! [B3: nat,A5: set_nat] :
( aa_set_nat_o(finite_finite_nat,A5)
=> ( ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),A5)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B3),X4) )
=> ( aa_set_nat_o(P,A5)
=> aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(B3),A5)) ) ) )
=> aa_set_nat_o(P,Aa) ) ) ) ).
% finite_linorder_min_induct
tff(fact_6142_finite__linorder__min__induct,axiom,
! [Aa: set_int,P: fun_set_int_o] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( aa_set_int_o(P,bot_bot_set_int)
=> ( ! [B3: int,A5: set_int] :
( aa_set_int_o(finite_finite_int,A5)
=> ( ! [X4: int] :
( aa_set_int_o(member_int(X4),A5)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,B3),X4) )
=> ( aa_set_int_o(P,A5)
=> aa_set_int_o(P,aa_set_int_set_int(insert_int(B3),A5)) ) ) )
=> aa_set_int_o(P,Aa) ) ) ) ).
% finite_linorder_min_induct
tff(fact_6143_finite__linorder__min__induct,axiom,
! [Aa: set_Code_integer,P: fun_se7289222521248229150eger_o] :
( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Aa)
=> ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
=> ( ! [B3: code_integer,A5: set_Code_integer] :
( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A5)
=> ( ! [X4: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(X4),A5)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B3),X4) )
=> ( aa_set5643387332196575111eger_o(P,A5)
=> aa_set5643387332196575111eger_o(P,insert_Code_integer(B3,A5)) ) ) )
=> aa_set5643387332196575111eger_o(P,Aa) ) ) ) ).
% finite_linorder_min_induct
tff(fact_6144_finite__linorder__min__induct,axiom,
! [Aa: set_Code_natural,P: fun_se3210475987521176083ural_o] :
( aa_set1564640798469522044ural_o(finite4840673202314680255atural,Aa)
=> ( aa_set1564640798469522044ural_o(P,bot_bo8448779972843232370atural)
=> ( ! [B3: code_natural,A5: set_Code_natural] :
( aa_set1564640798469522044ural_o(finite4840673202314680255atural,A5)
=> ( ! [X4: code_natural] :
( aa_set1564640798469522044ural_o(member_Code_natural(X4),A5)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B3),X4) )
=> ( aa_set1564640798469522044ural_o(P,A5)
=> aa_set1564640798469522044ural_o(P,insert_Code_natural(B3,A5)) ) ) )
=> aa_set1564640798469522044ural_o(P,Aa) ) ) ) ).
% finite_linorder_min_induct
tff(fact_6145_finite__ranking__induct,axiom,
! [S: set_o,P: fun_set_o_o,F: fun_o_Code_integer] :
( aa_set_o_o(finite_finite_o,S)
=> ( aa_set_o_o(P,bot_bot_set_o)
=> ( ! [X2: $o,S4: set_o] :
( aa_set_o_o(finite_finite_o,S4)
=> ( ! [Y3: $o] :
( aa_set_o_o(member_o((Y3)),S4)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(F,(Y3))),aa_o_Code_integer(F,(X2))) )
=> ( aa_set_o_o(P,S4)
=> aa_set_o_o(P,aa_set_o_set_o(insert_o((X2)),S4)) ) ) )
=> aa_set_o_o(P,S) ) ) ) ).
% finite_ranking_induct
tff(fact_6146_finite__ranking__induct,axiom,
! [S: set_nat,P: fun_set_nat_o,F: fun_nat_Code_integer] :
( aa_set_nat_o(finite_finite_nat,S)
=> ( aa_set_nat_o(P,bot_bot_set_nat)
=> ( ! [X2: nat,S4: set_nat] :
( aa_set_nat_o(finite_finite_nat,S4)
=> ( ! [Y3: nat] :
( aa_set_nat_o(member_nat(Y3),S4)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,Y3)),aa_nat_Code_integer(F,X2)) )
=> ( aa_set_nat_o(P,S4)
=> aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(X2),S4)) ) ) )
=> aa_set_nat_o(P,S) ) ) ) ).
% finite_ranking_induct
tff(fact_6147_finite__ranking__induct,axiom,
! [S: set_int,P: fun_set_int_o,F: fun_int_Code_integer] :
( aa_set_int_o(finite_finite_int,S)
=> ( aa_set_int_o(P,bot_bot_set_int)
=> ( ! [X2: int,S4: set_int] :
( aa_set_int_o(finite_finite_int,S4)
=> ( ! [Y3: int] :
( aa_set_int_o(member_int(Y3),S4)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,Y3)),aa_int_Code_integer(F,X2)) )
=> ( aa_set_int_o(P,S4)
=> aa_set_int_o(P,aa_set_int_set_int(insert_int(X2),S4)) ) ) )
=> aa_set_int_o(P,S) ) ) ) ).
% finite_ranking_induct
tff(fact_6148_finite__ranking__induct,axiom,
! [S: set_o,P: fun_set_o_o,F: fun_o_nat] :
( aa_set_o_o(finite_finite_o,S)
=> ( aa_set_o_o(P,bot_bot_set_o)
=> ( ! [X2: $o,S4: set_o] :
( aa_set_o_o(finite_finite_o,S4)
=> ( ! [Y3: $o] :
( aa_set_o_o(member_o((Y3)),S4)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(Y3))),aa_o_nat(F,(X2))) )
=> ( aa_set_o_o(P,S4)
=> aa_set_o_o(P,aa_set_o_set_o(insert_o((X2)),S4)) ) ) )
=> aa_set_o_o(P,S) ) ) ) ).
% finite_ranking_induct
tff(fact_6149_finite__ranking__induct,axiom,
! [S: set_nat,P: fun_set_nat_o,F: fun_nat_nat] :
( aa_set_nat_o(finite_finite_nat,S)
=> ( aa_set_nat_o(P,bot_bot_set_nat)
=> ( ! [X2: nat,S4: set_nat] :
( aa_set_nat_o(finite_finite_nat,S4)
=> ( ! [Y3: nat] :
( aa_set_nat_o(member_nat(Y3),S4)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Y3)),aa_nat_nat(F,X2)) )
=> ( aa_set_nat_o(P,S4)
=> aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(X2),S4)) ) ) )
=> aa_set_nat_o(P,S) ) ) ) ).
% finite_ranking_induct
tff(fact_6150_finite__ranking__induct,axiom,
! [S: set_int,P: fun_set_int_o,F: fun_int_nat] :
( aa_set_int_o(finite_finite_int,S)
=> ( aa_set_int_o(P,bot_bot_set_int)
=> ( ! [X2: int,S4: set_int] :
( aa_set_int_o(finite_finite_int,S4)
=> ( ! [Y3: int] :
( aa_set_int_o(member_int(Y3),S4)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,Y3)),aa_int_nat(F,X2)) )
=> ( aa_set_int_o(P,S4)
=> aa_set_int_o(P,aa_set_int_set_int(insert_int(X2),S4)) ) ) )
=> aa_set_int_o(P,S) ) ) ) ).
% finite_ranking_induct
tff(fact_6151_finite__ranking__induct,axiom,
! [S: set_o,P: fun_set_o_o,F: fun_o_int] :
( aa_set_o_o(finite_finite_o,S)
=> ( aa_set_o_o(P,bot_bot_set_o)
=> ( ! [X2: $o,S4: set_o] :
( aa_set_o_o(finite_finite_o,S4)
=> ( ! [Y3: $o] :
( aa_set_o_o(member_o((Y3)),S4)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(Y3))),aa_o_int(F,(X2))) )
=> ( aa_set_o_o(P,S4)
=> aa_set_o_o(P,aa_set_o_set_o(insert_o((X2)),S4)) ) ) )
=> aa_set_o_o(P,S) ) ) ) ).
% finite_ranking_induct
tff(fact_6152_finite__ranking__induct,axiom,
! [S: set_nat,P: fun_set_nat_o,F: fun_nat_int] :
( aa_set_nat_o(finite_finite_nat,S)
=> ( aa_set_nat_o(P,bot_bot_set_nat)
=> ( ! [X2: nat,S4: set_nat] :
( aa_set_nat_o(finite_finite_nat,S4)
=> ( ! [Y3: nat] :
( aa_set_nat_o(member_nat(Y3),S4)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Y3)),aa_nat_int(F,X2)) )
=> ( aa_set_nat_o(P,S4)
=> aa_set_nat_o(P,aa_set_nat_set_nat(insert_nat(X2),S4)) ) ) )
=> aa_set_nat_o(P,S) ) ) ) ).
% finite_ranking_induct
tff(fact_6153_finite__ranking__induct,axiom,
! [S: set_int,P: fun_set_int_o,F: fun_int_int] :
( aa_set_int_o(finite_finite_int,S)
=> ( aa_set_int_o(P,bot_bot_set_int)
=> ( ! [X2: int,S4: set_int] :
( aa_set_int_o(finite_finite_int,S4)
=> ( ! [Y3: int] :
( aa_set_int_o(member_int(Y3),S4)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,Y3)),aa_int_int(F,X2)) )
=> ( aa_set_int_o(P,S4)
=> aa_set_int_o(P,aa_set_int_set_int(insert_int(X2),S4)) ) ) )
=> aa_set_int_o(P,S) ) ) ) ).
% finite_ranking_induct
tff(fact_6154_finite__ranking__induct,axiom,
! [S: set_Pr1261947904930325089at_nat,P: fun_se4782206882984618170_nat_o,F: fun_Pr6026251430685297nteger] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
=> ( aa_set3702741264012733747_nat_o(P,bot_bo2099793752762293965at_nat)
=> ( ! [X2: product_prod_nat_nat,S4: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S4)
=> ( ! [Y3: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y3),S4)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Pro4764841431850000930nteger(F,Y3)),aa_Pro4764841431850000930nteger(F,X2)) )
=> ( aa_set3702741264012733747_nat_o(P,S4)
=> aa_set3702741264012733747_nat_o(P,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X2),S4)) ) ) )
=> aa_set3702741264012733747_nat_o(P,S) ) ) ) ).
% finite_ranking_induct
tff(fact_6155_ex__min__if__finite,axiom,
! [S: set_o] :
( aa_set_o_o(finite_finite_o,S)
=> ( ( S != bot_bot_set_o )
=> ? [X2: $o] :
( aa_set_o_o(member_o((X2)),S)
& ~ ? [Xa3: $o] :
( aa_set_o_o(member_o((Xa3)),S)
& aa_o_o(aa_o_fun_o_o(ord_less_o,(Xa3)),(X2)) ) ) ) ) ).
% ex_min_if_finite
tff(fact_6156_ex__min__if__finite,axiom,
! [S: set_assn] :
( finite_finite_assn(S)
=> ( ( S != bot_bot_set_assn )
=> ? [X2: assn] :
( member_assn(X2,S)
& ~ ? [Xa3: assn] :
( member_assn(Xa3,S)
& aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Xa3),X2) ) ) ) ) ).
% ex_min_if_finite
tff(fact_6157_ex__min__if__finite,axiom,
! [S: set_nat] :
( aa_set_nat_o(finite_finite_nat,S)
=> ( ( S != bot_bot_set_nat )
=> ? [X2: nat] :
( aa_set_nat_o(member_nat(X2),S)
& ~ ? [Xa3: nat] :
( aa_set_nat_o(member_nat(Xa3),S)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa3),X2) ) ) ) ) ).
% ex_min_if_finite
tff(fact_6158_ex__min__if__finite,axiom,
! [S: set_int] :
( aa_set_int_o(finite_finite_int,S)
=> ( ( S != bot_bot_set_int )
=> ? [X2: int] :
( aa_set_int_o(member_int(X2),S)
& ~ ? [Xa3: int] :
( aa_set_int_o(member_int(Xa3),S)
& aa_int_o(aa_int_fun_int_o(ord_less_int,Xa3),X2) ) ) ) ) ).
% ex_min_if_finite
tff(fact_6159_ex__min__if__finite,axiom,
! [S: set_Code_integer] :
( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
=> ( ( S != bot_bo3990330152332043303nteger )
=> ? [X2: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(X2),S)
& ~ ? [Xa3: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),S)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa3),X2) ) ) ) ) ).
% ex_min_if_finite
tff(fact_6160_ex__min__if__finite,axiom,
! [S: set_Code_natural] :
( aa_set1564640798469522044ural_o(finite4840673202314680255atural,S)
=> ( ( S != bot_bo8448779972843232370atural )
=> ? [X2: code_natural] :
( aa_set1564640798469522044ural_o(member_Code_natural(X2),S)
& ~ ? [Xa3: code_natural] :
( aa_set1564640798469522044ural_o(member_Code_natural(Xa3),S)
& aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Xa3),X2) ) ) ) ) ).
% ex_min_if_finite
tff(fact_6161_infinite__growing,axiom,
! [X5: set_o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X2: $o] :
( aa_set_o_o(member_o((X2)),X5)
=> ? [Xa3: $o] :
( aa_set_o_o(member_o((Xa3)),X5)
& aa_o_o(aa_o_fun_o_o(ord_less_o,(X2)),(Xa3)) ) )
=> ~ aa_set_o_o(finite_finite_o,X5) ) ) ).
% infinite_growing
tff(fact_6162_infinite__growing,axiom,
! [X5: set_nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( aa_set_nat_o(member_nat(X2),X5)
=> ? [Xa3: nat] :
( aa_set_nat_o(member_nat(Xa3),X5)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Xa3) ) )
=> ~ aa_set_nat_o(finite_finite_nat,X5) ) ) ).
% infinite_growing
tff(fact_6163_infinite__growing,axiom,
! [X5: set_int] :
( ( X5 != bot_bot_set_int )
=> ( ! [X2: int] :
( aa_set_int_o(member_int(X2),X5)
=> ? [Xa3: int] :
( aa_set_int_o(member_int(Xa3),X5)
& aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Xa3) ) )
=> ~ aa_set_int_o(finite_finite_int,X5) ) ) ).
% infinite_growing
tff(fact_6164_infinite__growing,axiom,
! [X5: set_Code_integer] :
( ( X5 != bot_bo3990330152332043303nteger )
=> ( ! [X2: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(X2),X5)
=> ? [Xa3: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),X5)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X2),Xa3) ) )
=> ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,X5) ) ) ).
% infinite_growing
tff(fact_6165_infinite__growing,axiom,
! [X5: set_Code_natural] :
( ( X5 != bot_bo8448779972843232370atural )
=> ( ! [X2: code_natural] :
( aa_set1564640798469522044ural_o(member_Code_natural(X2),X5)
=> ? [Xa3: code_natural] :
( aa_set1564640798469522044ural_o(member_Code_natural(Xa3),X5)
& aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X2),Xa3) ) )
=> ~ aa_set1564640798469522044ural_o(finite4840673202314680255atural,X5) ) ) ).
% infinite_growing
tff(fact_6166_uminus__rat_Oabs__eq,axiom,
! [X: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),X)
=> ( aa_rat_rat(uminus_uminus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,X))),aa_Pro6799258434465214121nt_int(product_snd_int_int,X))) ) ) ).
% uminus_rat.abs_eq
tff(fact_6167_plus__rat_Oabs__eq,axiom,
! [Xa: product_prod_int_int,X: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xa),Xa)
=> ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),X)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa)),aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,X)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X)))) ) ) ) ).
% plus_rat.abs_eq
tff(fact_6168_one__rat_Orsp,axiom,
aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)),aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)) ).
% one_rat.rsp
tff(fact_6169_zero__rat_Orsp,axiom,
aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int)),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int)) ).
% zero_rat.rsp
tff(fact_6170_times__rat_Oabs__eq,axiom,
! [Xa: product_prod_int_int,X: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xa),Xa)
=> ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),X)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa)),aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_fst_int_int,X))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X)))) ) ) ) ).
% times_rat.abs_eq
tff(fact_6171_inverse__rat_Oabs__eq,axiom,
! [X: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),X)
=> ( aa_rat_rat(inverse_inverse_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,
$ite(aa_Pro6799258434465214121nt_int(product_fst_int_int,X) = zero_zero_int,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,X)),aa_Pro6799258434465214121nt_int(product_fst_int_int,X)))) ) ) ).
% inverse_rat.abs_eq
tff(fact_6172_arg__min__if__finite_I2_J,axiom,
! [S: set_o,F: fun_o_assn] :
( aa_set_o_o(finite_finite_o,S)
=> ( ( S != bot_bot_set_o )
=> ~ ? [X4: $o] :
( aa_set_o_o(member_o((X4)),S)
& aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_o_assn(F,(X4))),aa_o_assn(F,aa_set_o_o(lattic7708394068118247271o_assn(F),S))) ) ) ) ).
% arg_min_if_finite(2)
tff(fact_6173_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: fun_nat_assn] :
( aa_set_nat_o(finite_finite_nat,S)
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( aa_set_nat_o(member_nat(X4),S)
& aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X4)),aa_nat_assn(F,lattic2187264146484958483t_assn(F,S))) ) ) ) ).
% arg_min_if_finite(2)
tff(fact_6174_arg__min__if__finite_I2_J,axiom,
! [S: set_int,F: fun_int_assn] :
( aa_set_int_o(finite_finite_int,S)
=> ( ( S != bot_bot_set_int )
=> ~ ? [X4: int] :
( aa_set_int_o(member_int(X4),S)
& aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,X4)),aa_int_assn(F,lattic3162799611691135855t_assn(F,S))) ) ) ) ).
% arg_min_if_finite(2)
tff(fact_6175_arg__min__if__finite_I2_J,axiom,
! [S: set_o,F: fun_o_nat] :
( aa_set_o_o(finite_finite_o,S)
=> ( ( S != bot_bot_set_o )
=> ~ ? [X4: $o] :
( aa_set_o_o(member_o((X4)),S)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(F,(X4))),aa_o_nat(F,aa_set_o_o(lattic2775856028456453135_o_nat(F),S))) ) ) ) ).
% arg_min_if_finite(2)
tff(fact_6176_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: fun_nat_nat] :
( aa_set_nat_o(finite_finite_nat,S)
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( aa_set_nat_o(member_nat(X4),S)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,lattic7446932960582359483at_nat(F,S))) ) ) ) ).
% arg_min_if_finite(2)
tff(fact_6177_arg__min__if__finite_I2_J,axiom,
! [S: set_int,F: fun_int_nat] :
( aa_set_int_o(finite_finite_int,S)
=> ( ( S != bot_bot_set_int )
=> ~ ? [X4: int] :
( aa_set_int_o(member_int(X4),S)
& aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X4)),aa_int_nat(F,lattic8446286672483414039nt_nat(F,S))) ) ) ) ).
% arg_min_if_finite(2)
tff(fact_6178_arg__min__if__finite_I2_J,axiom,
! [S: set_o,F: fun_o_int] :
( aa_set_o_o(finite_finite_o,S)
=> ( ( S != bot_bot_set_o )
=> ~ ? [X4: $o] :
( aa_set_o_o(member_o((X4)),S)
& aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(F,(X4))),aa_o_int(F,aa_set_o_o(lattic2773365557947402859_o_int(F),S))) ) ) ) ).
% arg_min_if_finite(2)
tff(fact_6179_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: fun_nat_int] :
( aa_set_nat_o(finite_finite_nat,S)
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( aa_set_nat_o(member_nat(X4),S)
& aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X4)),aa_nat_int(F,lattic7444442490073309207at_int(F,S))) ) ) ) ).
% arg_min_if_finite(2)
tff(fact_6180_arg__min__if__finite_I2_J,axiom,
! [S: set_int,F: fun_int_int] :
( aa_set_int_o(finite_finite_int,S)
=> ( ( S != bot_bot_set_int )
=> ~ ? [X4: int] :
( aa_set_int_o(member_int(X4),S)
& aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X4)),aa_int_int(F,lattic8443796201974363763nt_int(F,S))) ) ) ) ).
% arg_min_if_finite(2)
tff(fact_6181_arg__min__if__finite_I2_J,axiom,
! [S: set_o,F: fun_o_Code_integer] :
( aa_set_o_o(finite_finite_o,S)
=> ( ( S != bot_bot_set_o )
=> ~ ? [X4: $o] :
( aa_set_o_o(member_o((X4)),S)
& aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_o_Code_integer(F,(X4))),aa_o_Code_integer(F,aa_set_o_o(lattic8357703018590397596nteger(F),S))) ) ) ) ).
% arg_min_if_finite(2)
tff(fact_6182_arg__min__least,axiom,
! [S: set_o,Y: $o,F: fun_o_Code_integer] :
( aa_set_o_o(finite_finite_o,S)
=> ( ( S != bot_bot_set_o )
=> ( aa_set_o_o(member_o((Y)),S)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(F,aa_set_o_o(lattic8357703018590397596nteger(F),S))),aa_o_Code_integer(F,(Y))) ) ) ) ).
% arg_min_least
tff(fact_6183_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: fun_nat_Code_integer] :
( aa_set_nat_o(finite_finite_nat,S)
=> ( ( S != bot_bot_set_nat )
=> ( aa_set_nat_o(member_nat(Y),S)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,lattic6339492004331817544nteger(F,S))),aa_nat_Code_integer(F,Y)) ) ) ) ).
% arg_min_least
tff(fact_6184_arg__min__least,axiom,
! [S: set_int,Y: int,F: fun_int_Code_integer] :
( aa_set_int_o(finite_finite_int,S)
=> ( ( S != bot_bot_set_int )
=> ( aa_set_int_o(member_int(Y),S)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,lattic6711145564568662180nteger(F,S))),aa_int_Code_integer(F,Y)) ) ) ) ).
% arg_min_least
tff(fact_6185_arg__min__least,axiom,
! [S: set_o,Y: $o,F: fun_o_nat] :
( aa_set_o_o(finite_finite_o,S)
=> ( ( S != bot_bot_set_o )
=> ( aa_set_o_o(member_o((Y)),S)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,aa_set_o_o(lattic2775856028456453135_o_nat(F),S))),aa_o_nat(F,(Y))) ) ) ) ).
% arg_min_least
tff(fact_6186_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: fun_nat_nat] :
( aa_set_nat_o(finite_finite_nat,S)
=> ( ( S != bot_bot_set_nat )
=> ( aa_set_nat_o(member_nat(Y),S)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,lattic7446932960582359483at_nat(F,S))),aa_nat_nat(F,Y)) ) ) ) ).
% arg_min_least
tff(fact_6187_arg__min__least,axiom,
! [S: set_int,Y: int,F: fun_int_nat] :
( aa_set_int_o(finite_finite_int,S)
=> ( ( S != bot_bot_set_int )
=> ( aa_set_int_o(member_int(Y),S)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,lattic8446286672483414039nt_nat(F,S))),aa_int_nat(F,Y)) ) ) ) ).
% arg_min_least
tff(fact_6188_arg__min__least,axiom,
! [S: set_o,Y: $o,F: fun_o_int] :
( aa_set_o_o(finite_finite_o,S)
=> ( ( S != bot_bot_set_o )
=> ( aa_set_o_o(member_o((Y)),S)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,aa_set_o_o(lattic2773365557947402859_o_int(F),S))),aa_o_int(F,(Y))) ) ) ) ).
% arg_min_least
tff(fact_6189_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: fun_nat_int] :
( aa_set_nat_o(finite_finite_nat,S)
=> ( ( S != bot_bot_set_nat )
=> ( aa_set_nat_o(member_nat(Y),S)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,lattic7444442490073309207at_int(F,S))),aa_nat_int(F,Y)) ) ) ) ).
% arg_min_least
tff(fact_6190_arg__min__least,axiom,
! [S: set_int,Y: int,F: fun_int_int] :
( aa_set_int_o(finite_finite_int,S)
=> ( ( S != bot_bot_set_int )
=> ( aa_set_int_o(member_int(Y),S)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,lattic8443796201974363763nt_int(F,S))),aa_int_int(F,Y)) ) ) ) ).
% arg_min_least
tff(fact_6191_arg__min__least,axiom,
! [S: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,F: fun_Pr6026251430685297nteger] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
=> ( ( S != bot_bo2099793752762293965at_nat )
=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y),S)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Pro4764841431850000930nteger(F,lattic1349466628417373189nteger(F,S))),aa_Pro4764841431850000930nteger(F,Y)) ) ) ) ).
% arg_min_least
tff(fact_6192_finite__transitivity__chain,axiom,
! [Aa: set_fu3555266933746557007_nat_o,R: fun_fu4164567611050918574at_o_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( ! [X2: fun_Pr6460944699146269935_nat_o] : ~ aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,X2),X2)
=> ( ! [X2: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o,Z2: fun_Pr6460944699146269935_nat_o] :
( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,X2),Y2)
=> ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,Y2),Z2)
=> aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,X2),Z2) ) )
=> ( ! [X2: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),Aa)
=> ? [Y3: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y3),Aa)
& aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,X2),Y3) ) )
=> ( Aa = bot_bo8632255151191677115_nat_o ) ) ) ) ) ).
% finite_transitivity_chain
tff(fact_6193_finite__transitivity__chain,axiom,
! [Aa: set_Pr1261947904930325089at_nat,R: fun_Pr2028404973800837934_nat_o] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Aa)
=> ( ! [X2: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X2),X2)
=> ( ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat,Z2: product_prod_nat_nat] :
( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X2),Y2)
=> ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,Y2),Z2)
=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X2),Z2) ) )
=> ( ! [X2: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X2),Aa)
=> ? [Y3: product_prod_nat_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y3),Aa)
& aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X2),Y3) ) )
=> ( Aa = bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% finite_transitivity_chain
tff(fact_6194_finite__transitivity__chain,axiom,
! [Aa: set_o,R: fun_o_fun_o_o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ! [X2: $o] : ~ aa_o_o(aa_o_fun_o_o(R,(X2)),(X2))
=> ( ! [X2: $o,Y2: $o,Z2: $o] :
( aa_o_o(aa_o_fun_o_o(R,(X2)),(Y2))
=> ( aa_o_o(aa_o_fun_o_o(R,(Y2)),(Z2))
=> aa_o_o(aa_o_fun_o_o(R,(X2)),(Z2)) ) )
=> ( ! [X2: $o] :
( aa_set_o_o(member_o((X2)),Aa)
=> ? [Y3: $o] :
( aa_set_o_o(member_o((Y3)),Aa)
& aa_o_o(aa_o_fun_o_o(R,(X2)),(Y3)) ) )
=> ( Aa = bot_bot_set_o ) ) ) ) ) ).
% finite_transitivity_chain
tff(fact_6195_finite__transitivity__chain,axiom,
! [Aa: set_nat,R: fun_nat_fun_nat_o] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ! [X2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(R,X2),X2)
=> ( ! [X2: nat,Y2: nat,Z2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(R,X2),Y2)
=> ( aa_nat_o(aa_nat_fun_nat_o(R,Y2),Z2)
=> aa_nat_o(aa_nat_fun_nat_o(R,X2),Z2) ) )
=> ( ! [X2: nat] :
( aa_set_nat_o(member_nat(X2),Aa)
=> ? [Y3: nat] :
( aa_set_nat_o(member_nat(Y3),Aa)
& aa_nat_o(aa_nat_fun_nat_o(R,X2),Y3) ) )
=> ( Aa = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
tff(fact_6196_finite__transitivity__chain,axiom,
! [Aa: set_int,R: fun_int_fun_int_o] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ! [X2: int] : ~ aa_int_o(aa_int_fun_int_o(R,X2),X2)
=> ( ! [X2: int,Y2: int,Z2: int] :
( aa_int_o(aa_int_fun_int_o(R,X2),Y2)
=> ( aa_int_o(aa_int_fun_int_o(R,Y2),Z2)
=> aa_int_o(aa_int_fun_int_o(R,X2),Z2) ) )
=> ( ! [X2: int] :
( aa_set_int_o(member_int(X2),Aa)
=> ? [Y3: int] :
( aa_set_int_o(member_int(Y3),Aa)
& aa_int_o(aa_int_fun_int_o(R,X2),Y3) ) )
=> ( Aa = bot_bot_set_int ) ) ) ) ) ).
% finite_transitivity_chain
tff(fact_6197_Gcd__fin__0__iff,axiom,
! [Aa: set_int] :
( ( semiri4256215615220890538in_int(Aa) = zero_zero_int )
<=> ( aa_set_int_o(ord_less_eq_set_int(Aa),aa_set_int_set_int(insert_int(zero_zero_int),bot_bot_set_int))
& aa_set_int_o(finite_finite_int,Aa) ) ) ).
% Gcd_fin_0_iff
tff(fact_6198_Gcd__fin__0__iff,axiom,
! [Aa: set_nat] :
( ( semiri4258706085729940814in_nat(Aa) = zero_zero_nat )
<=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat))
& aa_set_nat_o(finite_finite_nat,Aa) ) ) ).
% Gcd_fin_0_iff
tff(fact_6199_greaterThan__Suc,axiom,
! [K: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_or1210151606488870762an_nat,K)),aa_set_nat_set_nat(insert_nat(aa_nat_nat(suc,K)),bot_bot_set_nat)) ).
% greaterThan_Suc
tff(fact_6200_Sup__fin_Oinsert__remove,axiom,
! [Aa: set_assn,X: assn] :
( finite_finite_assn(Aa)
=> ( lattic2150320897289308081n_assn(insert_assn(X,Aa)) = $ite(minus_minus_set_assn(Aa,insert_assn(X,bot_bot_set_assn)) = bot_bot_set_assn,X,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),lattic2150320897289308081n_assn(minus_minus_set_assn(Aa,insert_assn(X,bot_bot_set_assn))))) ) ) ).
% Sup_fin.insert_remove
tff(fact_6201_Sup__fin_Oinsert__remove,axiom,
! [Aa: set_set_nat,X: set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( lattic3835124923745554447et_nat(insert_set_nat(X,Aa)) = $ite(minus_2163939370556025621et_nat(Aa,insert_set_nat(X,bot_bot_set_set_nat)) = bot_bot_set_set_nat,X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),lattic3835124923745554447et_nat(minus_2163939370556025621et_nat(Aa,insert_set_nat(X,bot_bot_set_set_nat))))) ) ) ).
% Sup_fin.insert_remove
tff(fact_6202_Sup__fin_Oinsert__remove,axiom,
! [Aa: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat] :
( finite4768004253697144688at_nat(Aa)
=> ( lattic8394993312974875902at_nat(insert2290795007606087959at_nat(X,Aa)) = $ite(minus_5090130588299000206at_nat(Aa,insert2290795007606087959at_nat(X,bot_bo1534825418459847827at_nat)) = bot_bo1534825418459847827at_nat,X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),lattic8394993312974875902at_nat(minus_5090130588299000206at_nat(Aa,insert2290795007606087959at_nat(X,bot_bo1534825418459847827at_nat))))) ) ) ).
% Sup_fin.insert_remove
tff(fact_6203_Sup__fin_Oinsert__remove,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( aa_set_o_o(lattic1508158080041050831_fin_o,aa_set_o_set_o(insert_o((X)),Aa))
<=> $ite(minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(sup_sup_o,(X)),aa_set_o_o(lattic1508158080041050831_fin_o,minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))))) ) ) ).
% Sup_fin.insert_remove
tff(fact_6204_Sup__fin_Oinsert__remove,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( lattic1091506334969745077in_int(aa_set_int_set_int(insert_int(X),Aa)) = $ite(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int)) = bot_bot_set_int,X,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),lattic1091506334969745077in_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int))))) ) ) ).
% Sup_fin.insert_remove
tff(fact_6205_Sup__fin_Oinsert__remove,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( lattic1093996805478795353in_nat(aa_set_nat_set_nat(insert_nat(X),Aa)) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),lattic1093996805478795353in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))))) ) ) ).
% Sup_fin.insert_remove
tff(fact_6206_Sup__fin_Oremove,axiom,
! [Aa: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Aa)
=> ( lattic7388253182610234406_nat_o(Aa) = $ite(minus_6456595905716665654_nat_o(Aa,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = bot_bo8632255151191677115_nat_o,X,aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(X),lattic7388253182610234406_nat_o(minus_6456595905716665654_nat_o(Aa,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))))) ) ) ) ).
% Sup_fin.remove
tff(fact_6207_Sup__fin_Oremove,axiom,
! [Aa: set_assn,X: assn] :
( finite_finite_assn(Aa)
=> ( member_assn(X,Aa)
=> ( lattic2150320897289308081n_assn(Aa) = $ite(minus_minus_set_assn(Aa,insert_assn(X,bot_bot_set_assn)) = bot_bot_set_assn,X,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),lattic2150320897289308081n_assn(minus_minus_set_assn(Aa,insert_assn(X,bot_bot_set_assn))))) ) ) ) ).
% Sup_fin.remove
tff(fact_6208_Sup__fin_Oremove,axiom,
! [Aa: set_set_nat,X: set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( member_set_nat(X,Aa)
=> ( lattic3835124923745554447et_nat(Aa) = $ite(minus_2163939370556025621et_nat(Aa,insert_set_nat(X,bot_bot_set_set_nat)) = bot_bot_set_set_nat,X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),lattic3835124923745554447et_nat(minus_2163939370556025621et_nat(Aa,insert_set_nat(X,bot_bot_set_set_nat))))) ) ) ) ).
% Sup_fin.remove
tff(fact_6209_Sup__fin_Oremove,axiom,
! [Aa: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat] :
( finite4768004253697144688at_nat(Aa)
=> ( member6273874620674182256at_nat(X,Aa)
=> ( lattic8394993312974875902at_nat(Aa) = $ite(minus_5090130588299000206at_nat(Aa,insert2290795007606087959at_nat(X,bot_bo1534825418459847827at_nat)) = bot_bo1534825418459847827at_nat,X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),lattic8394993312974875902at_nat(minus_5090130588299000206at_nat(Aa,insert2290795007606087959at_nat(X,bot_bo1534825418459847827at_nat))))) ) ) ) ).
% Sup_fin.remove
tff(fact_6210_Sup__fin_Oremove,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( aa_set_o_o(member_o((X)),Aa)
=> ( aa_set_o_o(lattic1508158080041050831_fin_o,Aa)
<=> $ite(minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(sup_sup_o,(X)),aa_set_o_o(lattic1508158080041050831_fin_o,minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))))) ) ) ) ).
% Sup_fin.remove
tff(fact_6211_Sup__fin_Oremove,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( aa_set_int_o(member_int(X),Aa)
=> ( lattic1091506334969745077in_int(Aa) = $ite(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int)) = bot_bot_set_int,X,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),lattic1091506334969745077in_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int))))) ) ) ) ).
% Sup_fin.remove
tff(fact_6212_Sup__fin_Oremove,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( aa_set_nat_o(member_nat(X),Aa)
=> ( lattic1093996805478795353in_nat(Aa) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),lattic1093996805478795353in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))))) ) ) ) ).
% Sup_fin.remove
tff(fact_6213_Gcd__fin_Oempty,axiom,
semiri4258706085729940814in_nat(bot_bot_set_nat) = zero_zero_nat ).
% Gcd_fin.empty
tff(fact_6214_Gcd__fin_Oempty,axiom,
semiri4256215615220890538in_int(bot_bot_set_int) = zero_zero_int ).
% Gcd_fin.empty
tff(fact_6215_Sup__fin_Osingleton,axiom,
! [X: $o] :
( aa_set_o_o(lattic1508158080041050831_fin_o,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))
<=> (X) ) ).
% Sup_fin.singleton
tff(fact_6216_Sup__fin_Osingleton,axiom,
! [X: nat] : lattic1093996805478795353in_nat(aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) = X ).
% Sup_fin.singleton
tff(fact_6217_Sup__fin_Osingleton,axiom,
! [X: int] : lattic1091506334969745077in_int(aa_set_int_set_int(insert_int(X),bot_bot_set_int)) = X ).
% Sup_fin.singleton
tff(fact_6218_Gcd__fin_Oinfinite,axiom,
! [Aa: set_nat] :
( ~ aa_set_nat_o(finite_finite_nat,Aa)
=> ( semiri4258706085729940814in_nat(Aa) = one_one_nat ) ) ).
% Gcd_fin.infinite
tff(fact_6219_Gcd__fin_Oinfinite,axiom,
! [Aa: set_int] :
( ~ aa_set_int_o(finite_finite_int,Aa)
=> ( semiri4256215615220890538in_int(Aa) = one_one_int ) ) ).
% Gcd_fin.infinite
tff(fact_6220_is__unit__Gcd__fin__iff,axiom,
! [Aa: set_nat] :
( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,semiri4258706085729940814in_nat(Aa)),one_one_nat)
<=> ( semiri4258706085729940814in_nat(Aa) = one_one_nat ) ) ).
% is_unit_Gcd_fin_iff
tff(fact_6221_is__unit__Gcd__fin__iff,axiom,
! [Aa: set_int] :
( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,semiri4256215615220890538in_int(Aa)),one_one_int)
<=> ( semiri4256215615220890538in_int(Aa) = one_one_int ) ) ).
% is_unit_Gcd_fin_iff
tff(fact_6222_Sup__fin_Oinsert,axiom,
! [Aa: set_assn,X: assn] :
( finite_finite_assn(Aa)
=> ( ( Aa != bot_bot_set_assn )
=> ( lattic2150320897289308081n_assn(insert_assn(X,Aa)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),lattic2150320897289308081n_assn(Aa)) ) ) ) ).
% Sup_fin.insert
tff(fact_6223_Sup__fin_Oinsert,axiom,
! [Aa: set_set_nat,X: set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ( Aa != bot_bot_set_set_nat )
=> ( lattic3835124923745554447et_nat(insert_set_nat(X,Aa)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),lattic3835124923745554447et_nat(Aa)) ) ) ) ).
% Sup_fin.insert
tff(fact_6224_Sup__fin_Oinsert,axiom,
! [Aa: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat] :
( finite4768004253697144688at_nat(Aa)
=> ( ( Aa != bot_bo1534825418459847827at_nat )
=> ( lattic8394993312974875902at_nat(insert2290795007606087959at_nat(X,Aa)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),lattic8394993312974875902at_nat(Aa)) ) ) ) ).
% Sup_fin.insert
tff(fact_6225_Sup__fin_Oinsert,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> ( aa_set_o_o(lattic1508158080041050831_fin_o,aa_set_o_set_o(insert_o((X)),Aa))
<=> aa_o_o(aa_o_fun_o_o(sup_sup_o,(X)),aa_set_o_o(lattic1508158080041050831_fin_o,Aa)) ) ) ) ).
% Sup_fin.insert
tff(fact_6226_Sup__fin_Oinsert,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( lattic1093996805478795353in_nat(aa_set_nat_set_nat(insert_nat(X),Aa)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),lattic1093996805478795353in_nat(Aa)) ) ) ) ).
% Sup_fin.insert
tff(fact_6227_Sup__fin_Oinsert,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> ( lattic1091506334969745077in_int(aa_set_int_set_int(insert_int(X),Aa)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,X),lattic1091506334969745077in_int(Aa)) ) ) ) ).
% Sup_fin.insert
tff(fact_6228_greaterThan__non__empty,axiom,
! [X: int] : set_or1207661135979820486an_int(X) != bot_bot_set_int ).
% greaterThan_non_empty
tff(fact_6229_greaterThan__non__empty,axiom,
! [X: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,X) != bot_bot_set_nat ).
% greaterThan_non_empty
tff(fact_6230_Sup__fin_Oin__idem,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( aa_set_o_o(member_o((X)),Aa)
=> ( aa_o_o(aa_o_fun_o_o(sup_sup_o,(X)),aa_set_o_o(lattic1508158080041050831_fin_o,Aa))
<=> aa_set_o_o(lattic1508158080041050831_fin_o,Aa) ) ) ) ).
% Sup_fin.in_idem
tff(fact_6231_Sup__fin_Oin__idem,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( aa_set_int_o(member_int(X),Aa)
=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),lattic1091506334969745077in_int(Aa)) = lattic1091506334969745077in_int(Aa) ) ) ) ).
% Sup_fin.in_idem
tff(fact_6232_Sup__fin_Oin__idem,axiom,
! [Aa: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Aa)
=> ( aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(X),lattic7388253182610234406_nat_o(Aa)) = lattic7388253182610234406_nat_o(Aa) ) ) ) ).
% Sup_fin.in_idem
tff(fact_6233_Sup__fin_Oin__idem,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( aa_set_nat_o(member_nat(X),Aa)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),lattic1093996805478795353in_nat(Aa)) = lattic1093996805478795353in_nat(Aa) ) ) ) ).
% Sup_fin.in_idem
tff(fact_6234_Sup__fin_Oin__idem,axiom,
! [Aa: set_assn,X: assn] :
( finite_finite_assn(Aa)
=> ( member_assn(X,Aa)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),lattic2150320897289308081n_assn(Aa)) = lattic2150320897289308081n_assn(Aa) ) ) ) ).
% Sup_fin.in_idem
tff(fact_6235_Sup__fin_Oin__idem,axiom,
! [Aa: set_set_nat,X: set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( member_set_nat(X,Aa)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),lattic3835124923745554447et_nat(Aa)) = lattic3835124923745554447et_nat(Aa) ) ) ) ).
% Sup_fin.in_idem
tff(fact_6236_Sup__fin_Oin__idem,axiom,
! [Aa: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat] :
( finite4768004253697144688at_nat(Aa)
=> ( member6273874620674182256at_nat(X,Aa)
=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),lattic8394993312974875902at_nat(Aa)) = lattic8394993312974875902at_nat(Aa) ) ) ) ).
