TPTP Problem File: SLH0677^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FO_Theory_Rewriting/0082_FOR_Check/prob_00354_018133__18922006_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1632 ( 602 unt; 349 typ; 0 def)
% Number of atoms : 3728 (1231 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10224 ( 390 ~; 65 |; 203 &;7978 @)
% ( 0 <=>;1588 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 70 ( 69 usr)
% Number of type conns : 1137 (1137 >; 0 *; 0 +; 0 <<)
% Number of symbols : 283 ( 280 usr; 22 con; 0-3 aty)
% Number of variables : 3694 ( 363 ^;3212 !; 119 ?;3694 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:03:12.123
%------------------------------------------------------------------------------
% Could-be-implicit typings (69)
thf(ty_n_t__Product____Type__Oprod_I_062_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_M_062_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_Mt__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_J_J,type,
produc2070832938513523962tion_f: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_M_062_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J_Mt__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_J_J,type,
produc3878888619416440882tion_f: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_M_062_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_Mt__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J_J_J,type,
produc7762418835577779250_nat_f: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_Mt__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_J,type,
produc441656135839998061tion_f: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_M_062_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J_Mt__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J_J_J,type,
produc8574009693863894890_nat_f: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J_Mt__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_J,type,
produc724344743186279177tion_f: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_Mt__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J_J,type,
produc8927788648312868617_nat_f: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
produc2858277421346412199at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc6121120109295599847at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J_Mt__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J_J,type,
produc5143877011543624101_nat_f: $tType ).
thf(ty_n_t__Option__Ooption_It__List__Olist_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_J,type,
option2150321469529786326tion_f: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_J,type,
set_op6172961469967127676tion_f: $tType ).
thf(ty_n_t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J,type,
option3789488934265196358tion_f: $tType ).
thf(ty_n_t__Option__Ooption_It__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J_J,type,
option6825207169704394579rm_f_v: $tType ).
thf(ty_n_t__List__Olist_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J,type,
list_T5268601877343193350tion_f: $tType ).
thf(ty_n_t__Set__Oset_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J,type,
set_Tr6476182622925392812tion_f: $tType ).
thf(ty_n_t__Option__Ooption_It__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J,type,
option2233895434904908115rm_f_v: $tType ).
thf(ty_n_t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J,type,
option3296083141436081229rm_f_v: $tType ).
thf(ty_n_t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J,type,
tree_r733329426570293750tion_f: $tType ).
thf(ty_n_t__List__Olist_It__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J,type,
list_f1824981274722084755rm_f_v: $tType ).
thf(ty_n_t__FSet__Ofset_It__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J,type,
fset_f2722810715853128435rm_f_v: $tType ).
thf(ty_n_t__Set__Oset_It__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J,type,
set_fs7307227306443116653rm_f_v: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J,type,
list_s8746099396510718605rm_f_v: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J,type,
set_se4583834864486174823rm_f_v: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J_J,type,
list_P4363786793477243949term_f: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J_J,type,
set_Pr989862937836626183term_f: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
list_P4093298276913796397rm_f_v: $tType ).
thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
fset_P8018961893305114765rm_f_v: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
set_Pr8827868859434726151rm_f_v: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J,type,
produc7245736746747425831term_f: $tType ).
thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
option99490722083217462at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_Pr9093778441882193744at_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__FSet__Ofset_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J_J,type,
option5345500251201882370_f_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J_J,type,
set_op1790067033808154392_nat_f: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
produc8199716216217303280at_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
option4927543243414619207at_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P6011104703257516679at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
set_Pr958786334691620121nt_int: $tType ).
thf(ty_n_t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J,type,
option5916524851906092002_nat_f: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
list_P3903862279629787026_f_nat: $tType ).
thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
fset_P6228066233360383026_f_nat: $tType ).
thf(ty_n_t__FOR____Certificate__Orr2____rel_It__FOR____Certificate__Oftrs_J,type,
fOR_rr2_rel_FOR_ftrs: $tType ).
thf(ty_n_t__FOR____Certificate__Orr1____rel_It__FOR____Certificate__Oftrs_J,type,
fOR_rr1_rel_FOR_ftrs: $tType ).
thf(ty_n_t__Set__Oset_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J,type,
set_Tree_reg_nat_f: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
set_Pr5245412377734449720_f_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
product_prod_int_int: $tType ).
thf(ty_n_t__List__Olist_It__Ground____Terms__Ogterm_Itf__f_J_J,type,
list_Ground_gterm_f: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Real__Oreal_J_J,type,
set_option_real: $tType ).
thf(ty_n_t__Set__Oset_It__Ground____Terms__Ogterm_Itf__f_J_J,type,
set_Ground_gterm_f: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
set_option_nat: $tType ).
thf(ty_n_t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J,type,
tree_reg_nat_f: $tType ).