% Sup_fin.in_idem
tff(fact_6237_Sup__fin_OboundedE,axiom,
! [Aa: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( ( Aa != bot_bo8632255151191677115_nat_o )
=> ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,lattic7388253182610234406_nat_o(Aa)),X)
=> ! [A7: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A7),Aa)
=> aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,A7),X) ) ) ) ) ).
% Sup_fin.boundedE
tff(fact_6238_Sup__fin_OboundedE,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1508158080041050831_fin_o,Aa)),(X))
=> ! [A7: $o] :
( aa_set_o_o(member_o((A7)),Aa)
=> aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A7)),(X)) ) ) ) ) ).
% Sup_fin.boundedE
tff(fact_6239_Sup__fin_OboundedE,axiom,
! [Aa: set_assn,X: assn] :
( finite_finite_assn(Aa)
=> ( ( Aa != bot_bot_set_assn )
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,lattic2150320897289308081n_assn(Aa)),X)
=> ! [A7: assn] :
( member_assn(A7,Aa)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A7),X) ) ) ) ) ).
% Sup_fin.boundedE
tff(fact_6240_Sup__fin_OboundedE,axiom,
! [Aa: set_set_nat,X: set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ( Aa != bot_bot_set_set_nat )
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3835124923745554447et_nat(Aa)),X)
=> ! [A7: set_nat] :
( member_set_nat(A7,Aa)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A7),X) ) ) ) ) ).
% Sup_fin.boundedE
tff(fact_6241_Sup__fin_OboundedE,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic1093996805478795353in_nat(Aa)),X)
=> ! [A7: nat] :
( aa_set_nat_o(member_nat(A7),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A7),X) ) ) ) ) ).
% Sup_fin.boundedE
tff(fact_6242_Sup__fin_OboundedE,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic1091506334969745077in_int(Aa)),X)
=> ! [A7: int] :
( aa_set_int_o(member_int(A7),Aa)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A7),X) ) ) ) ) ).
% Sup_fin.boundedE
tff(fact_6243_Sup__fin_OboundedI,axiom,
! [Aa: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( ( Aa != bot_bo8632255151191677115_nat_o )
=> ( ! [A3: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),Aa)
=> aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,A3),X) )
=> aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,lattic7388253182610234406_nat_o(Aa)),X) ) ) ) ).
% Sup_fin.boundedI
tff(fact_6244_Sup__fin_OboundedI,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> ( ! [A3: $o] :
( aa_set_o_o(member_o((A3)),Aa)
=> aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A3)),(X)) )
=> aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1508158080041050831_fin_o,Aa)),(X)) ) ) ) ).
% Sup_fin.boundedI
tff(fact_6245_Sup__fin_OboundedI,axiom,
! [Aa: set_assn,X: assn] :
( finite_finite_assn(Aa)
=> ( ( Aa != bot_bot_set_assn )
=> ( ! [A3: assn] :
( member_assn(A3,Aa)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A3),X) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,lattic2150320897289308081n_assn(Aa)),X) ) ) ) ).
% Sup_fin.boundedI
tff(fact_6246_Sup__fin_OboundedI,axiom,
! [Aa: set_set_nat,X: set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ( Aa != bot_bot_set_set_nat )
=> ( ! [A3: set_nat] :
( member_set_nat(A3,Aa)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),X) )
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3835124923745554447et_nat(Aa)),X) ) ) ) ).
% Sup_fin.boundedI
tff(fact_6247_Sup__fin_OboundedI,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( ! [A3: nat] :
( aa_set_nat_o(member_nat(A3),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),X) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic1093996805478795353in_nat(Aa)),X) ) ) ) ).
% Sup_fin.boundedI
tff(fact_6248_Sup__fin_OboundedI,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> ( ! [A3: int] :
( aa_set_int_o(member_int(A3),Aa)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),X) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic1091506334969745077in_int(Aa)),X) ) ) ) ).
% Sup_fin.boundedI
tff(fact_6249_Sup__fin_Obounded__iff,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1508158080041050831_fin_o,Aa)),(X))
<=> ! [X3: $o] :
( aa_set_o_o(member_o((X3)),Aa)
=> aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X3)),(X)) ) ) ) ) ).
% Sup_fin.bounded_iff
tff(fact_6250_Sup__fin_Obounded__iff,axiom,
! [Aa: set_assn,X: assn] :
( finite_finite_assn(Aa)
=> ( ( Aa != bot_bot_set_assn )
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,lattic2150320897289308081n_assn(Aa)),X)
<=> ! [X3: assn] :
( member_assn(X3,Aa)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X3),X) ) ) ) ) ).
% Sup_fin.bounded_iff
tff(fact_6251_Sup__fin_Obounded__iff,axiom,
! [Aa: set_set_nat,X: set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ( Aa != bot_bot_set_set_nat )
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3835124923745554447et_nat(Aa)),X)
<=> ! [X3: set_nat] :
( member_set_nat(X3,Aa)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),X) ) ) ) ) ).
% Sup_fin.bounded_iff
tff(fact_6252_Sup__fin_Obounded__iff,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic1093996805478795353in_nat(Aa)),X)
<=> ! [X3: nat] :
( aa_set_nat_o(member_nat(X3),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),X) ) ) ) ) ).
% Sup_fin.bounded_iff
tff(fact_6253_Sup__fin_Obounded__iff,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic1091506334969745077in_int(Aa)),X)
<=> ! [X3: int] :
( aa_set_int_o(member_int(X3),Aa)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),X) ) ) ) ) ).
% Sup_fin.bounded_iff
tff(fact_6254_ivl__disj__int__one_I7_J,axiom,
! [L: $o,U: $o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,set_or8904488021354931149Most_o((L),(U))),set_or6416164934427428222Than_o((U))) = bot_bot_set_o ).
% ivl_disj_int_one(7)
tff(fact_6255_ivl__disj__int__one_I7_J,axiom,
! [L: int,U: int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,set_or1266510415728281911st_int(L,U)),set_or1207661135979820486an_int(U)) = bot_bot_set_int ).
% ivl_disj_int_one(7)
tff(fact_6256_ivl__disj__int__one_I7_J,axiom,
! [L: nat,U: nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)),aa_nat_set_nat(set_or1210151606488870762an_nat,U)) = bot_bot_set_nat ).
% ivl_disj_int_one(7)
tff(fact_6257_Sup__fin_Osubset__imp,axiom,
! [Aa: set_o,Ba: set_o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Aa),Ba)
=> ( ( Aa != bot_bot_set_o )
=> ( aa_set_o_o(finite_finite_o,Ba)
=> aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1508158080041050831_fin_o,Aa)),aa_set_o_o(lattic1508158080041050831_fin_o,Ba)) ) ) ) ).
% Sup_fin.subset_imp
tff(fact_6258_Sup__fin_Osubset__imp,axiom,
! [Aa: set_assn,Ba: set_assn] :
( ord_less_eq_set_assn(Aa,Ba)
=> ( ( Aa != bot_bot_set_assn )
=> ( finite_finite_assn(Ba)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,lattic2150320897289308081n_assn(Aa)),lattic2150320897289308081n_assn(Ba)) ) ) ) ).
% Sup_fin.subset_imp
tff(fact_6259_Sup__fin_Osubset__imp,axiom,
! [Aa: set_set_nat,Ba: set_set_nat] :
( ord_le6893508408891458716et_nat(Aa,Ba)
=> ( ( Aa != bot_bot_set_set_nat )
=> ( finite1152437895449049373et_nat(Ba)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3835124923745554447et_nat(Aa)),lattic3835124923745554447et_nat(Ba)) ) ) ) ).
% Sup_fin.subset_imp
tff(fact_6260_Sup__fin_Osubset__imp,axiom,
! [Aa: set_int,Ba: set_int] :
( aa_set_int_o(ord_less_eq_set_int(Aa),Ba)
=> ( ( Aa != bot_bot_set_int )
=> ( aa_set_int_o(finite_finite_int,Ba)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic1091506334969745077in_int(Aa)),lattic1091506334969745077in_int(Ba)) ) ) ) ).
% Sup_fin.subset_imp
tff(fact_6261_Sup__fin_Osubset__imp,axiom,
! [Aa: set_nat,Ba: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),Ba)
=> ( ( Aa != bot_bot_set_nat )
=> ( aa_set_nat_o(finite_finite_nat,Ba)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic1093996805478795353in_nat(Aa)),lattic1093996805478795353in_nat(Ba)) ) ) ) ).
% Sup_fin.subset_imp
tff(fact_6262_Sup__fin_Osubset,axiom,
! [Aa: set_assn,Ba: set_assn] :
( finite_finite_assn(Aa)
=> ( ( Ba != bot_bot_set_assn )
=> ( ord_less_eq_set_assn(Ba,Aa)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,lattic2150320897289308081n_assn(Ba)),lattic2150320897289308081n_assn(Aa)) = lattic2150320897289308081n_assn(Aa) ) ) ) ) ).
% Sup_fin.subset
tff(fact_6263_Sup__fin_Osubset,axiom,
! [Aa: set_set_nat,Ba: set_set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ( Ba != bot_bot_set_set_nat )
=> ( ord_le6893508408891458716et_nat(Ba,Aa)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,lattic3835124923745554447et_nat(Ba)),lattic3835124923745554447et_nat(Aa)) = lattic3835124923745554447et_nat(Aa) ) ) ) ) ).
% Sup_fin.subset
tff(fact_6264_Sup__fin_Osubset,axiom,
! [Aa: set_se5479201105039950375at_nat,Ba: set_se5479201105039950375at_nat] :
( finite4768004253697144688at_nat(Aa)
=> ( ( Ba != bot_bo1534825418459847827at_nat )
=> ( ord_le7966958417815640519at_nat(Ba,Aa)
=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,lattic8394993312974875902at_nat(Ba)),lattic8394993312974875902at_nat(Aa)) = lattic8394993312974875902at_nat(Aa) ) ) ) ) ).
% Sup_fin.subset
tff(fact_6265_Sup__fin_Osubset,axiom,
! [Aa: set_o,Ba: set_o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Ba != bot_bot_set_o )
=> ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Ba),Aa)
=> ( aa_o_o(aa_o_fun_o_o(sup_sup_o,aa_set_o_o(lattic1508158080041050831_fin_o,Ba)),aa_set_o_o(lattic1508158080041050831_fin_o,Aa))
<=> aa_set_o_o(lattic1508158080041050831_fin_o,Aa) ) ) ) ) ).
% Sup_fin.subset
tff(fact_6266_Sup__fin_Osubset,axiom,
! [Aa: set_int,Ba: set_int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Ba != bot_bot_set_int )
=> ( aa_set_int_o(ord_less_eq_set_int(Ba),Aa)
=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,lattic1091506334969745077in_int(Ba)),lattic1091506334969745077in_int(Aa)) = lattic1091506334969745077in_int(Aa) ) ) ) ) ).
% Sup_fin.subset
tff(fact_6267_Sup__fin_Osubset,axiom,
! [Aa: set_nat,Ba: set_nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Ba != bot_bot_set_nat )
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Ba),Aa)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,lattic1093996805478795353in_nat(Ba)),lattic1093996805478795353in_nat(Aa)) = lattic1093996805478795353in_nat(Aa) ) ) ) ) ).
% Sup_fin.subset
tff(fact_6268_Sup__fin_Oclosed,axiom,
! [Aa: set_fu3555266933746557007_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( ( Aa != bot_bo8632255151191677115_nat_o )
=> ( ! [X2: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(X2),Y2)),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X2),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(Y2),bot_bo8632255151191677115_nat_o)))
=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(lattic7388253182610234406_nat_o(Aa)),Aa) ) ) ) ).
% Sup_fin.closed
tff(fact_6269_Sup__fin_Oclosed,axiom,
! [Aa: set_assn] :
( finite_finite_assn(Aa)
=> ( ( Aa != bot_bot_set_assn )
=> ( ! [X2: assn,Y2: assn] : member_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X2),Y2),insert_assn(X2,insert_assn(Y2,bot_bot_set_assn)))
=> member_assn(lattic2150320897289308081n_assn(Aa),Aa) ) ) ) ).
% Sup_fin.closed
tff(fact_6270_Sup__fin_Oclosed,axiom,
! [Aa: set_set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ( Aa != bot_bot_set_set_nat )
=> ( ! [X2: set_nat,Y2: set_nat] : member_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),Y2),insert_set_nat(X2,insert_set_nat(Y2,bot_bot_set_set_nat)))
=> member_set_nat(lattic3835124923745554447et_nat(Aa),Aa) ) ) ) ).
% Sup_fin.closed
tff(fact_6271_Sup__fin_Oclosed,axiom,
! [Aa: set_se5479201105039950375at_nat] :
( finite4768004253697144688at_nat(Aa)
=> ( ( Aa != bot_bo1534825418459847827at_nat )
=> ( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : member6273874620674182256at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X2),Y2),insert2290795007606087959at_nat(X2,insert2290795007606087959at_nat(Y2,bot_bo1534825418459847827at_nat)))
=> member6273874620674182256at_nat(lattic8394993312974875902at_nat(Aa),Aa) ) ) ) ).
% Sup_fin.closed
tff(fact_6272_Sup__fin_Oclosed,axiom,
! [Aa: set_o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> ( ! [X2: $o,Y2: $o] : aa_set_o_o(member_o(aa_o_o(aa_o_fun_o_o(sup_sup_o,(X2)),(Y2))),aa_set_o_set_o(insert_o((X2)),aa_set_o_set_o(insert_o((Y2)),bot_bot_set_o)))
=> aa_set_o_o(member_o(aa_set_o_o(lattic1508158080041050831_fin_o,Aa)),Aa) ) ) ) ).
% Sup_fin.closed
tff(fact_6273_Sup__fin_Oclosed,axiom,
! [Aa: set_nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] : aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X2),Y2)),aa_set_nat_set_nat(insert_nat(X2),aa_set_nat_set_nat(insert_nat(Y2),bot_bot_set_nat)))
=> aa_set_nat_o(member_nat(lattic1093996805478795353in_nat(Aa)),Aa) ) ) ) ).
% Sup_fin.closed
tff(fact_6274_Sup__fin_Oclosed,axiom,
! [Aa: set_int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> ( ! [X2: int,Y2: int] : aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(sup_sup_int,X2),Y2)),aa_set_int_set_int(insert_int(X2),aa_set_int_set_int(insert_int(Y2),bot_bot_set_int)))
=> aa_set_int_o(member_int(lattic1091506334969745077in_int(Aa)),Aa) ) ) ) ).
% Sup_fin.closed
tff(fact_6275_Sup__fin_Oinsert__not__elem,axiom,
! [Aa: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Aa)
=> ( ( Aa != bot_bo8632255151191677115_nat_o )
=> ( lattic7388253182610234406_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),Aa)) = aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(X),lattic7388253182610234406_nat_o(Aa)) ) ) ) ) ).
% Sup_fin.insert_not_elem
tff(fact_6276_Sup__fin_Oinsert__not__elem,axiom,
! [Aa: set_assn,X: assn] :
( finite_finite_assn(Aa)
=> ( ~ member_assn(X,Aa)
=> ( ( Aa != bot_bot_set_assn )
=> ( lattic2150320897289308081n_assn(insert_assn(X,Aa)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),lattic2150320897289308081n_assn(Aa)) ) ) ) ) ).
% Sup_fin.insert_not_elem
tff(fact_6277_Sup__fin_Oinsert__not__elem,axiom,
! [Aa: set_set_nat,X: set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ~ member_set_nat(X,Aa)
=> ( ( Aa != bot_bot_set_set_nat )
=> ( lattic3835124923745554447et_nat(insert_set_nat(X,Aa)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),lattic3835124923745554447et_nat(Aa)) ) ) ) ) ).
% Sup_fin.insert_not_elem
tff(fact_6278_Sup__fin_Oinsert__not__elem,axiom,
! [Aa: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat] :
( finite4768004253697144688at_nat(Aa)
=> ( ~ member6273874620674182256at_nat(X,Aa)
=> ( ( Aa != bot_bo1534825418459847827at_nat )
=> ( lattic8394993312974875902at_nat(insert2290795007606087959at_nat(X,Aa)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),lattic8394993312974875902at_nat(Aa)) ) ) ) ) ).
% Sup_fin.insert_not_elem
tff(fact_6279_Sup__fin_Oinsert__not__elem,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ~ aa_set_o_o(member_o((X)),Aa)
=> ( ( Aa != bot_bot_set_o )
=> ( aa_set_o_o(lattic1508158080041050831_fin_o,aa_set_o_set_o(insert_o((X)),Aa))
<=> aa_o_o(aa_o_fun_o_o(sup_sup_o,(X)),aa_set_o_o(lattic1508158080041050831_fin_o,Aa)) ) ) ) ) ).
% Sup_fin.insert_not_elem
tff(fact_6280_Sup__fin_Oinsert__not__elem,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ~ aa_set_nat_o(member_nat(X),Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( lattic1093996805478795353in_nat(aa_set_nat_set_nat(insert_nat(X),Aa)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),lattic1093996805478795353in_nat(Aa)) ) ) ) ) ).
% Sup_fin.insert_not_elem
tff(fact_6281_Sup__fin_Oinsert__not__elem,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ~ aa_set_int_o(member_int(X),Aa)
=> ( ( Aa != bot_bot_set_int )
=> ( lattic1091506334969745077in_int(aa_set_int_set_int(insert_int(X),Aa)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,X),lattic1091506334969745077in_int(Aa)) ) ) ) ) ).
% Sup_fin.insert_not_elem
tff(fact_6282_Sup__fin_Ounion,axiom,
! [Aa: set_assn,Ba: set_assn] :
( finite_finite_assn(Aa)
=> ( ( Aa != bot_bot_set_assn )
=> ( finite_finite_assn(Ba)
=> ( ( Ba != bot_bot_set_assn )
=> ( lattic2150320897289308081n_assn(sup_sup_set_assn(Aa,Ba)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,lattic2150320897289308081n_assn(Aa)),lattic2150320897289308081n_assn(Ba)) ) ) ) ) ) ).
% Sup_fin.union
tff(fact_6283_Sup__fin_Ounion,axiom,
! [Aa: set_set_nat,Ba: set_set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ( Aa != bot_bot_set_set_nat )
=> ( finite1152437895449049373et_nat(Ba)
=> ( ( Ba != bot_bot_set_set_nat )
=> ( lattic3835124923745554447et_nat(sup_sup_set_set_nat(Aa,Ba)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,lattic3835124923745554447et_nat(Aa)),lattic3835124923745554447et_nat(Ba)) ) ) ) ) ) ).
% Sup_fin.union
tff(fact_6284_Sup__fin_Ounion,axiom,
! [Aa: set_se5479201105039950375at_nat,Ba: set_se5479201105039950375at_nat] :
( finite4768004253697144688at_nat(Aa)
=> ( ( Aa != bot_bo1534825418459847827at_nat )
=> ( finite4768004253697144688at_nat(Ba)
=> ( ( Ba != bot_bo1534825418459847827at_nat )
=> ( lattic8394993312974875902at_nat(sup_su474284620922681339at_nat(Aa,Ba)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,lattic8394993312974875902at_nat(Aa)),lattic8394993312974875902at_nat(Ba)) ) ) ) ) ) ).
% Sup_fin.union
tff(fact_6285_Sup__fin_Ounion,axiom,
! [Aa: set_o,Ba: set_o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> ( aa_set_o_o(finite_finite_o,Ba)
=> ( ( Ba != bot_bot_set_o )
=> ( aa_set_o_o(lattic1508158080041050831_fin_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Aa),Ba))
<=> aa_o_o(aa_o_fun_o_o(sup_sup_o,aa_set_o_o(lattic1508158080041050831_fin_o,Aa)),aa_set_o_o(lattic1508158080041050831_fin_o,Ba)) ) ) ) ) ) ).
% Sup_fin.union
tff(fact_6286_Sup__fin_Ounion,axiom,
! [Aa: set_nat,Ba: set_nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( aa_set_nat_o(finite_finite_nat,Ba)
=> ( ( Ba != bot_bot_set_nat )
=> ( lattic1093996805478795353in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Aa),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,lattic1093996805478795353in_nat(Aa)),lattic1093996805478795353in_nat(Ba)) ) ) ) ) ) ).
% Sup_fin.union
tff(fact_6287_Sup__fin_Ounion,axiom,
! [Aa: set_int,Ba: set_int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> ( aa_set_int_o(finite_finite_int,Ba)
=> ( ( Ba != bot_bot_set_int )
=> ( lattic1091506334969745077in_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Aa),Ba)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,lattic1091506334969745077in_int(Aa)),lattic1091506334969745077in_int(Ba)) ) ) ) ) ) ).
% Sup_fin.union
tff(fact_6288_enumerate__Suc_H,axiom,
! [S: set_nat,N: nat] : aa_nat_nat(infini8530281810654367211te_nat(S),aa_nat_nat(suc,N)) = aa_nat_nat(infini8530281810654367211te_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),aa_set_nat_set_nat(insert_nat(aa_nat_nat(infini8530281810654367211te_nat(S),zero_zero_nat)),bot_bot_set_nat))),N) ).
% enumerate_Suc'
tff(fact_6289_Inf__fin_Oremove,axiom,
! [Aa: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Aa)
=> ( lattic6089491107740071820_nat_o(Aa) = $ite(minus_6456595905716665654_nat_o(Aa,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = bot_bo8632255151191677115_nat_o,X,aa_fun2638059832795814268_nat_o(inf_in4424744116413743581_nat_o(X),lattic6089491107740071820_nat_o(minus_6456595905716665654_nat_o(Aa,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))))) ) ) ) ).
% Inf_fin.remove
tff(fact_6290_Inf__fin_Oremove,axiom,
! [Aa: set_set_nat,X: set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( member_set_nat(X,Aa)
=> ( lattic3014633134055518761et_nat(Aa) = $ite(minus_2163939370556025621et_nat(Aa,insert_set_nat(X,bot_bot_set_set_nat)) = bot_bot_set_set_nat,X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),lattic3014633134055518761et_nat(minus_2163939370556025621et_nat(Aa,insert_set_nat(X,bot_bot_set_set_nat))))) ) ) ) ).
% Inf_fin.remove
tff(fact_6291_Inf__fin_Oremove,axiom,
! [Aa: set_assn,X: assn] :
( finite_finite_assn(Aa)
=> ( member_assn(X,Aa)
=> ( lattic47131356835913163n_assn(Aa) = $ite(minus_minus_set_assn(Aa,insert_assn(X,bot_bot_set_assn)) = bot_bot_set_assn,X,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),lattic47131356835913163n_assn(minus_minus_set_assn(Aa,insert_assn(X,bot_bot_set_assn))))) ) ) ) ).
% Inf_fin.remove
tff(fact_6292_Inf__fin_Oremove,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( aa_set_o_o(member_o((X)),Aa)
=> ( aa_set_o_o(lattic4107685809792843317_fin_o,Aa)
<=> $ite(minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(inf_inf_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))))) ) ) ) ).
% Inf_fin.remove
tff(fact_6293_Inf__fin_Oremove,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( aa_set_int_o(member_int(X),Aa)
=> ( lattic5235898064620869839in_int(Aa) = $ite(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int)) = bot_bot_set_int,X,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),lattic5235898064620869839in_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int))))) ) ) ) ).
% Inf_fin.remove
tff(fact_6294_Inf__fin_Oremove,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( aa_set_nat_o(member_nat(X),Aa)
=> ( lattic5238388535129920115in_nat(Aa) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),lattic5238388535129920115in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))))) ) ) ) ).
% Inf_fin.remove
tff(fact_6295_Inf__fin_Oinsert__remove,axiom,
! [Aa: set_set_nat,X: set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( lattic3014633134055518761et_nat(insert_set_nat(X,Aa)) = $ite(minus_2163939370556025621et_nat(Aa,insert_set_nat(X,bot_bot_set_set_nat)) = bot_bot_set_set_nat,X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),lattic3014633134055518761et_nat(minus_2163939370556025621et_nat(Aa,insert_set_nat(X,bot_bot_set_set_nat))))) ) ) ).
% Inf_fin.insert_remove
tff(fact_6296_Inf__fin_Oinsert__remove,axiom,
! [Aa: set_assn,X: assn] :
( finite_finite_assn(Aa)
=> ( lattic47131356835913163n_assn(insert_assn(X,Aa)) = $ite(minus_minus_set_assn(Aa,insert_assn(X,bot_bot_set_assn)) = bot_bot_set_assn,X,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),lattic47131356835913163n_assn(minus_minus_set_assn(Aa,insert_assn(X,bot_bot_set_assn))))) ) ) ).
% Inf_fin.insert_remove
tff(fact_6297_Inf__fin_Oinsert__remove,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( aa_set_o_o(lattic4107685809792843317_fin_o,aa_set_o_set_o(insert_o((X)),Aa))
<=> $ite(minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(inf_inf_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))))) ) ) ).
% Inf_fin.insert_remove
tff(fact_6298_Inf__fin_Oinsert__remove,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( lattic5235898064620869839in_int(aa_set_int_set_int(insert_int(X),Aa)) = $ite(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int)) = bot_bot_set_int,X,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),lattic5235898064620869839in_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int))))) ) ) ).
% Inf_fin.insert_remove
tff(fact_6299_Inf__fin_Oinsert__remove,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( lattic5238388535129920115in_nat(aa_set_nat_set_nat(insert_nat(X),Aa)) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),lattic5238388535129920115in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))))) ) ) ).
% Inf_fin.insert_remove
tff(fact_6300_Inf__fin_Oinsert,axiom,
! [Aa: set_set_nat,X: set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ( Aa != bot_bot_set_set_nat )
=> ( lattic3014633134055518761et_nat(insert_set_nat(X,Aa)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),lattic3014633134055518761et_nat(Aa)) ) ) ) ).
% Inf_fin.insert
tff(fact_6301_Inf__fin_Oinsert,axiom,
! [Aa: set_assn,X: assn] :
( finite_finite_assn(Aa)
=> ( ( Aa != bot_bot_set_assn )
=> ( lattic47131356835913163n_assn(insert_assn(X,Aa)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),lattic47131356835913163n_assn(Aa)) ) ) ) ).
% Inf_fin.insert
tff(fact_6302_Inf__fin_Oinsert,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> ( aa_set_o_o(lattic4107685809792843317_fin_o,aa_set_o_set_o(insert_o((X)),Aa))
<=> aa_o_o(aa_o_fun_o_o(inf_inf_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,Aa)) ) ) ) ).
% Inf_fin.insert
tff(fact_6303_Inf__fin_Oinsert,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( lattic5238388535129920115in_nat(aa_set_nat_set_nat(insert_nat(X),Aa)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),lattic5238388535129920115in_nat(Aa)) ) ) ) ).
% Inf_fin.insert
tff(fact_6304_Inf__fin_Oinsert,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> ( lattic5235898064620869839in_int(aa_set_int_set_int(insert_int(X),Aa)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,X),lattic5235898064620869839in_int(Aa)) ) ) ) ).
% Inf_fin.insert
tff(fact_6305_image__mult__atLeastAtMost__if,axiom,
! [C: rat,X: rat,Y: rat] :
image_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),set_or633870826150836451st_rat(X,Y)) = $ite(
aa_rat_o(ord_less_rat(zero_zero_rat),C),
set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),Y)),
$ite(aa_rat_o(ord_less_eq_rat(X),Y),set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),Y),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),X)),bot_bot_set_rat) ) ).
% image_mult_atLeastAtMost_if
tff(fact_6306_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_atMost_nat,N)),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat)) ).
% atLeast1_atMost_eq_remove0
tff(fact_6307_ivl__disj__un__singleton_I5_J,axiom,
! [L: $o,U: $o] :
( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(L)),(U))
=> ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(insert_o((L)),bot_bot_set_o)),set_or8254209520273421544Most_o((L),(U))) = set_or8904488021354931149Most_o((L),(U)) ) ) ).
% ivl_disj_un_singleton(5)
tff(fact_6308_ivl__disj__un__singleton_I5_J,axiom,
! [L: code_integer,U: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),U)
=> ( sup_su848401254843788991nteger(insert_Code_integer(L,bot_bo3990330152332043303nteger),set_or2715278749043346189nteger(L,U)) = set_or189985376899183464nteger(L,U) ) ) ).
% ivl_disj_un_singleton(5)
tff(fact_6309_ivl__disj__un__singleton_I5_J,axiom,
! [L: int,U: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),U)
=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(insert_int(L),bot_bot_set_int)),set_or6656581121297822940st_int(L,U)) = set_or1266510415728281911st_int(L,U) ) ) ).
% ivl_disj_un_singleton(5)
tff(fact_6310_ivl__disj__un__singleton_I5_J,axiom,
! [L: nat,U: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),U)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(insert_nat(L),bot_bot_set_nat)),aa_nat_set_nat(set_or6659071591806873216st_nat(L),U)) = aa_nat_set_nat(set_or1269000886237332187st_nat(L),U) ) ) ).
% ivl_disj_un_singleton(5)
tff(fact_6311_image__is__empty,axiom,
! [F: fun_o_o,Aa: set_o] :
( ( image_o_o(F,Aa) = bot_bot_set_o )
<=> ( Aa = bot_bot_set_o ) ) ).
% image_is_empty
tff(fact_6312_image__is__empty,axiom,
! [F: fun_nat_o,Aa: set_nat] :
( ( image_nat_o(F,Aa) = bot_bot_set_o )
<=> ( Aa = bot_bot_set_nat ) ) ).
% image_is_empty
tff(fact_6313_image__is__empty,axiom,
! [F: fun_int_o,Aa: set_int] :
( ( image_int_o(F,Aa) = bot_bot_set_o )
<=> ( Aa = bot_bot_set_int ) ) ).
% image_is_empty
tff(fact_6314_image__is__empty,axiom,
! [F: fun_o_nat,Aa: set_o] :
( ( image_o_nat(F,Aa) = bot_bot_set_nat )
<=> ( Aa = bot_bot_set_o ) ) ).
% image_is_empty
tff(fact_6315_image__is__empty,axiom,
! [F: fun_nat_nat,Aa: set_nat] :
( ( aa_set_nat_set_nat(image_nat_nat(F),Aa) = bot_bot_set_nat )
<=> ( Aa = bot_bot_set_nat ) ) ).
% image_is_empty
tff(fact_6316_image__is__empty,axiom,
! [F: fun_int_nat,Aa: set_int] :
( ( image_int_nat(F,Aa) = bot_bot_set_nat )
<=> ( Aa = bot_bot_set_int ) ) ).
% image_is_empty
tff(fact_6317_image__is__empty,axiom,
! [F: fun_o_int,Aa: set_o] :
( ( image_o_int(F,Aa) = bot_bot_set_int )
<=> ( Aa = bot_bot_set_o ) ) ).
% image_is_empty
tff(fact_6318_image__is__empty,axiom,
! [F: fun_nat_int,Aa: set_nat] :
( ( image_nat_int(F,Aa) = bot_bot_set_int )
<=> ( Aa = bot_bot_set_nat ) ) ).
% image_is_empty
tff(fact_6319_image__is__empty,axiom,
! [F: fun_int_int,Aa: set_int] :
( ( image_int_int(F,Aa) = bot_bot_set_int )
<=> ( Aa = bot_bot_set_int ) ) ).
% image_is_empty
tff(fact_6320_image__is__empty,axiom,
! [F: fun_nat_set_nat,Aa: set_nat] :
( ( image_nat_set_nat(F,Aa) = bot_bot_set_set_nat )
<=> ( Aa = bot_bot_set_nat ) ) ).
% image_is_empty
tff(fact_6321_empty__is__image,axiom,
! [F: fun_o_o,Aa: set_o] :
( ( bot_bot_set_o = image_o_o(F,Aa) )
<=> ( Aa = bot_bot_set_o ) ) ).
% empty_is_image
tff(fact_6322_empty__is__image,axiom,
! [F: fun_nat_o,Aa: set_nat] :
( ( bot_bot_set_o = image_nat_o(F,Aa) )
<=> ( Aa = bot_bot_set_nat ) ) ).
% empty_is_image
tff(fact_6323_empty__is__image,axiom,
! [F: fun_int_o,Aa: set_int] :
( ( bot_bot_set_o = image_int_o(F,Aa) )
<=> ( Aa = bot_bot_set_int ) ) ).
% empty_is_image
tff(fact_6324_empty__is__image,axiom,
! [F: fun_o_nat,Aa: set_o] :
( ( bot_bot_set_nat = image_o_nat(F,Aa) )
<=> ( Aa = bot_bot_set_o ) ) ).
% empty_is_image
tff(fact_6325_empty__is__image,axiom,
! [F: fun_nat_nat,Aa: set_nat] :
( ( bot_bot_set_nat = aa_set_nat_set_nat(image_nat_nat(F),Aa) )
<=> ( Aa = bot_bot_set_nat ) ) ).
% empty_is_image
tff(fact_6326_empty__is__image,axiom,
! [F: fun_int_nat,Aa: set_int] :
( ( bot_bot_set_nat = image_int_nat(F,Aa) )
<=> ( Aa = bot_bot_set_int ) ) ).
% empty_is_image
tff(fact_6327_empty__is__image,axiom,
! [F: fun_o_int,Aa: set_o] :
( ( bot_bot_set_int = image_o_int(F,Aa) )
<=> ( Aa = bot_bot_set_o ) ) ).
% empty_is_image
tff(fact_6328_empty__is__image,axiom,
! [F: fun_nat_int,Aa: set_nat] :
( ( bot_bot_set_int = image_nat_int(F,Aa) )
<=> ( Aa = bot_bot_set_nat ) ) ).
% empty_is_image
tff(fact_6329_empty__is__image,axiom,
! [F: fun_int_int,Aa: set_int] :
( ( bot_bot_set_int = image_int_int(F,Aa) )
<=> ( Aa = bot_bot_set_int ) ) ).
% empty_is_image
tff(fact_6330_empty__is__image,axiom,
! [F: fun_nat_set_nat,Aa: set_nat] :
( ( bot_bot_set_set_nat = image_nat_set_nat(F,Aa) )
<=> ( Aa = bot_bot_set_nat ) ) ).
% empty_is_image
tff(fact_6331_image__empty,axiom,
! [F: fun_o_o] : image_o_o(F,bot_bot_set_o) = bot_bot_set_o ).
% image_empty
tff(fact_6332_image__empty,axiom,
! [F: fun_o_nat] : image_o_nat(F,bot_bot_set_o) = bot_bot_set_nat ).
% image_empty
tff(fact_6333_image__empty,axiom,
! [F: fun_o_int] : image_o_int(F,bot_bot_set_o) = bot_bot_set_int ).
% image_empty
tff(fact_6334_image__empty,axiom,
! [F: fun_nat_o] : image_nat_o(F,bot_bot_set_nat) = bot_bot_set_o ).
% image_empty
tff(fact_6335_image__empty,axiom,
! [F: fun_nat_nat] : aa_set_nat_set_nat(image_nat_nat(F),bot_bot_set_nat) = bot_bot_set_nat ).
% image_empty
tff(fact_6336_image__empty,axiom,
! [F: fun_nat_int] : image_nat_int(F,bot_bot_set_nat) = bot_bot_set_int ).
% image_empty
tff(fact_6337_image__empty,axiom,
! [F: fun_int_o] : image_int_o(F,bot_bot_set_int) = bot_bot_set_o ).
% image_empty
tff(fact_6338_image__empty,axiom,
! [F: fun_int_nat] : image_int_nat(F,bot_bot_set_int) = bot_bot_set_nat ).
% image_empty
tff(fact_6339_image__empty,axiom,
! [F: fun_int_int] : image_int_int(F,bot_bot_set_int) = bot_bot_set_int ).
% image_empty
tff(fact_6340_image__empty,axiom,
! [F: fun_nat_set_nat] : image_nat_set_nat(F,bot_bot_set_nat) = bot_bot_set_set_nat ).
% image_empty
tff(fact_6341_img__fst,axiom,
! [A: nat,B: nat,S: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),S)
=> aa_set_nat_o(member_nat(A),image_2486076414777270412at_nat(product_fst_nat_nat,S)) ) ).
% img_fst
tff(fact_6342_img__fst,axiom,
! [A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,S: set_Pr3486312109254157132nteger] :
( member8978266922497061677nteger(produc565448451787809096nteger(A,B),S)
=> member8464590939221061905e_term(A,image_5882744770105888791e_term(produc2481150572478215260nteger,S)) ) ).
% img_fst
tff(fact_6343_img__fst,axiom,
! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,S: set_Pr1312406793078187350et_nat] :
( member4513763875858039863et_nat(produc6017135503889078682et_nat(A,B),S)
=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),image_4444162529858428168_nat_o(produc5325900173150335662et_nat,S)) ) ).
% img_fst
tff(fact_6344_img__fst,axiom,
! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,S: set_Pr4040394263813967220et_nat] :
( member9174161442226468181et_nat(produc7291855603632349880et_nat(A,B),S)
=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),image_8400968272122294250_nat_o(produc4078497350495916492et_nat,S)) ) ).
% img_fst
tff(fact_6345_img__fst,axiom,
! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,S: set_Pr6700895561065273913nt_int] :
( member753276645541486746nt_int(produc7320969046608631221nt_int(A,B),S)
=> member1495369047980753794e_term(A,image_143812984558558581e_term(produc3130796691802200265nt_int,S)) ) ).
% img_fst
tff(fact_6346_img__fst,axiom,
! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,S: set_Pr4166680992137330533nt_int] :
( member5890090881338784710nt_int(produc4729913460834754785nt_int(A,B),S)
=> member4871225538671630166e_term(A,image_7516633138899002997e_term(produc4463008262350210037nt_int,S)) ) ).
% img_fst
tff(fact_6347_img__fst,axiom,
! [A: heap_e7401611519738050253t_unit,B: set_nat,S: set_Pr3948176798113811640et_nat] :
( member6260224972018164377et_nat(produc7507926704131184380et_nat(A,B),S)
=> member242389114213480694t_unit(A,image_6824039242672219652t_unit(produc1824681642469235216et_nat,S)) ) ).
% img_fst
tff(fact_6348_img__fst,axiom,
! [A: int,B: int,S: set_Pr958786334691620121nt_int] :
( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A),B),S)
=> aa_set_int_o(member_int(A),image_5042161079198086560nt_int(product_fst_int_int,S)) ) ).
% img_fst
tff(fact_6349_img__snd,axiom,
! [A: nat,B: nat,S: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),S)
=> aa_set_nat_o(member_nat(B),image_2486076414777270412at_nat(product_snd_nat_nat,S)) ) ).
% img_snd
tff(fact_6350_img__snd,axiom,
! [A: fun_Pr5098669352780829690e_term,B: produc8923325533196201883nteger,S: set_Pr3486312109254157132nteger] :
( member8978266922497061677nteger(produc565448451787809096nteger(A,B),S)
=> member157494554546826820nteger(B,image_2271979006234848446nteger(produc6589491691745714842nteger,S)) ) ).
% img_snd
tff(fact_6351_img__snd,axiom,
! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,S: set_Pr1312406793078187350et_nat] :
( member4513763875858039863et_nat(produc6017135503889078682et_nat(A,B),S)
=> member6260224972018164377et_nat(B,image_5643293374885392937et_nat(produc4960056325182568940et_nat,S)) ) ).
% img_snd
tff(fact_6352_img__snd,axiom,
! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,S: set_Pr4040394263813967220et_nat] :
( member9174161442226468181et_nat(produc7291855603632349880et_nat(A,B),S)
=> member4513763875858039863et_nat(B,image_1425188809386248805et_nat(produc9081944055658670602et_nat,S)) ) ).
% img_snd
tff(fact_6353_img__snd,axiom,
! [A: fun_Pr7763680713840462443e_term,B: product_prod_int_int,S: set_Pr6700895561065273913nt_int] :
( member753276645541486746nt_int(produc7320969046608631221nt_int(A,B),S)
=> member5262025264175285858nt_int(B,image_5003770246312495919nt_int(produc1557020426977177351nt_int,S)) ) ).
% img_snd
tff(fact_6354_img__snd,axiom,
! [A: fun_in5730379987311621695e_term,B: product_prod_int_int,S: set_Pr4166680992137330533nt_int] :
( member5890090881338784710nt_int(produc4729913460834754785nt_int(A,B),S)
=> member5262025264175285858nt_int(B,image_5733965904763621635nt_int(produc7629796867588368947nt_int,S)) ) ).
% img_snd
tff(fact_6355_img__snd,axiom,
! [A: heap_e7401611519738050253t_unit,B: set_nat,S: set_Pr3948176798113811640et_nat] :
( member6260224972018164377et_nat(produc7507926704131184380et_nat(A,B),S)
=> member_set_nat(B,image_6088997261112373647et_nat(produc8586169260539613262et_nat,S)) ) ).
% img_snd
tff(fact_6356_img__snd,axiom,
! [A: int,B: int,S: set_Pr958786334691620121nt_int] :
( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A),B),S)
=> aa_set_int_o(member_int(B),image_5042161079198086560nt_int(product_snd_int_int,S)) ) ).
% img_snd
tff(fact_6357_img__snd,axiom,
! [A: code_natural,B: produc7822875418678951345atural,S: set_Pr2352657477744078772atural] :
( member2108296798382401941atural(aa_Pro7165575656943818355atural(produc6639722614265839536atural(A),B),S)
=> member8280416476884352090atural(B,image_3837677093542026988atural(produc6591912806276919810atural,S)) ) ).
% img_snd
tff(fact_6358_pair__in__swap__image,axiom,
! [Y: nat,X: nat,Aa: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),X)),image_5168914502847457605at_nat(product_swap_nat_nat,Aa))
<=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),Aa) ) ).
% pair_in_swap_image
tff(fact_6359_pair__in__swap__image,axiom,
! [Y: product_prod_int_int,X: fun_in5730379987311621695e_term,Aa: set_Pr4166680992137330533nt_int] :
( member6566240764482910910e_term(produc7771867975113488801e_term(Y,X),image_6826431172304158557e_term(produc1545122483907238721nt_int,Aa))
<=> member5890090881338784710nt_int(produc4729913460834754785nt_int(X,Y),Aa) ) ).
% pair_in_swap_image
tff(fact_6360_pair__in__swap__image,axiom,
! [Y: fun_in5730379987311621695e_term,X: product_prod_int_int,Aa: set_Pr4417106977106297437e_term] :
( member5890090881338784710nt_int(produc4729913460834754785nt_int(Y,X),image_4281971784858038109nt_int(produc4587076998185972737e_term,Aa))
<=> member6566240764482910910e_term(produc7771867975113488801e_term(X,Y),Aa) ) ).
% pair_in_swap_image
tff(fact_6361_pair__in__swap__image,axiom,
! [Y: produc3658429121746597890et_nat,X: fun_Pr6460944699146269935_nat_o,Aa: set_Pr1312406793078187350et_nat] :
( member7677167966932745849_nat_o(produc6339119935501530516_nat_o(Y,X),image_1122556727974905353_nat_o(produc6102546028088763386et_nat,Aa))
<=> member4513763875858039863et_nat(produc6017135503889078682et_nat(X,Y),Aa) ) ).
% pair_in_swap_image
tff(fact_6362_pair__in__swap__image,axiom,
! [Y: fun_Pr6460944699146269935_nat_o,X: produc3658429121746597890et_nat,Aa: set_Pr3013394129895136920_nat_o] :
( member4513763875858039863et_nat(produc6017135503889078682et_nat(Y,X),image_2538532745900505353et_nat(produc6424530459701215220_nat_o,Aa))
<=> member7677167966932745849_nat_o(produc6339119935501530516_nat_o(X,Y),Aa) ) ).
% pair_in_swap_image
tff(fact_6363_pair__in__swap__image,axiom,
! [Y: produc8923325533196201883nteger,X: fun_Pr5098669352780829690e_term,Aa: set_Pr3486312109254157132nteger] :
( member7117455668553134555e_term(produc1742689375443011134e_term(Y,X),image_7918960756227866593e_term(produc9085244685421878184nteger,Aa))
<=> member8978266922497061677nteger(produc565448451787809096nteger(X,Y),Aa) ) ).
% pair_in_swap_image
tff(fact_6364_pair__in__swap__image,axiom,
! [Y: product_prod_int_int,X: fun_Pr7763680713840462443e_term,Aa: set_Pr6700895561065273913nt_int] :
( member1865511963043944682e_term(produc3622783088900447693e_term(Y,X),image_5010469593371777629e_term(produc8452261716800147477nt_int,Aa))
<=> member753276645541486746nt_int(produc7320969046608631221nt_int(X,Y),Aa) ) ).
% pair_in_swap_image
tff(fact_6365_pair__in__swap__image,axiom,
! [Y: fun_Pr5098669352780829690e_term,X: produc8923325533196201883nteger,Aa: set_Pr3944325338781952762e_term] :
( member8978266922497061677nteger(produc565448451787809096nteger(Y,X),image_3617234230432879585nteger(produc1039113572222304414e_term,Aa))
<=> member7117455668553134555e_term(produc1742689375443011134e_term(X,Y),Aa) ) ).
% pair_in_swap_image
tff(fact_6366_pair__in__swap__image,axiom,
! [Y: fun_Pr7763680713840462443e_term,X: product_prod_int_int,Aa: set_Pr3640028372902554249e_term] :
( member753276645541486746nt_int(produc7320969046608631221nt_int(Y,X),image_3062550852079041117nt_int(produc4754075759091963949e_term,Aa))
<=> member1865511963043944682e_term(produc3622783088900447693e_term(X,Y),Aa) ) ).
% pair_in_swap_image
tff(fact_6367_pair__in__swap__image,axiom,
! [Y: produc8799533983703829792et_nat,X: fun_Pr6460944699146269935_nat_o,Aa: set_Pr4040394263813967220et_nat] :
( member3696715088396795611_nat_o(produc6151749222514847990_nat_o(Y,X),image_2438217170757901833_nat_o(produc2593090504590949656et_nat,Aa))
<=> member9174161442226468181et_nat(produc7291855603632349880et_nat(X,Y),Aa) ) ).
% pair_in_swap_image
tff(fact_6368_greaterThanAtMost__empty,axiom,
! [L: $o,K: $o] :
( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(L)),(K))
=> ( set_or8254209520273421544Most_o((K),(L)) = bot_bot_set_o ) ) ).
% greaterThanAtMost_empty
tff(fact_6369_greaterThanAtMost__empty,axiom,
! [L: code_integer,K: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),K)
=> ( set_or2715278749043346189nteger(K,L) = bot_bo3990330152332043303nteger ) ) ).
% greaterThanAtMost_empty
tff(fact_6370_greaterThanAtMost__empty,axiom,
! [L: assn,K: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,L),K)
=> ( set_or2869495477957074904t_assn(K,L) = bot_bot_set_assn ) ) ).
% greaterThanAtMost_empty
tff(fact_6371_greaterThanAtMost__empty,axiom,
! [L: set_nat,K: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,L),K)
=> ( set_or7074010630789208630et_nat(K,L) = bot_bot_set_set_nat ) ) ).
% greaterThanAtMost_empty
tff(fact_6372_greaterThanAtMost__empty,axiom,
! [L: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),K)
=> ( aa_nat_set_nat(set_or6659071591806873216st_nat(K),L) = bot_bot_set_nat ) ) ).
% greaterThanAtMost_empty
tff(fact_6373_greaterThanAtMost__empty,axiom,
! [L: int,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
=> ( set_or6656581121297822940st_int(K,L) = bot_bot_set_int ) ) ).
% greaterThanAtMost_empty
tff(fact_6374_greaterThanAtMost__empty__iff,axiom,
! [K: $o,L: $o] :
( ( set_or8254209520273421544Most_o((K),(L)) = bot_bot_set_o )
<=> ~ aa_o_o(aa_o_fun_o_o(ord_less_o,(K)),(L)) ) ).
% greaterThanAtMost_empty_iff
tff(fact_6375_greaterThanAtMost__empty__iff,axiom,
! [K: assn,L: assn] :
( ( set_or2869495477957074904t_assn(K,L) = bot_bot_set_assn )
<=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,K),L) ) ).
% greaterThanAtMost_empty_iff
tff(fact_6376_greaterThanAtMost__empty__iff,axiom,
! [K: nat,L: nat] :
( ( aa_nat_set_nat(set_or6659071591806873216st_nat(K),L) = bot_bot_set_nat )
<=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) ) ).
% greaterThanAtMost_empty_iff
tff(fact_6377_greaterThanAtMost__empty__iff,axiom,
! [K: int,L: int] :
( ( set_or6656581121297822940st_int(K,L) = bot_bot_set_int )
<=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) ) ).
% greaterThanAtMost_empty_iff
tff(fact_6378_greaterThanAtMost__empty__iff,axiom,
! [K: code_integer,L: code_integer] :
( ( set_or2715278749043346189nteger(K,L) = bot_bo3990330152332043303nteger )
<=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),L) ) ).
% greaterThanAtMost_empty_iff
tff(fact_6379_greaterThanAtMost__empty__iff,axiom,
! [K: code_natural,L: code_natural] :
( ( set_or1538873900800063704atural(K,L) = bot_bo8448779972843232370atural )
<=> ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,K),L) ) ).
% greaterThanAtMost_empty_iff
tff(fact_6380_greaterThanAtMost__empty__iff2,axiom,
! [K: $o,L: $o] :
( ( bot_bot_set_o = set_or8254209520273421544Most_o((K),(L)) )
<=> ~ aa_o_o(aa_o_fun_o_o(ord_less_o,(K)),(L)) ) ).
% greaterThanAtMost_empty_iff2
tff(fact_6381_greaterThanAtMost__empty__iff2,axiom,
! [K: assn,L: assn] :
( ( bot_bot_set_assn = set_or2869495477957074904t_assn(K,L) )
<=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,K),L) ) ).
% greaterThanAtMost_empty_iff2
tff(fact_6382_greaterThanAtMost__empty__iff2,axiom,
! [K: nat,L: nat] :
( ( bot_bot_set_nat = aa_nat_set_nat(set_or6659071591806873216st_nat(K),L) )
<=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) ) ).
% greaterThanAtMost_empty_iff2
tff(fact_6383_greaterThanAtMost__empty__iff2,axiom,
! [K: int,L: int] :
( ( bot_bot_set_int = set_or6656581121297822940st_int(K,L) )
<=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) ) ).
% greaterThanAtMost_empty_iff2
tff(fact_6384_greaterThanAtMost__empty__iff2,axiom,
! [K: code_integer,L: code_integer] :
( ( bot_bo3990330152332043303nteger = set_or2715278749043346189nteger(K,L) )
<=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),L) ) ).
% greaterThanAtMost_empty_iff2
tff(fact_6385_greaterThanAtMost__empty__iff2,axiom,
! [K: code_natural,L: code_natural] :
( ( bot_bo8448779972843232370atural = set_or1538873900800063704atural(K,L) )
<=> ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,K),L) ) ).
% greaterThanAtMost_empty_iff2
tff(fact_6386_Inf__fin_Osingleton,axiom,
! [X: $o] :
( aa_set_o_o(lattic4107685809792843317_fin_o,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))
<=> (X) ) ).
% Inf_fin.singleton
tff(fact_6387_Inf__fin_Osingleton,axiom,
! [X: nat] : lattic5238388535129920115in_nat(aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) = X ).
% Inf_fin.singleton
tff(fact_6388_Inf__fin_Osingleton,axiom,
! [X: int] : lattic5235898064620869839in_int(aa_set_int_set_int(insert_int(X),bot_bot_set_int)) = X ).
% Inf_fin.singleton
tff(fact_6389_sup__Inf__absorb,axiom,
! [Aa: set_o,A: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( aa_set_o_o(member_o((A)),Aa)
=> ( aa_o_o(aa_o_fun_o_o(sup_sup_o,aa_set_o_o(lattic4107685809792843317_fin_o,Aa)),(A))
<=> (A) ) ) ) ).
% sup_Inf_absorb
tff(fact_6390_sup__Inf__absorb,axiom,
! [Aa: set_int,A: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( aa_set_int_o(member_int(A),Aa)
=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,lattic5235898064620869839in_int(Aa)),A) = A ) ) ) ).
% sup_Inf_absorb
tff(fact_6391_sup__Inf__absorb,axiom,
! [Aa: set_fu3555266933746557007_nat_o,A: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A),Aa)
=> ( aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(lattic6089491107740071820_nat_o(Aa)),A) = A ) ) ) ).
% sup_Inf_absorb
tff(fact_6392_sup__Inf__absorb,axiom,
! [Aa: set_nat,A: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( aa_set_nat_o(member_nat(A),Aa)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,lattic5238388535129920115in_nat(Aa)),A) = A ) ) ) ).
% sup_Inf_absorb
tff(fact_6393_sup__Inf__absorb,axiom,
! [Aa: set_assn,A: assn] :
( finite_finite_assn(Aa)
=> ( member_assn(A,Aa)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,lattic47131356835913163n_assn(Aa)),A) = A ) ) ) ).
% sup_Inf_absorb
tff(fact_6394_sup__Inf__absorb,axiom,
! [Aa: set_set_nat,A: set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( member_set_nat(A,Aa)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,lattic3014633134055518761et_nat(Aa)),A) = A ) ) ) ).
% sup_Inf_absorb
tff(fact_6395_sup__Inf__absorb,axiom,
! [Aa: set_se5479201105039950375at_nat,A: set_Pr4329608150637261639at_nat] :
( finite4768004253697144688at_nat(Aa)
=> ( member6273874620674182256at_nat(A,Aa)
=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,lattic6378378157708517732at_nat(Aa)),A) = A ) ) ) ).
% sup_Inf_absorb
tff(fact_6396_atMost__0,axiom,
aa_nat_set_nat(set_ord_atMost_nat,zero_zero_nat) = aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat) ).
% atMost_0
tff(fact_6397_image__mult__atLeastAtMost,axiom,
! [D2: rat,A: rat,B: rat] :
( aa_rat_o(ord_less_rat(zero_zero_rat),D2)
=> ( image_rat_rat(aa_rat_fun_rat_rat(times_times_rat,D2),set_or633870826150836451st_rat(A,B)) = set_or633870826150836451st_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,D2),A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,D2),B)) ) ) ).
% image_mult_atLeastAtMost
tff(fact_6398_ivl__disj__int__one_I3_J,axiom,
! [L: $o,U: $o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,set_ord_atMost_o((L))),set_or8254209520273421544Most_o((L),(U))) = bot_bot_set_o ).
% ivl_disj_int_one(3)
tff(fact_6399_ivl__disj__int__one_I3_J,axiom,
! [L: int,U: int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,set_ord_atMost_int(L)),set_or6656581121297822940st_int(L,U)) = bot_bot_set_int ).
% ivl_disj_int_one(3)
tff(fact_6400_ivl__disj__int__one_I3_J,axiom,
! [L: nat,U: nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_ord_atMost_nat,L)),aa_nat_set_nat(set_or6659071591806873216st_nat(L),U)) = bot_bot_set_nat ).
% ivl_disj_int_one(3)
tff(fact_6401_not__empty__eq__Iic__eq__empty,axiom,
! [H2: $o] : bot_bot_set_o != set_ord_atMost_o((H2)) ).
% not_empty_eq_Iic_eq_empty
tff(fact_6402_not__empty__eq__Iic__eq__empty,axiom,
! [H2: int] : bot_bot_set_int != set_ord_atMost_int(H2) ).
% not_empty_eq_Iic_eq_empty
tff(fact_6403_not__empty__eq__Iic__eq__empty,axiom,
! [H2: nat] : bot_bot_set_nat != aa_nat_set_nat(set_ord_atMost_nat,H2) ).
% not_empty_eq_Iic_eq_empty
tff(fact_6404_Inf__fin_Ohom__commute,axiom,
! [H2: fun_set_nat_set_nat,N3: set_set_nat] :
( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_set_nat(H2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),Y2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(H2,X2)),aa_set_nat_set_nat(H2,Y2))
=> ( finite1152437895449049373et_nat(N3)
=> ( ( N3 != bot_bot_set_set_nat )
=> ( aa_set_nat_set_nat(H2,lattic3014633134055518761et_nat(N3)) = lattic3014633134055518761et_nat(image_7916887816326733075et_nat(H2,N3)) ) ) ) ) ).
% Inf_fin.hom_commute
tff(fact_6405_Inf__fin_Ohom__commute,axiom,
! [H2: fun_assn_assn,N3: set_assn] :
( ! [X2: assn,Y2: assn] : aa_assn_assn(H2,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X2),Y2)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(H2,X2)),aa_assn_assn(H2,Y2))
=> ( finite_finite_assn(N3)
=> ( ( N3 != bot_bot_set_assn )
=> ( aa_assn_assn(H2,lattic47131356835913163n_assn(N3)) = lattic47131356835913163n_assn(image_assn_assn(H2,N3)) ) ) ) ) ).
% Inf_fin.hom_commute
tff(fact_6406_Inf__fin_Ohom__commute,axiom,
! [H2: fun_o_o,N3: set_o] :
( ! [X2: $o,Y2: $o] :
( aa_o_o(H2,aa_o_o(aa_o_fun_o_o(inf_inf_o,(X2)),(Y2)))
<=> aa_o_o(aa_o_fun_o_o(inf_inf_o,aa_o_o(H2,(X2))),aa_o_o(H2,(Y2))) )
=> ( aa_set_o_o(finite_finite_o,N3)
=> ( ( N3 != bot_bot_set_o )
=> ( aa_o_o(H2,aa_set_o_o(lattic4107685809792843317_fin_o,N3))
<=> aa_set_o_o(lattic4107685809792843317_fin_o,image_o_o(H2,N3)) ) ) ) ) ).
% Inf_fin.hom_commute
tff(fact_6407_Inf__fin_Ohom__commute,axiom,
! [H2: fun_nat_nat,N3: set_nat] :
( ! [X2: nat,Y2: nat] : aa_nat_nat(H2,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X2),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(H2,X2)),aa_nat_nat(H2,Y2))
=> ( aa_set_nat_o(finite_finite_nat,N3)
=> ( ( N3 != bot_bot_set_nat )
=> ( aa_nat_nat(H2,lattic5238388535129920115in_nat(N3)) = lattic5238388535129920115in_nat(aa_set_nat_set_nat(image_nat_nat(H2),N3)) ) ) ) ) ).
% Inf_fin.hom_commute
tff(fact_6408_Inf__fin_Ohom__commute,axiom,
! [H2: fun_int_int,N3: set_int] :
( ! [X2: int,Y2: int] : aa_int_int(H2,aa_int_int(aa_int_fun_int_int(inf_inf_int,X2),Y2)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(H2,X2)),aa_int_int(H2,Y2))
=> ( aa_set_int_o(finite_finite_int,N3)
=> ( ( N3 != bot_bot_set_int )
=> ( aa_int_int(H2,lattic5235898064620869839in_int(N3)) = lattic5235898064620869839in_int(image_int_int(H2,N3)) ) ) ) ) ).
% Inf_fin.hom_commute
tff(fact_6409_in__fst__imageE,axiom,
! [X: nat,S: set_Pr1261947904930325089at_nat] :
( aa_set_nat_o(member_nat(X),image_2486076414777270412at_nat(product_fst_nat_nat,S))
=> ~ ! [Y2: nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y2)),S) ) ).
% in_fst_imageE
tff(fact_6410_in__fst__imageE,axiom,
! [X: fun_Pr5098669352780829690e_term,S: set_Pr3486312109254157132nteger] :
( member8464590939221061905e_term(X,image_5882744770105888791e_term(produc2481150572478215260nteger,S))
=> ~ ! [Y2: produc8923325533196201883nteger] : ~ member8978266922497061677nteger(produc565448451787809096nteger(X,Y2),S) ) ).
% in_fst_imageE
tff(fact_6411_in__fst__imageE,axiom,
! [X: fun_Pr6460944699146269935_nat_o,S: set_Pr1312406793078187350et_nat] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),image_4444162529858428168_nat_o(produc5325900173150335662et_nat,S))
=> ~ ! [Y2: produc3658429121746597890et_nat] : ~ member4513763875858039863et_nat(produc6017135503889078682et_nat(X,Y2),S) ) ).
% in_fst_imageE
tff(fact_6412_in__fst__imageE,axiom,
! [X: fun_Pr6460944699146269935_nat_o,S: set_Pr4040394263813967220et_nat] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),image_8400968272122294250_nat_o(produc4078497350495916492et_nat,S))
=> ~ ! [Y2: produc8799533983703829792et_nat] : ~ member9174161442226468181et_nat(produc7291855603632349880et_nat(X,Y2),S) ) ).
% in_fst_imageE
tff(fact_6413_in__fst__imageE,axiom,
! [X: fun_Pr7763680713840462443e_term,S: set_Pr6700895561065273913nt_int] :
( member1495369047980753794e_term(X,image_143812984558558581e_term(produc3130796691802200265nt_int,S))
=> ~ ! [Y2: product_prod_int_int] : ~ member753276645541486746nt_int(produc7320969046608631221nt_int(X,Y2),S) ) ).
% in_fst_imageE
tff(fact_6414_in__fst__imageE,axiom,
! [X: fun_in5730379987311621695e_term,S: set_Pr4166680992137330533nt_int] :
( member4871225538671630166e_term(X,image_7516633138899002997e_term(produc4463008262350210037nt_int,S))
=> ~ ! [Y2: product_prod_int_int] : ~ member5890090881338784710nt_int(produc4729913460834754785nt_int(X,Y2),S) ) ).
% in_fst_imageE
tff(fact_6415_in__fst__imageE,axiom,
! [X: heap_e7401611519738050253t_unit,S: set_Pr3948176798113811640et_nat] :
( member242389114213480694t_unit(X,image_6824039242672219652t_unit(produc1824681642469235216et_nat,S))
=> ~ ! [Y2: set_nat] : ~ member6260224972018164377et_nat(produc7507926704131184380et_nat(X,Y2),S) ) ).
% in_fst_imageE
tff(fact_6416_in__fst__imageE,axiom,
! [X: int,S: set_Pr958786334691620121nt_int] :
( aa_set_int_o(member_int(X),image_5042161079198086560nt_int(product_fst_int_int,S))
=> ~ ! [Y2: int] : ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y2),S) ) ).
% in_fst_imageE
tff(fact_6417_in__snd__imageE,axiom,
! [Y: nat,S: set_Pr1261947904930325089at_nat] :
( aa_set_nat_o(member_nat(Y),image_2486076414777270412at_nat(product_snd_nat_nat,S))
=> ~ ! [X2: nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y)),S) ) ).
% in_snd_imageE
tff(fact_6418_in__snd__imageE,axiom,
! [Y: produc8923325533196201883nteger,S: set_Pr3486312109254157132nteger] :
( member157494554546826820nteger(Y,image_2271979006234848446nteger(produc6589491691745714842nteger,S))
=> ~ ! [X2: fun_Pr5098669352780829690e_term] : ~ member8978266922497061677nteger(produc565448451787809096nteger(X2,Y),S) ) ).
% in_snd_imageE
tff(fact_6419_in__snd__imageE,axiom,
! [Y: produc3658429121746597890et_nat,S: set_Pr1312406793078187350et_nat] :
( member6260224972018164377et_nat(Y,image_5643293374885392937et_nat(produc4960056325182568940et_nat,S))
=> ~ ! [X2: fun_Pr6460944699146269935_nat_o] : ~ member4513763875858039863et_nat(produc6017135503889078682et_nat(X2,Y),S) ) ).
% in_snd_imageE
tff(fact_6420_in__snd__imageE,axiom,
! [Y: produc8799533983703829792et_nat,S: set_Pr4040394263813967220et_nat] :
( member4513763875858039863et_nat(Y,image_1425188809386248805et_nat(produc9081944055658670602et_nat,S))
=> ~ ! [X2: fun_Pr6460944699146269935_nat_o] : ~ member9174161442226468181et_nat(produc7291855603632349880et_nat(X2,Y),S) ) ).
% in_snd_imageE
tff(fact_6421_in__snd__imageE,axiom,
! [Y: product_prod_int_int,S: set_Pr6700895561065273913nt_int] :
( member5262025264175285858nt_int(Y,image_5003770246312495919nt_int(produc1557020426977177351nt_int,S))
=> ~ ! [X2: fun_Pr7763680713840462443e_term] : ~ member753276645541486746nt_int(produc7320969046608631221nt_int(X2,Y),S) ) ).
% in_snd_imageE
tff(fact_6422_in__snd__imageE,axiom,
! [Y: product_prod_int_int,S: set_Pr4166680992137330533nt_int] :
( member5262025264175285858nt_int(Y,image_5733965904763621635nt_int(produc7629796867588368947nt_int,S))
=> ~ ! [X2: fun_in5730379987311621695e_term] : ~ member5890090881338784710nt_int(produc4729913460834754785nt_int(X2,Y),S) ) ).
% in_snd_imageE
tff(fact_6423_in__snd__imageE,axiom,
! [Y: set_nat,S: set_Pr3948176798113811640et_nat] :
( member_set_nat(Y,image_6088997261112373647et_nat(produc8586169260539613262et_nat,S))
=> ~ ! [X2: heap_e7401611519738050253t_unit] : ~ member6260224972018164377et_nat(produc7507926704131184380et_nat(X2,Y),S) ) ).
% in_snd_imageE
tff(fact_6424_in__snd__imageE,axiom,
! [Y: int,S: set_Pr958786334691620121nt_int] :
( aa_set_int_o(member_int(Y),image_5042161079198086560nt_int(product_snd_int_int,S))
=> ~ ! [X2: int] : ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Y),S) ) ).
% in_snd_imageE
tff(fact_6425_in__snd__imageE,axiom,
! [Y: produc7822875418678951345atural,S: set_Pr2352657477744078772atural] :
( member8280416476884352090atural(Y,image_3837677093542026988atural(produc6591912806276919810atural,S))
=> ~ ! [X2: code_natural] : ~ member2108296798382401941atural(aa_Pro7165575656943818355atural(produc6639722614265839536atural(X2),Y),S) ) ).
% in_snd_imageE
tff(fact_6426_ivl__disj__int__two_I6_J,axiom,
! [L: $o,M2: $o,U: $o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,set_or8254209520273421544Most_o((L),(M2))),set_or8254209520273421544Most_o((M2),(U))) = bot_bot_set_o ).
% ivl_disj_int_two(6)
tff(fact_6427_ivl__disj__int__two_I6_J,axiom,
! [L: nat,M2: nat,U: nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_or6659071591806873216st_nat(L),M2)),aa_nat_set_nat(set_or6659071591806873216st_nat(M2),U)) = bot_bot_set_nat ).
% ivl_disj_int_two(6)
tff(fact_6428_ivl__disj__int__two_I6_J,axiom,
! [L: int,M2: int,U: int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,set_or6656581121297822940st_int(L,M2)),set_or6656581121297822940st_int(M2,U)) = bot_bot_set_int ).
% ivl_disj_int_two(6)
tff(fact_6429_atMost__eq__UNIV__iff,axiom,
! [X: assn] :
( ( set_ord_atMost_assn(X) = top_top_set_assn )
<=> ( X = top_top_assn ) ) ).
% atMost_eq_UNIV_iff
tff(fact_6430_atMost__eq__UNIV__iff,axiom,
! [X: set_o] :
( ( set_ord_atMost_set_o(X) = top_top_set_set_o )
<=> ( X = top_top_set_o ) ) ).
% atMost_eq_UNIV_iff
tff(fact_6431_atMost__eq__UNIV__iff,axiom,
! [X: set_nat] :
( ( set_or4236626031148496127et_nat(X) = top_top_set_set_nat )
<=> ( X = top_top_set_nat ) ) ).
% atMost_eq_UNIV_iff
tff(fact_6432_atMost__eq__UNIV__iff,axiom,
! [X: set_Product_unit] :
( ( set_or2827140217781692084t_unit(X) = top_to1767297665138865437t_unit )
<=> ( X = top_to1996260823553986621t_unit ) ) ).
% atMost_eq_UNIV_iff
tff(fact_6433_atMost__eq__UNIV__iff,axiom,
! [X: $o] :
( ( set_ord_atMost_o((X)) = top_top_set_o )
<=> ( (X)
<=> top_top_o ) ) ).
% atMost_eq_UNIV_iff
tff(fact_6434_atMost__eq__UNIV__iff,axiom,
! [X: product_unit] :
( ( set_or8621721900777396692t_unit(X) = top_to1996260823553986621t_unit )
<=> ( X = top_top_Product_unit ) ) ).
% atMost_eq_UNIV_iff
tff(fact_6435_the__elem__image__unique,axiom,
! [Aa: set_nat,F: fun_nat_nat,X: nat] :
( ( Aa != bot_bot_set_nat )
=> ( ! [Y2: nat] :
( aa_set_nat_o(member_nat(Y2),Aa)
=> ( aa_nat_nat(F,Y2) = aa_nat_nat(F,X) ) )
=> ( the_elem_nat(aa_set_nat_set_nat(image_nat_nat(F),Aa)) = aa_nat_nat(F,X) ) ) ) ).
% the_elem_image_unique
tff(fact_6436_the__elem__image__unique,axiom,
! [Aa: set_nat,F: fun_nat_set_nat,X: nat] :
( ( Aa != bot_bot_set_nat )
=> ( ! [Y2: nat] :
( aa_set_nat_o(member_nat(Y2),Aa)
=> ( aa_nat_set_nat(F,Y2) = aa_nat_set_nat(F,X) ) )
=> ( the_elem_set_nat(image_nat_set_nat(F,Aa)) = aa_nat_set_nat(F,X) ) ) ) ).
% the_elem_image_unique
tff(fact_6437_range__eq__singletonD,axiom,
! [F: fun_o_o,A: $o,X: $o] :
( ( image_o_o(F,top_top_set_o) = aa_set_o_set_o(insert_o((A)),bot_bot_set_o) )
=> ( aa_o_o(F,(X))
<=> (A) ) ) ).
% range_eq_singletonD
tff(fact_6438_range__eq__singletonD,axiom,
! [F: fun_o_nat,A: nat,X: $o] :
( ( image_o_nat(F,top_top_set_o) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) )
=> ( aa_o_nat(F,(X)) = A ) ) ).
% range_eq_singletonD
tff(fact_6439_range__eq__singletonD,axiom,
! [F: fun_o_int,A: int,X: $o] :
( ( image_o_int(F,top_top_set_o) = aa_set_int_set_int(insert_int(A),bot_bot_set_int) )
=> ( aa_o_int(F,(X)) = A ) ) ).
% range_eq_singletonD
tff(fact_6440_range__eq__singletonD,axiom,
! [F: fun_nat_o,A: $o,X: nat] :
( ( image_nat_o(F,top_top_set_nat) = aa_set_o_set_o(insert_o((A)),bot_bot_set_o) )
=> ( aa_nat_o(F,X)
<=> (A) ) ) ).
% range_eq_singletonD
tff(fact_6441_range__eq__singletonD,axiom,
! [F: fun_nat_nat,A: nat,X: nat] :
( ( aa_set_nat_set_nat(image_nat_nat(F),top_top_set_nat) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) )
=> ( aa_nat_nat(F,X) = A ) ) ).
% range_eq_singletonD
tff(fact_6442_range__eq__singletonD,axiom,
! [F: fun_nat_int,A: int,X: nat] :
( ( image_nat_int(F,top_top_set_nat) = aa_set_int_set_int(insert_int(A),bot_bot_set_int) )
=> ( aa_nat_int(F,X) = A ) ) ).
% range_eq_singletonD
tff(fact_6443_range__eq__singletonD,axiom,
! [F: fun_Product_unit_o,A: $o,X: product_unit] :
( ( image_Product_unit_o(F,top_to1996260823553986621t_unit) = aa_set_o_set_o(insert_o((A)),bot_bot_set_o) )
=> ( aa_Product_unit_o(F,X)
<=> (A) ) ) ).
% range_eq_singletonD
tff(fact_6444_range__eq__singletonD,axiom,
! [F: fun_Product_unit_nat,A: nat,X: product_unit] :
( ( image_875570014554754200it_nat(F,top_to1996260823553986621t_unit) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) )
=> ( aa_Product_unit_nat(F,X) = A ) ) ).
% range_eq_singletonD
tff(fact_6445_range__eq__singletonD,axiom,
! [F: fun_Product_unit_int,A: int,X: product_unit] :
( ( image_873079544045703924it_int(F,top_to1996260823553986621t_unit) = aa_set_int_set_int(insert_int(A),bot_bot_set_int) )
=> ( aa_Product_unit_int(F,X) = A ) ) ).
% range_eq_singletonD
tff(fact_6446_range__eq__singletonD,axiom,
! [F: fun_nat_set_nat,A: set_nat,X: nat] :
( ( image_nat_set_nat(F,top_top_set_nat) = insert_set_nat(A,bot_bot_set_set_nat) )
=> ( aa_nat_set_nat(F,X) = A ) ) ).
% range_eq_singletonD
tff(fact_6447_fst__image__mp,axiom,
! [Aa: set_Pr3486312109254157132nteger,Ba: set_fu8280898353253485872e_term,X: fun_Pr5098669352780829690e_term,Y: produc8923325533196201883nteger] :
( ord_le8538105397850660240e_term(image_5882744770105888791e_term(produc2481150572478215260nteger,Aa),Ba)
=> ( member8978266922497061677nteger(produc565448451787809096nteger(X,Y),Aa)
=> member8464590939221061905e_term(X,Ba) ) ) ).
% fst_image_mp
tff(fact_6448_fst__image__mp,axiom,
! [Aa: set_Pr1312406793078187350et_nat,Ba: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat] :
( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(image_4444162529858428168_nat_o(produc5325900173150335662et_nat,Aa)),Ba)
=> ( member4513763875858039863et_nat(produc6017135503889078682et_nat(X,Y),Aa)
=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Ba) ) ) ).
% fst_image_mp
tff(fact_6449_fst__image__mp,axiom,
! [Aa: set_Pr4040394263813967220et_nat,Ba: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat] :
( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(image_8400968272122294250_nat_o(produc4078497350495916492et_nat,Aa)),Ba)
=> ( member9174161442226468181et_nat(produc7291855603632349880et_nat(X,Y),Aa)
=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Ba) ) ) ).
% fst_image_mp
tff(fact_6450_fst__image__mp,axiom,
! [Aa: set_Pr6700895561065273913nt_int,Ba: set_fu5653557704672872225e_term,X: fun_Pr7763680713840462443e_term,Y: product_prod_int_int] :
( ord_le792320400390739841e_term(image_143812984558558581e_term(produc3130796691802200265nt_int,Aa),Ba)
=> ( member753276645541486746nt_int(produc7320969046608631221nt_int(X,Y),Aa)
=> member1495369047980753794e_term(X,Ba) ) ) ).
% fst_image_mp
tff(fact_6451_fst__image__mp,axiom,
! [Aa: set_Pr4166680992137330533nt_int,Ba: set_fu3629540756087427317e_term,X: fun_in5730379987311621695e_term,Y: product_prod_int_int] :
( ord_le2047893094613950805e_term(image_7516633138899002997e_term(produc4463008262350210037nt_int,Aa),Ba)
=> ( member5890090881338784710nt_int(produc4729913460834754785nt_int(X,Y),Aa)
=> member4871225538671630166e_term(X,Ba) ) ) ).
% fst_image_mp
tff(fact_6452_fst__image__mp,axiom,
! [Aa: set_Pr3948176798113811640et_nat,Ba: set_he8644174446132254637t_unit,X: heap_e7401611519738050253t_unit,Y: set_nat] :
( ord_le4903435234951335757t_unit(image_6824039242672219652t_unit(produc1824681642469235216et_nat,Aa),Ba)
=> ( member6260224972018164377et_nat(produc7507926704131184380et_nat(X,Y),Aa)
=> member242389114213480694t_unit(X,Ba) ) ) ).
% fst_image_mp
tff(fact_6453_fst__image__mp,axiom,
! [Aa: set_Pr958786334691620121nt_int,Ba: set_int,X: int,Y: int] :
( aa_set_int_o(ord_less_eq_set_int(image_5042161079198086560nt_int(product_fst_int_int,Aa)),Ba)
=> ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),Aa)
=> aa_set_int_o(member_int(X),Ba) ) ) ).
% fst_image_mp
tff(fact_6454_fst__image__mp,axiom,
! [Aa: set_Pr1261947904930325089at_nat,Ba: set_nat,X: nat,Y: nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_2486076414777270412at_nat(product_fst_nat_nat,Aa)),Ba)
=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),Aa)
=> aa_set_nat_o(member_nat(X),Ba) ) ) ).
% fst_image_mp
tff(fact_6455_snd__image__mp,axiom,
! [Aa: set_Pr3486312109254157132nteger,Ba: set_Pr4811707699266497531nteger,X: fun_Pr5098669352780829690e_term,Y: produc8923325533196201883nteger] :
( ord_le3725938330318615451nteger(image_2271979006234848446nteger(produc6589491691745714842nteger,Aa),Ba)
=> ( member8978266922497061677nteger(produc565448451787809096nteger(X,Y),Aa)
=> member157494554546826820nteger(Y,Ba) ) ) ).
% snd_image_mp
tff(fact_6456_snd__image__mp,axiom,
! [Aa: set_Pr1312406793078187350et_nat,Ba: set_Pr3948176798113811640et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat] :
( ord_le8090042543978450200et_nat(image_5643293374885392937et_nat(produc4960056325182568940et_nat,Aa),Ba)
=> ( member4513763875858039863et_nat(produc6017135503889078682et_nat(X,Y),Aa)
=> member6260224972018164377et_nat(Y,Ba) ) ) ).
% snd_image_mp
tff(fact_6457_snd__image__mp,axiom,
! [Aa: set_Pr4040394263813967220et_nat,Ba: set_Pr1312406793078187350et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat] :
( ord_le3919382008627836342et_nat(image_1425188809386248805et_nat(produc9081944055658670602et_nat,Aa),Ba)
=> ( member9174161442226468181et_nat(produc7291855603632349880et_nat(X,Y),Aa)
=> member4513763875858039863et_nat(Y,Ba) ) ) ).
% snd_image_mp
tff(fact_6458_snd__image__mp,axiom,
! [Aa: set_Pr6700895561065273913nt_int,Ba: set_Pr958786334691620121nt_int,X: fun_Pr7763680713840462443e_term,Y: product_prod_int_int] :
( ord_le2843351958646193337nt_int(image_5003770246312495919nt_int(produc1557020426977177351nt_int,Aa),Ba)
=> ( member753276645541486746nt_int(produc7320969046608631221nt_int(X,Y),Aa)
=> member5262025264175285858nt_int(Y,Ba) ) ) ).
% snd_image_mp
tff(fact_6459_snd__image__mp,axiom,
! [Aa: set_Pr4166680992137330533nt_int,Ba: set_Pr958786334691620121nt_int,X: fun_in5730379987311621695e_term,Y: product_prod_int_int] :
( ord_le2843351958646193337nt_int(image_5733965904763621635nt_int(produc7629796867588368947nt_int,Aa),Ba)
=> ( member5890090881338784710nt_int(produc4729913460834754785nt_int(X,Y),Aa)
=> member5262025264175285858nt_int(Y,Ba) ) ) ).
% snd_image_mp
tff(fact_6460_snd__image__mp,axiom,
! [Aa: set_Pr3948176798113811640et_nat,Ba: set_set_nat,X: heap_e7401611519738050253t_unit,Y: set_nat] :
( ord_le6893508408891458716et_nat(image_6088997261112373647et_nat(produc8586169260539613262et_nat,Aa),Ba)
=> ( member6260224972018164377et_nat(produc7507926704131184380et_nat(X,Y),Aa)
=> member_set_nat(Y,Ba) ) ) ).
% snd_image_mp
tff(fact_6461_snd__image__mp,axiom,
! [Aa: set_Pr958786334691620121nt_int,Ba: set_int,X: int,Y: int] :
( aa_set_int_o(ord_less_eq_set_int(image_5042161079198086560nt_int(product_snd_int_int,Aa)),Ba)
=> ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),Aa)
=> aa_set_int_o(member_int(Y),Ba) ) ) ).
% snd_image_mp
tff(fact_6462_snd__image__mp,axiom,
! [Aa: set_Pr2352657477744078772atural,Ba: set_Pr7564037697178849041atural,X: code_natural,Y: produc7822875418678951345atural] :
( ord_le6478268328230966961atural(image_3837677093542026988atural(produc6591912806276919810atural,Aa),Ba)
=> ( member2108296798382401941atural(aa_Pro7165575656943818355atural(produc6639722614265839536atural(X),Y),Aa)
=> member8280416476884352090atural(Y,Ba) ) ) ).
% snd_image_mp
tff(fact_6463_snd__image__mp,axiom,
! [Aa: set_Pr1261947904930325089at_nat,Ba: set_nat,X: nat,Y: nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_2486076414777270412at_nat(product_snd_nat_nat,Aa)),Ba)
=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),Aa)
=> aa_set_nat_o(member_nat(Y),Ba) ) ) ).
% snd_image_mp
tff(fact_6464_in__image__insert__iff,axiom,
! [Ba: set_se5479201105039950375at_nat,X: produc3843707927480180839at_nat,Aa: set_Pr4329608150637261639at_nat] :
( ! [C5: set_Pr4329608150637261639at_nat] :
( member6273874620674182256at_nat(C5,Ba)
=> ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),C5) )
=> ( member6273874620674182256at_nat(Aa,image_4303274092540115909at_nat(insert9069300056098147895at_nat(X),Ba))
<=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),Aa)
& member6273874620674182256at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,Aa),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat)),Ba) ) ) ) ).
% in_image_insert_iff
tff(fact_6465_in__image__insert__iff,axiom,
! [Ba: set_se6690607124073039791_nat_o,X: fun_Pr6460944699146269935_nat_o,Aa: set_fu3555266933746557007_nat_o] :
( ! [C5: set_fu3555266933746557007_nat_o] :
( member361801372707117048_nat_o(C5,Ba)
=> ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),C5) )
=> ( member361801372707117048_nat_o(Aa,image_7577113955783394245_nat_o(insert4923026906511862463_nat_o(X),Ba))
<=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Aa)
& member361801372707117048_nat_o(minus_6456595905716665654_nat_o(Aa,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)),Ba) ) ) ) ).