thf(ty_n_t__List__Olist_It__FOR____Certificate__Oftrs_J,type,
list_FOR_ftrs: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_I_Eo_J_J,type,
set_option_o: $tType ).
thf(ty_n_t__Option__Ooption_It__Real__Oreal_J,type,
option_real: $tType ).
thf(ty_n_t__Ground____Terms__Ogterm_Itf__f_J,type,
ground_gterm_f: $tType ).
thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
option_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
option_int: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Option__Ooption_I_Eo_J,type,
option_o: $tType ).
thf(ty_n_t__List__Olist_I_Eo_J,type,
list_o: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (280)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
archim6058952711729229775r_real: real > int ).
thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
archim8280529875227126926d_real: real > int ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__FSet__Ofset_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
bNF_Ca4696266947080128040_f_nat: set_Pr1261947904930325089at_nat > ( nat > fset_P6228066233360383026_f_nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__Int__Oint,type,
bNF_Ca966259857504369954at_int: set_Pr1261947904930325089at_nat > ( nat > int ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_Ca968750328013420230at_nat: set_Pr1261947904930325089at_nat > ( nat > nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__Real__Oreal,type,
bNF_Ca9191250440166129314t_real: set_Pr1261947904930325089at_nat > ( nat > real ) > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_Eo,type,
complete_Inf_Inf_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Option__Ooption_I_Eo_J,type,
comple2387459607550929125tion_o: set_option_o > option_o ).
thf(sy_c_FOR__Certificate_Orr1__rel_OR1Proj_001t__FOR____Certificate__Oftrs,type,
fOR_rr305201390985280028R_ftrs: nat > fOR_rr2_rel_FOR_ftrs > fOR_rr1_rel_FOR_ftrs ).
thf(sy_c_FOR__Check_Ois__to__trs_H_001tf__f_001tf__v,type,
fOR_is_to_trs_f_v: list_f1824981274722084755rm_f_v > list_FOR_ftrs > option2233895434904908115rm_f_v ).
thf(sy_c_FOR__Check_OliftO1_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J,type,
fOR_li2611159955866944838tion_f: ( tree_r733329426570293750tion_f > tree_r733329426570293750tion_f ) > option3789488934265196358tion_f > option3789488934265196358tion_f ).
thf(sy_c_FOR__Check_OliftO1_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J,type,
fOR_li967653975282842594_nat_f: ( tree_r733329426570293750tion_f > tree_reg_nat_f ) > option3789488934265196358tion_f > option5916524851906092002_nat_f ).
thf(sy_c_FOR__Check_OliftO1_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J,type,
fOR_li259870005745783778tion_f: ( tree_reg_nat_f > tree_r733329426570293750tion_f ) > option5916524851906092002_nat_f > option3789488934265196358tion_f ).
thf(sy_c_FOR__Check_OliftO1_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J,type,
fOR_li1500713569213856382_nat_f: ( tree_reg_nat_f > tree_reg_nat_f ) > option5916524851906092002_nat_f > option5916524851906092002_nat_f ).
thf(sy_c_FOR__Check_OliftO2_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J,type,
fOR_li3518057619300867550tion_f: ( tree_r733329426570293750tion_f > tree_r733329426570293750tion_f > tree_r733329426570293750tion_f ) > option3789488934265196358tion_f > option3789488934265196358tion_f > option3789488934265196358tion_f ).
thf(sy_c_FOR__Check_OliftO2_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J,type,
fOR_li3014203053008925818_nat_f: ( tree_r733329426570293750tion_f > tree_r733329426570293750tion_f > tree_reg_nat_f ) > option3789488934265196358tion_f > option3789488934265196358tion_f > option5916524851906092002_nat_f ).
thf(sy_c_FOR__Check_OliftO2_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J,type,
fOR_li2306419083471867002tion_f: ( tree_r733329426570293750tion_f > tree_reg_nat_f > tree_r733329426570293750tion_f ) > option3789488934265196358tion_f > option5916524851906092002_nat_f > option3789488934265196358tion_f ).
thf(sy_c_FOR__Check_OliftO2_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J,type,
fOR_li2310238201417421590_nat_f: ( tree_r733329426570293750tion_f > tree_reg_nat_f > tree_reg_nat_f ) > option3789488934265196358tion_f > option5916524851906092002_nat_f > option5916524851906092002_nat_f ).
thf(sy_c_FOR__Check_OliftO2_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J,type,
fOR_li5786706929538884730tion_f: ( tree_reg_nat_f > tree_r733329426570293750tion_f > tree_r733329426570293750tion_f ) > option5916524851906092002_nat_f > option3789488934265196358tion_f > option3789488934265196358tion_f ).
thf(sy_c_FOR__Check_OliftO2_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J,type,
fOR_li6346090120295541526_nat_f: ( tree_reg_nat_f > tree_r733329426570293750tion_f > tree_reg_nat_f ) > option5916524851906092002_nat_f > option3789488934265196358tion_f > option5916524851906092002_nat_f ).