% in_image_insert_iff
tff(fact_6466_in__image__insert__iff,axiom,
! [Ba: set_se7855581050983116737at_nat,X: product_prod_nat_nat,Aa: set_Pr1261947904930325089at_nat] :
( ! [C5: set_Pr1261947904930325089at_nat] :
( member2643936169264416010at_nat(C5,Ba)
=> ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),C5) )
=> ( member2643936169264416010at_nat(Aa,image_3684629450409544005at_nat(insert8211810215607154385at_nat(X),Ba))
<=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),Aa)
& member2643936169264416010at_nat(minus_1356011639430497352at_nat(Aa,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat)),Ba) ) ) ) ).
% in_image_insert_iff
tff(fact_6467_in__image__insert__iff,axiom,
! [Ba: set_set_o,X: $o,Aa: set_o] :
( ! [C5: set_o] :
( member_set_o(C5,Ba)
=> ~ aa_set_o_o(member_o((X)),C5) )
=> ( member_set_o(Aa,image_set_o_set_o(insert_o((X)),Ba))
<=> ( aa_set_o_o(member_o((X)),Aa)
& member_set_o(minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o)),Ba) ) ) ) ).
% in_image_insert_iff
tff(fact_6468_in__image__insert__iff,axiom,
! [Ba: set_set_int,X: int,Aa: set_int] :
( ! [C5: set_int] :
( member_set_int(C5,Ba)
=> ~ aa_set_int_o(member_int(X),C5) )
=> ( member_set_int(Aa,image_524474410958335435et_int(insert_int(X),Ba))
<=> ( aa_set_int_o(member_int(X),Aa)
& member_set_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int)),Ba) ) ) ) ).
% in_image_insert_iff
tff(fact_6469_in__image__insert__iff,axiom,
! [Ba: set_set_nat,X: nat,Aa: set_nat] :
( ! [C5: set_nat] :
( member_set_nat(C5,Ba)
=> ~ aa_set_nat_o(member_nat(X),C5) )
=> ( member_set_nat(Aa,image_7916887816326733075et_nat(insert_nat(X),Ba))
<=> ( aa_set_nat_o(member_nat(X),Aa)
& member_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)),Ba) ) ) ) ).
% in_image_insert_iff
tff(fact_6470_Ioc__disjoint,axiom,
! [A: $o,B: $o,C: $o,D2: $o] :
( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,set_or8254209520273421544Most_o((A),(B))),set_or8254209520273421544Most_o((C),(D2))) = bot_bot_set_o )
<=> ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(B)),(A))
| aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(D2)),(C))
| aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(B)),(C))
| aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(D2)),(A)) ) ) ).
% Ioc_disjoint
tff(fact_6471_Ioc__disjoint,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( inf_in1364745209274528805nteger(set_or2715278749043346189nteger(A,B),set_or2715278749043346189nteger(C,D2)) = bot_bo3990330152332043303nteger )
<=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,D2),C)
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
| aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,D2),A) ) ) ).
% Ioc_disjoint
tff(fact_6472_Ioc__disjoint,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_or6659071591806873216st_nat(A),B)),aa_nat_set_nat(set_or6659071591806873216st_nat(C),D2)) = bot_bot_set_nat )
<=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D2),C)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
| aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D2),A) ) ) ).
% Ioc_disjoint
tff(fact_6473_Ioc__disjoint,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,set_or6656581121297822940st_int(A,B)),set_or6656581121297822940st_int(C,D2)) = bot_bot_set_int )
<=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
| aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D2),C)
| aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
| aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D2),A) ) ) ).
% Ioc_disjoint
tff(fact_6474_ivl__disj__int__two_I8_J,axiom,
! [L: $o,M2: $o,U: $o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,set_or8904488021354931149Most_o((L),(M2))),set_or8254209520273421544Most_o((M2),(U))) = bot_bot_set_o ).
% ivl_disj_int_two(8)
tff(fact_6475_ivl__disj__int__two_I8_J,axiom,
! [L: int,M2: int,U: int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,set_or1266510415728281911st_int(L,M2)),set_or6656581121297822940st_int(M2,U)) = bot_bot_set_int ).
% ivl_disj_int_two(8)
tff(fact_6476_ivl__disj__int__two_I8_J,axiom,
! [L: nat,M2: nat,U: nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),M2)),aa_nat_set_nat(set_or6659071591806873216st_nat(M2),U)) = bot_bot_set_nat ).
% ivl_disj_int_two(8)
tff(fact_6477_Inf__fin_Obounded__iff,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,Aa))
<=> ! [X3: $o] :
( aa_set_o_o(member_o((X3)),Aa)
=> aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(X3)) ) ) ) ) ).
% Inf_fin.bounded_iff
tff(fact_6478_Inf__fin_Obounded__iff,axiom,
! [Aa: set_assn,X: assn] :
( finite_finite_assn(Aa)
=> ( ( Aa != bot_bot_set_assn )
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),lattic47131356835913163n_assn(Aa))
<=> ! [X3: assn] :
( member_assn(X3,Aa)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),X3) ) ) ) ) ).
% Inf_fin.bounded_iff
tff(fact_6479_Inf__fin_Obounded__iff,axiom,
! [Aa: set_set_nat,X: set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ( Aa != bot_bot_set_set_nat )
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),lattic3014633134055518761et_nat(Aa))
<=> ! [X3: set_nat] :
( member_set_nat(X3,Aa)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),X3) ) ) ) ) ).
% Inf_fin.bounded_iff
tff(fact_6480_Inf__fin_Obounded__iff,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),lattic5238388535129920115in_nat(Aa))
<=> ! [X3: nat] :
( aa_set_nat_o(member_nat(X3),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),X3) ) ) ) ) ).
% Inf_fin.bounded_iff
tff(fact_6481_Inf__fin_Obounded__iff,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic5235898064620869839in_int(Aa))
<=> ! [X3: int] :
( aa_set_int_o(member_int(X3),Aa)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),X3) ) ) ) ) ).
% Inf_fin.bounded_iff
tff(fact_6482_Inf__fin_OboundedI,axiom,
! [Aa: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( ( Aa != bot_bo8632255151191677115_nat_o )
=> ( ! [A3: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),Aa)
=> aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X),A3) )
=> aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X),lattic6089491107740071820_nat_o(Aa)) ) ) ) ).
% Inf_fin.boundedI
tff(fact_6483_Inf__fin_OboundedI,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> ( ! [A3: $o] :
( aa_set_o_o(member_o((A3)),Aa)
=> aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(A3)) )
=> aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,Aa)) ) ) ) ).
% Inf_fin.boundedI
tff(fact_6484_Inf__fin_OboundedI,axiom,
! [Aa: set_assn,X: assn] :
( finite_finite_assn(Aa)
=> ( ( Aa != bot_bot_set_assn )
=> ( ! [A3: assn] :
( member_assn(A3,Aa)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),A3) )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),lattic47131356835913163n_assn(Aa)) ) ) ) ).
% Inf_fin.boundedI
tff(fact_6485_Inf__fin_OboundedI,axiom,
! [Aa: set_set_nat,X: set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ( Aa != bot_bot_set_set_nat )
=> ( ! [A3: set_nat] :
( member_set_nat(A3,Aa)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A3) )
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),lattic3014633134055518761et_nat(Aa)) ) ) ) ).
% Inf_fin.boundedI
tff(fact_6486_Inf__fin_OboundedI,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( ! [A3: nat] :
( aa_set_nat_o(member_nat(A3),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A3) )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),lattic5238388535129920115in_nat(Aa)) ) ) ) ).
% Inf_fin.boundedI
tff(fact_6487_Inf__fin_OboundedI,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> ( ! [A3: int] :
( aa_set_int_o(member_int(A3),Aa)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A3) )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic5235898064620869839in_int(Aa)) ) ) ) ).
% Inf_fin.boundedI
tff(fact_6488_Inf__fin_OboundedE,axiom,
! [Aa: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( ( Aa != bot_bo8632255151191677115_nat_o )
=> ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X),lattic6089491107740071820_nat_o(Aa))
=> ! [A7: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A7),Aa)
=> aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X),A7) ) ) ) ) ).
% Inf_fin.boundedE
tff(fact_6489_Inf__fin_OboundedE,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,Aa))
=> ! [A7: $o] :
( aa_set_o_o(member_o((A7)),Aa)
=> aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(A7)) ) ) ) ) ).
% Inf_fin.boundedE
tff(fact_6490_Inf__fin_OboundedE,axiom,
! [Aa: set_assn,X: assn] :
( finite_finite_assn(Aa)
=> ( ( Aa != bot_bot_set_assn )
=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),lattic47131356835913163n_assn(Aa))
=> ! [A7: assn] :
( member_assn(A7,Aa)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),A7) ) ) ) ) ).
% Inf_fin.boundedE
tff(fact_6491_Inf__fin_OboundedE,axiom,
! [Aa: set_set_nat,X: set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ( Aa != bot_bot_set_set_nat )
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),lattic3014633134055518761et_nat(Aa))
=> ! [A7: set_nat] :
( member_set_nat(A7,Aa)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A7) ) ) ) ) ).
% Inf_fin.boundedE
tff(fact_6492_Inf__fin_OboundedE,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),lattic5238388535129920115in_nat(Aa))
=> ! [A7: nat] :
( aa_set_nat_o(member_nat(A7),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A7) ) ) ) ) ).
% Inf_fin.boundedE
tff(fact_6493_Inf__fin_OboundedE,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic5235898064620869839in_int(Aa))
=> ! [A7: int] :
( aa_set_int_o(member_int(A7),Aa)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A7) ) ) ) ) ).
% Inf_fin.boundedE
tff(fact_6494_ivl__disj__int__one_I5_J,axiom,
! [L: $o,U: $o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,set_or8254209520273421544Most_o((L),(U))),set_or6416164934427428222Than_o((U))) = bot_bot_set_o ).
% ivl_disj_int_one(5)
tff(fact_6495_ivl__disj__int__one_I5_J,axiom,
! [L: int,U: int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,set_or6656581121297822940st_int(L,U)),set_or1207661135979820486an_int(U)) = bot_bot_set_int ).
% ivl_disj_int_one(5)
tff(fact_6496_ivl__disj__int__one_I5_J,axiom,
! [L: nat,U: nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_or6659071591806873216st_nat(L),U)),aa_nat_set_nat(set_or1210151606488870762an_nat,U)) = bot_bot_set_nat ).
% ivl_disj_int_one(5)
tff(fact_6497_Sup__fin_Ohom__commute,axiom,
! [H2: fun_assn_assn,N3: set_assn] :
( ! [X2: assn,Y2: assn] : aa_assn_assn(H2,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X2),Y2)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(H2,X2)),aa_assn_assn(H2,Y2))
=> ( finite_finite_assn(N3)
=> ( ( N3 != bot_bot_set_assn )
=> ( aa_assn_assn(H2,lattic2150320897289308081n_assn(N3)) = lattic2150320897289308081n_assn(image_assn_assn(H2,N3)) ) ) ) ) ).
% Sup_fin.hom_commute
tff(fact_6498_Sup__fin_Ohom__commute,axiom,
! [H2: fun_set_nat_set_nat,N3: set_set_nat] :
( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_set_nat(H2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),Y2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(H2,X2)),aa_set_nat_set_nat(H2,Y2))
=> ( finite1152437895449049373et_nat(N3)
=> ( ( N3 != bot_bot_set_set_nat )
=> ( aa_set_nat_set_nat(H2,lattic3835124923745554447et_nat(N3)) = lattic3835124923745554447et_nat(image_7916887816326733075et_nat(H2,N3)) ) ) ) ) ).
% Sup_fin.hom_commute
tff(fact_6499_Sup__fin_Ohom__commute,axiom,
! [H2: fun_se6846035212209387651at_nat,N3: set_se5479201105039950375at_nat] :
( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(H2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X2),Y2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(H2,X2)),aa_set4883462369968598652at_nat(H2,Y2))
=> ( finite4768004253697144688at_nat(N3)
=> ( ( N3 != bot_bo1534825418459847827at_nat )
=> ( aa_set4883462369968598652at_nat(H2,lattic8394993312974875902at_nat(N3)) = lattic8394993312974875902at_nat(image_4303274092540115909at_nat(H2,N3)) ) ) ) ) ).
% Sup_fin.hom_commute
tff(fact_6500_Sup__fin_Ohom__commute,axiom,
! [H2: fun_o_o,N3: set_o] :
( ! [X2: $o,Y2: $o] :
( aa_o_o(H2,aa_o_o(aa_o_fun_o_o(sup_sup_o,(X2)),(Y2)))
<=> aa_o_o(aa_o_fun_o_o(sup_sup_o,aa_o_o(H2,(X2))),aa_o_o(H2,(Y2))) )
=> ( aa_set_o_o(finite_finite_o,N3)
=> ( ( N3 != bot_bot_set_o )
=> ( aa_o_o(H2,aa_set_o_o(lattic1508158080041050831_fin_o,N3))
<=> aa_set_o_o(lattic1508158080041050831_fin_o,image_o_o(H2,N3)) ) ) ) ) ).
% Sup_fin.hom_commute
tff(fact_6501_Sup__fin_Ohom__commute,axiom,
! [H2: fun_nat_nat,N3: set_nat] :
( ! [X2: nat,Y2: nat] : aa_nat_nat(H2,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X2),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(H2,X2)),aa_nat_nat(H2,Y2))
=> ( aa_set_nat_o(finite_finite_nat,N3)
=> ( ( N3 != bot_bot_set_nat )
=> ( aa_nat_nat(H2,lattic1093996805478795353in_nat(N3)) = lattic1093996805478795353in_nat(aa_set_nat_set_nat(image_nat_nat(H2),N3)) ) ) ) ) ).
% Sup_fin.hom_commute
tff(fact_6502_Sup__fin_Ohom__commute,axiom,
! [H2: fun_int_int,N3: set_int] :
( ! [X2: int,Y2: int] : aa_int_int(H2,aa_int_int(aa_int_fun_int_int(sup_sup_int,X2),Y2)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(H2,X2)),aa_int_int(H2,Y2))
=> ( aa_set_int_o(finite_finite_int,N3)
=> ( ( N3 != bot_bot_set_int )
=> ( aa_int_int(H2,lattic1091506334969745077in_int(N3)) = lattic1091506334969745077in_int(image_int_int(H2,N3)) ) ) ) ) ).
% Sup_fin.hom_commute
tff(fact_6503_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $o,B: $o] : set_or8254209520273421544Most_o((A),(B)) = minus_minus_set_o(set_or8904488021354931149Most_o((A),(B)),aa_set_o_set_o(insert_o((A)),bot_bot_set_o)) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_6504_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: int,B: int] : set_or6656581121297822940st_int(A,B) = minus_minus_set_int(set_or1266510415728281911st_int(A,B),aa_set_int_set_int(insert_int(A),bot_bot_set_int)) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_6505_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: nat,B: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(A),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_6506_Inf__fin_Osubset__imp,axiom,
! [Aa: set_o,Ba: set_o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Aa),Ba)
=> ( ( Aa != bot_bot_set_o )
=> ( aa_set_o_o(finite_finite_o,Ba)
=> aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic4107685809792843317_fin_o,Ba)),aa_set_o_o(lattic4107685809792843317_fin_o,Aa)) ) ) ) ).
% Inf_fin.subset_imp
tff(fact_6507_Inf__fin_Osubset__imp,axiom,
! [Aa: set_assn,Ba: set_assn] :
( ord_less_eq_set_assn(Aa,Ba)
=> ( ( Aa != bot_bot_set_assn )
=> ( finite_finite_assn(Ba)
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,lattic47131356835913163n_assn(Ba)),lattic47131356835913163n_assn(Aa)) ) ) ) ).
% Inf_fin.subset_imp
tff(fact_6508_Inf__fin_Osubset__imp,axiom,
! [Aa: set_set_nat,Ba: set_set_nat] :
( ord_le6893508408891458716et_nat(Aa,Ba)
=> ( ( Aa != bot_bot_set_set_nat )
=> ( finite1152437895449049373et_nat(Ba)
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3014633134055518761et_nat(Ba)),lattic3014633134055518761et_nat(Aa)) ) ) ) ).
% Inf_fin.subset_imp
tff(fact_6509_Inf__fin_Osubset__imp,axiom,
! [Aa: set_int,Ba: set_int] :
( aa_set_int_o(ord_less_eq_set_int(Aa),Ba)
=> ( ( Aa != bot_bot_set_int )
=> ( aa_set_int_o(finite_finite_int,Ba)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic5235898064620869839in_int(Ba)),lattic5235898064620869839in_int(Aa)) ) ) ) ).
% Inf_fin.subset_imp
tff(fact_6510_Inf__fin_Osubset__imp,axiom,
! [Aa: set_nat,Ba: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),Ba)
=> ( ( Aa != bot_bot_set_nat )
=> ( aa_set_nat_o(finite_finite_nat,Ba)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic5238388535129920115in_nat(Ba)),lattic5238388535129920115in_nat(Aa)) ) ) ) ).
% Inf_fin.subset_imp
tff(fact_6511_Inf__fin_Osubset,axiom,
! [Aa: set_set_nat,Ba: set_set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ( Ba != bot_bot_set_set_nat )
=> ( ord_le6893508408891458716et_nat(Ba,Aa)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,lattic3014633134055518761et_nat(Ba)),lattic3014633134055518761et_nat(Aa)) = lattic3014633134055518761et_nat(Aa) ) ) ) ) ).
% Inf_fin.subset
tff(fact_6512_Inf__fin_Osubset,axiom,
! [Aa: set_assn,Ba: set_assn] :
( finite_finite_assn(Aa)
=> ( ( Ba != bot_bot_set_assn )
=> ( ord_less_eq_set_assn(Ba,Aa)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,lattic47131356835913163n_assn(Ba)),lattic47131356835913163n_assn(Aa)) = lattic47131356835913163n_assn(Aa) ) ) ) ) ).
% Inf_fin.subset
tff(fact_6513_Inf__fin_Osubset,axiom,
! [Aa: set_o,Ba: set_o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Ba != bot_bot_set_o )
=> ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Ba),Aa)
=> ( aa_o_o(aa_o_fun_o_o(inf_inf_o,aa_set_o_o(lattic4107685809792843317_fin_o,Ba)),aa_set_o_o(lattic4107685809792843317_fin_o,Aa))
<=> aa_set_o_o(lattic4107685809792843317_fin_o,Aa) ) ) ) ) ).
% Inf_fin.subset
tff(fact_6514_Inf__fin_Osubset,axiom,
! [Aa: set_int,Ba: set_int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Ba != bot_bot_set_int )
=> ( aa_set_int_o(ord_less_eq_set_int(Ba),Aa)
=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,lattic5235898064620869839in_int(Ba)),lattic5235898064620869839in_int(Aa)) = lattic5235898064620869839in_int(Aa) ) ) ) ) ).
% Inf_fin.subset
tff(fact_6515_Inf__fin_Osubset,axiom,
! [Aa: set_nat,Ba: set_nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Ba != bot_bot_set_nat )
=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Ba),Aa)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,lattic5238388535129920115in_nat(Ba)),lattic5238388535129920115in_nat(Aa)) = lattic5238388535129920115in_nat(Aa) ) ) ) ) ).
% Inf_fin.subset
tff(fact_6516_Inf__fin_Oclosed,axiom,
! [Aa: set_fu3555266933746557007_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( ( Aa != bot_bo8632255151191677115_nat_o )
=> ( ! [X2: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_fun2638059832795814268_nat_o(inf_in4424744116413743581_nat_o(X2),Y2)),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X2),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(Y2),bot_bo8632255151191677115_nat_o)))
=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(lattic6089491107740071820_nat_o(Aa)),Aa) ) ) ) ).
% Inf_fin.closed
tff(fact_6517_Inf__fin_Oclosed,axiom,
! [Aa: set_set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ( Aa != bot_bot_set_set_nat )
=> ( ! [X2: set_nat,Y2: set_nat] : member_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),Y2),insert_set_nat(X2,insert_set_nat(Y2,bot_bot_set_set_nat)))
=> member_set_nat(lattic3014633134055518761et_nat(Aa),Aa) ) ) ) ).
% Inf_fin.closed
tff(fact_6518_Inf__fin_Oclosed,axiom,
! [Aa: set_assn] :
( finite_finite_assn(Aa)
=> ( ( Aa != bot_bot_set_assn )
=> ( ! [X2: assn,Y2: assn] : member_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X2),Y2),insert_assn(X2,insert_assn(Y2,bot_bot_set_assn)))
=> member_assn(lattic47131356835913163n_assn(Aa),Aa) ) ) ) ).
% Inf_fin.closed
tff(fact_6519_Inf__fin_Oclosed,axiom,
! [Aa: set_o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> ( ! [X2: $o,Y2: $o] : aa_set_o_o(member_o(aa_o_o(aa_o_fun_o_o(inf_inf_o,(X2)),(Y2))),aa_set_o_set_o(insert_o((X2)),aa_set_o_set_o(insert_o((Y2)),bot_bot_set_o)))
=> aa_set_o_o(member_o(aa_set_o_o(lattic4107685809792843317_fin_o,Aa)),Aa) ) ) ) ).
% Inf_fin.closed
tff(fact_6520_Inf__fin_Oclosed,axiom,
! [Aa: set_nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] : aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X2),Y2)),aa_set_nat_set_nat(insert_nat(X2),aa_set_nat_set_nat(insert_nat(Y2),bot_bot_set_nat)))
=> aa_set_nat_o(member_nat(lattic5238388535129920115in_nat(Aa)),Aa) ) ) ) ).
% Inf_fin.closed
tff(fact_6521_Inf__fin_Oclosed,axiom,
! [Aa: set_int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> ( ! [X2: int,Y2: int] : aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(inf_inf_int,X2),Y2)),aa_set_int_set_int(insert_int(X2),aa_set_int_set_int(insert_int(Y2),bot_bot_set_int)))
=> aa_set_int_o(member_int(lattic5235898064620869839in_int(Aa)),Aa) ) ) ) ).
% Inf_fin.closed
tff(fact_6522_Inf__fin_Oinsert__not__elem,axiom,
! [Aa: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Aa)
=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Aa)
=> ( ( Aa != bot_bo8632255151191677115_nat_o )
=> ( lattic6089491107740071820_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),Aa)) = aa_fun2638059832795814268_nat_o(inf_in4424744116413743581_nat_o(X),lattic6089491107740071820_nat_o(Aa)) ) ) ) ) ).
% Inf_fin.insert_not_elem
tff(fact_6523_Inf__fin_Oinsert__not__elem,axiom,
! [Aa: set_set_nat,X: set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ~ member_set_nat(X,Aa)
=> ( ( Aa != bot_bot_set_set_nat )
=> ( lattic3014633134055518761et_nat(insert_set_nat(X,Aa)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),lattic3014633134055518761et_nat(Aa)) ) ) ) ) ).
% Inf_fin.insert_not_elem
tff(fact_6524_Inf__fin_Oinsert__not__elem,axiom,
! [Aa: set_assn,X: assn] :
( finite_finite_assn(Aa)
=> ( ~ member_assn(X,Aa)
=> ( ( Aa != bot_bot_set_assn )
=> ( lattic47131356835913163n_assn(insert_assn(X,Aa)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),lattic47131356835913163n_assn(Aa)) ) ) ) ) ).
% Inf_fin.insert_not_elem
tff(fact_6525_Inf__fin_Oinsert__not__elem,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ~ aa_set_o_o(member_o((X)),Aa)
=> ( ( Aa != bot_bot_set_o )
=> ( aa_set_o_o(lattic4107685809792843317_fin_o,aa_set_o_set_o(insert_o((X)),Aa))
<=> aa_o_o(aa_o_fun_o_o(inf_inf_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,Aa)) ) ) ) ) ).
% Inf_fin.insert_not_elem
tff(fact_6526_Inf__fin_Oinsert__not__elem,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ~ aa_set_nat_o(member_nat(X),Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( lattic5238388535129920115in_nat(aa_set_nat_set_nat(insert_nat(X),Aa)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),lattic5238388535129920115in_nat(Aa)) ) ) ) ) ).
% Inf_fin.insert_not_elem
tff(fact_6527_Inf__fin_Oinsert__not__elem,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ~ aa_set_int_o(member_int(X),Aa)
=> ( ( Aa != bot_bot_set_int )
=> ( lattic5235898064620869839in_int(aa_set_int_set_int(insert_int(X),Aa)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,X),lattic5235898064620869839in_int(Aa)) ) ) ) ) ).
% Inf_fin.insert_not_elem
tff(fact_6528_Inf__fin_Ounion,axiom,
! [Aa: set_set_nat,Ba: set_set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ( Aa != bot_bot_set_set_nat )
=> ( finite1152437895449049373et_nat(Ba)
=> ( ( Ba != bot_bot_set_set_nat )
=> ( lattic3014633134055518761et_nat(sup_sup_set_set_nat(Aa,Ba)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,lattic3014633134055518761et_nat(Aa)),lattic3014633134055518761et_nat(Ba)) ) ) ) ) ) ).
% Inf_fin.union
tff(fact_6529_Inf__fin_Ounion,axiom,
! [Aa: set_assn,Ba: set_assn] :
( finite_finite_assn(Aa)
=> ( ( Aa != bot_bot_set_assn )
=> ( finite_finite_assn(Ba)
=> ( ( Ba != bot_bot_set_assn )
=> ( lattic47131356835913163n_assn(sup_sup_set_assn(Aa,Ba)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,lattic47131356835913163n_assn(Aa)),lattic47131356835913163n_assn(Ba)) ) ) ) ) ) ).
% Inf_fin.union
tff(fact_6530_Inf__fin_Ounion,axiom,
! [Aa: set_o,Ba: set_o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> ( aa_set_o_o(finite_finite_o,Ba)
=> ( ( Ba != bot_bot_set_o )
=> ( aa_set_o_o(lattic4107685809792843317_fin_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Aa),Ba))
<=> aa_o_o(aa_o_fun_o_o(inf_inf_o,aa_set_o_o(lattic4107685809792843317_fin_o,Aa)),aa_set_o_o(lattic4107685809792843317_fin_o,Ba)) ) ) ) ) ) ).
% Inf_fin.union
tff(fact_6531_Inf__fin_Ounion,axiom,
! [Aa: set_nat,Ba: set_nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( aa_set_nat_o(finite_finite_nat,Ba)
=> ( ( Ba != bot_bot_set_nat )
=> ( lattic5238388535129920115in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Aa),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,lattic5238388535129920115in_nat(Aa)),lattic5238388535129920115in_nat(Ba)) ) ) ) ) ) ).
% Inf_fin.union
tff(fact_6532_Inf__fin_Ounion,axiom,
! [Aa: set_int,Ba: set_int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> ( aa_set_int_o(finite_finite_int,Ba)
=> ( ( Ba != bot_bot_set_int )
=> ( lattic5235898064620869839in_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Aa),Ba)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,lattic5235898064620869839in_int(Aa)),lattic5235898064620869839in_int(Ba)) ) ) ) ) ) ).
% Inf_fin.union
tff(fact_6533_Inf__fin__le__Sup__fin,axiom,
! [Aa: set_o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic4107685809792843317_fin_o,Aa)),aa_set_o_o(lattic1508158080041050831_fin_o,Aa)) ) ) ).
% Inf_fin_le_Sup_fin
tff(fact_6534_Inf__fin__le__Sup__fin,axiom,
! [Aa: set_assn] :
( finite_finite_assn(Aa)
=> ( ( Aa != bot_bot_set_assn )
=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,lattic47131356835913163n_assn(Aa)),lattic2150320897289308081n_assn(Aa)) ) ) ).
% Inf_fin_le_Sup_fin
tff(fact_6535_Inf__fin__le__Sup__fin,axiom,
! [Aa: set_set_nat] :
( finite1152437895449049373et_nat(Aa)
=> ( ( Aa != bot_bot_set_set_nat )
=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3014633134055518761et_nat(Aa)),lattic3835124923745554447et_nat(Aa)) ) ) ).
% Inf_fin_le_Sup_fin
tff(fact_6536_Inf__fin__le__Sup__fin,axiom,
! [Aa: set_nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lattic5238388535129920115in_nat(Aa)),lattic1093996805478795353in_nat(Aa)) ) ) ).
% Inf_fin_le_Sup_fin
tff(fact_6537_Inf__fin__le__Sup__fin,axiom,
! [Aa: set_int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic5235898064620869839in_int(Aa)),lattic1091506334969745077in_int(Aa)) ) ) ).
% Inf_fin_le_Sup_fin
tff(fact_6538_ivl__disj__un__singleton_I4_J,axiom,
! [L: $o,U: $o] :
( aa_o_o(aa_o_fun_o_o(ord_less_o,(L)),(U))
=> ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,set_or1716231572884733764Than_o((L),(U))),aa_set_o_set_o(insert_o((U)),bot_bot_set_o)) = set_or8254209520273421544Most_o((L),(U)) ) ) ).
% ivl_disj_un_singleton(4)
tff(fact_6539_ivl__disj__un__singleton_I4_J,axiom,
! [L: nat,U: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,L),U)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)),aa_set_nat_set_nat(insert_nat(U),bot_bot_set_nat)) = aa_nat_set_nat(set_or6659071591806873216st_nat(L),U) ) ) ).
% ivl_disj_un_singleton(4)
tff(fact_6540_ivl__disj__un__singleton_I4_J,axiom,
! [L: int,U: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,L),U)
=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,set_or5832277885323065728an_int(L,U)),aa_set_int_set_int(insert_int(U),bot_bot_set_int)) = set_or6656581121297822940st_int(L,U) ) ) ).
% ivl_disj_un_singleton(4)
tff(fact_6541_ivl__disj__un__singleton_I4_J,axiom,
! [L: code_integer,U: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,L),U)
=> ( sup_su848401254843788991nteger(set_or4266950643985792945nteger(L,U),insert_Code_integer(U,bot_bo3990330152332043303nteger)) = set_or2715278749043346189nteger(L,U) ) ) ).
% ivl_disj_un_singleton(4)
tff(fact_6542_ivl__disj__un__singleton_I4_J,axiom,
! [L: code_natural,U: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,L),U)
=> ( sup_su5306851075354978058atural(set_or3090545795742510460atural(L,U),insert_Code_natural(U,bot_bo8448779972843232370atural)) = set_or1538873900800063704atural(L,U) ) ) ).
% ivl_disj_un_singleton(4)
tff(fact_6543_prod_Ocl__ivl__Suc,axiom,
! [G: fun_nat_Code_integer,M2: nat,N: nat] :
groups3455450783089532116nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M2),one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3455450783089532116nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_Code_integer(G,aa_nat_nat(suc,N)))) ).
% prod.cl_ivl_Suc
tff(fact_6544_prod_Ocl__ivl__Suc,axiom,
! [G: fun_nat_assn,M2: nat,N: nat] :
groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M2),one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_assn(G,aa_nat_nat(suc,N)))) ).
% prod.cl_ivl_Suc
tff(fact_6545_prod_Ocl__ivl__Suc,axiom,
! [G: fun_nat_rat,M2: nat,N: nat] :
groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M2),one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_rat(G,aa_nat_nat(suc,N)))) ).
% prod.cl_ivl_Suc
tff(fact_6546_prod_Ocl__ivl__Suc,axiom,
! [G: fun_nat_nat,M2: nat,N: nat] :
groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M2),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_nat(G,aa_nat_nat(suc,N)))) ).
% prod.cl_ivl_Suc
tff(fact_6547_prod_Ocl__ivl__Suc,axiom,
! [G: fun_nat_int,M2: nat,N: nat] :
groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M2),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_int(G,aa_nat_nat(suc,N)))) ).
% prod.cl_ivl_Suc
tff(fact_6548_atLeastLessThan__nat__numeral,axiom,
! [M2: nat,K: num] :
aa_nat_set_nat(set_or4665077453230672383an_nat(M2),numeral_numeral_nat(K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),pred_numeral(K)),aa_set_nat_set_nat(insert_nat(pred_numeral(K)),aa_nat_set_nat(set_or4665077453230672383an_nat(M2),pred_numeral(K))),bot_bot_set_nat) ).
% atLeastLessThan_nat_numeral
tff(fact_6549_atLeast__Suc,axiom,
! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_atLeast_nat,K)),aa_set_nat_set_nat(insert_nat(K),bot_bot_set_nat)) ).
% atLeast_Suc
tff(fact_6550_Min_Oremove,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( aa_set_o_o(member_o((X)),Aa)
=> ( aa_set_o_o(lattic1973801136483472281_Min_o,Aa)
<=> $ite(minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(ord_min_o,(X)),aa_set_o_o(lattic1973801136483472281_Min_o,minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))))) ) ) ) ).
% Min.remove
tff(fact_6551_Min_Oremove,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( aa_set_int_o(member_int(X),Aa)
=> ( lattic8718645017227715691in_int(Aa) = $ite(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int)) = bot_bot_set_int,X,aa_int_int(aa_int_fun_int_int(ord_min_int,X),lattic8718645017227715691in_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int))))) ) ) ) ).
% Min.remove
tff(fact_6552_Min_Oremove,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( aa_set_nat_o(member_nat(X),Aa)
=> ( lattic8721135487736765967in_nat(Aa) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),lattic8721135487736765967in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))))) ) ) ) ).
% Min.remove
tff(fact_6553_Min_Oinsert__remove,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( aa_set_o_o(lattic1973801136483472281_Min_o,aa_set_o_set_o(insert_o((X)),Aa))
<=> $ite(minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(ord_min_o,(X)),aa_set_o_o(lattic1973801136483472281_Min_o,minus_minus_set_o(Aa,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))))) ) ) ).
% Min.insert_remove
tff(fact_6554_Min_Oinsert__remove,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( lattic8718645017227715691in_int(aa_set_int_set_int(insert_int(X),Aa)) = $ite(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int)) = bot_bot_set_int,X,aa_int_int(aa_int_fun_int_int(ord_min_int,X),lattic8718645017227715691in_int(minus_minus_set_int(Aa,aa_set_int_set_int(insert_int(X),bot_bot_set_int))))) ) ) ).
% Min.insert_remove
tff(fact_6555_Min_Oinsert__remove,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( lattic8721135487736765967in_nat(aa_set_nat_set_nat(insert_nat(X),Aa)) = $ite(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),lattic8721135487736765967in_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Aa),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))))) ) ) ).
% Min.insert_remove
tff(fact_6556_gbinomial__code,axiom,
! [A: rat,K: nat] :
aa_nat_rat(gbinomial_rat(A),K) = $ite(K = zero_zero_nat,one_one_rat,divide_divide_rat(set_fo1949268297981939178at_rat(aTP_Lamm_ae(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat),one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,K))) ).
% gbinomial_code
tff(fact_6557_bool__assn__proper_I2_J,axiom,
aa_fun1675921005893230629at_o_o(proper,aTP_Lamm_af) ).
% bool_assn_proper(2)
tff(fact_6558_bool__assn__proper_I3_J,axiom,
! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o] :
( aa_fun1675921005893230629at_o_o(proper,P)
=> ( aa_fun1675921005893230629at_o_o(proper,Q)
=> aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(aTP_Lamm_ag(P),Q)) ) ) ).
% bool_assn_proper(3)
tff(fact_6559_bool__assn__proper_I4_J,axiom,
! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o] :
( aa_fun1675921005893230629at_o_o(proper,P)
=> ( aa_fun1675921005893230629at_o_o(proper,Q)
=> aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(aTP_Lamm_ah(P),Q)) ) ) ).
% bool_assn_proper(4)
tff(fact_6560_singleton__conv2,axiom,
! [A: produc3843707927480180839at_nat] : collec6321179662152712658at_nat(fequal7949547063978738364at_nat(A)) = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),bot_bo228742789529271731at_nat) ).
% singleton_conv2
tff(fact_6561_singleton__conv2,axiom,
! [A: fun_Pr6460944699146269935_nat_o] : collec6407654713642994138_nat_o(aa_fun3380769697021226975at_o_o(fequal1890233700291625924_nat_o,A)) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A),bot_bo8632255151191677115_nat_o) ).
% singleton_conv2
tff(fact_6562_singleton__conv2,axiom,
! [A: product_prod_nat_nat] : collec3392354462482085612at_nat(aa_Pro989595756709598047_nat_o(fequal9188845342289841878at_nat,A)) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),bot_bo2099793752762293965at_nat) ).
% singleton_conv2
tff(fact_6563_singleton__conv2,axiom,
! [A: $o] : collect_o(aa_o_fun_o_o(fequal_o,(A))) = aa_set_o_set_o(insert_o((A)),bot_bot_set_o) ).
% singleton_conv2
tff(fact_6564_singleton__conv2,axiom,
! [A: nat] : collect_nat(aa_nat_fun_nat_o(fequal_nat,A)) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ).
% singleton_conv2
tff(fact_6565_singleton__conv2,axiom,
! [A: int] : collect_int(aa_int_fun_int_o(fequal_int,A)) = aa_set_int_set_int(insert_int(A),bot_bot_set_int) ).
% singleton_conv2
tff(fact_6566_singleton__conv,axiom,
! [A: produc3843707927480180839at_nat] : collec6321179662152712658at_nat(aTP_Lamm_ai(A)) = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),bot_bo228742789529271731at_nat) ).
% singleton_conv
tff(fact_6567_singleton__conv,axiom,
! [A: fun_Pr6460944699146269935_nat_o] : collec6407654713642994138_nat_o(aa_fun3380769697021226975at_o_o(aTP_Lamm_aj,A)) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A),bot_bo8632255151191677115_nat_o) ).
% singleton_conv
tff(fact_6568_singleton__conv,axiom,
! [A: product_prod_nat_nat] : collec3392354462482085612at_nat(aa_Pro989595756709598047_nat_o(aTP_Lamm_ak,A)) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),bot_bo2099793752762293965at_nat) ).
% singleton_conv
tff(fact_6569_singleton__conv,axiom,
! [A: $o] : collect_o(aa_o_fun_o_o(aTP_Lamm_al,(A))) = aa_set_o_set_o(insert_o((A)),bot_bot_set_o) ).
% singleton_conv
tff(fact_6570_singleton__conv,axiom,
! [A: nat] : collect_nat(aa_nat_fun_nat_o(aTP_Lamm_am,A)) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ).
% singleton_conv
tff(fact_6571_singleton__conv,axiom,
! [A: int] : collect_int(aa_int_fun_int_o(aTP_Lamm_an,A)) = aa_set_int_set_int(insert_int(A),bot_bot_set_int) ).
% singleton_conv
tff(fact_6572_Collect__const,axiom,
! [P: $o] :
collec6407654713642994138_nat_o(aTP_Lamm_ao((P))) = $ite((P),top_to9044565752620676511_nat_o,bot_bo8632255151191677115_nat_o) ).
% Collect_const
tff(fact_6573_Collect__const,axiom,
! [P: $o] :
collec3392354462482085612at_nat(aTP_Lamm_ap((P))) = $ite((P),top_to4669805908274784177at_nat,bot_bo2099793752762293965at_nat) ).
% Collect_const
tff(fact_6574_Collect__const,axiom,
! [P: $o] :
collect_int(aTP_Lamm_aq((P))) = $ite((P),top_top_set_int,bot_bot_set_int) ).
% Collect_const
tff(fact_6575_Collect__const,axiom,
! [P: $o] :
collect_o(aa_o_fun_o_o(aTP_Lamm_ar,(P))) = $ite((P),top_top_set_o,bot_bot_set_o) ).
% Collect_const
tff(fact_6576_Collect__const,axiom,
! [P: $o] :
collect_nat(aTP_Lamm_as((P))) = $ite((P),top_top_set_nat,bot_bot_set_nat) ).
% Collect_const
tff(fact_6577_Collect__const,axiom,
! [P: $o] :
collect_Product_unit(aTP_Lamm_at((P))) = $ite((P),top_to1996260823553986621t_unit,bot_bo3957492148770167129t_unit) ).
% Collect_const
tff(fact_6578_bool__assn__proper_I5_J,axiom,
! [P: fun_Pr6460944699146269935_nat_o] :
( aa_fun1675921005893230629at_o_o(proper,P)
=> aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(aTP_Lamm_au,P)) ) ).
% bool_assn_proper(5)
tff(fact_6579_atLeastLessThan__empty,axiom,
! [B: $o,A: $o] :
( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(B)),(A))
=> ( set_or7139685690850216873Than_o((A),(B)) = bot_bot_set_o ) ) ).
% atLeastLessThan_empty
tff(fact_6580_atLeastLessThan__empty,axiom,
! [B: code_integer,A: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
=> ( set_or8404916559141939852nteger(A,B) = bot_bo3990330152332043303nteger ) ) ).
% atLeastLessThan_empty
tff(fact_6581_atLeastLessThan__empty,axiom,
! [B: assn,A: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
=> ( set_or5523502641901747543n_assn(A,B) = bot_bot_set_assn ) ) ).
% atLeastLessThan_empty
tff(fact_6582_atLeastLessThan__empty,axiom,
! [B: set_nat,A: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
=> ( set_or3540276404033026485et_nat(A,B) = bot_bot_set_set_nat ) ) ).
% atLeastLessThan_empty
tff(fact_6583_atLeastLessThan__empty,axiom,
! [B: int,A: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
=> ( set_or4662586982721622107an_int(A,B) = bot_bot_set_int ) ) ).
% atLeastLessThan_empty
tff(fact_6584_atLeastLessThan__empty,axiom,
! [B: nat,A: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
=> ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = bot_bot_set_nat ) ) ).
% atLeastLessThan_empty
tff(fact_6585_atLeastLessThan__empty__iff2,axiom,
! [A: $o,B: $o] :
( ( bot_bot_set_o = set_or7139685690850216873Than_o((A),(B)) )
<=> ~ aa_o_o(aa_o_fun_o_o(ord_less_o,(A)),(B)) ) ).
% atLeastLessThan_empty_iff2
tff(fact_6586_atLeastLessThan__empty__iff2,axiom,
! [A: assn,B: assn] :
( ( bot_bot_set_assn = set_or5523502641901747543n_assn(A,B) )
<=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ).
% atLeastLessThan_empty_iff2
tff(fact_6587_atLeastLessThan__empty__iff2,axiom,
! [A: int,B: int] :
( ( bot_bot_set_int = set_or4662586982721622107an_int(A,B) )
<=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).
% atLeastLessThan_empty_iff2
tff(fact_6588_atLeastLessThan__empty__iff2,axiom,
! [A: code_integer,B: code_integer] :
( ( bot_bo3990330152332043303nteger = set_or8404916559141939852nteger(A,B) )
<=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).
% atLeastLessThan_empty_iff2
tff(fact_6589_atLeastLessThan__empty__iff2,axiom,
! [A: code_natural,B: code_natural] :
( ( bot_bo8448779972843232370atural = set_or7228511710898657367atural(A,B) )
<=> ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B) ) ).
% atLeastLessThan_empty_iff2
tff(fact_6590_atLeastLessThan__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat = aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) )
<=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).
% atLeastLessThan_empty_iff2
tff(fact_6591_atLeastLessThan__empty__iff,axiom,
! [A: $o,B: $o] :
( ( set_or7139685690850216873Than_o((A),(B)) = bot_bot_set_o )
<=> ~ aa_o_o(aa_o_fun_o_o(ord_less_o,(A)),(B)) ) ).
% atLeastLessThan_empty_iff
tff(fact_6592_atLeastLessThan__empty__iff,axiom,
! [A: assn,B: assn] :
( ( set_or5523502641901747543n_assn(A,B) = bot_bot_set_assn )
<=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ).
% atLeastLessThan_empty_iff
tff(fact_6593_atLeastLessThan__empty__iff,axiom,
! [A: int,B: int] :
( ( set_or4662586982721622107an_int(A,B) = bot_bot_set_int )
<=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).
% atLeastLessThan_empty_iff
tff(fact_6594_atLeastLessThan__empty__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( set_or8404916559141939852nteger(A,B) = bot_bo3990330152332043303nteger )
<=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).
% atLeastLessThan_empty_iff
tff(fact_6595_atLeastLessThan__empty__iff,axiom,
! [A: code_natural,B: code_natural] :
( ( set_or7228511710898657367atural(A,B) = bot_bo8448779972843232370atural )
<=> ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A),B) ) ).
% atLeastLessThan_empty_iff
tff(fact_6596_atLeastLessThan__empty__iff,axiom,
! [A: nat,B: nat] :
( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = bot_bot_set_nat )
<=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).
% atLeastLessThan_empty_iff
tff(fact_6597_greaterThanLessThan__empty,axiom,
! [L: $o,K: $o] :
( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(L)),(K))
=> ( set_or1716231572884733764Than_o((K),(L)) = bot_bot_set_o ) ) ).
% greaterThanLessThan_empty
tff(fact_6598_greaterThanLessThan__empty,axiom,
! [L: code_integer,K: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),K)
=> ( set_or4266950643985792945nteger(K,L) = bot_bo3990330152332043303nteger ) ) ).
% greaterThanLessThan_empty
tff(fact_6599_greaterThanLessThan__empty,axiom,
! [L: assn,K: assn] :
( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,L),K)
=> ( set_or4250685708352838780n_assn(K,L) = bot_bot_set_assn ) ) ).
% greaterThanLessThan_empty
tff(fact_6600_greaterThanLessThan__empty,axiom,
! [L: set_nat,K: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,L),K)
=> ( set_or8625682525731655386et_nat(K,L) = bot_bot_set_set_nat ) ) ).
% greaterThanLessThan_empty
tff(fact_6601_greaterThanLessThan__empty,axiom,
! [L: nat,K: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),K)
=> ( aa_nat_set_nat(set_or5834768355832116004an_nat(K),L) = bot_bot_set_nat ) ) ).
% greaterThanLessThan_empty
tff(fact_6602_greaterThanLessThan__empty,axiom,
! [L: int,K: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
=> ( set_or5832277885323065728an_int(K,L) = bot_bot_set_int ) ) ).
% greaterThanLessThan_empty
tff(fact_6603_Min__singleton,axiom,
! [X: $o] :
( aa_set_o_o(lattic1973801136483472281_Min_o,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))
<=> (X) ) ).
% Min_singleton
tff(fact_6604_Min__singleton,axiom,
! [X: nat] : lattic8721135487736765967in_nat(aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)) = X ).
% Min_singleton
tff(fact_6605_Min__singleton,axiom,
! [X: int] : lattic8718645017227715691in_int(aa_set_int_set_int(insert_int(X),bot_bot_set_int)) = X ).
% Min_singleton
tff(fact_6606_Min_Obounded__iff,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic1973801136483472281_Min_o,Aa))
<=> ! [X3: $o] :
( aa_set_o_o(member_o((X3)),Aa)
=> aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(X3)) ) ) ) ) ).
% Min.bounded_iff
tff(fact_6607_Min_Obounded__iff,axiom,
! [Aa: set_Code_integer,X: code_integer] :
( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Aa)
=> ( ( Aa != bot_bo3990330152332043303nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),lattic1063845414844153500nteger(Aa))
<=> ! [X3: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(X3),Aa)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),X3) ) ) ) ) ).
% Min.bounded_iff
tff(fact_6608_Min_Obounded__iff,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),lattic8721135487736765967in_nat(Aa))
<=> ! [X3: nat] :
( aa_set_nat_o(member_nat(X3),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),X3) ) ) ) ) ).
% Min.bounded_iff
tff(fact_6609_Min_Obounded__iff,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic8718645017227715691in_int(Aa))
<=> ! [X3: int] :
( aa_set_int_o(member_int(X3),Aa)
=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),X3) ) ) ) ) ).
% Min.bounded_iff
tff(fact_6610_Min__gr__iff,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> ( aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),aa_set_o_o(lattic1973801136483472281_Min_o,Aa))
<=> ! [X3: $o] :
( aa_set_o_o(member_o((X3)),Aa)
=> aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),(X3)) ) ) ) ) ).
% Min_gr_iff
tff(fact_6611_Min__gr__iff,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),lattic8721135487736765967in_nat(Aa))
<=> ! [X3: nat] :
( aa_set_nat_o(member_nat(X3),Aa)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X3) ) ) ) ) ).
% Min_gr_iff
tff(fact_6612_Min__gr__iff,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),lattic8718645017227715691in_int(Aa))
<=> ! [X3: int] :
( aa_set_int_o(member_int(X3),Aa)
=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),X3) ) ) ) ) ).
% Min_gr_iff
tff(fact_6613_Min__gr__iff,axiom,
! [Aa: set_Code_integer,X: code_integer] :
( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Aa)
=> ( ( Aa != bot_bo3990330152332043303nteger )
=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),lattic1063845414844153500nteger(Aa))
<=> ! [X3: code_integer] :
( aa_set5643387332196575111eger_o(member_Code_integer(X3),Aa)
=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),X3) ) ) ) ) ).
% Min_gr_iff
tff(fact_6614_Min__gr__iff,axiom,
! [Aa: set_Code_natural,X: code_natural] :
( aa_set1564640798469522044ural_o(finite4840673202314680255atural,Aa)
=> ( ( Aa != bot_bo8448779972843232370atural )
=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),lattic9110812603455646823atural(Aa))
<=> ! [X3: code_natural] :
( aa_set1564640798469522044ural_o(member_Code_natural(X3),Aa)
=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),X3) ) ) ) ) ).
% Min_gr_iff
tff(fact_6615_prod_OatMost__Suc,axiom,
! [G: fun_nat_Code_integer,N: nat] : groups3455450783089532116nteger(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3455450783089532116nteger(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_Code_integer(G,aa_nat_nat(suc,N))) ).
% prod.atMost_Suc
tff(fact_6616_prod_OatMost__Suc,axiom,
! [G: fun_nat_assn,N: nat] : groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_assn(G,aa_nat_nat(suc,N))) ).
% prod.atMost_Suc
tff(fact_6617_prod_OatMost__Suc,axiom,
! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).
% prod.atMost_Suc
tff(fact_6618_prod_OatMost__Suc,axiom,
! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).
% prod.atMost_Suc
tff(fact_6619_prod_OatMost__Suc,axiom,
! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).
% prod.atMost_Suc
tff(fact_6620_range__constant,axiom,
! [X: $o] : image_o_o(aa_o_fun_o_o(aTP_Lamm_ar,(X)),top_top_set_o) = aa_set_o_set_o(insert_o((X)),bot_bot_set_o) ).
% range_constant
tff(fact_6621_range__constant,axiom,
! [X: nat] : image_o_nat(aTP_Lamm_av(X),top_top_set_o) = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) ).
% range_constant
tff(fact_6622_range__constant,axiom,
! [X: int] : image_o_int(aTP_Lamm_aw(X),top_top_set_o) = aa_set_int_set_int(insert_int(X),bot_bot_set_int) ).
% range_constant
tff(fact_6623_range__constant,axiom,
! [X: $o] : image_nat_o(aTP_Lamm_as((X)),top_top_set_nat) = aa_set_o_set_o(insert_o((X)),bot_bot_set_o) ).
% range_constant
tff(fact_6624_range__constant,axiom,
! [X: nat] : aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ax,X)),top_top_set_nat) = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) ).
% range_constant
tff(fact_6625_range__constant,axiom,
! [X: int] : image_nat_int(aTP_Lamm_ay(X),top_top_set_nat) = aa_set_int_set_int(insert_int(X),bot_bot_set_int) ).
% range_constant
tff(fact_6626_range__constant,axiom,
! [X: $o] : image_Product_unit_o(aTP_Lamm_at((X)),top_to1996260823553986621t_unit) = aa_set_o_set_o(insert_o((X)),bot_bot_set_o) ).
% range_constant
tff(fact_6627_range__constant,axiom,
! [X: nat] : image_875570014554754200it_nat(aTP_Lamm_az(X),top_to1996260823553986621t_unit) = aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) ).
% range_constant
tff(fact_6628_range__constant,axiom,
! [X: int] : image_873079544045703924it_int(aTP_Lamm_ba(X),top_to1996260823553986621t_unit) = aa_set_int_set_int(insert_int(X),bot_bot_set_int) ).
% range_constant
tff(fact_6629_range__constant,axiom,
! [X: set_nat] : image_nat_set_nat(aTP_Lamm_bb(X),top_top_set_nat) = insert_set_nat(X,bot_bot_set_set_nat) ).
% range_constant
tff(fact_6630_atLeastLessThan__singleton,axiom,
! [M2: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M2),aa_nat_nat(suc,M2)) = aa_set_nat_set_nat(insert_nat(M2),bot_bot_set_nat) ).
% atLeastLessThan_singleton
tff(fact_6631_Min__const,axiom,
! [Aa: set_nat,C: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( lattic8721135487736765967in_nat(aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ax,C)),Aa)) = C ) ) ) ).
% Min_const
tff(fact_6632_Min__insert,axiom,
! [Aa: set_o,X: $o] :
( aa_set_o_o(finite_finite_o,Aa)
=> ( ( Aa != bot_bot_set_o )
=> ( aa_set_o_o(lattic1973801136483472281_Min_o,aa_set_o_set_o(insert_o((X)),Aa))
<=> aa_o_o(aa_o_fun_o_o(ord_min_o,(X)),aa_set_o_o(lattic1973801136483472281_Min_o,Aa)) ) ) ) ).
% Min_insert
tff(fact_6633_Min__insert,axiom,
! [Aa: set_nat,X: nat] :
( aa_set_nat_o(finite_finite_nat,Aa)
=> ( ( Aa != bot_bot_set_nat )
=> ( lattic8721135487736765967in_nat(aa_set_nat_set_nat(insert_nat(X),Aa)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),lattic8721135487736765967in_nat(Aa)) ) ) ) ).
% Min_insert
tff(fact_6634_Min__insert,axiom,
! [Aa: set_int,X: int] :
( aa_set_int_o(finite_finite_int,Aa)
=> ( ( Aa != bot_bot_set_int )
=> ( lattic8718645017227715691in_int(aa_set_int_set_int(insert_int(X),Aa)) = aa_int_int(aa_int_fun_int_int(ord_min_int,X),lattic8718645017227715691in_int(Aa)) ) ) ) ).
% Min_insert
tff(fact_6635_prod_Oop__ivl__Suc,axiom,
! [G: fun_nat_Code_integer,M2: nat,N: nat] :
groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2),one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))),aa_nat_Code_integer(G,N))) ).
% prod.op_ivl_Suc
tff(fact_6636_prod_Oop__ivl__Suc,axiom,
! [G: fun_nat_assn,M2: nat,N: nat] :
groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2),one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))),aa_nat_assn(G,N))) ).
% prod.op_ivl_Suc
tff(fact_6637_prod_Oop__ivl__Suc,axiom,
! [G: fun_nat_rat,M2: nat,N: nat] :
groups73079841787564623at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2),one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))),aa_nat_rat(G,N))) ).
% prod.op_ivl_Suc
tff(fact_6638_prod_Oop__ivl__Suc,axiom,
! [G: fun_nat_nat,M2: nat,N: nat] :
groups708209901874060359at_nat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))),aa_nat_nat(G,N))) ).
% prod.op_ivl_Suc
tff(fact_6639_prod_Oop__ivl__Suc,axiom,
! [G: fun_nat_int,M2: nat,N: nat] :
groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))),aa_nat_int(G,N))) ).
% prod.op_ivl_Suc
tff(fact_6640_lambda__zero,axiom,
aTP_Lamm_bc = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger) ).
% lambda_zero
tff(fact_6641_lambda__zero,axiom,
aTP_Lamm_bd = aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat) ).
% lambda_zero
tff(fact_6642_lambda__zero,axiom,
aTP_Lamm_be = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).
% lambda_zero
tff(fact_6643_lambda__zero,axiom,
aTP_Lamm_bf = aa_int_fun_int_int(times_times_int,zero_zero_int) ).
% lambda_zero
tff(fact_6644_top__empty__eq2,axiom,
! [X4: nat,Xa3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(top_to5288702739685986426_nat_o,X4),Xa3)
<=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Xa3)),top_to4669805908274784177at_nat) ) ).
% top_empty_eq2
tff(fact_6645_top__empty__eq2,axiom,
! [X4: fun_Pr5098669352780829690e_term,Xa3: produc8923325533196201883nteger] :
( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(top_to2784585791485011495eger_o,X4),Xa3)
<=> member8978266922497061677nteger(produc565448451787809096nteger(X4,Xa3),top_to7525899545246753276nteger) ) ).
% top_empty_eq2
tff(fact_6646_top__empty__eq2,axiom,
! [X4: fun_Pr6460944699146269935_nat_o,Xa3: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(top_to2712223222597888051_nat_o,X4),Xa3)
<=> member4513763875858039863et_nat(produc6017135503889078682et_nat(X4,Xa3),top_to2987642378616949766et_nat) ) ).
% top_empty_eq2
tff(fact_6647_top__empty__eq2,axiom,
! [X4: fun_Pr6460944699146269935_nat_o,Xa3: produc8799533983703829792et_nat] :
( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(top_to3067105749023872149_nat_o,X4),Xa3)
<=> member9174161442226468181et_nat(produc7291855603632349880et_nat(X4,Xa3),top_to2648389616742571044et_nat) ) ).
% top_empty_eq2
tff(fact_6648_top__empty__eq2,axiom,
! [X4: fun_Pr7763680713840462443e_term,Xa3: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(top_to1218662295448195450_int_o,X4),Xa3)
<=> member753276645541486746nt_int(produc7320969046608631221nt_int(X4,Xa3),top_to6107225058432391401nt_int) ) ).
% top_empty_eq2
tff(fact_6649_top__empty__eq2,axiom,
! [X4: fun_in5730379987311621695e_term,Xa3: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(top_to1089409542825796430_int_o,X4),Xa3)
<=> member5890090881338784710nt_int(produc4729913460834754785nt_int(X4,Xa3),top_to1212741241995312661nt_int) ) ).
% top_empty_eq2
tff(fact_6650_inf__Int__eq2,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,X4: nat,Xa3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(inf_in375002649321709592_nat_o(aTP_Lamm_bg(R),aTP_Lamm_bg(S)),X4),Xa3)
<=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Xa3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,R),S)) ) ).
% inf_Int_eq2
tff(fact_6651_inf__Int__eq2,axiom,
! [R: set_Pr3486312109254157132nteger,S: set_Pr3486312109254157132nteger,X4: fun_Pr5098669352780829690e_term,Xa3: produc8923325533196201883nteger] :
( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(inf_in7430926233015486473eger_o(aTP_Lamm_bh(R),aTP_Lamm_bh(S)),X4),Xa3)
<=> member8978266922497061677nteger(produc565448451787809096nteger(X4,Xa3),inf_in6300790993181727198nteger(R,S)) ) ).
% inf_Int_eq2
tff(fact_6652_inf__Int__eq2,axiom,
! [R: set_Pr1312406793078187350et_nat,S: set_Pr1312406793078187350et_nat,X4: fun_Pr6460944699146269935_nat_o,Xa3: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(inf_in3476382694143140117_nat_o(aTP_Lamm_bi(R),aTP_Lamm_bi(S)),X4),Xa3)
<=> member4513763875858039863et_nat(produc6017135503889078682et_nat(X4,Xa3),inf_in4791168559842171624et_nat(R,S)) ) ).
% inf_Int_eq2
tff(fact_6653_inf__Int__eq2,axiom,
! [R: set_Pr4040394263813967220et_nat,S: set_Pr4040394263813967220et_nat,X4: fun_Pr6460944699146269935_nat_o,Xa3: produc8799533983703829792et_nat] :
( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(inf_in6333540121576287351_nat_o(aTP_Lamm_bj(R),aTP_Lamm_bj(S)),X4),Xa3)
<=> member9174161442226468181et_nat(produc7291855603632349880et_nat(X4,Xa3),inf_in6821596156192049670et_nat(R,S)) ) ).
% inf_Int_eq2
tff(fact_6654_inf__Int__eq2,axiom,
! [R: set_Pr6700895561065273913nt_int,S: set_Pr6700895561065273913nt_int,X4: fun_Pr7763680713840462443e_term,Xa3: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(inf_in132845422080707420_int_o(aTP_Lamm_bk(R),aTP_Lamm_bk(S)),X4),Xa3)
<=> member753276645541486746nt_int(produc7320969046608631221nt_int(X4,Xa3),inf_in3653535833467300043nt_int(R,S)) ) ).
% inf_Int_eq2
tff(fact_6655_inf__Int__eq2,axiom,
! [R: set_Pr4166680992137330533nt_int,S: set_Pr4166680992137330533nt_int,X4: fun_in5730379987311621695e_term,Xa3: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(inf_in4973581681881435440_int_o(aTP_Lamm_bl(R),aTP_Lamm_bl(S)),X4),Xa3)
<=> member5890090881338784710nt_int(produc4729913460834754785nt_int(X4,Xa3),inf_in8549327947507203575nt_int(R,S)) ) ).
% inf_Int_eq2
tff(fact_6656_pred__equals__eq2,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X3: nat,Xa2: nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa2)),R)
<=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa2)),S) )
<=> ( R = S ) ) ).
% pred_equals_eq2
tff(fact_6657_pred__equals__eq2,axiom,
! [R: set_Pr3486312109254157132nteger,S: set_Pr3486312109254157132nteger] :
( ! [X3: fun_Pr5098669352780829690e_term,Xa2: produc8923325533196201883nteger] :
( member8978266922497061677nteger(produc565448451787809096nteger(X3,Xa2),R)
<=> member8978266922497061677nteger(produc565448451787809096nteger(X3,Xa2),S) )
<=> ( R = S ) ) ).
% pred_equals_eq2
tff(fact_6658_pred__equals__eq2,axiom,
! [R: set_Pr1312406793078187350et_nat,S: set_Pr1312406793078187350et_nat] :
( ! [X3: fun_Pr6460944699146269935_nat_o,Xa2: produc3658429121746597890et_nat] :
( member4513763875858039863et_nat(produc6017135503889078682et_nat(X3,Xa2),R)
<=> member4513763875858039863et_nat(produc6017135503889078682et_nat(X3,Xa2),S) )
<=> ( R = S ) ) ).
% pred_equals_eq2
tff(fact_6659_pred__equals__eq2,axiom,
! [R: set_Pr4040394263813967220et_nat,S: set_Pr4040394263813967220et_nat] :
( ! [X3: fun_Pr6460944699146269935_nat_o,Xa2: produc8799533983703829792et_nat] :
( member9174161442226468181et_nat(produc7291855603632349880et_nat(X3,Xa2),R)
<=> member9174161442226468181et_nat(produc7291855603632349880et_nat(X3,Xa2),S) )
<=> ( R = S ) ) ).
% pred_equals_eq2
tff(fact_6660_pred__equals__eq2,axiom,
! [R: set_Pr6700895561065273913nt_int,S: set_Pr6700895561065273913nt_int] :
( ! [X3: fun_Pr7763680713840462443e_term,Xa2: product_prod_int_int] :
( member753276645541486746nt_int(produc7320969046608631221nt_int(X3,Xa2),R)
<=> member753276645541486746nt_int(produc7320969046608631221nt_int(X3,Xa2),S) )
<=> ( R = S ) ) ).
% pred_equals_eq2
tff(fact_6661_pred__equals__eq2,axiom,
! [R: set_Pr4166680992137330533nt_int,S: set_Pr4166680992137330533nt_int] :
( ! [X3: fun_in5730379987311621695e_term,Xa2: product_prod_int_int] :
( member5890090881338784710nt_int(produc4729913460834754785nt_int(X3,Xa2),R)
<=> member5890090881338784710nt_int(produc4729913460834754785nt_int(X3,Xa2),S) )
<=> ( R = S ) ) ).
% pred_equals_eq2
tff(fact_6662_sup__Un__eq2,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,X4: nat,Xa3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(sup_su8083968640029086462_nat_o(aTP_Lamm_bg(R),aTP_Lamm_bg(S)),X4),Xa3)
<=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Xa3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,R),S)) ) ).
% sup_Un_eq2
tff(fact_6663_sup__Un__eq2,axiom,
! [R: set_Pr3486312109254157132nteger,S: set_Pr3486312109254157132nteger,X4: fun_Pr5098669352780829690e_term,Xa3: produc8923325533196201883nteger] :
( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(sup_su8446958762992054435eger_o(aTP_Lamm_bh(R),aTP_Lamm_bh(S)),X4),Xa3)
<=> member8978266922497061677nteger(produc565448451787809096nteger(X4,Xa3),sup_su4503389074446631032nteger(R,S)) ) ).
% sup_Un_eq2
tff(fact_6664_sup__Un__eq2,axiom,
! [R: set_Pr1312406793078187350et_nat,S: set_Pr1312406793078187350et_nat,X4: fun_Pr6460944699146269935_nat_o,Xa3: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(sup_su4970993554803434671_nat_o(aTP_Lamm_bi(R),aTP_Lamm_bi(S)),X4),Xa3)
<=> member4513763875858039863et_nat(produc6017135503889078682et_nat(X4,Xa3),sup_su2387659076810923138et_nat(R,S)) ) ).
% sup_Un_eq2
tff(fact_6665_sup__Un__eq2,axiom,
! [R: set_Pr4040394263813967220et_nat,S: set_Pr4040394263813967220et_nat,X4: fun_Pr6460944699146269935_nat_o,Xa3: produc8799533983703829792et_nat] :
( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(sup_su6780828561269846801_nat_o(aTP_Lamm_bj(R),aTP_Lamm_bj(S)),X4),Xa3)
<=> member9174161442226468181et_nat(produc7291855603632349880et_nat(X4,Xa3),sup_su5858552540197408416et_nat(R,S)) ) ).
% sup_Un_eq2
tff(fact_6666_sup__Un__eq2,axiom,
! [R: set_Pr6700895561065273913nt_int,S: set_Pr6700895561065273913nt_int,X4: fun_Pr7763680713840462443e_term,Xa3: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(sup_su2175524065394371574_int_o(aTP_Lamm_bk(R),aTP_Lamm_bk(S)),X4),Xa3)
<=> member753276645541486746nt_int(produc7320969046608631221nt_int(X4,Xa3),sup_su2017857570560100197nt_int(R,S)) ) ).
% sup_Un_eq2
tff(fact_6667_sup__Un__eq2,axiom,
! [R: set_Pr4166680992137330533nt_int,S: set_Pr4166680992137330533nt_int,X4: fun_in5730379987311621695e_term,Xa3: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(sup_su7230269426179251658_int_o(aTP_Lamm_bl(R),aTP_Lamm_bl(S)),X4),Xa3)
<=> member5890090881338784710nt_int(produc4729913460834754785nt_int(X4,Xa3),sup_su7513280395484302481nt_int(R,S)) ) ).
% sup_Un_eq2
tff(fact_6668_sup__Un__eq2,axiom,
! [R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat,X4: set_Pr1261947904930325089at_nat,Xa3: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(sup_su5424986631308460290_nat_o(aTP_Lamm_bm(R),aTP_Lamm_bm(S)),X4),Xa3)
<=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X4,Xa3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,R),S)) ) ).
% sup_Un_eq2
tff(fact_6669_Set_Oempty__def,axiom,
bot_bo8632255151191677115_nat_o = collec6407654713642994138_nat_o(aTP_Lamm_bn) ).
% Set.empty_def
tff(fact_6670_Set_Oempty__def,axiom,
bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(aTP_Lamm_bo) ).
% Set.empty_def
tff(fact_6671_Set_Oempty__def,axiom,
bot_bot_set_o = collect_o(aTP_Lamm_bp) ).
% Set.empty_def
tff(fact_6672_Set_Oempty__def,axiom,
bot_bot_set_nat = collect_nat(aTP_Lamm_bq) ).
% Set.empty_def
tff(fact_6673_Set_Oempty__def,axiom,
bot_bot_set_int = collect_int(aTP_Lamm_br) ).
% Set.empty_def
tff(fact_6674_bot__empty__eq2,axiom,
! [X4: fun_Pr5098669352780829690e_term,Xa3: produc8923325533196201883nteger] :
( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(bot_bo4567088895781559819eger_o,X4),Xa3)
<=> member8978266922497061677nteger(produc565448451787809096nteger(X4,Xa3),bot_bo9050247954524366304nteger) ) ).
% bot_empty_eq2
tff(fact_6675_bot__empty__eq2,axiom,
! [X4: fun_Pr6460944699146269935_nat_o,Xa3: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(bot_bo3509460380234399255_nat_o,X4),Xa3)
<=> member4513763875858039863et_nat(produc6017135503889078682et_nat(X4,Xa3),bot_bo1493351289776401898et_nat) ) ).
% bot_empty_eq2
tff(fact_6676_bot__empty__eq2,axiom,
! [X4: fun_Pr6460944699146269935_nat_o,Xa3: produc8799533983703829792et_nat] :
( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(bot_bo5031374944739956857_nat_o,X4),Xa3)
<=> member9174161442226468181et_nat(produc7291855603632349880et_nat(X4,Xa3),bot_bo6121529163330120712et_nat) ) ).
% bot_empty_eq2
tff(fact_6677_bot__empty__eq2,axiom,
! [X4: fun_Pr7763680713840462443e_term,Xa3: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(bot_bo1117279750869432670_int_o,X4),Xa3)
<=> member753276645541486746nt_int(produc7320969046608631221nt_int(X4,Xa3),bot_bo2501116328598523085nt_int) ) ).
% bot_empty_eq2
tff(fact_6678_bot__empty__eq2,axiom,
! [X4: fun_in5730379987311621695e_term,Xa3: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(bot_bo6524246659364023090_int_o,X4),Xa3)
<=> member5890090881338784710nt_int(produc4729913460834754785nt_int(X4,Xa3),bot_bo5189594446080201209nt_int) ) ).
% bot_empty_eq2
tff(fact_6679_bot__empty__eq2,axiom,
! [X4: nat,Xa3: nat] :
( aa_nat_o(aa_nat_fun_nat_o(bot_bo1118997454112379798_nat_o,X4),Xa3)
<=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Xa3)),bot_bo2099793752762293965at_nat) ) ).
% bot_empty_eq2
tff(fact_6680_lambda__one,axiom,
aTP_Lamm_bs = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer) ).
% lambda_one
tff(fact_6681_lambda__one,axiom,
aTP_Lamm_bt = aa_ass954072293579886259n_assn(times_times_assn,one_one_assn) ).
% lambda_one
tff(fact_6682_lambda__one,axiom,
aTP_Lamm_bu = aa_rat_fun_rat_rat(times_times_rat,one_one_rat) ).
% lambda_one
tff(fact_6683_lambda__one,axiom,
aTP_Lamm_bv = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).
% lambda_one
tff(fact_6684_lambda__one,axiom,
aTP_Lamm_bw = aa_int_fun_int_int(times_times_int,one_one_int) ).
% lambda_one
tff(fact_6685_Collect__conv__if2,axiom,
! [A: produc3843707927480180839at_nat,P: fun_Pr3059359313727237556_nat_o] :
collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_bx(A),P)) = $ite(aa_Pro5947980084102686253_nat_o(P,A),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),bot_bo228742789529271731at_nat),bot_bo228742789529271731at_nat) ).
% Collect_conv_if2
tff(fact_6686_Collect__conv__if2,axiom,
! [A: fun_Pr6460944699146269935_nat_o,P: fun_fu5135969500543783852at_o_o] :
collec6407654713642994138_nat_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_by(A),P)) = $ite(aa_fun1675921005893230629at_o_o(P,A),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A),bot_bo8632255151191677115_nat_o),bot_bo8632255151191677115_nat_o) ).
% Collect_conv_if2
tff(fact_6687_Collect__conv__if2,axiom,
! [A: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_bz(A),P)) = $ite(aa_Pro6150637604445964947_nat_o(P,A),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),bot_bo2099793752762293965at_nat),bot_bo2099793752762293965at_nat) ).
% Collect_conv_if2
tff(fact_6688_Collect__conv__if2,axiom,
! [A: $o,P: fun_o_o] :
collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ca((A)),P)) = $ite(aa_o_o(P,(A)),aa_set_o_set_o(insert_o((A)),bot_bot_set_o),bot_bot_set_o) ).
% Collect_conv_if2
tff(fact_6689_Collect__conv__if2,axiom,
! [A: nat,P: fun_nat_o] :
collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cb(A),P)) = $ite(aa_nat_o(P,A),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat),bot_bot_set_nat) ).
% Collect_conv_if2
tff(fact_6690_Collect__conv__if2,axiom,
! [A: int,P: fun_int_o] :
collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cc(A),P)) = $ite(aa_int_o(P,A),aa_set_int_set_int(insert_int(A),bot_bot_set_int),bot_bot_set_int) ).
% Collect_conv_if2
tff(fact_6691_Collect__conv__if,axiom,
! [A: produc3843707927480180839at_nat,P: fun_Pr3059359313727237556_nat_o] :
collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_cd(A),P)) = $ite(aa_Pro5947980084102686253_nat_o(P,A),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),bot_bo228742789529271731at_nat),bot_bo228742789529271731at_nat) ).
% Collect_conv_if
tff(fact_6692_Collect__conv__if,axiom,
! [A: fun_Pr6460944699146269935_nat_o,P: fun_fu5135969500543783852at_o_o] :
collec6407654713642994138_nat_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_ce(A),P)) = $ite(aa_fun1675921005893230629at_o_o(P,A),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A),bot_bo8632255151191677115_nat_o),bot_bo8632255151191677115_nat_o) ).
% Collect_conv_if
tff(fact_6693_Collect__conv__if,axiom,
! [A: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_cf(A),P)) = $ite(aa_Pro6150637604445964947_nat_o(P,A),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),bot_bo2099793752762293965at_nat),bot_bo2099793752762293965at_nat) ).
% Collect_conv_if
tff(fact_6694_Collect__conv__if,axiom,
! [A: $o,P: fun_o_o] :
collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_cg((A)),P)) = $ite(aa_o_o(P,(A)),aa_set_o_set_o(insert_o((A)),bot_bot_set_o),bot_bot_set_o) ).
% Collect_conv_if
tff(fact_6695_Collect__conv__if,axiom,
! [A: nat,P: fun_nat_o] :
collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ch(A),P)) = $ite(aa_nat_o(P,A),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat),bot_bot_set_nat) ).
% Collect_conv_if
tff(fact_6696_Collect__conv__if,axiom,
! [A: int,P: fun_int_o] :
collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ci(A),P)) = $ite(aa_int_o(P,A),aa_set_int_set_int(insert_int(A),bot_bot_set_int),bot_bot_set_int) ).
% Collect_conv_if
tff(fact_6697_pred__subset__eq2,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ord_le768339454706049738_nat_o(aTP_Lamm_bg(R),aTP_Lamm_bg(S))
<=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(R),S) ) ).
% pred_subset_eq2
tff(fact_6698_pred__subset__eq2,axiom,
! [R: set_Pr3486312109254157132nteger,S: set_Pr3486312109254157132nteger] :
( ord_le4825817444062779351eger_o(aTP_Lamm_bh(R),aTP_Lamm_bh(S))
<=> ord_le2932375828722672556nteger(R,S) ) ).
% pred_subset_eq2
tff(fact_6699_pred__subset__eq2,axiom,
! [R: set_Pr1312406793078187350et_nat,S: set_Pr1312406793078187350et_nat] :
( ord_le3580378406483493347_nat_o(aTP_Lamm_bi(R),aTP_Lamm_bi(S))
<=> ord_le3919382008627836342et_nat(R,S) ) ).
% pred_subset_eq2
tff(fact_6700_pred__subset__eq2,axiom,
! [R: set_Pr4040394263813967220et_nat,S: set_Pr4040394263813967220et_nat] :
( ord_le2834457679758710341_nat_o(aTP_Lamm_bj(R),aTP_Lamm_bj(S))
<=> ord_le5189643068899110356et_nat(R,S) ) ).
% pred_subset_eq2
tff(fact_6701_pred__subset__eq2,axiom,
! [R: set_Pr6700895561065273913nt_int,S: set_Pr6700895561065273913nt_int] :
( ord_le4806730579060245290_int_o(aTP_Lamm_bk(R),aTP_Lamm_bk(S))
<=> ord_le3990897393281686169nt_int(R,S) ) ).
% pred_subset_eq2
tff(fact_6702_pred__subset__eq2,axiom,
! [R: set_Pr4166680992137330533nt_int,S: set_Pr4166680992137330533nt_int] :
( ord_le2876595303158420734_int_o(aTP_Lamm_bl(R),aTP_Lamm_bl(S))
<=> ord_le3197784583289274309nt_int(R,S) ) ).
% pred_subset_eq2
tff(fact_6703_prod_OatLeastLessThan__concat,axiom,
! [M2: nat,N: nat,P5: nat,G: fun_nat_Code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),P5)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))),groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(N),P5))) = groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),P5)) ) ) ) ).
% prod.atLeastLessThan_concat
tff(fact_6704_prod_OatLeastLessThan__concat,axiom,
! [M2: nat,N: nat,P5: nat,G: fun_nat_assn] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),P5)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))),groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(N),P5))) = groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),P5)) ) ) ) ).
% prod.atLeastLessThan_concat
tff(fact_6705_prod_OatLeastLessThan__concat,axiom,
! [M2: nat,N: nat,P5: nat,G: fun_nat_rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),P5)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(N),P5))) = groups73079841787564623at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),P5)) ) ) ) ).
% prod.atLeastLessThan_concat
tff(fact_6706_prod_OatLeastLessThan__concat,axiom,
! [M2: nat,N: nat,P5: nat,G: fun_nat_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),P5)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))),groups708209901874060359at_nat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(N),P5))) = groups708209901874060359at_nat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),P5)) ) ) ) ).
% prod.atLeastLessThan_concat
tff(fact_6707_prod_OatLeastLessThan__concat,axiom,
! [M2: nat,N: nat,P5: nat,G: fun_nat_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),P5)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(N),P5))) = groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),P5)) ) ) ) ).
% prod.atLeastLessThan_concat
tff(fact_6708_ivl__disj__int__one_I8_J,axiom,
! [L: $o,U: $o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,set_or7139685690850216873Than_o((L),(U))),set_ord_atLeast_o((U))) = bot_bot_set_o ).
% ivl_disj_int_one(8)
tff(fact_6709_ivl__disj__int__one_I8_J,axiom,
! [L: int,U: int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,set_or4662586982721622107an_int(L,U)),set_ord_atLeast_int(U)) = bot_bot_set_int ).
% ivl_disj_int_one(8)
tff(fact_6710_ivl__disj__int__one_I8_J,axiom,
! [L: nat,U: nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(L),U)),aa_nat_set_nat(set_ord_atLeast_nat,U)) = bot_bot_set_nat ).
% ivl_disj_int_one(8)
tff(fact_6711_ivl__disj__int__two_I1_J,axiom,
! [L: $o,M2: $o,U: $o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,set_or1716231572884733764Than_o((L),(M2))),set_or7139685690850216873Than_o((M2),(U))) = bot_bot_set_o ).
% ivl_disj_int_two(1)
tff(fact_6712_ivl__disj__int__two_I1_J,axiom,
! [L: int,M2: int,U: int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,set_or5832277885323065728an_int(L,M2)),set_or4662586982721622107an_int(M2,U)) = bot_bot_set_int ).
% ivl_disj_int_two(1)
tff(fact_6713_ivl__disj__int__two_I1_J,axiom,
! [L: nat,M2: nat,U: nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_or5834768355832116004an_nat(L),M2)),aa_nat_set_nat(set_or4665077453230672383an_nat(M2),U)) = bot_bot_set_nat ).
% ivl_disj_int_two(1)
tff(fact_6714_ivl__disj__int__one_I6_J,axiom,
! [L: $o,U: $o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(inf_inf_set_o,set_or1716231572884733764Than_o((L),(U))),set_ord_atLeast_o((U))) = bot_bot_set_o ).
% ivl_disj_int_one(6)
tff(fact_6715_ivl__disj__int__one_I6_J,axiom,
! [L: int,U: int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,set_or5832277885323065728an_int(L,U)),set_ord_atLeast_int(U)) = bot_bot_set_int ).
% ivl_disj_int_one(6)
tff(fact_6716_ivl__disj__int__one_I6_J,axiom,
! [L: nat,U: nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)),aa_nat_set_nat(set_ord_atLeast_nat,U)) = bot_bot_set_nat ).
% ivl_disj_int_one(6)
tff(fact_6717_prod_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: fun_nat_Code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3455450783089532116nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_Code_integer(G,aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,M2)),groups3455450783089532116nteger(aTP_Lamm_cj(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).
% prod.Suc_reindex_ivl
tff(fact_6718_prod_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: fun_nat_assn] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_assn(G,aa_nat_nat(suc,N))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,M2)),groups6906906614972039071t_assn(aTP_Lamm_ck(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).
% prod.Suc_reindex_ivl
tff(fact_6719_prod_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: fun_nat_rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,M2)),groups73079841787564623at_rat(aTP_Lamm_cl(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).
% prod.Suc_reindex_ivl
tff(fact_6720_prod_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: fun_nat_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M2)),groups708209901874060359at_nat(aTP_Lamm_cm(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).
% prod.Suc_reindex_ivl
tff(fact_6721_prod_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: fun_nat_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))),aa_nat_int(G,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M2)),groups705719431365010083at_int(aTP_Lamm_cn(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N))) ) ) ).
% prod.Suc_reindex_ivl
tff(fact_6722_Min__add__commute,axiom,
! [S: set_o,F: fun_o_Code_integer,K: code_integer] :
( aa_set_o_o(finite_finite_o,S)
=> ( ( S != bot_bot_set_o )
=> ( lattic1063845414844153500nteger(image_o_Code_integer(aa_Cod4043041029439962285nteger(aTP_Lamm_co(F),K),S)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,lattic1063845414844153500nteger(image_o_Code_integer(F,S))),K) ) ) ) ).
% Min_add_commute
tff(fact_6723_Min__add__commute,axiom,
! [S: set_o,F: fun_o_rat,K: rat] :
( aa_set_o_o(finite_finite_o,S)
=> ( ( S != bot_bot_set_o )
=> ( lattic8086005427650270231in_rat(image_o_rat(aa_rat_fun_o_rat(aTP_Lamm_cp(F),K),S)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,lattic8086005427650270231in_rat(image_o_rat(F,S))),K) ) ) ) ).
% Min_add_commute
tff(fact_6724_Min__add__commute,axiom,
! [S: set_o,F: fun_o_nat,K: nat] :
( aa_set_o_o(finite_finite_o,S)
=> ( ( S != bot_bot_set_o )
=> ( lattic8721135487736765967in_nat(image_o_nat(aa_nat_fun_o_nat(aTP_Lamm_cq(F),K),S)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,lattic8721135487736765967in_nat(image_o_nat(F,S))),K) ) ) ) ).