thf(sy_c_FOR__Check_OliftO2_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J,type,
fOR_li5638306150758482710tion_f: ( tree_reg_nat_f > tree_reg_nat_f > tree_r733329426570293750tion_f ) > option5916524851906092002_nat_f > option5916524851906092002_nat_f > option3789488934265196358tion_f ).
thf(sy_c_FOR__Check_OliftO2_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J,type,
fOR_li3434644546498476466_nat_f: ( tree_reg_nat_f > tree_reg_nat_f > tree_reg_nat_f ) > option5916524851906092002_nat_f > option5916524851906092002_nat_f > option5916524851906092002_nat_f ).
thf(sy_c_FOR__Check_Orr1__of__rr1__rel_001tf__f_001tf__v,type,
fOR_rr7289491001293628373el_f_v: fset_P6228066233360383026_f_nat > list_f1824981274722084755rm_f_v > fOR_rr1_rel_FOR_ftrs > option5916524851906092002_nat_f ).
thf(sy_c_FOR__Check_Orr2__of__rr2__rel_001tf__f_001tf__v,type,
fOR_rr7226795042121552277el_f_v: fset_P6228066233360383026_f_nat > list_f1824981274722084755rm_f_v > fOR_rr2_rel_FOR_ftrs > option3789488934265196358tion_f ).
thf(sy_c_FOR__Check_Osimplify__reg_001t__Nat__Onat_001t__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J,type,
fOR_si5451137711280541013tion_f: tree_r733329426570293750tion_f > tree_r733329426570293750tion_f ).
thf(sy_c_FOR__Check_Osimplify__reg_001t__Nat__Onat_001tf__f,type,
fOR_si5663490455393327601_nat_f: tree_reg_nat_f > tree_reg_nat_f ).
thf(sy_c_FOR__Semantics_Oeval__rr1__rel_001tf__f_001tf__v,type,
fOR_eval_rr1_rel_f_v: set_Pr5245412377734449720_f_nat > list_s8746099396510718605rm_f_v > fOR_rr1_rel_FOR_ftrs > set_Ground_gterm_f ).
thf(sy_c_FOR__Semantics_Oeval__rr2__rel_001tf__f_001tf__v,type,
fOR_eval_rr2_rel_f_v: set_Pr5245412377734449720_f_nat > list_s8746099396510718605rm_f_v > fOR_rr2_rel_FOR_ftrs > set_Pr989862937836626183term_f ).
thf(sy_c_FOR__Semantics_Ois__to__trs_001tf__f_001tf__v,type,
fOR_is_to_trs_f_v2: list_s8746099396510718605rm_f_v > list_FOR_ftrs > set_Pr8827868859434726151rm_f_v ).
thf(sy_c_FSet_Ofset_Ofset_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
fset_f1410452810862158076rm_f_v: fset_f2722810715853128435rm_f_v > set_fs7307227306443116653rm_f_v ).
thf(sy_c_FSet_Ofset_Ofset_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
fset_P4617584883882644886rm_f_v: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v ).
thf(sy_c_FSet_Ofset_Ofset_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
fset_P3576968334923099475_f_nat: fset_P6228066233360383026_f_nat > set_Pr5245412377734449720_f_nat ).
thf(sy_c_FSet_Ofset__of__list_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
fset_o5782745410740424019rm_f_v: list_f1824981274722084755rm_f_v > fset_f2722810715853128435rm_f_v ).
thf(sy_c_FSet_Ofset__of__list_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
fset_o4970844032613833069rm_f_v: list_P4093298276913796397rm_f_v > fset_P8018961893305114765rm_f_v ).
thf(sy_c_FSet_Ofset__of__list_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
fset_o8009517685352940092_f_nat: list_P3903862279629787026_f_nat > fset_P6228066233360383026_f_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__List__Olist_It__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J,type,
if_lis137525736101459289rm_f_v: $o > list_f1824981274722084755rm_f_v > list_f1824981274722084755rm_f_v > list_f1824981274722084755rm_f_v ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Option__Ooption_I_Eo_J,type,
if_option_o: $o > option_o > option_o > option_o ).
thf(sy_c_If_001t__Option__Ooption_It__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J_J,type,
if_opt1743852810818088729rm_f_v: $o > option6825207169704394579rm_f_v > option6825207169704394579rm_f_v > option6825207169704394579rm_f_v ).
thf(sy_c_If_001t__Option__Ooption_It__List__Olist_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_J,type,
if_opt7165310518185895824tion_f: $o > option2150321469529786326tion_f > option2150321469529786326tion_f > option2150321469529786326tion_f ).
thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > set_Pr958786334691620121nt_int ).
thf(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_LV__to__GTT_Offunas__trs_001tf__f_001tf__v,type,
lV_to_ffunas_trs_f_v: fset_P8018961893305114765rm_f_v > fset_P6228066233360383026_f_nat ).
thf(sy_c_LV__to__GTT_Olv__trs_001tf__f_001tf__v,type,
lV_to_lv_trs_f_v: set_Pr8827868859434726151rm_f_v > $o ).
thf(sy_c_List_Ocoset_001_Eo,type,
coset_o: list_o > set_o ).
thf(sy_c_List_Ocoset_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
coset_7382170490471766784rm_f_v: list_f1824981274722084755rm_f_v > set_fs7307227306443116653rm_f_v ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Ocoset_001t__Real__Oreal,type,
coset_real: list_real > set_real ).
thf(sy_c_List_Ocount__list_001_Eo,type,
count_list_o: list_o > $o > nat ).
thf(sy_c_List_Ocount__list_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
count_1704377189486916105rm_f_v: list_f1824981274722084755rm_f_v > fset_P8018961893305114765rm_f_v > nat ).
thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
count_list_nat: list_nat > nat > nat ).
thf(sy_c_List_Ocount__list_001t__Real__Oreal,type,
count_list_real: list_real > real > nat ).
thf(sy_c_List_Ocount__list_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
count_4977700719192441475rm_f_v: list_s8746099396510718605rm_f_v > set_Pr8827868859434726151rm_f_v > nat ).
thf(sy_c_List_Odistinct_001_Eo,type,
distinct_o: list_o > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
distin6923225563576452346at_nat: list_P6011104703257516679at_nat > $o ).
thf(sy_c_List_Odistinct_001t__Real__Oreal,type,
distinct_real: list_real > $o ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001_Eo_001_Eo,type,
linord6472470733373143810ey_o_o: ( $o > $o ) > $o > list_o > list_o ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001_Eo,type,
linord8610820856414126925_f_v_o: ( fset_P8018961893305114765rm_f_v > $o ) > fset_P8018961893305114765rm_f_v > list_f1824981274722084755rm_f_v > list_f1824981274722084755rm_f_v ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001t__Nat__Onat,type,
linord7013379258955989851_v_nat: ( fset_P8018961893305114765rm_f_v > nat ) > fset_P8018961893305114765rm_f_v > list_f1824981274722084755rm_f_v > list_f1824981274722084755rm_f_v ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001t__Real__Oreal,type,
linord405245991147262135v_real: ( fset_P8018961893305114765rm_f_v > real ) > fset_P8018961893305114765rm_f_v > list_f1824981274722084755rm_f_v > list_f1824981274722084755rm_f_v ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord1921536354676448932at_nat: ( nat > nat ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__Real__Oreal_001t__Real__Oreal,type,
linord1891625487229344476l_real: ( real > real ) > real > list_real > list_real ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001_Eo_001_Eo,type,
linord5141348845282165115ey_o_o: ( $o > $o ) > $o > list_o > list_o ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001_Eo,type,
linord9188204533028590406_f_v_o: ( fset_P8018961893305114765rm_f_v > $o ) > fset_P8018961893305114765rm_f_v > list_f1824981274722084755rm_f_v > list_f1824981274722084755rm_f_v ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001t__Nat__Onat,type,
linord45516334601593762_v_nat: ( fset_P8018961893305114765rm_f_v > nat ) > fset_P8018961893305114765rm_f_v > list_f1824981274722084755rm_f_v > list_f1824981274722084755rm_f_v ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001t__Real__Oreal,type,
linord3420383945269706110v_real: ( fset_P8018961893305114765rm_f_v > real ) > fset_P8018961893305114765rm_f_v > list_f1824981274722084755rm_f_v > list_f1824981274722084755rm_f_v ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord8961336180081300637at_nat: ( nat > nat ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Real__Oreal_001t__Real__Oreal,type,
linord1674302359176591317l_real: ( real > real ) > real > list_real > list_real ).
thf(sy_c_List_Olist_Omap_001_Eo_001_Eo,type,
map_o_o: ( $o > $o ) > list_o > list_o ).
thf(sy_c_List_Olist_Omap_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001_Eo,type,
map_fs1183804113021257231_f_v_o: ( fset_P8018961893305114765rm_f_v > $o ) > list_f1824981274722084755rm_f_v > list_o ).
thf(sy_c_List_Olist_Omap_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
map_fs3095252337778551172rm_f_v: ( fset_P8018961893305114765rm_f_v > fset_P8018961893305114765rm_f_v ) > list_f1824981274722084755rm_f_v > list_f1824981274722084755rm_f_v ).
thf(sy_c_List_Olist_Omap_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001t__Nat__Onat,type,
map_fs2439897442395436313_v_nat: ( fset_P8018961893305114765rm_f_v > nat ) > list_f1824981274722084755rm_f_v > list_nat ).
thf(sy_c_List_Olist_Omap_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001t__Real__Oreal,type,
map_fs3787659953648829301v_real: ( fset_P8018961893305114765rm_f_v > real ) > list_f1824981274722084755rm_f_v > list_real ).