% Min_add_commute
tff(fact_6725_Min__add__commute,axiom,
! [S: set_o,F: fun_o_int,K: int] :
( aa_set_o_o(finite_finite_o,S)
=> ( ( S != bot_bot_set_o )
=> ( lattic8718645017227715691in_int(image_o_int(aa_int_fun_o_int(aTP_Lamm_cr(F),K),S)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,lattic8718645017227715691in_int(image_o_int(F,S))),K) ) ) ) ).
% Min_add_commute
tff(fact_6726_Min__add__commute,axiom,
! [S: set_nat,F: fun_nat_Code_integer,K: code_integer] :
( aa_set_nat_o(finite_finite_nat,S)
=> ( ( S != bot_bot_set_nat )
=> ( lattic1063845414844153500nteger(image_1215581382706833972nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_cs(F),K),S)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,lattic1063845414844153500nteger(image_1215581382706833972nteger(F,S))),K) ) ) ) ).
% Min_add_commute
tff(fact_6727_Min__add__commute,axiom,
! [S: set_nat,F: fun_nat_rat,K: rat] :
( aa_set_nat_o(finite_finite_nat,S)
=> ( ( S != bot_bot_set_nat )
=> ( lattic8086005427650270231in_rat(image_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_ct(F),K),S)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,lattic8086005427650270231in_rat(image_nat_rat(F,S))),K) ) ) ) ).
% Min_add_commute
tff(fact_6728_Min__add__commute,axiom,
! [S: set_nat,F: fun_nat_nat,K: nat] :
( aa_set_nat_o(finite_finite_nat,S)
=> ( ( S != bot_bot_set_nat )
=> ( lattic8721135487736765967in_nat(aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cu(F),K)),S)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,lattic8721135487736765967in_nat(aa_set_nat_set_nat(image_nat_nat(F),S))),K) ) ) ) ).
% Min_add_commute
tff(fact_6729_Min__add__commute,axiom,
! [S: set_nat,F: fun_nat_int,K: int] :
( aa_set_nat_o(finite_finite_nat,S)
=> ( ( S != bot_bot_set_nat )
=> ( lattic8718645017227715691in_int(image_nat_int(aa_int_fun_nat_int(aTP_Lamm_cv(F),K),S)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,lattic8718645017227715691in_int(image_nat_int(F,S))),K) ) ) ) ).
% Min_add_commute
tff(fact_6730_Min__add__commute,axiom,
! [S: set_int,F: fun_int_Code_integer,K: code_integer] :
( aa_set_int_o(finite_finite_int,S)
=> ( ( S != bot_bot_set_int )
=> ( lattic1063845414844153500nteger(image_1587234942943678608nteger(aa_Cod1949805079358391713nteger(aTP_Lamm_cw(F),K),S)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,lattic1063845414844153500nteger(image_1587234942943678608nteger(F,S))),K) ) ) ) ).
% Min_add_commute
tff(fact_6731_Min__add__commute,axiom,
! [S: set_int,F: fun_int_rat,K: rat] :
( aa_set_int_o(finite_finite_int,S)
=> ( ( S != bot_bot_set_int )
=> ( lattic8086005427650270231in_rat(image_int_rat(aa_rat_fun_int_rat(aTP_Lamm_cx(F),K),S)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,lattic8086005427650270231in_rat(image_int_rat(F,S))),K) ) ) ) ).
% Min_add_commute
tff(fact_6732_prod_OatMost__Suc__shift,axiom,
! [G: fun_nat_Code_integer,N: nat] : groups3455450783089532116nteger(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,zero_zero_nat)),groups3455450783089532116nteger(aTP_Lamm_cj(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).
% prod.atMost_Suc_shift
tff(fact_6733_prod_OatMost__Suc__shift,axiom,
! [G: fun_nat_assn,N: nat] : groups6906906614972039071t_assn(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,zero_zero_nat)),groups6906906614972039071t_assn(aTP_Lamm_ck(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).
% prod.atMost_Suc_shift
tff(fact_6734_prod_OatMost__Suc__shift,axiom,
! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_cl(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).
% prod.atMost_Suc_shift
tff(fact_6735_prod_OatMost__Suc__shift,axiom,
! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),groups708209901874060359at_nat(aTP_Lamm_cm(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).
% prod.atMost_Suc_shift
tff(fact_6736_prod_OatMost__Suc__shift,axiom,
! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_cn(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).
% prod.atMost_Suc_shift
tff(fact_6737_bot__assn__def,axiom,
bot_bot_assn = aa_fun5526454687433275355o_assn(abs_assn,aTP_Lamm_af) ).
% bot_assn_def
tff(fact_6738_prod__atLeastAtMost__code,axiom,
! [F: fun_nat_Code_integer,A: nat,B: nat] : groups3455450783089532116nteger(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_Cod1783974652533396728nteger(set_fo1084959871951514735nteger(aTP_Lamm_cy(F),A,B),one_one_Code_integer) ).
% prod_atLeastAtMost_code
tff(fact_6739_prod__atLeastAtMost__code,axiom,
! [F: fun_nat_assn,A: nat,B: nat] : groups6906906614972039071t_assn(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_assn_assn(set_fo1959793692361082170t_assn(aTP_Lamm_cz(F),A,B),one_one_assn) ).
% prod_atLeastAtMost_code
tff(fact_6740_prod__atLeastAtMost__code,axiom,
! [F: fun_nat_rat,A: nat,B: nat] : groups73079841787564623at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_fo1949268297981939178at_rat(aTP_Lamm_da(F),A,B,one_one_rat) ).
% prod_atLeastAtMost_code
tff(fact_6741_prod__atLeastAtMost__code,axiom,
! [F: fun_nat_nat,A: nat,B: nat] : groups708209901874060359at_nat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_db(F),A),B),one_one_nat) ).
% prod_atLeastAtMost_code
tff(fact_6742_prod__atLeastAtMost__code,axiom,
! [F: fun_nat_int,A: nat,B: nat] : groups705719431365010083at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_dc(F),A,B),one_one_int) ).
% prod_atLeastAtMost_code
tff(fact_6743_not__empty__eq__Ici__eq__empty,axiom,
! [L: $o] : bot_bot_set_o != set_ord_atLeast_o((L)) ).
% not_empty_eq_Ici_eq_empty
tff(fact_6744_not__empty__eq__Ici__eq__empty,axiom,
! [L: int] : bot_bot_set_int != set_ord_atLeast_int(L) ).
% not_empty_eq_Ici_eq_empty
tff(fact_6745_not__empty__eq__Ici__eq__empty,axiom,
! [L: nat] : bot_bot_set_nat != aa_nat_set_nat(set_ord_atLeast_nat,L) ).
% not_empty_eq_Ici_eq_empty
tff(fact_6746_image__constant,axiom,
! [X: $o,Aa: set_o,C: $o] :
( aa_set_o_o(member_o((X)),Aa)
=> ( image_o_o(aa_o_fun_o_o(aTP_Lamm_ar,(C)),Aa) = aa_set_o_set_o(insert_o((C)),bot_bot_set_o) ) ) ).
% image_constant
tff(fact_6747_image__constant,axiom,
! [X: nat,Aa: set_nat,C: $o] :
( aa_set_nat_o(member_nat(X),Aa)
=> ( image_nat_o(aTP_Lamm_as((C)),Aa) = aa_set_o_set_o(insert_o((C)),bot_bot_set_o) ) ) ).
% image_constant
tff(fact_6748_image__constant,axiom,
! [X: int,Aa: set_int,C: $o] :
( aa_set_int_o(member_int(X),Aa)
=> ( image_int_o(aTP_Lamm_aq((C)),Aa) = aa_set_o_set_o(insert_o((C)),bot_bot_set_o) ) ) ).
% image_constant
tff(fact_6749_image__constant,axiom,
! [X: $o,Aa: set_o,C: nat] :
( aa_set_o_o(member_o((X)),Aa)
=> ( image_o_nat(aTP_Lamm_av(C),Aa) = aa_set_nat_set_nat(insert_nat(C),bot_bot_set_nat) ) ) ).
% image_constant
tff(fact_6750_image__constant,axiom,
! [X: nat,Aa: set_nat,C: nat] :
( aa_set_nat_o(member_nat(X),Aa)
=> ( aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ax,C)),Aa) = aa_set_nat_set_nat(insert_nat(C),bot_bot_set_nat) ) ) ).
% image_constant
tff(fact_6751_image__constant,axiom,
! [X: int,Aa: set_int,C: nat] :
( aa_set_int_o(member_int(X),Aa)
=> ( image_int_nat(aTP_Lamm_dd(C),Aa) = aa_set_nat_set_nat(insert_nat(C),bot_bot_set_nat) ) ) ).
% image_constant
tff(fact_6752_image__constant,axiom,
! [X: $o,Aa: set_o,C: int] :
( aa_set_o_o(member_o((X)),Aa)
=> ( image_o_int(aTP_Lamm_aw(C),Aa) = aa_set_int_set_int(insert_int(C),bot_bot_set_int) ) ) ).
% image_constant
tff(fact_6753_image__constant,axiom,
! [X: nat,Aa: set_nat,C: int] :
( aa_set_nat_o(member_nat(X),Aa)
=> ( image_nat_int(aTP_Lamm_ay(C),Aa) = aa_set_int_set_int(insert_int(C),bot_bot_set_int) ) ) ).
% image_constant
tff(fact_6754_image__constant,axiom,
! [X: int,Aa: set_int,C: int] :
( aa_set_int_o(member_int(X),Aa)
=> ( image_int_int(aa_int_fun_int_int(aTP_Lamm_de,C),Aa) = aa_set_int_set_int(insert_int(C),bot_bot_set_int) ) ) ).
% image_constant
tff(fact_6755_image__constant,axiom,
! [X: nat,Aa: set_nat,C: set_nat] :
( aa_set_nat_o(member_nat(X),Aa)
=> ( image_nat_set_nat(aTP_Lamm_bb(C),Aa) = insert_set_nat(C,bot_bot_set_set_nat) ) ) ).
% image_constant
tff(fact_6756_image__constant__conv,axiom,
! [C: $o,Aa: set_o] :
image_o_o(aa_o_fun_o_o(aTP_Lamm_ar,(C)),Aa) = $ite(Aa = bot_bot_set_o,bot_bot_set_o,aa_set_o_set_o(insert_o((C)),bot_bot_set_o)) ).
% image_constant_conv
tff(fact_6757_image__constant__conv,axiom,
! [C: nat,Aa: set_o] :
image_o_nat(aTP_Lamm_av(C),Aa) = $ite(Aa = bot_bot_set_o,bot_bot_set_nat,aa_set_nat_set_nat(insert_nat(C),bot_bot_set_nat)) ).
% image_constant_conv
tff(fact_6758_image__constant__conv,axiom,
! [C: int,Aa: set_o] :
image_o_int(aTP_Lamm_aw(C),Aa) = $ite(Aa = bot_bot_set_o,bot_bot_set_int,aa_set_int_set_int(insert_int(C),bot_bot_set_int)) ).
% image_constant_conv
tff(fact_6759_image__constant__conv,axiom,
! [C: $o,Aa: set_nat] :
image_nat_o(aTP_Lamm_as((C)),Aa) = $ite(Aa = bot_bot_set_nat,bot_bot_set_o,aa_set_o_set_o(insert_o((C)),bot_bot_set_o)) ).
% image_constant_conv
tff(fact_6760_image__constant__conv,axiom,
! [C: nat,Aa: set_nat] :
aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ax,C)),Aa) = $ite(Aa = bot_bot_set_nat,bot_bot_set_nat,aa_set_nat_set_nat(insert_nat(C),bot_bot_set_nat)) ).
% image_constant_conv
tff(fact_6761_image__constant__conv,axiom,
! [C: int,Aa: set_nat] :
image_nat_int(aTP_Lamm_ay(C),Aa) = $ite(Aa = bot_bot_set_nat,bot_bot_set_int,aa_set_int_set_int(insert_int(C),bot_bot_set_int)) ).
% image_constant_conv
tff(fact_6762_image__constant__conv,axiom,
! [C: $o,Aa: set_int] :
image_int_o(aTP_Lamm_aq((C)),Aa) = $ite(Aa = bot_bot_set_int,bot_bot_set_o,aa_set_o_set_o(insert_o((C)),bot_bot_set_o)) ).
% image_constant_conv
tff(fact_6763_image__constant__conv,axiom,
! [C: nat,Aa: set_int] :
image_int_nat(aTP_Lamm_dd(C),Aa) = $ite(Aa = bot_bot_set_int,bot_bot_set_nat,aa_set_nat_set_nat(insert_nat(C),bot_bot_set_nat)) ).
% image_constant_conv
tff(fact_6764_image__constant__conv,axiom,
! [C: int,Aa: set_int] :
image_int_int(aa_int_fun_int_int(aTP_Lamm_de,C),Aa) = $ite(Aa = bot_bot_set_int,bot_bot_set_int,aa_set_int_set_int(insert_int(C),bot_bot_set_int)) ).
% image_constant_conv
tff(fact_6765_image__constant__conv,axiom,
! [C: set_nat,Aa: set_nat] :
image_nat_set_nat(aTP_Lamm_bb(C),Aa) = $ite(Aa = bot_bot_set_nat,bot_bot_set_set_nat,insert_set_nat(C,bot_bot_set_set_nat)) ).
% image_constant_conv
tff(fact_6766_gbinomial__mult__fact,axiom,
! [K: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(gbinomial_rat(A),K)) = groups73079841787564623at_rat(aTP_Lamm_df(A),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ).
% gbinomial_mult_fact
tff(fact_6767_gbinomial__mult__fact_H,axiom,
! [A: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(semiri773545260158071498ct_rat,K)) = groups73079841787564623at_rat(aTP_Lamm_df(A),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ).
% gbinomial_mult_fact'
tff(fact_6768_prod_OatLeast0__lessThan__Suc,axiom,
! [G: fun_nat_Code_integer,N: nat] : groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_Code_integer(G,N)) ).
% prod.atLeast0_lessThan_Suc
tff(fact_6769_prod_OatLeast0__lessThan__Suc,axiom,
! [G: fun_nat_assn,N: nat] : groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_assn(G,N)) ).
% prod.atLeast0_lessThan_Suc
tff(fact_6770_prod_OatLeast0__lessThan__Suc,axiom,
! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_rat(G,N)) ).
% prod.atLeast0_lessThan_Suc
tff(fact_6771_prod_OatLeast0__lessThan__Suc,axiom,
! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_nat(G,N)) ).
% prod.atLeast0_lessThan_Suc
tff(fact_6772_prod_OatLeast0__lessThan__Suc,axiom,
! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_int(G,N)) ).
% prod.atLeast0_lessThan_Suc
tff(fact_6773_prod_OatLeast__Suc__lessThan,axiom,
! [M2: nat,N: nat,G: fun_nat_Code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
=> ( groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,M2)),groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M2)),N))) ) ) ).
% prod.atLeast_Suc_lessThan
tff(fact_6774_prod_OatLeast__Suc__lessThan,axiom,
! [M2: nat,N: nat,G: fun_nat_assn] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
=> ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,M2)),groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M2)),N))) ) ) ).
% prod.atLeast_Suc_lessThan
tff(fact_6775_prod_OatLeast__Suc__lessThan,axiom,
! [M2: nat,N: nat,G: fun_nat_rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
=> ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,M2)),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M2)),N))) ) ) ).
% prod.atLeast_Suc_lessThan
tff(fact_6776_prod_OatLeast__Suc__lessThan,axiom,
! [M2: nat,N: nat,G: fun_nat_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
=> ( groups708209901874060359at_nat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M2)),groups708209901874060359at_nat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M2)),N))) ) ) ).
% prod.atLeast_Suc_lessThan
tff(fact_6777_prod_OatLeast__Suc__lessThan,axiom,
! [M2: nat,N: nat,G: fun_nat_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
=> ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M2)),groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M2)),N))) ) ) ).
% prod.atLeast_Suc_lessThan
tff(fact_6778_prod_OatLeastLessThan__Suc,axiom,
! [A: nat,B: nat,G: fun_nat_Code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_Code_integer(G,B)) ) ) ).
% prod.atLeastLessThan_Suc
tff(fact_6779_prod_OatLeastLessThan__Suc,axiom,
! [A: nat,B: nat,G: fun_nat_assn] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_assn(G,B)) ) ) ).
% prod.atLeastLessThan_Suc
tff(fact_6780_prod_OatLeastLessThan__Suc,axiom,
! [A: nat,B: nat,G: fun_nat_rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_rat(G,B)) ) ) ).
% prod.atLeastLessThan_Suc
tff(fact_6781_prod_OatLeastLessThan__Suc,axiom,
! [A: nat,B: nat,G: fun_nat_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( groups708209901874060359at_nat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_nat(G,B)) ) ) ).
% prod.atLeastLessThan_Suc
tff(fact_6782_prod_OatLeastLessThan__Suc,axiom,
! [A: nat,B: nat,G: fun_nat_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_int(G,B)) ) ) ).
% prod.atLeastLessThan_Suc
tff(fact_6783_prod_Olast__plus,axiom,
! [M2: nat,N: nat,G: fun_nat_Code_integer] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( groups3455450783089532116nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(G,N)),groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))) ) ) ).
% prod.last_plus
tff(fact_6784_prod_Olast__plus,axiom,
! [M2: nat,N: nat,G: fun_nat_assn] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(G,N)),groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))) ) ) ).
% prod.last_plus
tff(fact_6785_prod_Olast__plus,axiom,
! [M2: nat,N: nat,G: fun_nat_rat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,N)),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))) ) ) ).
% prod.last_plus
tff(fact_6786_prod_Olast__plus,axiom,
! [M2: nat,N: nat,G: fun_nat_nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,N)),groups708209901874060359at_nat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))) ) ) ).
% prod.last_plus
tff(fact_6787_prod_Olast__plus,axiom,
! [M2: nat,N: nat,G: fun_nat_int] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,N)),groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))) ) ) ).
% prod.last_plus
tff(fact_6788_image__minus__const__atLeastLessThan__nat,axiom,
! [C: nat,X: nat,Y: nat] :
aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dg,C)),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = $ite(
aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y),
aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)),
$ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat),bot_bot_set_nat) ) ).
% image_minus_const_atLeastLessThan_nat
tff(fact_6789_sup__assn__def,axiom,
! [P: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,P),Q) = aa_fun5526454687433275355o_assn(abs_assn,aa_ass4243947775541419017_nat_o(aTP_Lamm_dh(P),Q)) ).
% sup_assn_def
tff(fact_6790_inf__assn__def,axiom,
! [P: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P),Q) = aa_fun5526454687433275355o_assn(abs_assn,aa_ass4243947775541419017_nat_o(aTP_Lamm_di(P),Q)) ).
% inf_assn_def
tff(fact_6791_numeral__code_I3_J,axiom,
! [N: num] :
numeral_numeral_nat(aa_num_num(bit1,N)) = $let(
m: nat,
m:= numeral_numeral_nat(N),
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m)),one_one_nat) ) ).
% numeral_code(3)
tff(fact_6792_numeral__code_I3_J,axiom,
! [N: num] :
numeral_numeral_int(aa_num_num(bit1,N)) = $let(
m: int,
m:= numeral_numeral_int(N),
aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,m),m)),one_one_int) ) ).
% numeral_code(3)
tff(fact_6793_numeral__code_I3_J,axiom,
! [N: num] :
numeral_numeral_rat(aa_num_num(bit1,N)) = $let(
m: rat,
m:= numeral_numeral_rat(N),
aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m),m)),one_one_rat) ) ).
% numeral_code(3)
tff(fact_6794_numeral__code_I3_J,axiom,
! [N: num] :
numera6620942414471956472nteger(aa_num_num(bit1,N)) = $let(
m: code_integer,
m:= numera6620942414471956472nteger(N),
aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,m),m)),one_one_Code_integer) ) ).
% numeral_code(3)
tff(fact_6795_numeral__code_I3_J,axiom,
! [N: num] :
numera5444537566228673987atural(aa_num_num(bit1,N)) = $let(
m: code_natural,
m:= numera5444537566228673987atural(N),
aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(m),m)),one_one_Code_natural) ) ).
% numeral_code(3)
tff(fact_6796_power__numeral__even,axiom,
! [Z: code_integer,W: num] :
aa_nat_Code_integer(power_8256067586552552935nteger(Z),numeral_numeral_nat(aa_num_num(bit0,W))) = $let(
w: code_integer,
w:= aa_nat_Code_integer(power_8256067586552552935nteger(Z),numeral_numeral_nat(W)),
aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,w),w) ) ).
% power_numeral_even
tff(fact_6797_power__numeral__even,axiom,
! [Z: assn,W: num] :
aa_nat_assn(power_power_assn(Z),numeral_numeral_nat(aa_num_num(bit0,W))) = $let(
w: assn,
w:= aa_nat_assn(power_power_assn(Z),numeral_numeral_nat(W)),
aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,w),w) ) ).
% power_numeral_even
tff(fact_6798_power__numeral__even,axiom,
! [Z: rat,W: num] :
aa_nat_rat(power_power_rat(Z),numeral_numeral_nat(aa_num_num(bit0,W))) = $let(
w: rat,
w:= aa_nat_rat(power_power_rat(Z),numeral_numeral_nat(W)),
aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,w),w) ) ).
% power_numeral_even
tff(fact_6799_power__numeral__even,axiom,
! [Z: nat,W: num] :
aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),numeral_numeral_nat(aa_num_num(bit0,W))) = $let(
w: nat,
w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),numeral_numeral_nat(W)),
aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,w),w) ) ).
% power_numeral_even
tff(fact_6800_power__numeral__even,axiom,
! [Z: int,W: num] :
aa_nat_int(power_power_int(Z),numeral_numeral_nat(aa_num_num(bit0,W))) = $let(
w: int,
w:= aa_nat_int(power_power_int(Z),numeral_numeral_nat(W)),
aa_int_int(aa_int_fun_int_int(times_times_int,w),w) ) ).
% power_numeral_even
tff(fact_6801_power__numeral__odd,axiom,
! [Z: code_integer,W: num] :
aa_nat_Code_integer(power_8256067586552552935nteger(Z),numeral_numeral_nat(aa_num_num(bit1,W))) = $let(
w: code_integer,
w:= aa_nat_Code_integer(power_8256067586552552935nteger(Z),numeral_numeral_nat(W)),
aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),w)),w) ) ).
% power_numeral_odd
tff(fact_6802_power__numeral__odd,axiom,
! [Z: assn,W: num] :
aa_nat_assn(power_power_assn(Z),numeral_numeral_nat(aa_num_num(bit1,W))) = $let(
w: assn,
w:= aa_nat_assn(power_power_assn(Z),numeral_numeral_nat(W)),
aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Z),w)),w) ) ).
% power_numeral_odd
tff(fact_6803_power__numeral__odd,axiom,
! [Z: rat,W: num] :
aa_nat_rat(power_power_rat(Z),numeral_numeral_nat(aa_num_num(bit1,W))) = $let(
w: rat,
w:= aa_nat_rat(power_power_rat(Z),numeral_numeral_nat(W)),
aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),w)),w) ) ).
% power_numeral_odd
tff(fact_6804_power__numeral__odd,axiom,
! [Z: nat,W: num] :
aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),numeral_numeral_nat(aa_num_num(bit1,W))) = $let(
w: nat,
w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),numeral_numeral_nat(W)),
aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),w)),w) ) ).
% power_numeral_odd
tff(fact_6805_power__numeral__odd,axiom,
! [Z: int,W: num] :
aa_nat_int(power_power_int(Z),numeral_numeral_nat(aa_num_num(bit1,W))) = $let(
w: int,
w:= aa_nat_int(power_power_int(Z),numeral_numeral_nat(W)),
aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),w)),w) ) ).
% power_numeral_odd
tff(fact_6806_prod_Ohead__if,axiom,
! [G: fun_nat_Code_integer,M2: nat,N: nat] :
groups3455450783089532116nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2),one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))),aa_nat_Code_integer(G,N))) ).
% prod.head_if
tff(fact_6807_prod_Ohead__if,axiom,
! [G: fun_nat_assn,M2: nat,N: nat] :
groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2),one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))),aa_nat_assn(G,N))) ).
% prod.head_if
tff(fact_6808_prod_Ohead__if,axiom,
! [G: fun_nat_rat,M2: nat,N: nat] :
groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2),one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))),aa_nat_rat(G,N))) ).
% prod.head_if
tff(fact_6809_prod_Ohead__if,axiom,
! [G: fun_nat_nat,M2: nat,N: nat] :
groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))),aa_nat_nat(G,N))) ).
% prod.head_if
tff(fact_6810_prod_Ohead__if,axiom,
! [G: fun_nat_int,M2: nat,N: nat] :
groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N))),aa_nat_int(G,N))) ).
% prod.head_if
tff(fact_6811_bot_Oordering__top__axioms,axiom,
aa_set3702741264012733747_nat_o(orderi2172309028950807442at_nat(aTP_Lamm_dj,aTP_Lamm_dk),bot_bo2099793752762293965at_nat) ).
% bot.ordering_top_axioms
tff(fact_6812_bot_Oordering__top__axioms,axiom,
aa_set_o_o(ordering_top_set_o(aTP_Lamm_dl,aTP_Lamm_dm),bot_bot_set_o) ).
% bot.ordering_top_axioms
tff(fact_6813_bot_Oordering__top__axioms,axiom,
aa_set_int_o(ordering_top_set_int(aTP_Lamm_dn,aTP_Lamm_do),bot_bot_set_int) ).
% bot.ordering_top_axioms
tff(fact_6814_bot_Oordering__top__axioms,axiom,
aa_assn_o(ordering_top_assn(aTP_Lamm_dp,aTP_Lamm_dq),bot_bot_assn) ).
% bot.ordering_top_axioms
tff(fact_6815_bot_Oordering__top__axioms,axiom,
aa_set_nat_o(ordering_top_set_nat(aTP_Lamm_dr,aTP_Lamm_ds),bot_bot_set_nat) ).
% bot.ordering_top_axioms
tff(fact_6816_bot_Oordering__top__axioms,axiom,
aa_nat_o(ordering_top_nat(aTP_Lamm_dt,aTP_Lamm_du),bot_bot_nat) ).
% bot.ordering_top_axioms
tff(fact_6817_ivl__disj__un__singleton_I3_J,axiom,
! [L: $o,U: $o] :
( aa_o_o(aa_o_fun_o_o(ord_less_o,(L)),(U))
=> ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(insert_o((L)),bot_bot_set_o)),set_or1716231572884733764Than_o((L),(U))) = set_or7139685690850216873Than_o((L),(U)) ) ) ).
% ivl_disj_un_singleton(3)
tff(fact_6818_ivl__disj__un__singleton_I3_J,axiom,
! [L: int,U: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,L),U)
=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(insert_int(L),bot_bot_set_int)),set_or5832277885323065728an_int(L,U)) = set_or4662586982721622107an_int(L,U) ) ) ).
% ivl_disj_un_singleton(3)
tff(fact_6819_ivl__disj__un__singleton_I3_J,axiom,
! [L: code_integer,U: code_integer] :
( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,L),U)
=> ( sup_su848401254843788991nteger(insert_Code_integer(L,bot_bo3990330152332043303nteger),set_or4266950643985792945nteger(L,U)) = set_or8404916559141939852nteger(L,U) ) ) ).
% ivl_disj_un_singleton(3)
tff(fact_6820_ivl__disj__un__singleton_I3_J,axiom,
! [L: code_natural,U: code_natural] :
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,L),U)
=> ( sup_su5306851075354978058atural(insert_Code_natural(L,bot_bo8448779972843232370atural),set_or3090545795742510460atural(L,U)) = set_or7228511710898657367atural(L,U) ) ) ).
% ivl_disj_un_singleton(3)
tff(fact_6821_ivl__disj__un__singleton_I3_J,axiom,
! [L: nat,U: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,L),U)
=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(insert_nat(L),bot_bot_set_nat)),aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)) = aa_nat_set_nat(set_or4665077453230672383an_nat(L),U) ) ) ).
% ivl_disj_un_singleton(3)
tff(fact_6822_prod_Oin__pairs,axiom,
! [G: fun_nat_assn,M2: nat,N: nat] : groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M2)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = groups6906906614972039071t_assn(aTP_Lamm_dv(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).
% prod.in_pairs
tff(fact_6823_prod_Oin__pairs,axiom,
! [G: fun_nat_rat,M2: nat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M2)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = groups73079841787564623at_rat(aTP_Lamm_dw(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).
% prod.in_pairs
tff(fact_6824_prod_Oin__pairs,axiom,
! [G: fun_nat_nat,M2: nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M2)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = groups708209901874060359at_nat(aTP_Lamm_dx(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).
% prod.in_pairs
tff(fact_6825_prod_Oin__pairs,axiom,
! [G: fun_nat_int,M2: nat,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),M2)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))) = groups705719431365010083at_int(aTP_Lamm_dy(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N)) ).
% prod.in_pairs
tff(fact_6826_atLeastLessThan0,axiom,
! [M2: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M2),zero_zero_nat) = bot_bot_set_nat ).
% atLeastLessThan0
tff(fact_6827_uminus__assn__def,axiom,
! [P: assn] : aa_assn_assn(uminus_uminus_assn,P) = aa_fun5526454687433275355o_assn(abs_assn,aTP_Lamm_dz(P)) ).
% uminus_assn_def
tff(fact_6828_atLeastLessThanSuc,axiom,
! [M2: nat,N: nat] :
aa_nat_set_nat(set_or4665077453230672383an_nat(M2),aa_nat_nat(suc,N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N),aa_set_nat_set_nat(insert_nat(N),aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N)),bot_bot_set_nat) ).
% atLeastLessThanSuc
tff(fact_6829_normalize__def,axiom,
! [P5: product_prod_int_int] :
normalize(P5) = $ite(
aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,P5)),
$let(
a2: int,
a2:= aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P5)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P5)),
aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P5)),a2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P5)),a2)) ),
$ite(
aa_Pro6799258434465214121nt_int(product_snd_int_int,P5) = zero_zero_int,
aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),
$let(
a2: int,
a2:= aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P5)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P5))),
aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P5)),a2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P5)),a2)) ) ) ) ).
% normalize_def
tff(fact_6830_divmod__step__int__def,axiom,
! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_ea(L),Qr) ).
% divmod_step_int_def
tff(fact_6831_lessThan__0,axiom,
aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).
% lessThan_0
tff(fact_6832_lessThan__empty__iff,axiom,
! [N: nat] :
( ( aa_nat_set_nat(set_ord_lessThan_nat,N) = bot_bot_set_nat )
<=> ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
tff(fact_6833_rat__abs__code,axiom,
! [P5: rat] : quotient_of(abs_abs_rat(P5)) = produc4245557441103728435nt_int(aTP_Lamm_eb,quotient_of(P5)) ).
% rat_abs_code
tff(fact_6834_rat__uminus__code,axiom,
! [P5: rat] : quotient_of(aa_rat_rat(uminus_uminus_rat,P5)) = produc4245557441103728435nt_int(aTP_Lamm_ec,quotient_of(P5)) ).
% rat_uminus_code
tff(fact_6835_rat__minus__code,axiom,
! [P5: rat,Q4: rat] : quotient_of(minus_minus_rat(P5,Q4)) = produc4245557441103728435nt_int(aTP_Lamm_ee(Q4),quotient_of(P5)) ).
% rat_minus_code
tff(fact_6836_rat__divide__code,axiom,
! [P5: rat,Q4: rat] : quotient_of(divide_divide_rat(P5,Q4)) = produc4245557441103728435nt_int(aTP_Lamm_eg(Q4),quotient_of(P5)) ).
% rat_divide_code
tff(fact_6837_rat__times__code,axiom,
! [P5: rat,Q4: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P5),Q4)) = produc4245557441103728435nt_int(aTP_Lamm_ei(Q4),quotient_of(P5)) ).
% rat_times_code
tff(fact_6838_divmod__nat__if,axiom,
! [M2: nat,N: nat] :
divmod_nat(M2,N) = $ite(
( ( N = zero_zero_nat )
| aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N) ),
aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M2),
aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ej),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N),N)) ) ).
% divmod_nat_if
tff(fact_6839_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,N)),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat)) ).
% atLeast1_lessThan_eq_remove0
tff(fact_6840_rat__plus__code,axiom,
! [P5: rat,Q4: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,P5),Q4)) = produc4245557441103728435nt_int(aTP_Lamm_el(Q4),quotient_of(P5)) ).
% rat_plus_code
tff(fact_6841_divmod__step__nat__def,axiom,
! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_em(L)),Qr) ).
% divmod_step_nat_def
tff(fact_6842_rat__inverse__code,axiom,
! [P5: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P5)) = produc4245557441103728435nt_int(aTP_Lamm_en,quotient_of(P5)) ).
% rat_inverse_code
tff(fact_6843_divmod__step__integer__def,axiom,
! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_eo(L),Qr) ).
% divmod_step_integer_def
tff(fact_6844_inverse__rat_Otransfer,axiom,
aa_fun_rat_rat_o(aa_fun382765352565706371_rat_o(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_ep),inverse_inverse_rat) ).
% inverse_rat.transfer
tff(fact_6845_divmod__integer_H__def,axiom,
! [M2: num,N: num] : unique3479559517661332726nteger(M2,N) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(M2)),numera6620942414471956472nteger(N)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(M2)),numera6620942414471956472nteger(N))) ).
% divmod_integer'_def
tff(fact_6846_gcd__nat_Osemilattice__neutr__axioms,axiom,
aa_nat_o(semila9081495762789891438tr_nat(gcd_gcd_nat),zero_zero_nat) ).
% gcd_nat.semilattice_neutr_axioms
tff(fact_6847_Gcd__in,axiom,
! [Aa: set_nat] :
( ! [A3: nat,B3: nat] :
( aa_set_nat_o(member_nat(A3),Aa)
=> ( aa_set_nat_o(member_nat(B3),Aa)
=> aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A3),B3)),Aa) ) )
=> ( ( Aa != bot_bot_set_nat )
=> aa_set_nat_o(member_nat(gcd_Gcd_nat(Aa)),Aa) ) ) ).
% Gcd_in
tff(fact_6848_uminus__rat_Otransfer,axiom,
aa_fun_rat_rat_o(aa_fun382765352565706371_rat_o(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_eq),uminus_uminus_rat) ).
% uminus_rat.transfer
tff(fact_6849_gcd__nat_Opelims,axiom,
! [X: nat,Xa: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa) = Y )
=> ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa))
=> ~ ( ( Y = $ite(Xa = zero_zero_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa))) )
=> ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)) ) ) ) ).
% gcd_nat.pelims
tff(fact_6850_plus__rat_Otransfer,axiom,
bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_er,plus_plus_rat) ).
% plus_rat.transfer
tff(fact_6851_times__rat_Otransfer,axiom,
bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_es,times_times_rat) ).
% times_rat.transfer
tff(fact_6852_times__int_Oabs__eq,axiom,
! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_eu),Xa),X)) ).
% times_int.abs_eq
tff(fact_6853_eq__Abs__Integ,axiom,
! [Z: int] :
~ ! [X2: nat,Y2: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2)) ).
% eq_Abs_Integ
tff(fact_6854_Fract_Orsp,axiom,
bNF_re4808202430149235895nt_int(fequal_int,bNF_re6250860962936578807nt_int(fequal_int,ratrel),aTP_Lamm_ev,aTP_Lamm_ev) ).
% Fract.rsp
tff(fact_6855_times__rat_Orsp,axiom,
bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_es,aTP_Lamm_es) ).
% times_rat.rsp
tff(fact_6856_plus__int_Otransfer,axiom,
bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_ex),plus_plus_int) ).
% plus_int.transfer
tff(fact_6857_minus__int_Otransfer,axiom,
bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_ez),minus_minus_int) ).
% minus_int.transfer
tff(fact_6858_plus__rat_Orsp,axiom,
bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_er,aTP_Lamm_er) ).
% plus_rat.rsp
tff(fact_6859_times__int_Otransfer,axiom,
bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_eu),times_times_int) ).
% times_int.transfer
tff(fact_6860_zero__int__def,axiom,
zero_zero_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).
% zero_int_def
tff(fact_6861_int__def,axiom,
! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N),zero_zero_nat)) ).
% int_def
tff(fact_6862_int__transfer,axiom,
bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_fa,semiri1314217659103216013at_int) ).
% int_transfer
tff(fact_6863_uminus__int_Otransfer,axiom,
aa_fun_int_int_o(aa_fun856177901615811547_int_o(bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc2626176000494625587at_nat(aTP_Lamm_fb)),uminus_uminus_int) ).
% uminus_int.transfer
tff(fact_6864_uminus__int_Oabs__eq,axiom,
! [X: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_fb),X)) ).
% uminus_int.abs_eq
tff(fact_6865_one__int__def,axiom,
one_one_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).
% one_int_def
tff(fact_6866_uminus__rat_Orsp,axiom,
aa_fun2650603632653508759_int_o(aa_fun3734617511155415725_int_o(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_eq),aTP_Lamm_eq) ).
% uminus_rat.rsp
tff(fact_6867_plus__int_Oabs__eq,axiom,
! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_ex),Xa),X)) ).
% plus_int.abs_eq
tff(fact_6868_minus__int_Oabs__eq,axiom,
! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_ez),Xa),X)) ).
% minus_int.abs_eq
tff(fact_6869_inverse__rat_Orsp,axiom,
aa_fun2650603632653508759_int_o(aa_fun3734617511155415725_int_o(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_ep),aTP_Lamm_ep) ).
% inverse_rat.rsp
tff(fact_6870_bit__cut__integer__def,axiom,
! [K: code_integer] : code_bit_cut_integer(K) = produc6677183202524767010eger_o(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(K),numera6620942414471956472nteger(aa_num_num(bit0,one))),~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),K)) ).
% bit_cut_integer_def
tff(fact_6871_divmod__integer__def,axiom,
! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(K),L),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(K),L)) ).
% divmod_integer_def
tff(fact_6872_times__int_Orsp,axiom,
bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_eu),produc4528391901023682385at_nat(aTP_Lamm_eu)) ).
% times_int.rsp
tff(fact_6873_intrel__iff,axiom,
! [X: nat,Y: nat,U: nat,V: nat] :
( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V))
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),V) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,U),Y) ) ) ).
% intrel_iff
tff(fact_6874_zero__int_Orsp,axiom,
aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).
% zero_int.rsp
tff(fact_6875_uminus__int_Orsp,axiom,
aa_fun3538214411680374423_nat_o(aa_fun6920661532540581549_nat_o(bNF_re2241393799969408733at_nat(intrel,intrel),produc2626176000494625587at_nat(aTP_Lamm_fb)),produc2626176000494625587at_nat(aTP_Lamm_fb)) ).
% uminus_int.rsp
tff(fact_6876_one__int_Orsp,axiom,
aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).
% one_int.rsp
tff(fact_6877_minus__int_Orsp,axiom,
bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_ez),produc4528391901023682385at_nat(aTP_Lamm_ez)) ).
% minus_int.rsp
tff(fact_6878_plus__int_Orsp,axiom,
bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_ex),produc4528391901023682385at_nat(aTP_Lamm_ex)) ).
% plus_int.rsp
tff(fact_6879_Sup__nat__empty,axiom,
complete_Sup_Sup_nat(bot_bot_set_nat) = zero_zero_nat ).
% Sup_nat_empty
tff(fact_6880_Sup__nat__def,axiom,
! [X5: set_nat] :
complete_Sup_Sup_nat(X5) = $ite(X5 = bot_bot_set_nat,zero_zero_nat,lattic8265883725875713057ax_nat(X5)) ).
% Sup_nat_def
tff(fact_6881_bit__cut__integer__code,axiom,
! [K: code_integer] :
code_bit_cut_integer(K) = $ite(K = zero_z3403309356797280102nteger,produc6677183202524767010eger_o(zero_z3403309356797280102nteger,$false),produc9125791028180074456eger_o(aTP_Lamm_fc(K),code_divmod_abs(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).
% bit_cut_integer_code
tff(fact_6882_UNIV__bool,axiom,
top_top_set_o = aa_set_o_set_o(insert_o($false),aa_set_o_set_o(insert_o($true),bot_bot_set_o)) ).
% UNIV_bool
tff(fact_6883_divmod__abs__code_I6_J,axiom,
! [J: code_integer] : code_divmod_abs(zero_z3403309356797280102nteger,J) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ).
% divmod_abs_code(6)
tff(fact_6884_divmod__abs__code_I5_J,axiom,
! [J: code_integer] : code_divmod_abs(J,zero_z3403309356797280102nteger) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,J)) ).
% divmod_abs_code(5)
tff(fact_6885_divmod__abs__def,axiom,
! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L))) ).
% divmod_abs_def
tff(fact_6886_divmod__integer__code,axiom,
! [K: code_integer,L: code_integer] :
code_divmod_integer(K,L) = $ite(
K = zero_z3403309356797280102nteger,
produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
$ite(
aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L),
$ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_fd(L),code_divmod_abs(K,L))),
$ite(
L = zero_z3403309356797280102nteger,
produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
$ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_fe(L),code_divmod_abs(K,L)))) ) ) ) ).