thf(sy_c_List_Olist_Omap_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
map_fs8602507653405230974rm_f_v: ( fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v ) > list_f1824981274722084755rm_f_v > list_s8746099396510718605rm_f_v ).
thf(sy_c_List_Olist_Omap_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J,type,
map_fs7137784065334668863tion_f: ( fset_P8018961893305114765rm_f_v > tree_r733329426570293750tion_f ) > list_f1824981274722084755rm_f_v > list_T5268601877343193350tion_f ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J_001t__Ground____Terms__Ogterm_Itf__f_J,type,
map_Pr4088600317852009361term_f: ( produc7245736746747425831term_f > ground_gterm_f ) > list_P4363786793477243949term_f > list_Ground_gterm_f ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
map_Pr3938374229010428429at_nat: ( product_prod_nat_nat > nat ) > list_P6011104703257516679at_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Real__Oreal,type,
map_real_real: ( real > real ) > list_real > list_real ).
thf(sy_c_List_Olist_Oset_001_Eo,type,
set_o2: list_o > set_o ).
thf(sy_c_List_Olist_Oset_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
set_fs7270820277574336546rm_f_v: list_f1824981274722084755rm_f_v > set_fs7307227306443116653rm_f_v ).
thf(sy_c_List_Olist_Oset_001t__Ground____Terms__Ogterm_Itf__f_J,type,
set_Ground_gterm_f2: list_Ground_gterm_f > set_Ground_gterm_f ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J,type,
set_Pr1901606489578307004term_f: list_P4363786793477243949term_f > set_Pr989862937836626183term_f ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
set_Pr817814403484925884rm_f_v: list_P4093298276913796397rm_f_v > set_Pr8827868859434726151rm_f_v ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
set_Pr7102205019285007021_f_nat: list_P3903862279629787026_f_nat > set_Pr5245412377734449720_f_nat ).
thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
set_real2: list_real > set_real ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
set_se722422988665441564rm_f_v: list_s8746099396510718605rm_f_v > set_se4583834864486174823rm_f_v ).
thf(sy_c_List_Omap__tailrec_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
map_ta9055086340571775688rm_f_v: ( fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v ) > list_f1824981274722084755rm_f_v > list_s8746099396510718605rm_f_v ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Option_Ooption_ONone_001_Eo,type,
none_o: option_o ).
thf(sy_c_Option_Ooption_ONone_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J,type,
none_l3004429212090885614rm_f_v: option6825207169704394579rm_f_v ).
thf(sy_c_Option_Ooption_ONone_001t__List__Olist_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J,type,
none_l7671671938703071973tion_f: option2150321469529786326tion_f ).
thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
none_P3005240565499088977at_nat: option99490722083217462at_nat ).
thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
none_s1889411087592137448rm_f_v: option3296083141436081229rm_f_v ).
thf(sy_c_Option_Ooption_ONone_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J,type,
none_T5277256714714431317tion_f: option3789488934265196358tion_f ).
thf(sy_c_Option_Ooption_ONone_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J,type,
none_Tree_reg_nat_f: option5916524851906092002_nat_f ).
thf(sy_c_Option_Ooption_OSome_001_Eo,type,
some_o: $o > option_o ).
thf(sy_c_Option_Ooption_OSome_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
some_f4918923380007242610rm_f_v: fset_P8018961893305114765rm_f_v > option2233895434904908115rm_f_v ).
thf(sy_c_Option_Ooption_OSome_001t__FSet__Ofset_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
some_f8740016242142974861_f_nat: fset_P6228066233360383026_f_nat > option5345500251201882370_f_nat ).
thf(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
some_int: int > option_int ).
thf(sy_c_Option_Ooption_OSome_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J,type,
some_l530080162245194098rm_f_v: list_s8746099396510718605rm_f_v > option6825207169704394579rm_f_v ).
thf(sy_c_Option_Ooption_OSome_001t__List__Olist_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J,type,
some_l8507658222393836641tion_f: list_T5268601877343193350tion_f > option2150321469529786326tion_f ).
thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
some_nat: nat > option_nat ).
thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
some_P4370590140854848469at_nat: produc8199716216217303280at_nat > option99490722083217462at_nat ).
thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
thf(sy_c_Option_Ooption_OSome_001t__Real__Oreal,type,
some_real: real > option_real ).
thf(sy_c_Option_Ooption_OSome_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J,type,
some_T4055341017772447441tion_f: tree_r733329426570293750tion_f > option3789488934265196358tion_f ).
thf(sy_c_Option_Ooption_OSome_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J,type,
some_Tree_reg_nat_f: tree_reg_nat_f > option5916524851906092002_nat_f ).
thf(sy_c_Option_Ooption_Othe_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
the_se7542970557029289371rm_f_v: option3296083141436081229rm_f_v > set_Pr8827868859434726151rm_f_v ).
thf(sy_c_Option_Ooption_Othe_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J,type,
the_Tr3488611493235794018tion_f: option3789488934265196358tion_f > tree_r733329426570293750tion_f ).
thf(sy_c_Option_Ooption_Othe_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J,type,
the_Tree_reg_nat_f: option5916524851906092002_nat_f > tree_reg_nat_f ).
thf(sy_c_Option_Othese_001_Eo,type,
these_o: set_option_o > set_o ).
thf(sy_c_Option_Othese_001t__Nat__Onat,type,
these_nat: set_option_nat > set_nat ).
thf(sy_c_Option_Othese_001t__Real__Oreal,type,
these_real: set_option_real > set_real ).
thf(sy_c_Option_Othese_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J,type,
these_4693336772737898105tion_f: set_op6172961469967127676tion_f > set_Tr6476182622925392812tion_f ).
thf(sy_c_Option_Othese_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J,type,
these_Tree_reg_nat_f: set_op1790067033808154392_nat_f > set_Tree_reg_nat_f ).
thf(sy_c_Option__Monad_OmapM_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
option5007661016126702777rm_f_v: ( fset_P8018961893305114765rm_f_v > option3296083141436081229rm_f_v ) > list_f1824981274722084755rm_f_v > option6825207169704394579rm_f_v ).
thf(sy_c_Option__Monad_OmapM_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J,type,
option5852021067674027716tion_f: ( fset_P8018961893305114765rm_f_v > option3789488934265196358tion_f ) > list_f1824981274722084755rm_f_v > option2150321469529786326tion_f ).
thf(sy_c_Orderings_Oord__class_Oless_001_Eo,type,
ord_less_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__FSet__Ofset_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
ord_le7711977203798163590_f_nat: fset_P6228066233360383026_f_nat > fset_P6228066233360383026_f_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Int__Oint_J,type,
ord_less_option_int: option_int > option_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Nat__Onat_J,type,
ord_less_option_nat: option_nat > option_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Real__Oreal_J,type,
ord_less_option_real: option_real > option_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_Eo_J_J,type,
ord_le5974136086019949911_nat_o: ( ( nat > nat ) > nat > $o ) > ( ( nat > nat ) > nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_Eo_J,type,
ord_less_eq_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J,type,
ord_le7606603380504677011rm_f_v: fset_f2722810715853128435rm_f_v > fset_f2722810715853128435rm_f_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
ord_le4587745213494032429rm_f_v: fset_P8018961893305114765rm_f_v > fset_P8018961893305114765rm_f_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
ord_le1552505484586773650_f_nat: fset_P6228066233360383026_f_nat > fset_P6228066233360383026_f_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__FSet__Ofset_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J_J,type,
ord_le1231092944882968930_f_nat: option5345500251201882370_f_nat > option5345500251201882370_f_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Int__Oint_J,type,
ord_le1736525451366464988on_int: option_int > option_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Nat__Onat_J,type,
ord_le5914376470875661696on_nat: option_nat > option_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Real__Oreal_J,type,
ord_le8614940839814719452n_real: option_real > option_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J,type,
ord_le2293696477246793741rm_f_v: set_fs7307227306443116653rm_f_v > set_fs7307227306443116653rm_f_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Ground____Terms__Ogterm_Itf__f_J_J,type,
ord_le2735537439747282356term_f: set_Ground_gterm_f > set_Ground_gterm_f > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le3678578370064672496at_nat: set_Pr9093778441882193744at_nat > set_Pr9093778441882193744at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J_J,type,
ord_le263819222746101927term_f: set_Pr989862937836626183term_f > set_Pr989862937836626183term_f > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
ord_le4559761987009501863rm_f_v: set_Pr8827868859434726151rm_f_v > set_Pr8827868859434726151rm_f_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
ord_le8976984241387448984_f_nat: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__FSet__Ofset_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
order_5949861133752320921_f_nat: ( fset_P6228066233360383026_f_nat > $o ) > fset_P6228066233360383026_f_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Int__Oint,type,
order_Greatest_int: ( int > $o ) > int ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal,type,
order_Greatest_real: ( real > $o ) > real ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivides__aux_001t__Int__Oint,type,
unique5329631941980267465ux_int: product_prod_int_int > $o ).
thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivides__aux_001t__Nat__Onat,type,
unique5332122412489317741ux_nat: product_prod_nat_nat > $o ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc72220940542539688at_nat: ( nat > nat ) > nat > produc8199716216217303280at_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_M_062_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_Mt__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_J,type,
produc7984526239635384938tion_f: ( tree_r733329426570293750tion_f > tree_r733329426570293750tion_f > $o ) > produc441656135839998061tion_f > produc2070832938513523962tion_f ).
thf(sy_c_Product__Type_OPair_001_062_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_M_062_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_Mt__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J_J,type,
produc1791491351022178210_nat_f: ( tree_r733329426570293750tion_f > tree_reg_nat_f > $o ) > produc8927788648312868617_nat_f > produc7762418835577779250_nat_f ).