% divmod_integer_code
tff(fact_6887_Gcd__eq__Max,axiom,
! [M4: set_nat] :
( aa_set_nat_o(finite_finite_nat,M4)
=> ( ( M4 != bot_bot_set_nat )
=> ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M4)
=> ( gcd_Gcd_nat(M4) = lattic8265883725875713057ax_nat(comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_fg,M4))) ) ) ) ) ).
% Gcd_eq_Max
tff(fact_6888_INT__greaterThan__UNIV,axiom,
comple7806235888213564991et_nat(image_nat_set_nat(set_or1210151606488870762an_nat,top_top_set_nat)) = bot_bot_set_nat ).
% INT_greaterThan_UNIV
tff(fact_6889_Inf__nat__def1,axiom,
! [K5: set_nat] :
( ( K5 != bot_bot_set_nat )
=> aa_set_nat_o(member_nat(complete_Inf_Inf_nat(K5)),K5) ) ).
% Inf_nat_def1
tff(fact_6890_divmod__integer__eq__cases,axiom,
! [K: code_integer,L: code_integer] :
code_divmod_integer(K,L) = $ite(
K = zero_z3403309356797280102nteger,
produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
$ite(
L = zero_z3403309356797280102nteger,
produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),
$ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_fh(L),code_divmod_abs(K,L)))) ) ) ).
% divmod_integer_eq_cases
tff(fact_6891_Rep__unit__induct,axiom,
! [Y: $o,P: fun_o_o] :
( aa_set_o_o(member_o((Y)),aa_set_o_set_o(insert_o($true),bot_bot_set_o))
=> ( ! [X2: product_unit] : aa_o_o(P,aa_Product_unit_o(product_Rep_unit,X2))
=> aa_o_o(P,(Y)) ) ) ).
% Rep_unit_induct
tff(fact_6892_Abs__unit__inject,axiom,
! [X: $o,Y: $o] :
( aa_set_o_o(member_o((X)),aa_set_o_set_o(insert_o($true),bot_bot_set_o))
=> ( aa_set_o_o(member_o((Y)),aa_set_o_set_o(insert_o($true),bot_bot_set_o))
=> ( ( aa_o_Product_unit(product_Abs_unit,(X)) = aa_o_Product_unit(product_Abs_unit,(Y)) )
<=> ( (X)
<=> (Y) ) ) ) ) ).
% Abs_unit_inject
tff(fact_6893_Abs__unit__induct,axiom,
! [P: fun_Product_unit_o,X: product_unit] :
( ! [Y2: $o] :
( aa_set_o_o(member_o((Y2)),aa_set_o_set_o(insert_o($true),bot_bot_set_o))
=> aa_Product_unit_o(P,aa_o_Product_unit(product_Abs_unit,(Y2))) )
=> aa_Product_unit_o(P,X) ) ).
% Abs_unit_induct
tff(fact_6894_Abs__unit__inverse,axiom,
! [Y: $o] :
( aa_set_o_o(member_o((Y)),aa_set_o_set_o(insert_o($true),bot_bot_set_o))
=> ( aa_Product_unit_o(product_Rep_unit,aa_o_Product_unit(product_Abs_unit,(Y)))
<=> (Y) ) ) ).
% Abs_unit_inverse
tff(fact_6895_Rep__unit,axiom,
! [X: product_unit] : aa_set_o_o(member_o(aa_Product_unit_o(product_Rep_unit,X)),aa_set_o_set_o(insert_o($true),bot_bot_set_o)) ).
% Rep_unit
tff(fact_6896_Abs__unit__cases,axiom,
! [X: product_unit] :
~ ! [Y2: $o] :
( ( X = aa_o_Product_unit(product_Abs_unit,(Y2)) )
=> ~ aa_set_o_o(member_o((Y2)),aa_set_o_set_o(insert_o($true),bot_bot_set_o)) ) ).
% Abs_unit_cases
tff(fact_6897_Rep__unit__cases,axiom,
! [Y: $o] :
( aa_set_o_o(member_o((Y)),aa_set_o_set_o(insert_o($true),bot_bot_set_o))
=> ~ ! [X2: product_unit] :
( (Y)
<=> ~ aa_Product_unit_o(product_Rep_unit,X2) ) ) ).
% Rep_unit_cases
tff(fact_6898_type__definition__unit,axiom,
aa_set_o_o(type_d6188575255521822967unit_o(product_Rep_unit,product_Abs_unit),aa_set_o_set_o(insert_o($true),bot_bot_set_o)) ).
% type_definition_unit
tff(fact_6899_pair__lessI2,axiom,
! [A: nat,B: nat,S2: nat,T: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S2),T)
=> member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ) ).
% pair_lessI2
tff(fact_6900_pair__less__iff1,axiom,
! [X: nat,Y: nat,Z: nat] :
( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z)),fun_pair_less)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ).
% pair_less_iff1
tff(fact_6901_pair__lessI1,axiom,
! [A: nat,B: nat,S2: nat,T: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ).
% pair_lessI1
tff(fact_6902_smin__insertI,axiom,
! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),XS)
=> ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_min_strict)
=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Y),YS))),fun_min_strict) ) ) ) ).
% smin_insertI
tff(fact_6903_pair__leqI2,axiom,
! [A: nat,B: nat,S2: nat,T: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S2),T)
=> member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ) ).
% pair_leqI2
tff(fact_6904_smin__emptyI,axiom,
! [X5: set_Pr1261947904930325089at_nat] :
( ( X5 != bot_bo2099793752762293965at_nat )
=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,bot_bo2099793752762293965at_nat)),fun_min_strict) ) ).
% smin_emptyI
tff(fact_6905_pair__leqI1,axiom,
! [A: nat,B: nat,S2: nat,T: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
=> member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ).
% pair_leqI1
tff(fact_6906_wmax__insertI,axiom,
! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y),YS)
=> ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_max_weak)
=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),XS),YS)),fun_max_weak) ) ) ) ).
% wmax_insertI
tff(fact_6907_wmin__insertI,axiom,
! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),XS)
=> ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_min_weak)
=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Y),YS))),fun_min_weak) ) ) ) ).
% wmin_insertI
tff(fact_6908_smax__insertI,axiom,
! [Y: product_prod_nat_nat,Y5: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X5: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y),Y5)
=> ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Y5)),fun_max_strict)
=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),X5),Y5)),fun_max_strict) ) ) ) ).
% smax_insertI
tff(fact_6909_max__weak__def,axiom,
fun_max_weak = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,max_ex8135407076693332796at_nat(fun_pair_leq)),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat)),bot_bo228742789529271731at_nat)) ).
% max_weak_def
tff(fact_6910_min__weak__def,axiom,
fun_min_weak = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,min_ex6901939911449802026at_nat(fun_pair_leq)),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat)),bot_bo228742789529271731at_nat)) ).
% min_weak_def
tff(fact_6911_max__rpair__set,axiom,
fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_max_strict,fun_max_weak)) ).
% max_rpair_set
tff(fact_6912_smax__emptyI,axiom,
! [Y5: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Y5)
=> ( ( Y5 != bot_bo2099793752762293965at_nat )
=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,Y5)),fun_max_strict) ) ) ).
% smax_emptyI
tff(fact_6913_wmin__emptyI,axiom,
! [X5: set_Pr1261947904930325089at_nat] : aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,bot_bo2099793752762293965at_nat)),fun_min_weak) ).
% wmin_emptyI
tff(fact_6914_wmax__emptyI,axiom,
! [X5: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,X5)
=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,X5)),fun_max_weak) ) ).
% wmax_emptyI
tff(fact_6915_min__rpair__set,axiom,
fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_min_strict,fun_min_weak)) ).
% min_rpair_set
tff(fact_6916_pred__nat__trancl__eq__le,axiom,
! [M2: nat,N: nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M2),N)),transi2905341329935302413cl_nat(pred_nat))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ).
% pred_nat_trancl_eq_le
tff(fact_6917_inverse__rat__def,axiom,
inverse_inverse_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_ep) ).
% inverse_rat_def
tff(fact_6918_less__eq,axiom,
! [M2: nat,N: nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M2),N)),transi6264000038957366511cl_nat(pred_nat))
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N) ) ).
% less_eq
tff(fact_6919_uminus__rat__def,axiom,
uminus_uminus_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_eq) ).
% uminus_rat_def
tff(fact_6920_plus__rat__def,axiom,
plus_plus_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_er) ).
% plus_rat_def
tff(fact_6921_times__rat__def,axiom,
times_times_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_es) ).
% times_rat_def
tff(fact_6922_less__than__iff,axiom,
! [X: nat,Y: nat] :
( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),less_than)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ).
% less_than_iff
tff(fact_6923_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
( ( nat_prod_decode_aux(X,Xa) = Y )
=> ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa))
=> ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa)),nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),aa_nat_nat(suc,X)))) )
=> ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)) ) ) ) ).
% prod_decode_aux.pelims
tff(fact_6924_prod__decode__aux_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [K2: nat,M3: nat] : X != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K2),M3) ).
% prod_decode_aux.cases
tff(fact_6925_prod__decode__aux_Osimps,axiom,
! [K: nat,M2: nat] :
nat_prod_decode_aux(K,M2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),K),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),M2)),nat_prod_decode_aux(aa_nat_nat(suc,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,K)))) ).
% prod_decode_aux.simps
tff(fact_6926_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
( ( nat_prod_decode_aux(X,Xa) = Y )
=> ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa)),nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),aa_nat_nat(suc,X)))) ) ) ).
% prod_decode_aux.elims
tff(fact_6927_set__encode__empty,axiom,
nat_set_encode(bot_bot_set_nat) = zero_zero_nat ).
% set_encode_empty
tff(fact_6928_normalize__stable,axiom,
! [Q4: int,P5: int] :
( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q4)
=> ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P5),Q4)
=> ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P5),Q4)) = aa_int7462565939943104719nt_int(product_Pair_int_int(P5),Q4) ) ) ) ).
% normalize_stable
tff(fact_6929_quotient__of__coprime,axiom,
! [R2: rat,P5: int,Q4: int] :
( ( quotient_of(R2) = aa_int7462565939943104719nt_int(product_Pair_int_int(P5),Q4) )
=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P5),Q4) ) ).
% quotient_of_coprime
tff(fact_6930_normalize__coprime,axiom,
! [R2: product_prod_int_int,P5: int,Q4: int] :
( ( normalize(R2) = aa_int7462565939943104719nt_int(product_Pair_int_int(P5),Q4) )
=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P5),Q4) ) ).
% normalize_coprime
tff(fact_6931_max__nat_Osemilattice__neutr__axioms,axiom,
aa_nat_o(semila9081495762789891438tr_nat(ord_max_nat),zero_zero_nat) ).
% max_nat.semilattice_neutr_axioms
tff(fact_6932_set__decode__zero,axiom,
aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).
% set_decode_zero
tff(fact_6933_Fract_Otransfer,axiom,
bNF_re2486238095283335024nt_rat(fequal_int,bNF_re2214769303045360666nt_rat(fequal_int,pcr_rat),aTP_Lamm_ev,fract) ).
% Fract.transfer
tff(fact_6934_quotient__of__eq,axiom,
! [A: int,B: int,P5: int,Q4: int] :
( ( quotient_of(aa_int_rat(aa_int_fun_int_rat(fract,A),B)) = aa_int7462565939943104719nt_int(product_Pair_int_int(P5),Q4) )
=> ( aa_int_rat(aa_int_fun_int_rat(fract,P5),Q4) = aa_int_rat(aa_int_fun_int_rat(fract,A),B) ) ) ).
% quotient_of_eq
tff(fact_6935_normalize__eq,axiom,
! [A: int,B: int,P5: int,Q4: int] :
( ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(A),B)) = aa_int7462565939943104719nt_int(product_Pair_int_int(P5),Q4) )
=> ( aa_int_rat(aa_int_fun_int_rat(fract,P5),Q4) = aa_int_rat(aa_int_fun_int_rat(fract,A),B) ) ) ).
% normalize_eq
tff(fact_6936_quotient__of__Fract,axiom,
! [A: int,B: int] : quotient_of(aa_int_rat(aa_int_fun_int_rat(fract,A),B)) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(A),B)) ).
% quotient_of_Fract
tff(fact_6937_Fract_Oabs__eq,axiom,
! [Xa: int,X: int] :
aa_int_rat(aa_int_fun_int_rat(fract,Xa),X) = aa_Pro6166618844887768661nt_rat(abs_Rat,
$ite(X = zero_zero_int,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(Xa),X))) ).
% Fract.abs_eq
tff(fact_6938_smsI,axiom,
! [Aa: multis2468970476368604999at_nat,Ba: multis2468970476368604999at_nat,Z4: multis2468970476368604999at_nat] :
( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(Aa),set_ms8126754132646256062at_nat(Ba))),fun_max_strict)
=> member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z4),Aa),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z4),Ba)),ms_strict) ) ).
% smsI
tff(fact_6939_wmsI,axiom,
! [Aa: multis2468970476368604999at_nat,Ba: multis2468970476368604999at_nat,Z4: multis2468970476368604999at_nat] :
( ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(Aa),set_ms8126754132646256062at_nat(Ba))),fun_max_strict)
| ( ( Aa = zero_z1048942125864253310at_nat )
& ( Ba = zero_z1048942125864253310at_nat ) ) )
=> member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z4),Aa),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z4),Ba)),ms_weak) ) ).
% wmsI
tff(fact_6940_ms__reduction__pair,axiom,
fun_re7357418987779152907at_nat(produc5245064249948416855at_nat(ms_strict,ms_weak)) ).
% ms_reduction_pair
tff(fact_6941_ms__weakI1,axiom,
! [Z4: multis2468970476368604999at_nat,Z5: multis2468970476368604999at_nat,Aa: multis2468970476368604999at_nat,Ba: multis2468970476368604999at_nat] :
( pw_leq(Z4,Z5)
=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(Aa),set_ms8126754132646256062at_nat(Ba))),fun_max_strict)
=> member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z4),Aa),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z5),Ba)),ms_weak) ) ) ).
% ms_weakI1
tff(fact_6942_ms__strictI,axiom,
! [Z4: multis2468970476368604999at_nat,Z5: multis2468970476368604999at_nat,Aa: multis2468970476368604999at_nat,Ba: multis2468970476368604999at_nat] :
( pw_leq(Z4,Z5)
=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(Aa),set_ms8126754132646256062at_nat(Ba))),fun_max_strict)
=> member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z4),Aa),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z5),Ba)),ms_strict) ) ) ).
% ms_strictI
tff(fact_6943_pw__leq__split,axiom,
! [X5: multis2468970476368604999at_nat,Y5: multis2468970476368604999at_nat] :
( pw_leq(X5,Y5)
=> ? [A5: multis2468970476368604999at_nat,B7: multis2468970476368604999at_nat,Z6: multis2468970476368604999at_nat] :
( ( X5 = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,A5),Z6) )
& ( Y5 = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,B7),Z6) )
& ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A5),set_ms8126754132646256062at_nat(B7))),fun_max_strict)
| ( ( B7 = zero_z1048942125864253310at_nat )
& ( A5 = zero_z1048942125864253310at_nat ) ) ) ) ) ).
% pw_leq_split
tff(fact_6944_ms__weakI2,axiom,
! [Z4: multis2468970476368604999at_nat,Z5: multis2468970476368604999at_nat] :
( pw_leq(Z4,Z5)
=> member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z4),zero_z1048942125864253310at_nat),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,Z5),zero_z1048942125864253310at_nat)),ms_weak) ) ).
% ms_weakI2
tff(fact_6945_pw__leq_Ocases,axiom,
! [A1: multis2468970476368604999at_nat,A22: multis2468970476368604999at_nat] :
( pw_leq(A1,A22)
=> ( ( ( A1 = zero_z1048942125864253310at_nat )
=> ( A22 != zero_z1048942125864253310at_nat ) )
=> ~ ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat,X6: multis2468970476368604999at_nat] :
( ( A1 = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,add_ms2612439473150266591at_nat(X2,zero_z1048942125864253310at_nat)),X6) )
=> ! [Y6: multis2468970476368604999at_nat] :
( ( A22 = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,add_ms2612439473150266591at_nat(Y2,zero_z1048942125864253310at_nat)),Y6) )
=> ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X2,Y2),fun_pair_leq)
=> ~ pw_leq(X6,Y6) ) ) ) ) ) ).
% pw_leq.cases
tff(fact_6946_pw__leq__lstep,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
=> pw_leq(add_ms2612439473150266591at_nat(X,zero_z1048942125864253310at_nat),add_ms2612439473150266591at_nat(Y,zero_z1048942125864253310at_nat)) ) ).
% pw_leq_lstep
tff(fact_6947_pw__leq__step,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,X5: multis2468970476368604999at_nat,Y5: multis2468970476368604999at_nat] :
( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
=> ( pw_leq(X5,Y5)
=> pw_leq(aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,add_ms2612439473150266591at_nat(X,zero_z1048942125864253310at_nat)),X5),aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,add_ms2612439473150266591at_nat(Y,zero_z1048942125864253310at_nat)),Y5)) ) ) ).
% pw_leq_step
tff(fact_6948_pw__leq_Osimps,axiom,
! [A1: multis2468970476368604999at_nat,A22: multis2468970476368604999at_nat] :
( pw_leq(A1,A22)
<=> ( ( ( A1 = zero_z1048942125864253310at_nat )
& ( A22 = zero_z1048942125864253310at_nat ) )
| ? [X3: product_prod_nat_nat,Y4: product_prod_nat_nat,X7: multis2468970476368604999at_nat,Y7: multis2468970476368604999at_nat] :
( ( A1 = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,add_ms2612439473150266591at_nat(X3,zero_z1048942125864253310at_nat)),X7) )
& ( A22 = aa_mul5754855110707352700at_nat(aa_mul5680541068881757582at_nat(plus_p7104986032573967614at_nat,add_ms2612439473150266591at_nat(Y4,zero_z1048942125864253310at_nat)),Y7) )
& member8206827879206165904at_nat(produc6161850002892822231at_nat(X3,Y4),fun_pair_leq)
& pw_leq(X7,Y7) ) ) ) ).
% pw_leq.simps
tff(fact_6949_exhaustive__integer_H_Ocases,axiom,
! [X: produc7509114972854644512nteger] :
~ ! [F2: fun_Co3460408610320954864e_term,D: code_integer,I3: code_integer] : X != produc7264323995540650962nteger(F2,produc1086072967326762835nteger(D,I3)) ).
% exhaustive_integer'.cases
tff(fact_6950_exhaustive__int_H_Ocases,axiom,
! [X: produc4788131091073775919nt_int] :
~ ! [F2: fun_in5730379987311621695e_term,D: int,I3: int] : X != produc4729913460834754785nt_int(F2,aa_int7462565939943104719nt_int(product_Pair_int_int(D),I3)) ).
% exhaustive_int'.cases
tff(fact_6951_full__exhaustive__int_H_Ocases,axiom,
! [X: produc3043004577601930755nt_int] :
~ ! [F2: fun_Pr7763680713840462443e_term,D: int,I3: int] : X != produc7320969046608631221nt_int(F2,aa_int7462565939943104719nt_int(product_Pair_int_int(D),I3)) ).
% full_exhaustive_int'.cases
tff(fact_6952_full__exhaustive__integer_H_Ocases,axiom,
! [X: produc2453742332425399446nteger] :
~ ! [F2: fun_Pr5098669352780829690e_term,D: code_integer,I3: code_integer] : X != produc565448451787809096nteger(F2,produc1086072967326762835nteger(D,I3)) ).
% full_exhaustive_integer'.cases
tff(fact_6953_Lcm__nat__empty,axiom,
gcd_Lcm_nat(bot_bot_set_nat) = one_one_nat ).
% Lcm_nat_empty
tff(fact_6954_Lcm__eq__Max__nat,axiom,
! [M4: set_nat] :
( aa_set_nat_o(finite_finite_nat,M4)
=> ( ( M4 != bot_bot_set_nat )
=> ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M4)
=> ( ! [M3: nat,N4: nat] :
( aa_set_nat_o(member_nat(M3),M4)
=> ( aa_set_nat_o(member_nat(N4),M4)
=> aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M3),N4)),M4) ) )
=> ( gcd_Lcm_nat(M4) = lattic8265883725875713057ax_nat(M4) ) ) ) ) ) ).
% Lcm_eq_Max_nat
tff(fact_6955_Lcm__in__lcm__closed__set__nat,axiom,
! [M4: set_nat] :
( aa_set_nat_o(finite_finite_nat,M4)
=> ( ( M4 != bot_bot_set_nat )
=> ( ! [M3: nat,N4: nat] :
( aa_set_nat_o(member_nat(M3),M4)
=> ( aa_set_nat_o(member_nat(N4),M4)
=> aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M3),N4)),M4) ) )
=> aa_set_nat_o(member_nat(gcd_Lcm_nat(M4)),M4) ) ) ) ).
% Lcm_in_lcm_closed_set_nat
tff(fact_6956_in__range_Osimps,axiom,
! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As))
<=> ! [X3: nat] :
( aa_set_nat_o(member_nat(X3),As)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),lim_Product_unit(H2)) ) ) ).
% in_range.simps
tff(fact_6957_in__range_Oelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat,Y: $o] :
( ( aa_Pro653082453696054616_nat_o(in_range,X)
<=> (Y) )
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( X = produc7507926704131184380et_nat(H3,As3) )
=> ( (Y)
<=> ~ ! [X3: nat] :
( aa_set_nat_o(member_nat(X3),As3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),lim_Product_unit(H3)) ) ) ) ) ).
% in_range.elims(1)
tff(fact_6958_in__range_Oelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(in_range,X)
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( X = produc7507926704131184380et_nat(H3,As3) )
=> ~ ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),As3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),lim_Product_unit(H3)) ) ) ) ).
% in_range.elims(2)
tff(fact_6959_in__range_Oelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat] :
( ~ aa_Pro653082453696054616_nat_o(in_range,X)
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( X = produc7507926704131184380et_nat(H3,As3) )
=> ! [X2: nat] :
( aa_set_nat_o(member_nat(X2),As3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),lim_Product_unit(H3)) ) ) ) ).
% in_range.elims(3)
tff(fact_6960_ctwo__Cnotzero,axiom,
( ~ member444158400953824016od_o_o(produc763777882069021527od_o_o(bNF_Cardinal_ctwo,bNF_Cardinal_czero_o),bNF_We2654380646378065620so_o_o)
& bNF_Ca8331644756375544342r_on_o(field_o(bNF_Cardinal_ctwo),bNF_Cardinal_ctwo) ) ).
% ctwo_Cnotzero
tff(fact_6961_ctwo__ordLess__natLeq,axiom,
member4095101504841534314at_nat(produc8517790099723286449at_nat(bNF_Cardinal_ctwo,bNF_Ca8665028551170535155natLeq),bNF_We8182288985678559134_o_nat) ).
% ctwo_ordLess_natLeq
tff(fact_6962_card__of__Field__natLeq,axiom,
aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(bNF_Ca3793111618940312692of_nat(field_nat(bNF_Ca8665028551170535155natLeq)),bNF_Ca8665028551170535155natLeq)),bNF_We5258908940166488438at_nat) ).
% card_of_Field_natLeq
tff(fact_6963_card__of__nat,axiom,
aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(bNF_Ca3793111618940312692of_nat(top_top_set_nat),bNF_Ca8665028551170535155natLeq)),bNF_We5258908940166488438at_nat) ).
% card_of_nat
tff(fact_6964_small__lazy_H_Ocases,axiom,
! [X: product_prod_int_int] :
~ ! [D: int,I3: int] : X != aa_int7462565939943104719nt_int(product_Pair_int_int(D),I3) ).
% small_lazy'.cases
tff(fact_6965_le__prod__encode__2,axiom,
! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),nat_prod_encode(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))) ).
% le_prod_encode_2
tff(fact_6966_le__prod__encode__1,axiom,
! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),nat_prod_encode(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))) ).
% le_prod_encode_1
tff(fact_6967_upto_Opsimps,axiom,
! [I: int,J: int] :
( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(I),J))
=> ( upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ) ) ).
% upto.psimps
tff(fact_6968_upto_Opelims,axiom,
! [X: int,Xa: int,Y: list_int] :
( ( upto(X,Xa) = Y )
=> ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa))
=> ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa)),nil_int) )
=> ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa)) ) ) ) ).
% upto.pelims
tff(fact_6969_list__encode_Oelims,axiom,
! [X: list_nat,Y: nat] :
( ( nat_list_encode(X) = Y )
=> ( ( ( X = nil_nat )
=> ( Y != zero_zero_nat ) )
=> ~ ! [X2: nat,Xs: list_nat] :
( ( X = cons_nat(X2,Xs) )
=> ( Y != aa_nat_nat(suc,nat_prod_encode(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),nat_list_encode(Xs)))) ) ) ) ) ).
% list_encode.elims
tff(fact_6970_list__encode_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] : nat_list_encode(cons_nat(X,Xs2)) = aa_nat_nat(suc,nat_prod_encode(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),nat_list_encode(Xs2)))) ).
% list_encode.simps(2)
tff(fact_6971_list__encode_Opelims,axiom,
! [X: list_nat,Y: nat] :
( ( nat_list_encode(X) = Y )
=> ( accp_list_nat(nat_list_encode_rel,X)
=> ( ( ( X = nil_nat )
=> ( ( Y = zero_zero_nat )
=> ~ accp_list_nat(nat_list_encode_rel,nil_nat) ) )
=> ~ ! [X2: nat,Xs: list_nat] :
( ( X = cons_nat(X2,Xs) )
=> ( ( Y = aa_nat_nat(suc,nat_prod_encode(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),nat_list_encode(Xs)))) )
=> ~ accp_list_nat(nat_list_encode_rel,cons_nat(X2,Xs)) ) ) ) ) ) ).
% list_encode.pelims
tff(fact_6972_sum__list__upt,axiom,
! [M2: nat,N: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
=> ( groups4561878855575611511st_nat(upt(M2,N)) = groups3542108847815614940at_nat(aTP_Lamm_bv,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N)) ) ) ).
% sum_list_upt
tff(fact_6973_Fract__def,axiom,
fract = map_fu7945985449245211511nt_rat(id_int,map_fu3424225382358772769nt_rat(id_int,abs_Rat),aTP_Lamm_ev) ).
% Fract_def
tff(fact_6974_relH__def,axiom,
! [As: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
( aa_hea7994211679605476167unit_o(relH(As,H2),H5)
<=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As))
& aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H5,As))
& ! [T3: typerep,X3: nat] :
( aa_set_nat_o(member_nat(X3),As)
=> ( ( aa_nat_nat(refs_Product_unit(H2,T3),X3) = aa_nat_nat(refs_Product_unit(H5,T3),X3) )
& ( arrays_Product_unit(H2,T3,X3) = arrays_Product_unit(H5,T3,X3) ) ) ) ) ) ).
% relH_def
tff(fact_6975_in__range_Opelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat,Y: $o] :
( ( aa_Pro653082453696054616_nat_o(in_range,X)
<=> (Y) )
=> ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),X)
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( X = produc7507926704131184380et_nat(H3,As3) )
=> ( ( (Y)
<=> ! [X3: nat] :
( aa_set_nat_o(member_nat(X3),As3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),lim_Product_unit(H3)) ) )
=> ~ aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),produc7507926704131184380et_nat(H3,As3)) ) ) ) ) ).
% in_range.pelims(1)
tff(fact_6976_in__range_Opelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(in_range,X)
=> ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),X)
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( X = produc7507926704131184380et_nat(H3,As3) )
=> ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),produc7507926704131184380et_nat(H3,As3))
=> ~ ! [X4: nat] :
( aa_set_nat_o(member_nat(X4),As3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),lim_Product_unit(H3)) ) ) ) ) ) ).
% in_range.pelims(2)
tff(fact_6977_in__range_Opelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat] :
( ~ aa_Pro653082453696054616_nat_o(in_range,X)
=> ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),X)
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( X = produc7507926704131184380et_nat(H3,As3) )
=> ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),produc7507926704131184380et_nat(H3,As3))
=> ! [X2: nat] :
( aa_set_nat_o(member_nat(X2),As3)
=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),lim_Product_unit(H3)) ) ) ) ) ) ).
% in_range.pelims(3)
tff(fact_6978_one__assn__raw_Opelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat,Y: $o] :
( ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
<=> (Y) )
=> ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),X)
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( X = produc7507926704131184380et_nat(H3,As3) )
=> ( ( (Y)
<=> ( As3 = bot_bot_set_nat ) )
=> ~ aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),produc7507926704131184380et_nat(H3,As3)) ) ) ) ) ).
% one_assn_raw.pelims(1)
tff(fact_6979_one__assn__raw_Opelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
=> ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),X)
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( X = produc7507926704131184380et_nat(H3,As3) )
=> ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),produc7507926704131184380et_nat(H3,As3))
=> ( As3 != bot_bot_set_nat ) ) ) ) ) ).
% one_assn_raw.pelims(2)
tff(fact_6980_one__assn__raw_Opelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat] :
( ~ aa_Pro653082453696054616_nat_o(one_assn_raw,X)
=> ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),X)
=> ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( X = produc7507926704131184380et_nat(H3,As3) )
=> ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),produc7507926704131184380et_nat(H3,As3))
=> ( As3 = bot_bot_set_nat ) ) ) ) ) ).
% one_assn_raw.pelims(3)
tff(fact_6981_times__int__def,axiom,
times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_eu)) ).
% times_int_def
tff(fact_6982_plus__int__def,axiom,
plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_ex)) ).
% plus_int_def
tff(fact_6983_minus__int__def,axiom,
minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_ez)) ).
% minus_int_def
tff(fact_6984_uminus__int__def,axiom,
uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_fb)) ).
% uminus_int_def
tff(fact_6985_next_Osimps,axiom,
! [V: code_natural,W: code_natural] :
aa_Pro7165575656943818355atural(next,produc3574140220909816553atural(V,W)) = $let(
v: code_natural,
v:= aa_Cod4272041906116450446atural(minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(modulo8411746178871703098atural(V),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))))))))))))))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(V),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))))))))))))),
$let(
w2: code_natural,
w2:= aa_Cod4272041906116450446atural(minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(modulo8411746178871703098atural(W),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))))))))))))))),aa_Cod4272041906116450446atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(divide5121882707175180666atural(W),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one)))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))),
aa_Pro7165575656943818355atural(produc6639722614265839536atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(aa_Cod4272041906116450446atural(minus_shift(numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),v),aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(w2),one_one_Code_natural))),one_one_Code_natural)),produc3574140220909816553atural(v,w2)) ) ) ).
% next.simps
tff(fact_6986_log_Ocases,axiom,
! [X: produc7822875418678951345atural] :
~ ! [B3: code_natural,I3: code_natural] : X != produc3574140220909816553atural(B3,I3) ).
% log.cases
tff(fact_6987_exhaustive__natural_H_Ocases,axiom,
! [X: produc7476732711998919211atural] :
~ ! [F2: fun_Co6386927884524458043e_term,D: code_natural,I3: code_natural] : X != produc3379161622255323613atural(F2,produc3574140220909816553atural(D,I3)) ).
% exhaustive_natural'.cases
tff(fact_6988_full__exhaustive__natural_H_Ocases,axiom,
! [X: produc9140211490168418807atural] :
~ ! [F2: fun_Pr285225112257423215e_term,D: code_natural,I3: code_natural] : X != produc3399137497101226409atural(F2,produc3574140220909816553atural(D,I3)) ).
% full_exhaustive_natural'.cases
tff(fact_6989_split__seed__def,axiom,
! [S2: produc7822875418678951345atural] : split_seed(S2) = produc8282080750456430313atural(aTP_Lamm_fj(S2),S2) ).
% split_seed_def
tff(fact_6990_list__decode_Opinduct,axiom,
! [A0: nat,P: fun_nat_o] :
( aa_nat_o(accp_nat(nat_list_decode_rel),A0)
=> ( ( aa_nat_o(accp_nat(nat_list_decode_rel),zero_zero_nat)
=> aa_nat_o(P,zero_zero_nat) )
=> ( ! [N4: nat] :
( aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N4))
=> ( ! [X4: nat,Y3: nat] :
( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) = nat_prod_decode(N4) )
=> aa_nat_o(P,Y3) )
=> aa_nat_o(P,aa_nat_nat(suc,N4)) ) )
=> aa_nat_o(P,A0) ) ) ) ).
% list_decode.pinduct
tff(fact_6991_log_Opelims,axiom,
! [X: code_natural,Xa: code_natural,Y: code_natural] :
( ( aa_Cod4272041906116450446atural(log(X),Xa) = Y )
=> ( aa_Pro6073791466322389475ural_o(accp_P8126237942716283194atural(log_rel),produc3574140220909816553atural(X,Xa))
=> ~ ( ( Y = $ite(
( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,X),one_one_Code_natural)
| aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Xa),X) ),
one_one_Code_natural,
aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(one_one_Code_natural),aa_Cod4272041906116450446atural(log(X),aa_Cod4272041906116450446atural(divide5121882707175180666atural(Xa),X))) ) )
=> ~ aa_Pro6073791466322389475ural_o(accp_P8126237942716283194atural(log_rel),produc3574140220909816553atural(X,Xa)) ) ) ) ).
% log.pelims
tff(fact_6992_Random_Orange__def,axiom,
! [K: code_natural] : range(K) = produc5538323210962509403atural(iterat8892046348760725948atural(aa_Cod4272041906116450446atural(log(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))))))))))))))))))))),K),aTP_Lamm_fl,one_one_Code_natural),aTP_Lamm_fm(K)) ).
% Random.range_def
tff(fact_6993_cone__def,axiom,
bNF_Cardinal_cone = bNF_Ca7083678892733797993t_unit(insert_Product_unit(product_Unity,bot_bo3957492148770167129t_unit)) ).
% cone_def
tff(fact_6994_UNIV__unit,axiom,
top_to1996260823553986621t_unit = insert_Product_unit(product_Unity,bot_bo3957492148770167129t_unit) ).
% UNIV_unit
tff(fact_6995_bot__unit__def,axiom,
bot_bot_Product_unit = product_Unity ).
% bot_unit_def
tff(fact_6996_or__not__num__neg_Opelims,axiom,
! [X: num,Xa: num,Y: num] :
( ( bit_or_not_num_neg(X,Xa) = Y )
=> ( accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(X,Xa))
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( ( Y = one )
=> ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(one,one)) ) ) )
=> ( ( ( X = one )
=> ! [M3: num] :
( ( Xa = aa_num_num(bit0,M3) )
=> ( ( Y = aa_num_num(bit1,M3) )
=> ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(one,aa_num_num(bit0,M3))) ) ) )
=> ( ( ( X = one )
=> ! [M3: num] :
( ( Xa = aa_num_num(bit1,M3) )
=> ( ( Y = aa_num_num(bit1,M3) )
=> ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(one,aa_num_num(bit1,M3))) ) ) )
=> ( ! [N4: num] :
( ( X = aa_num_num(bit0,N4) )
=> ( ( Xa = one )
=> ( ( Y = aa_num_num(bit0,one) )
=> ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N4),one)) ) ) )
=> ( ! [N4: num] :
( ( X = aa_num_num(bit0,N4) )
=> ! [M3: num] :
( ( Xa = aa_num_num(bit0,M3) )
=> ( ( Y = bitM(bit_or_not_num_neg(N4,M3)) )
=> ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N4),aa_num_num(bit0,M3))) ) ) )
=> ( ! [N4: num] :
( ( X = aa_num_num(bit0,N4) )
=> ! [M3: num] :
( ( Xa = aa_num_num(bit1,M3) )
=> ( ( Y = aa_num_num(bit0,bit_or_not_num_neg(N4,M3)) )
=> ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N4),aa_num_num(bit1,M3))) ) ) )
=> ( ! [N4: num] :
( ( X = aa_num_num(bit1,N4) )
=> ( ( Xa = one )
=> ( ( Y = one )
=> ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N4),one)) ) ) )
=> ( ! [N4: num] :
( ( X = aa_num_num(bit1,N4) )
=> ! [M3: num] :
( ( Xa = aa_num_num(bit0,M3) )
=> ( ( Y = bitM(bit_or_not_num_neg(N4,M3)) )
=> ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N4),aa_num_num(bit0,M3))) ) ) )
=> ~ ! [N4: num] :
( ( X = aa_num_num(bit1,N4) )
=> ! [M3: num] :
( ( Xa = aa_num_num(bit1,M3) )
=> ( ( Y = bitM(bit_or_not_num_neg(N4,M3)) )
=> ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N4),aa_num_num(bit1,M3))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.pelims
tff(fact_6997_and__num_Opelims,axiom,
! [X: num,Xa: num,Y: option_num] :
( ( bit_un7362597486090784418nd_num(X,Xa) = Y )
=> ( accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(X,Xa))
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( ( Y = some_num(one) )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,one)) ) ) )
=> ( ( ( X = one )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit0,N4) )
=> ( ( Y = none_num )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,aa_num_num(bit0,N4))) ) ) )
=> ( ( ( X = one )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit1,N4) )
=> ( ( Y = some_num(one) )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,aa_num_num(bit1,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ( ( Xa = one )
=> ( ( Y = none_num )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M3),one)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit0,N4) )
=> ( ( Y = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M3,N4)) )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit1,N4) )
=> ( ( Y = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M3,N4)) )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ( ( Xa = one )
=> ( ( Y = some_num(one) )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M3),one)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit0,N4) )
=> ( ( Y = map_option_num_num(bit0,bit_un7362597486090784418nd_num(M3,N4)) )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N4))) ) ) )
=> ~ ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit1,N4) )
=> ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_fn,bit_un7362597486090784418nd_num(M3,N4)) )
=> ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N4))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
tff(fact_6998_and__not__num_Opelims,axiom,
! [X: num,Xa: num,Y: option_num] :
( ( bit_and_not_num(X,Xa) = Y )
=> ( accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(X,Xa))
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( ( Y = none_num )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,one)) ) ) )
=> ( ( ( X = one )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit0,N4) )
=> ( ( Y = some_num(one) )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,aa_num_num(bit0,N4))) ) ) )
=> ( ( ( X = one )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit1,N4) )
=> ( ( Y = none_num )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,aa_num_num(bit1,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ( ( Xa = one )
=> ( ( Y = some_num(aa_num_num(bit0,M3)) )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M3),one)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit0,N4) )
=> ( ( Y = map_option_num_num(bit0,bit_and_not_num(M3,N4)) )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit1,N4) )
=> ( ( Y = map_option_num_num(bit0,bit_and_not_num(M3,N4)) )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ( ( Xa = one )
=> ( ( Y = some_num(aa_num_num(bit0,M3)) )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M3),one)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit0,N4) )
=> ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_fn,bit_and_not_num(M3,N4)) )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N4))) ) ) )
=> ~ ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit1,N4) )
=> ( ( Y = map_option_num_num(bit0,bit_and_not_num(M3,N4)) )
=> ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N4))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
tff(fact_6999_xor__num_Opelims,axiom,
! [X: num,Xa: num,Y: option_num] :
( ( bit_un2480387367778600638or_num(X,Xa) = Y )
=> ( accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(X,Xa))
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( ( Y = none_num )
=> ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,one)) ) ) )
=> ( ( ( X = one )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit0,N4) )
=> ( ( Y = some_num(aa_num_num(bit1,N4)) )
=> ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,aa_num_num(bit0,N4))) ) ) )
=> ( ( ( X = one )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit1,N4) )
=> ( ( Y = some_num(aa_num_num(bit0,N4)) )
=> ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,aa_num_num(bit1,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ( ( Xa = one )
=> ( ( Y = some_num(aa_num_num(bit1,M3)) )
=> ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M3),one)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit0,N4) )
=> ( ( Y = map_option_num_num(bit0,bit_un2480387367778600638or_num(M3,N4)) )
=> ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit1,N4) )
=> ( ( Y = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M3,N4))) )
=> ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ( ( Xa = one )
=> ( ( Y = some_num(aa_num_num(bit0,M3)) )
=> ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M3),one)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit0,N4) )
=> ( ( Y = some_num(case_option_num_num(one,bit1,bit_un2480387367778600638or_num(M3,N4))) )
=> ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N4))) ) ) )
=> ~ ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit1,N4) )
=> ( ( Y = map_option_num_num(bit0,bit_un2480387367778600638or_num(M3,N4)) )
=> ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N4))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.pelims
tff(fact_7000_take__bit__num__code,axiom,
! [N: nat,M2: num] : bit_take_bit_num(N,M2) = produc478579273971653890on_num(aTP_Lamm_fs,product_Pair_nat_num(N,M2)) ).