thf(sy_c_Product__Type_OPair_001_062_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_M_062_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J_Mt__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_J,type,
produc230842789923892130tion_f: ( tree_reg_nat_f > tree_r733329426570293750tion_f > $o ) > produc724344743186279177tion_f > produc3878888619416440882tion_f ).
thf(sy_c_Product__Type_OPair_001_062_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_M_062_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J_Mt__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J_J,type,
produc822877007188873434_nat_f: ( tree_reg_nat_f > tree_reg_nat_f > $o ) > produc5143877011543624101_nat_f > produc8574009693863894890_nat_f ).
thf(sy_c_Product__Type_OPair_001t__Ground____Terms__Ogterm_Itf__f_J_001t__Ground____Terms__Ogterm_Itf__f_J,type,
produc3560254623552331287term_f: ground_gterm_f > ground_gterm_f > produc7245736746747425831term_f ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
produc254078196866515351at_nat: option99490722083217462at_nat > option99490722083217462at_nat > produc2858277421346412199at_nat ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J,type,
produc6849691629296390053tion_f: option3789488934265196358tion_f > option3789488934265196358tion_f > produc441656135839998061tion_f ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J,type,
produc7608536704123013185_nat_f: option3789488934265196358tion_f > option5916524851906092002_nat_f > produc8927788648312868617_nat_f ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J,type,
produc751580840900675137tion_f: option5916524851906092002_nat_f > option3789488934265196358tion_f > produc724344743186279177tion_f ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J,type,
produc8844834930725908189_nat_f: option5916524851906092002_nat_f > option5916524851906092002_nat_f > produc5143877011543624101_nat_f ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc6156676138143019412at_nat: produc8199716216217303280at_nat > nat > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Ground____Terms__Ogterm_Itf__f_J_001t__Ground____Terms__Ogterm_Itf__f_J,type,
produc1239122367099457539term_f: produc7245736746747425831term_f > ground_gterm_f ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_RRn__Automata_ORR1__spec_001t__Nat__Onat_001tf__f,type,
rRn_RR1_spec_nat_f: tree_reg_nat_f > set_Ground_gterm_f > $o ).
thf(sy_c_RRn__Automata_ORR2__spec_001t__Nat__Onat_001tf__f_001tf__f,type,
rRn_RR2_spec_nat_f_f: tree_r733329426570293750tion_f > set_Pr989862937836626183term_f > $o ).
thf(sy_c_RRn__Automata_Oproj__1__reg_001t__Nat__Onat_001tf__f_001t__Option__Ooption_Itf__f_J,type,
rRn_pr8053562776630354921tion_f: tree_r733329426570293750tion_f > tree_reg_nat_f ).
thf(sy_c_Seq_Oinf__concat__simple,type,
inf_concat_simple: ( nat > nat ) > nat > product_prod_nat_nat ).
thf(sy_c_Set_OBall_001_Eo,type,
ball_o: set_o > ( $o > $o ) > $o ).
thf(sy_c_Set_OBall_001t__Nat__Onat,type,
ball_nat: set_nat > ( nat > $o ) > $o ).
thf(sy_c_Set_OBall_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
ball_P4917565930384053347nt_int: set_Pr958786334691620121nt_int > ( product_prod_int_int > $o ) > $o ).
thf(sy_c_Set_OBall_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
ball_P8096063237992195499at_nat: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > $o ) > $o ).
thf(sy_c_Set_OBall_001t__Real__Oreal,type,
ball_real: set_real > ( real > $o ) > $o ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J,type,
collec5662084080931014299tion_f: ( option3789488934265196358tion_f > $o ) > set_op6172961469967127676tion_f ).
thf(sy_c_Set_OCollect_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J,type,
collec6098388481013462839_nat_f: ( option5916524851906092002_nat_f > $o ) > set_op1790067033808154392_nat_f ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J,type,
collec3209569126566747026term_f: ( produc7245736746747425831term_f > $o ) > set_Pr989862937836626183term_f ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Option__Ooption_I_Eo_J,type,
image_o_option_o: ( $o > option_o ) > set_o > set_option_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Real__Oreal,type,
image_o_real: ( $o > real ) > set_o > set_real ).
thf(sy_c_Set_Oimage_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001_Eo,type,
image_7665057491163335760_f_v_o: ( fset_P8018961893305114765rm_f_v > $o ) > set_fs7307227306443116653rm_f_v > set_o ).
thf(sy_c_Set_Oimage_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
image_1909965058460517701rm_f_v: ( fset_P8018961893305114765rm_f_v > fset_P8018961893305114765rm_f_v ) > set_fs7307227306443116653rm_f_v > set_fs7307227306443116653rm_f_v ).
thf(sy_c_Set_Oimage_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001t__Nat__Onat,type,
image_3972722278727975192_v_nat: ( fset_P8018961893305114765rm_f_v > nat ) > set_fs7307227306443116653rm_f_v > set_nat ).