% take_bit_num_code
tff(fact_7001_or__num_Opelims,axiom,
! [X: num,Xa: num,Y: num] :
( ( bit_un6697907153464112080or_num(X,Xa) = Y )
=> ( accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(X,Xa))
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( ( Y = one )
=> ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(one,one)) ) ) )
=> ( ( ( X = one )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit0,N4) )
=> ( ( Y = aa_num_num(bit1,N4) )
=> ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(one,aa_num_num(bit0,N4))) ) ) )
=> ( ( ( X = one )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit1,N4) )
=> ( ( Y = aa_num_num(bit1,N4) )
=> ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(one,aa_num_num(bit1,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ( ( Xa = one )
=> ( ( Y = aa_num_num(bit1,M3) )
=> ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit0,M3),one)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit0,N4) )
=> ( ( Y = aa_num_num(bit0,bit_un6697907153464112080or_num(M3,N4)) )
=> ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit1,N4) )
=> ( ( Y = aa_num_num(bit1,bit_un6697907153464112080or_num(M3,N4)) )
=> ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ( ( Xa = one )
=> ( ( Y = aa_num_num(bit1,M3) )
=> ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit1,M3),one)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit0,N4) )
=> ( ( Y = aa_num_num(bit1,bit_un6697907153464112080or_num(M3,N4)) )
=> ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N4))) ) ) )
=> ~ ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N4: num] :
( ( Xa = aa_num_num(bit1,N4) )
=> ( ( Y = aa_num_num(bit1,bit_un6697907153464112080or_num(M3,N4)) )
=> ~ accp_P3113834385874906142um_num(bit_un4773296044027857193um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N4))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_num.pelims
tff(fact_7002_unit__pred__cases,axiom,
! [P: fun_pr1768013098239751566unit_o,Q: pred_Product_unit] :
( aa_pre4150682718667570679unit_o(P,bot_bo2717538794563056311t_unit)
=> ( aa_pre4150682718667570679unit_o(P,single_Product_unit(product_Unity))
=> aa_pre4150682718667570679unit_o(P,Q) ) ) ).
% unit_pred_cases
tff(fact_7003_if__pred__eq,axiom,
! [B: $o] :
if_pred((B)) = $ite((B),single_Product_unit(product_Unity),bot_bo2717538794563056311t_unit) ).
% if_pred_eq
tff(fact_7004_Random__Pred_Onot__randompred__def,axiom,
! [P: fun_Pr160617923510306311atural,X4: produc7822875418678951345atural] : aa_Pro6676232280479689400atural(random6974930770145893639ompred(P),X4) = produc6665183775751917029atural(aTP_Lamm_ft,aa_Pro6676232280479689400atural(P,X4)) ).
% Random_Pred.not_randompred_def
tff(fact_7005_not__pred__eq,axiom,
! [P: pred_Product_unit] :
not_pred(P) = $ite(aa_Product_unit_o(eval_Product_unit(P),product_Unity),bot_bo2717538794563056311t_unit,single_Product_unit(product_Unity)) ).
% not_pred_eq
tff(fact_7006_wait__def,axiom,
! [N: nat] : heap_Time_wait(N) = heap_T6183433275982383450t_unit(aTP_Lamm_fu(N)) ).
% wait_def
tff(fact_7007_eventually__prod__sequentially,axiom,
! [P: fun_Pr4796943543331071514_nat_o] :
( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
<=> ? [N5: nat] :
! [M: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),M)
=> ! [N2: nat] :
( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N2)
=> aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N2),M)) ) ) ) ).
% eventually_prod_sequentially
tff(fact_7008_trivial__limit__sequentially,axiom,
at_top_nat != bot_bot_filter_nat ).
% trivial_limit_sequentially
tff(fact_7009_random__aux__set_Ocases,axiom,
! [X: produc7822875418678951345atural] :
( ! [J2: code_natural] : X != produc3574140220909816553atural(zero_z2226904508553997617atural,J2)
=> ~ ! [I3: code_natural,J2: code_natural] : X != produc3574140220909816553atural(aa_Cod4272041906116450446atural(code_Suc,I3),J2) ) ).
% random_aux_set.cases
tff(fact_7010_nth__item_Opinduct,axiom,
! [A0: nat,P: fun_nat_o] :
( aa_nat_o(accp_nat(nth_item_rel),A0)
=> ( ( aa_nat_o(accp_nat(nth_item_rel),zero_zero_nat)
=> aa_nat_o(P,zero_zero_nat) )
=> ( ! [N4: nat] :
( aa_nat_o(accp_nat(nth_item_rel),aa_nat_nat(suc,N4))
=> ( ! [A7: nat,Aa2: nat] :
( ( nat_sum_decode(N4) = sum_Inl_nat_nat(A7) )
=> ( ( nat_sum_decode(A7) = sum_Inl_nat_nat(Aa2) )
=> aa_nat_o(P,Aa2) ) )
=> ( ! [A7: nat,B8: nat] :
( ( nat_sum_decode(N4) = sum_Inl_nat_nat(A7) )
=> ( ( nat_sum_decode(A7) = sum_Inr_nat_nat(B8) )
=> aa_nat_o(P,B8) ) )
=> ( ! [B8: nat,Ba2: nat,X4: nat,Y3: nat] :
( ( nat_sum_decode(N4) = sum_Inr_nat_nat(B8) )
=> ( ( nat_sum_decode(B8) = sum_Inr_nat_nat(Ba2) )
=> ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) = nat_prod_decode(Ba2) )
=> aa_nat_o(P,X4) ) ) )
=> ( ! [B8: nat,Ba2: nat,X4: nat,Y3: nat] :
( ( nat_sum_decode(N4) = sum_Inr_nat_nat(B8) )
=> ( ( nat_sum_decode(B8) = sum_Inr_nat_nat(Ba2) )
=> ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) = nat_prod_decode(Ba2) )
=> aa_nat_o(P,Y3) ) ) )
=> aa_nat_o(P,aa_nat_nat(suc,N4)) ) ) ) ) )
=> aa_nat_o(P,A0) ) ) ) ).
% nth_item.pinduct
tff(fact_7011_ATP_Olambda__1,axiom,
! [Uu: product_prod_int_int] :
aa_Pro6970504127699319548nt_int(aTP_Lamm_ep,Uu) = $ite(aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu) = zero_zero_int,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu))) ).
% ATP.lambda_1
tff(fact_7012_ATP_Olambda__2,axiom,
! [Uu: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_eq,Uu) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu))),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu)) ).
% ATP.lambda_2
tff(fact_7013_ATP_Olambda__3,axiom,
! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_fa,Uu) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),zero_zero_nat) ).
% ATP.lambda_3
tff(fact_7014_ATP_Olambda__4,axiom,
! [Uu: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_fl,Uu) = produc5538323210962509403atural(next,aTP_Lamm_fk(Uu)) ).
% ATP.lambda_4
tff(fact_7015_ATP_Olambda__5,axiom,
! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_fg,Uu) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ff,Uu)) ).
% ATP.lambda_5
tff(fact_7016_ATP_Olambda__6,axiom,
! [Uu: num] : aa_num_option_num(aTP_Lamm_fn,Uu) = some_num(aa_num_num(bit1,Uu)) ).
% ATP.lambda_6
tff(fact_7017_ATP_Olambda__7,axiom,
! [Uu: num] : aa_num_option_num(aTP_Lamm_fo,Uu) = some_num(aa_num_num(bit0,Uu)) ).
% ATP.lambda_7
tff(fact_7018_ATP_Olambda__8,axiom,
! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_ec,Uu) = product_Pair_int_int(aa_int_int(uminus_uminus_int,Uu)) ).
% ATP.lambda_8
tff(fact_7019_ATP_Olambda__9,axiom,
! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_eb,Uu) = product_Pair_int_int(aa_int_int(abs_abs_int,Uu)) ).
% ATP.lambda_9
tff(fact_7020_ATP_Olambda__10,axiom,
! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_ej,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).
% ATP.lambda_10
tff(fact_7021_ATP_Olambda__11,axiom,
! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_fr(Uu),Uua) = case_num_option_num(some_num(one),aTP_Lamm_fp(Uua),aTP_Lamm_fq(Uua),Uu) ).
% ATP.lambda_11
tff(fact_7022_ATP_Olambda__12,axiom,
! [Uu: pred_Product_unit,Uua: produc7822875418678951345atural] :
aa_Pro6676232280479689400atural(aa_pre2304647123621358054atural(aTP_Lamm_ft,Uu),Uua) = $ite(aa_Product_unit_o(eval_Product_unit(Uu),product_Unity),aa_Pro6676232280479689400atural(produc5069803637994805237atural(bot_bo2717538794563056311t_unit),Uua),aa_Pro6676232280479689400atural(produc5069803637994805237atural(single_Product_unit(product_Unity)),Uua)) ).
% ATP.lambda_12
tff(fact_7023_ATP_Olambda__13,axiom,
! [Uu: int,Uua: int] :
aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_en,Uu),Uua) = $ite(Uu = zero_zero_int,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Uu)),Uua)),aa_int_int(abs_abs_int,Uu))) ).
% ATP.lambda_13
tff(fact_7024_ATP_Olambda__14,axiom,
! [Uu: int,Uua: int] :
aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ev,Uu),Uua) = $ite(Uua = zero_zero_int,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(Uu),Uua)) ).
% ATP.lambda_14
tff(fact_7025_ATP_Olambda__15,axiom,
! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_fs,Uu),Uua) = case_nat_option_num(none_num,aTP_Lamm_fr(Uua),Uu) ).
% ATP.lambda_15
tff(fact_7026_ATP_Olambda__16,axiom,
! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_fp(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_fo,bit_take_bit_num(Uu,Uua)) ).
% ATP.lambda_16
tff(fact_7027_ATP_Olambda__17,axiom,
! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_er,Uu),Uua) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua))) ).
% ATP.lambda_17
tff(fact_7028_ATP_Olambda__18,axiom,
! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_es,Uu),Uua) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua))) ).
% ATP.lambda_18
tff(fact_7029_ATP_Olambda__19,axiom,
! [Uu: fun_nat_assn,Uua: nat] : aa_nat_assn(aTP_Lamm_dv(Uu),Uua) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua))),aa_nat_assn(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)))) ).
% ATP.lambda_19
tff(fact_7030_ATP_Olambda__20,axiom,
! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_dw(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)))) ).
% ATP.lambda_20
tff(fact_7031_ATP_Olambda__21,axiom,
! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_dx(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)))) ).
% ATP.lambda_21
tff(fact_7032_ATP_Olambda__22,axiom,
! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_dy(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)))) ).
% ATP.lambda_22
tff(fact_7033_ATP_Olambda__23,axiom,
! [Uu: assn,Uua: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aTP_Lamm_dz(Uu),Uua)
<=> ( aa_Pro653082453696054616_nat_o(in_range,Uua)
& ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uu),Uua) ) ) ).
% ATP.lambda_23
tff(fact_7034_ATP_Olambda__24,axiom,
! [Uu: fun_Pr6460944699146269935_nat_o,Uua: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aTP_Lamm_au,Uu),Uua)
<=> ( aa_Pro653082453696054616_nat_o(in_range,Uua)
& ~ aa_Pro653082453696054616_nat_o(Uu,Uua) ) ) ).
% ATP.lambda_24
tff(fact_7035_ATP_Olambda__25,axiom,
! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_df(Uu),Uua) = minus_minus_rat(Uu,aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).
% ATP.lambda_25
tff(fact_7036_ATP_Olambda__26,axiom,
! [Uu: assn,Uua: assn] :
( aa_assn_o(aa_assn_fun_assn_o(aTP_Lamm_dp,Uu),Uua)
<=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Uua),Uu) ) ).
% ATP.lambda_26
tff(fact_7037_ATP_Olambda__27,axiom,
! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_dj,Uu),Uua)
<=> aa_set3702741264012733747_nat_o(ord_le3146513528884898305at_nat(Uua),Uu) ) ).
% ATP.lambda_27
tff(fact_7038_ATP_Olambda__28,axiom,
! [Uu: set_o,Uua: set_o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(aTP_Lamm_dl,Uu),Uua)
<=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,Uua),Uu) ) ).
% ATP.lambda_28
tff(fact_7039_ATP_Olambda__29,axiom,
! [Uu: set_nat,Uua: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_dr,Uu),Uua)
<=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Uua),Uu) ) ).
% ATP.lambda_29
tff(fact_7040_ATP_Olambda__30,axiom,
! [Uu: set_int,Uua: set_int] :
( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_dn,Uu),Uua)
<=> aa_set_int_o(ord_less_eq_set_int(Uua),Uu) ) ).
% ATP.lambda_30
tff(fact_7041_ATP_Olambda__31,axiom,
! [Uu: nat,Uua: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_dt,Uu),Uua)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).
% ATP.lambda_31
tff(fact_7042_ATP_Olambda__32,axiom,
! [Uu: assn,Uua: assn] :
( aa_assn_o(aa_assn_fun_assn_o(aTP_Lamm_dq,Uu),Uua)
<=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Uua),Uu) ) ).
% ATP.lambda_32
tff(fact_7043_ATP_Olambda__33,axiom,
! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_dk,Uu),Uua)
<=> aa_set3702741264012733747_nat_o(ord_le7866589430770878221at_nat(Uua),Uu) ) ).
% ATP.lambda_33
tff(fact_7044_ATP_Olambda__34,axiom,
! [Uu: set_o,Uua: set_o] :
( aa_set_o_o(aa_set_o_fun_set_o_o(aTP_Lamm_dm,Uu),Uua)
<=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,Uua),Uu) ) ).
% ATP.lambda_34
tff(fact_7045_ATP_Olambda__35,axiom,
! [Uu: set_nat,Uua: set_nat] :
( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_ds,Uu),Uua)
<=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Uua),Uu) ) ).
% ATP.lambda_35
tff(fact_7046_ATP_Olambda__36,axiom,
! [Uu: set_int,Uua: set_int] :
( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_do,Uu),Uua)
<=> aa_set_int_o(ord_less_set_int(Uua),Uu) ) ).
% ATP.lambda_36
tff(fact_7047_ATP_Olambda__37,axiom,
! [Uu: nat,Uua: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_du,Uu),Uua)
<=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).
% ATP.lambda_37
tff(fact_7048_ATP_Olambda__38,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dg,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).
% ATP.lambda_38
tff(fact_7049_ATP_Olambda__39,axiom,
! [Uu: nat,Uua: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ff,Uu),Uua)
<=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).
% ATP.lambda_39
tff(fact_7050_ATP_Olambda__40,axiom,
! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_fb,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).
% ATP.lambda_40
tff(fact_7051_ATP_Olambda__41,axiom,
! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
( aa_Pro6150637604445964947_nat_o(aTP_Lamm_a(Uu),Uua)
<=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uua),Uu) ) ).
% ATP.lambda_41
tff(fact_7052_ATP_Olambda__42,axiom,
! [Uu: set_o,Uua: $o] :
( aa_o_o(aTP_Lamm_aa(Uu),(Uua))
<=> aa_set_o_o(member_o((Uua)),Uu) ) ).
% ATP.lambda_42
tff(fact_7053_ATP_Olambda__43,axiom,
! [Uu: set_nat,Uua: nat] :
( aa_nat_o(aTP_Lamm_ad(Uu),Uua)
<=> aa_set_nat_o(member_nat(Uua),Uu) ) ).
% ATP.lambda_43
tff(fact_7054_ATP_Olambda__44,axiom,
! [Uu: set_int,Uua: int] :
( aa_int_o(aTP_Lamm_ab(Uu),Uua)
<=> aa_set_int_o(member_int(Uua),Uu) ) ).
% ATP.lambda_44
tff(fact_7055_ATP_Olambda__45,axiom,
! [Uu: set_fu3555266933746557007_nat_o,Uua: fun_Pr6460944699146269935_nat_o] :
( aa_fun1675921005893230629at_o_o(aTP_Lamm_ac(Uu),Uua)
<=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Uua),Uu) ) ).
% ATP.lambda_45
tff(fact_7056_ATP_Olambda__46,axiom,
! [Uu: produc3843707927480180839at_nat,Uua: produc3843707927480180839at_nat] :
( aa_Pro5947980084102686253_nat_o(aTP_Lamm_ai(Uu),Uua)
<=> ( Uua = Uu ) ) ).
% ATP.lambda_46
tff(fact_7057_ATP_Olambda__47,axiom,
! [Uu: product_prod_nat_nat,Uua: product_prod_nat_nat] :
( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aTP_Lamm_ak,Uu),Uua)
<=> ( Uua = Uu ) ) ).
% ATP.lambda_47
tff(fact_7058_ATP_Olambda__48,axiom,
! [Uu: $o,Uua: $o] :
( aa_o_o(aa_o_fun_o_o(aTP_Lamm_al,(Uu)),(Uua))
<=> ( (Uua)
<=> (Uu) ) ) ).
% ATP.lambda_48
tff(fact_7059_ATP_Olambda__49,axiom,
! [Uu: nat,Uua: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_am,Uu),Uua)
<=> ( Uua = Uu ) ) ).
% ATP.lambda_49
tff(fact_7060_ATP_Olambda__50,axiom,
! [Uu: int,Uua: int] :
( aa_int_o(aa_int_fun_int_o(aTP_Lamm_an,Uu),Uua)
<=> ( Uua = Uu ) ) ).
% ATP.lambda_50
tff(fact_7061_ATP_Olambda__51,axiom,
! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_Pr6460944699146269935_nat_o] :
( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(aTP_Lamm_aj,Uu),Uua)
<=> ( Uua = Uu ) ) ).
% ATP.lambda_51
tff(fact_7062_ATP_Olambda__52,axiom,
! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_cj(Uu),Uua) = aa_nat_Code_integer(Uu,aa_nat_nat(suc,Uua)) ).
% ATP.lambda_52
tff(fact_7063_ATP_Olambda__53,axiom,
! [Uu: fun_nat_assn,Uua: nat] : aa_nat_assn(aTP_Lamm_ck(Uu),Uua) = aa_nat_assn(Uu,aa_nat_nat(suc,Uua)) ).
% ATP.lambda_53
tff(fact_7064_ATP_Olambda__54,axiom,
! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_cl(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).
% ATP.lambda_54
tff(fact_7065_ATP_Olambda__55,axiom,
! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_cm(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).
% ATP.lambda_55
tff(fact_7066_ATP_Olambda__56,axiom,
! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_cn(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).
% ATP.lambda_56
tff(fact_7067_ATP_Olambda__57,axiom,
! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_fq(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu,Uua))) ).
% ATP.lambda_57
tff(fact_7068_ATP_Olambda__58,axiom,
! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_ez,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ey(Uu),Uua)) ).
% ATP.lambda_58
tff(fact_7069_ATP_Olambda__59,axiom,
! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_ex,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ew(Uu),Uua)) ).
% ATP.lambda_59
tff(fact_7070_ATP_Olambda__60,axiom,
! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_eu,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_et(Uu),Uua)) ).
% ATP.lambda_60
tff(fact_7071_ATP_Olambda__61,axiom,
! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_fk(Uu),Uua) = produc6639722614265839536atural(aa_Cod4272041906116450446atural(plus_p4538020629002901425atural(Uua),aa_Cod4272041906116450446atural(times_2397367101498566445atural(Uu),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))))))))))))))))))))))) ).
% ATP.lambda_61
tff(fact_7072_ATP_Olambda__62,axiom,
! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_ae(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(Uu,aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).
% ATP.lambda_62
tff(fact_7073_ATP_Olambda__63,axiom,
! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_fm(Uu),Uua) = produc6639722614265839536atural(aa_Cod4272041906116450446atural(modulo8411746178871703098atural(Uua),Uu)) ).
% ATP.lambda_63
tff(fact_7074_ATP_Olambda__64,axiom,
! [Uu: nat,Uua: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(aTP_Lamm_fu(Uu),Uua) = some_P1914260805536162275it_nat(produc7133225469290080770it_nat(product_Unity,produc584006145561248582it_nat(Uua,Uu))) ).
% ATP.lambda_64
tff(fact_7075_ATP_Olambda__65,axiom,
! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_cy(Uu),Uua) = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uu,Uua)) ).
% ATP.lambda_65
tff(fact_7076_ATP_Olambda__66,axiom,
! [Uu: fun_nat_assn,Uua: nat] : aa_nat_fun_assn_assn(aTP_Lamm_cz(Uu),Uua) = aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Uu,Uua)) ).
% ATP.lambda_66
tff(fact_7077_ATP_Olambda__67,axiom,
! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_da(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uua)) ).
% ATP.lambda_67
tff(fact_7078_ATP_Olambda__68,axiom,
! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_db(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ).
% ATP.lambda_68
tff(fact_7079_ATP_Olambda__69,axiom,
! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_dc(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)) ).
% ATP.lambda_69
tff(fact_7080_ATP_Olambda__70,axiom,
! [Uu: num,Uua: code_integer,Uub: code_integer] :
aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_eo(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu)),Uub),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Uub),numera6620942414471956472nteger(Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),Uub)) ).
% ATP.lambda_70
tff(fact_7081_ATP_Olambda__71,axiom,
! [Uu: num,Uua: nat,Uub: nat] :
aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_em(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(Uu)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),numeral_numeral_nat(Uu))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)),Uub)) ).
% ATP.lambda_71
tff(fact_7082_ATP_Olambda__72,axiom,
! [Uu: num,Uua: int,Uub: int] :
aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ea(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(Uu)),Uub),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),Uua)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),numeral_numeral_int(Uu))),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),Uua)),Uub)) ).
% ATP.lambda_72
tff(fact_7083_ATP_Olambda__73,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_fe(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu)),Uub))) ).
% ATP.lambda_73
tff(fact_7084_ATP_Olambda__74,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_fh(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Uu)),Uub))) ).
% ATP.lambda_74
tff(fact_7085_ATP_Olambda__75,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_fd(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Uu),Uub))) ).
% ATP.lambda_75
tff(fact_7086_ATP_Olambda__76,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_fc(Uu),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),Uub)),Uub = one_one_Code_integer) ).
% ATP.lambda_76
tff(fact_7087_ATP_Olambda__77,axiom,
! [Uu: produc7822875418678951345atural,Uua: code_natural,Uub: code_natural] : aa_Cod6808739660676082561atural(aa_Cod4292056611156568242atural(aTP_Lamm_fj(Uu),Uua),Uub) = produc8282080750456430313atural(aa_Cod8811915439870959971atural(aTP_Lamm_fi(Uua),Uub),aa_Pro3673693039631929973atural(produc6591912806276919810atural,aa_Pro7165575656943818355atural(next,Uu))) ).
% ATP.lambda_77
tff(fact_7088_ATP_Olambda__78,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_el(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ek(Uua),Uub),quotient_of(Uu)) ).
% ATP.lambda_78
tff(fact_7089_ATP_Olambda__79,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ei(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_eh(Uua),Uub),quotient_of(Uu)) ).
% ATP.lambda_79
tff(fact_7090_ATP_Olambda__80,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_eg(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ef(Uua),Uub),quotient_of(Uu)) ).
% ATP.lambda_80
tff(fact_7091_ATP_Olambda__81,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ee(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ed(Uua),Uub),quotient_of(Uu)) ).
% ATP.lambda_81
tff(fact_7092_ATP_Olambda__82,axiom,
! [Uu: assn,Uua: assn,Uub: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(aTP_Lamm_dh(Uu),Uua),Uub)
<=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uu),Uub)
| aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uua),Uub) ) ) ).
% ATP.lambda_82
tff(fact_7093_ATP_Olambda__83,axiom,
! [Uu: assn,Uua: assn,Uub: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(aTP_Lamm_di(Uu),Uua),Uub)
<=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uu),Uub)
& aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uua),Uub) ) ) ).
% ATP.lambda_83
tff(fact_7094_ATP_Olambda__84,axiom,
! [Uu: produc3843707927480180839at_nat,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_bx(Uu),Uua),Uub)
<=> ( ( Uu = Uub )
& aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).
% ATP.lambda_84
tff(fact_7095_ATP_Olambda__85,axiom,
! [Uu: product_prod_nat_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_bz(Uu),Uua),Uub)
<=> ( ( Uu = Uub )
& aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).
% ATP.lambda_85
tff(fact_7096_ATP_Olambda__86,axiom,
! [Uu: $o,Uua: fun_o_o,Uub: $o] :
( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ca((Uu)),Uua),(Uub))
<=> ( ( (Uu)
<=> (Uub) )
& aa_o_o(Uua,(Uub)) ) ) ).
% ATP.lambda_86
tff(fact_7097_ATP_Olambda__87,axiom,
! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_cb(Uu),Uua),Uub)
<=> ( ( Uu = Uub )
& aa_nat_o(Uua,Uub) ) ) ).
% ATP.lambda_87
tff(fact_7098_ATP_Olambda__88,axiom,
! [Uu: int,Uua: fun_int_o,Uub: int] :
( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_cc(Uu),Uua),Uub)
<=> ( ( Uu = Uub )
& aa_int_o(Uua,Uub) ) ) ).
% ATP.lambda_88
tff(fact_7099_ATP_Olambda__89,axiom,
! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_fu5135969500543783852at_o_o,Uub: fun_Pr6460944699146269935_nat_o] :
( aa_fun1675921005893230629at_o_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_by(Uu),Uua),Uub)
<=> ( ( Uu = Uub )
& aa_fun1675921005893230629at_o_o(Uua,Uub) ) ) ).
% ATP.lambda_89
tff(fact_7100_ATP_Olambda__90,axiom,
! [Uu: produc3843707927480180839at_nat,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_cd(Uu),Uua),Uub)
<=> ( ( Uub = Uu )
& aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).
% ATP.lambda_90
tff(fact_7101_ATP_Olambda__91,axiom,
! [Uu: product_prod_nat_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_cf(Uu),Uua),Uub)
<=> ( ( Uub = Uu )
& aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).
% ATP.lambda_91
tff(fact_7102_ATP_Olambda__92,axiom,
! [Uu: $o,Uua: fun_o_o,Uub: $o] :
( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_cg((Uu)),Uua),(Uub))
<=> ( ( (Uub)
<=> (Uu) )
& aa_o_o(Uua,(Uub)) ) ) ).
% ATP.lambda_92
tff(fact_7103_ATP_Olambda__93,axiom,
! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ch(Uu),Uua),Uub)
<=> ( ( Uub = Uu )
& aa_nat_o(Uua,Uub) ) ) ).
% ATP.lambda_93
tff(fact_7104_ATP_Olambda__94,axiom,
! [Uu: int,Uua: fun_int_o,Uub: int] :
( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ci(Uu),Uua),Uub)
<=> ( ( Uub = Uu )
& aa_int_o(Uua,Uub) ) ) ).
% ATP.lambda_94
tff(fact_7105_ATP_Olambda__95,axiom,
! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_fu5135969500543783852at_o_o,Uub: fun_Pr6460944699146269935_nat_o] :
( aa_fun1675921005893230629at_o_o(aa_fun6267365381853603094at_o_o(aTP_Lamm_ce(Uu),Uua),Uub)
<=> ( ( Uub = Uu )
& aa_fun1675921005893230629at_o_o(Uua,Uub) ) ) ).
% ATP.lambda_95
tff(fact_7106_ATP_Olambda__96,axiom,
! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: set_Pr1261947904930325089at_nat] :
( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_bm(Uu),Uua),Uub)
<=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(Uua,Uub)),Uu) ) ).
% ATP.lambda_96
tff(fact_7107_ATP_Olambda__97,axiom,
! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bg(Uu),Uua),Uub)
<=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)),Uu) ) ).
% ATP.lambda_97
tff(fact_7108_ATP_Olambda__98,axiom,
! [Uu: set_Pr3486312109254157132nteger,Uua: fun_Pr5098669352780829690e_term,Uub: produc8923325533196201883nteger] :
( aa_Pro4926521453222654329eger_o(aa_fun4505422719058378424eger_o(aTP_Lamm_bh(Uu),Uua),Uub)
<=> member8978266922497061677nteger(produc565448451787809096nteger(Uua,Uub),Uu) ) ).
% ATP.lambda_98
tff(fact_7109_ATP_Olambda__99,axiom,
! [Uu: set_Pr1312406793078187350et_nat,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aTP_Lamm_bi(Uu),Uua),Uub)
<=> member4513763875858039863et_nat(produc6017135503889078682et_nat(Uua,Uub),Uu) ) ).
% ATP.lambda_99
tff(fact_7110_ATP_Olambda__100,axiom,
! [Uu: set_Pr4040394263813967220et_nat,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc8799533983703829792et_nat] :
( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(aTP_Lamm_bj(Uu),Uua),Uub)
<=> member9174161442226468181et_nat(produc7291855603632349880et_nat(Uua,Uub),Uu) ) ).
% ATP.lambda_100
tff(fact_7111_ATP_Olambda__101,axiom,
! [Uu: set_Pr6700895561065273913nt_int,Uua: fun_Pr7763680713840462443e_term,Uub: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(aa_fun3538162476766722827_int_o(aTP_Lamm_bk(Uu),Uua),Uub)
<=> member753276645541486746nt_int(produc7320969046608631221nt_int(Uua,Uub),Uu) ) ).
% ATP.lambda_101
tff(fact_7112_ATP_Olambda__102,axiom,
! [Uu: set_Pr4166680992137330533nt_int,Uua: fun_in5730379987311621695e_term,Uub: product_prod_int_int] :
( aa_Pro4592068669816069595_int_o(aa_fun586359091965693151_int_o(aTP_Lamm_bl(Uu),Uua),Uub)
<=> member5890090881338784710nt_int(produc4729913460834754785nt_int(Uua,Uub),Uu) ) ).
% ATP.lambda_102
tff(fact_7113_ATP_Olambda__103,axiom,
! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aTP_Lamm_ag(Uu),Uua),Uub)
<=> ( aa_Pro653082453696054616_nat_o(Uu,Uub)
| aa_Pro653082453696054616_nat_o(Uua,Uub) ) ) ).
% ATP.lambda_103
tff(fact_7114_ATP_Olambda__104,axiom,
! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aTP_Lamm_ah(Uu),Uua),Uub)
<=> ( aa_Pro653082453696054616_nat_o(Uu,Uub)
& aa_Pro653082453696054616_nat_o(Uua,Uub) ) ) ).
% ATP.lambda_104
tff(fact_7115_ATP_Olambda__105,axiom,
! [Uu: fun_o_Code_integer,Uua: code_integer,Uub: $o] : aa_o_Code_integer(aa_Cod4043041029439962285nteger(aTP_Lamm_co(Uu),Uua),(Uub)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_o_Code_integer(Uu,(Uub))),Uua) ).
% ATP.lambda_105
tff(fact_7116_ATP_Olambda__106,axiom,
! [Uu: fun_o_rat,Uua: rat,Uub: $o] : aa_o_rat(aa_rat_fun_o_rat(aTP_Lamm_cp(Uu),Uua),(Uub)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(Uu,(Uub))),Uua) ).
% ATP.lambda_106
tff(fact_7117_ATP_Olambda__107,axiom,
! [Uu: fun_o_nat,Uua: nat,Uub: $o] : aa_o_nat(aa_nat_fun_o_nat(aTP_Lamm_cq(Uu),Uua),(Uub)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(Uu,(Uub))),Uua) ).
% ATP.lambda_107
tff(fact_7118_ATP_Olambda__108,axiom,
! [Uu: fun_o_int,Uua: int,Uub: $o] : aa_o_int(aa_int_fun_o_int(aTP_Lamm_cr(Uu),Uua),(Uub)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(Uu,(Uub))),Uua) ).
% ATP.lambda_108
tff(fact_7119_ATP_Olambda__109,axiom,
! [Uu: fun_nat_Code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_cs(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(Uu,Uub)),Uua) ).
% ATP.lambda_109
tff(fact_7120_ATP_Olambda__110,axiom,
! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_ct(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,Uub)),Uua) ).
% ATP.lambda_110
tff(fact_7121_ATP_Olambda__111,axiom,
! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cu(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uub)),Uua) ).
% ATP.lambda_111
tff(fact_7122_ATP_Olambda__112,axiom,
! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_cv(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uub)),Uua) ).
% ATP.lambda_112
tff(fact_7123_ATP_Olambda__113,axiom,
! [Uu: fun_int_Code_integer,Uua: code_integer,Uub: int] : aa_int_Code_integer(aa_Cod1949805079358391713nteger(aTP_Lamm_cw(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_int_Code_integer(Uu,Uub)),Uua) ).
% ATP.lambda_113
tff(fact_7124_ATP_Olambda__114,axiom,
! [Uu: fun_int_rat,Uua: rat,Uub: int] : aa_int_rat(aa_rat_fun_int_rat(aTP_Lamm_cx(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(Uu,Uub)),Uua) ).
% ATP.lambda_114
tff(fact_7125_ATP_Olambda__115,axiom,
! [Uu: code_natural,Uua: code_natural,Uub: code_natural,Uuc: code_natural] : aa_Cod6808739660676082561atural(aa_Cod4292056611156568242atural(aa_Cod8811915439870959971atural(aTP_Lamm_fi(Uu),Uua),Uub),Uuc) = produc4480994950612372183atural(produc3574140220909816553atural(aa_Cod4272041906116450446atural(inc_shift(numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))))))))))))))))))))),Uu),Uuc),produc3574140220909816553atural(Uub,aa_Cod4272041906116450446atural(inc_shift(numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))))))))))))))))))))),Uua))) ).
% ATP.lambda_115
tff(fact_7126_ATP_Olambda__116,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_et(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).
% ATP.lambda_116
tff(fact_7127_ATP_Olambda__117,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ew(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)) ).
% ATP.lambda_117
tff(fact_7128_ATP_Olambda__118,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ey(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).
% ATP.lambda_118
tff(fact_7129_ATP_Olambda__119,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ed(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).
% ATP.lambda_119
tff(fact_7130_ATP_Olambda__120,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ek(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).
% ATP.lambda_120
tff(fact_7131_ATP_Olambda__121,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_eh(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uub)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).
% ATP.lambda_121
tff(fact_7132_ATP_Olambda__122,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ef(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ).
% ATP.lambda_122
tff(fact_7133_ATP_Olambda__123,axiom,
! [Uu: $o,Uua: product_unit] :
( aa_Product_unit_o(aTP_Lamm_at((Uu)),Uua)
<=> (Uu) ) ).
% ATP.lambda_123
tff(fact_7134_ATP_Olambda__124,axiom,
! [Uu: $o,Uua: product_prod_nat_nat] :
( aa_Pro6150637604445964947_nat_o(aTP_Lamm_ap((Uu)),Uua)
<=> (Uu) ) ).
% ATP.lambda_124
tff(fact_7135_ATP_Olambda__125,axiom,
! [Uu: $o,Uua: $o] :
( aa_o_o(aa_o_fun_o_o(aTP_Lamm_ar,(Uu)),(Uua))
<=> (Uu) ) ).
% ATP.lambda_125
tff(fact_7136_ATP_Olambda__126,axiom,
! [Uu: $o,Uua: nat] :
( aa_nat_o(aTP_Lamm_as((Uu)),Uua)
<=> (Uu) ) ).
% ATP.lambda_126
tff(fact_7137_ATP_Olambda__127,axiom,
! [Uu: $o,Uua: int] :
( aa_int_o(aTP_Lamm_aq((Uu)),Uua)
<=> (Uu) ) ).
% ATP.lambda_127
tff(fact_7138_ATP_Olambda__128,axiom,
! [Uu: $o,Uua: fun_Pr6460944699146269935_nat_o] :
( aa_fun1675921005893230629at_o_o(aTP_Lamm_ao((Uu)),Uua)
<=> (Uu) ) ).
% ATP.lambda_128
tff(fact_7139_ATP_Olambda__129,axiom,
! [Uu: set_nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_bb(Uu),Uua) = Uu ).
% ATP.lambda_129
tff(fact_7140_ATP_Olambda__130,axiom,
! [Uu: nat,Uua: product_unit] : aa_Product_unit_nat(aTP_Lamm_az(Uu),Uua) = Uu ).
% ATP.lambda_130
tff(fact_7141_ATP_Olambda__131,axiom,
! [Uu: nat,Uua: $o] : aa_o_nat(aTP_Lamm_av(Uu),(Uua)) = Uu ).
% ATP.lambda_131
tff(fact_7142_ATP_Olambda__132,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ax,Uu),Uua) = Uu ).
% ATP.lambda_132
tff(fact_7143_ATP_Olambda__133,axiom,
! [Uu: nat,Uua: int] : aa_int_nat(aTP_Lamm_dd(Uu),Uua) = Uu ).
% ATP.lambda_133
tff(fact_7144_ATP_Olambda__134,axiom,
! [Uu: int,Uua: product_unit] : aa_Product_unit_int(aTP_Lamm_ba(Uu),Uua) = Uu ).
% ATP.lambda_134
tff(fact_7145_ATP_Olambda__135,axiom,
! [Uu: int,Uua: $o] : aa_o_int(aTP_Lamm_aw(Uu),(Uua)) = Uu ).
% ATP.lambda_135
tff(fact_7146_ATP_Olambda__136,axiom,
! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_ay(Uu),Uua) = Uu ).
% ATP.lambda_136
tff(fact_7147_ATP_Olambda__137,axiom,
! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_de,Uu),Uua) = Uu ).
% ATP.lambda_137
tff(fact_7148_ATP_Olambda__138,axiom,
! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_bs,Uu) = Uu ).
% ATP.lambda_138
tff(fact_7149_ATP_Olambda__139,axiom,
! [Uu: assn] : aa_assn_assn(aTP_Lamm_bt,Uu) = Uu ).
% ATP.lambda_139
tff(fact_7150_ATP_Olambda__140,axiom,
! [Uu: rat] : aa_rat_rat(aTP_Lamm_bu,Uu) = Uu ).
% ATP.lambda_140
tff(fact_7151_ATP_Olambda__141,axiom,
! [Uu: nat] : aa_nat_nat(aTP_Lamm_bv,Uu) = Uu ).
% ATP.lambda_141
tff(fact_7152_ATP_Olambda__142,axiom,
! [Uu: int] : aa_int_int(aTP_Lamm_bw,Uu) = Uu ).
% ATP.lambda_142
tff(fact_7153_ATP_Olambda__143,axiom,
! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_bc,Uu) = zero_z3403309356797280102nteger ).
% ATP.lambda_143
tff(fact_7154_ATP_Olambda__144,axiom,
! [Uu: rat] : aa_rat_rat(aTP_Lamm_bd,Uu) = zero_zero_rat ).
% ATP.lambda_144
tff(fact_7155_ATP_Olambda__145,axiom,
! [Uu: nat] : aa_nat_nat(aTP_Lamm_be,Uu) = zero_zero_nat ).
% ATP.lambda_145
tff(fact_7156_ATP_Olambda__146,axiom,
! [Uu: int] : aa_int_int(aTP_Lamm_bf,Uu) = zero_zero_int ).
% ATP.lambda_146
tff(fact_7157_ATP_Olambda__147,axiom,
! [Uu: produc3658429121746597890et_nat] :
( aa_Pro653082453696054616_nat_o(aTP_Lamm_af,Uu)
<=> $false ) ).
% ATP.lambda_147
tff(fact_7158_ATP_Olambda__148,axiom,
! [Uu: product_prod_nat_nat] :
( aa_Pro6150637604445964947_nat_o(aTP_Lamm_bo,Uu)
<=> $false ) ).
% ATP.lambda_148
tff(fact_7159_ATP_Olambda__149,axiom,
! [Uu: $o] :
( aa_o_o(aTP_Lamm_bp,(Uu))
<=> $false ) ).
% ATP.lambda_149
tff(fact_7160_ATP_Olambda__150,axiom,
! [Uu: nat] :
( aa_nat_o(aTP_Lamm_bq,Uu)
<=> $false ) ).
% ATP.lambda_150
tff(fact_7161_ATP_Olambda__151,axiom,
! [Uu: int] :
( aa_int_o(aTP_Lamm_br,Uu)
<=> $false ) ).
% ATP.lambda_151
tff(fact_7162_ATP_Olambda__152,axiom,
! [Uu: fun_Pr6460944699146269935_nat_o] :
( aa_fun1675921005893230629at_o_o(aTP_Lamm_bn,Uu)
<=> $false ) ).
% ATP.lambda_152
% Helper facts (12)
tff(help_fequal_2_1_fequal_001_Eo_T,axiom,
! [X: $o,Y: $o] :
( ( (X) != (Y) )
| aa_o_o(aa_o_fun_o_o(fequal_o,(X)),(Y)) ) ).
tff(help_fequal_1_1_fequal_001_Eo_T,axiom,
! [X: $o,Y: $o] :
( ~ aa_o_o(aa_o_fun_o_o(fequal_o,(X)),(Y))
| ( (X) = (Y) ) ) ).
tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( X != Y )
| aa_int_o(aa_int_fun_int_o(fequal_int,X),Y) ) ).
tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X),Y)
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
| aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X),Y) ) ).
tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X),Y)
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( X != Y )
| aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(fequal9188845342289841878at_nat,X),Y) ) ).
tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ~ aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(fequal9188845342289841878at_nat,X),Y)
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_T,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
( ( X != Y )
| aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(fequal1890233700291625924_nat_o,X),Y) ) ).
tff(help_fequal_1_1_fequal_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_T,axiom,
! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
( ~ aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(fequal1890233700291625924_nat_o,X),Y)
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
( ( X != Y )
| aa_Pro5947980084102686253_nat_o(fequal7949547063978738364at_nat(X),Y) ) ).
tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
( ~ aa_Pro5947980084102686253_nat_o(fequal7949547063978738364at_nat(X),Y)
| ( X = Y ) ) ).
% Conjectures (1)
tff(conj_0,conjecture,
aa_assn_o(entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,a),b)),c)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,a),c)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,b),c))) ).
%------------------------------------------------------------------------------