thf(sy_c_Set_Oimage_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001t__Real__Oreal,type,
image_7709105692213329908v_real: ( fset_P8018961893305114765rm_f_v > real ) > set_fs7307227306443116653rm_f_v > set_real ).
thf(sy_c_Set_Oimage_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
image_1525103038981075903rm_f_v: ( fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v ) > set_fs7307227306443116653rm_f_v > set_se4583834864486174823rm_f_v ).
thf(sy_c_Set_Oimage_001t__Ground____Terms__Ogterm_Itf__f_J_001t__Ground____Terms__Ogterm_Itf__f_J,type,
image_8650110728916761733term_f: ( ground_gterm_f > ground_gterm_f ) > set_Ground_gterm_f > set_Ground_gterm_f ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J,type,
image_7469968060601971523tion_f: ( option3789488934265196358tion_f > tree_r733329426570293750tion_f ) > set_op6172961469967127676tion_f > set_Tr6476182622925392812tion_f ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J_001t__Ground____Terms__Ogterm_Itf__f_J,type,
image_6328483948524962770term_f: ( produc7245736746747425831term_f > ground_gterm_f ) > set_Pr989862937836626183term_f > set_Ground_gterm_f ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J_001t__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J,type,
image_2537294313875852229term_f: ( produc7245736746747425831term_f > produc7245736746747425831term_f ) > set_Pr989862937836626183term_f > set_Pr989862937836626183term_f ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
image_2135063354759101220_int_o: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int > set_o ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Nat__Onat,type,
image_5044651549707136836nt_nat: ( product_prod_int_int > nat ) > set_Pr958786334691620121nt_int > set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Real__Oreal,type,
image_2737096548053899296t_real: ( product_prod_int_int > real ) > set_Pr958786334691620121nt_int > set_real ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo,type,
image_3693632289388996572_nat_o: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_2486076414777270412at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Real__Oreal,type,
image_2732955339748730216t_real: ( product_prod_nat_nat > real ) > set_Pr1261947904930325089at_nat > set_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001_Eo,type,
image_real_o: ( real > $o ) > set_real > set_o ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Nat__Onat,type,
image_real_nat: ( real > nat ) > set_real > set_nat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set_Oimage_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J,type,
image_8199543339563177155tion_f: ( tree_r733329426570293750tion_f > option3789488934265196358tion_f ) > set_Tr6476182622925392812tion_f > set_op6172961469967127676tion_f ).
thf(sy_c_Set_Oimage_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J,type,
image_868921236233539067_nat_f: ( tree_reg_nat_f > option5916524851906092002_nat_f ) > set_Tree_reg_nat_f > set_op1790067033808154392_nat_f ).
thf(sy_c_Tree__Automata_Oeps__free__reg_001t__Nat__Onat_001t__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J,type,
tree_e1535168946329719739tion_f: tree_r733329426570293750tion_f > tree_r733329426570293750tion_f ).
thf(sy_c_Tree__Automata_Oeps__free__reg_001t__Nat__Onat_001tf__f,type,
tree_e3620564068972927063_nat_f: tree_reg_nat_f > tree_reg_nat_f ).
thf(sy_c_Tree__Automata_Orelabel__reg_001t__Nat__Onat_001t__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J,type,
tree_r1168672525086783368tion_f: tree_r733329426570293750tion_f > tree_r733329426570293750tion_f ).
thf(sy_c_Tree__Automata_Orelabel__reg_001t__Nat__Onat_001tf__f,type,
tree_r7387757160296218660_nat_f: tree_reg_nat_f > tree_reg_nat_f ).
thf(sy_c_Tree__Automata_Otrim__reg_001t__Nat__Onat_001t__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J,type,
tree_t6100411961293590077tion_f: tree_r733329426570293750tion_f > tree_r733329426570293750tion_f ).
thf(sy_c_Tree__Automata_Otrim__reg_001t__Nat__Onat_001tf__f,type,
tree_trim_reg_nat_f: tree_reg_nat_f > tree_reg_nat_f ).
thf(sy_c_Utils_Ofunas__trs_001tf__f_001tf__v_001tf__v,type,
funas_trs_f_v_v: set_Pr8827868859434726151rm_f_v > set_Pr5245412377734449720_f_nat ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
member6790519936504491446rm_f_v: fset_P8018961893305114765rm_f_v > set_fs7307227306443116653rm_f_v > $o ).
thf(sy_c_member_001t__Ground____Terms__Ogterm_Itf__f_J,type,
member5261315044688711901term_f: ground_gterm_f > set_Ground_gterm_f > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Option__Ooption_I_Eo_J,type,
member_option_o: option_o > set_option_o > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Nat__Onat_J,type,
member_option_nat: option_nat > set_option_nat > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Real__Oreal_J,type,
member_option_real: option_real > set_option_real > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J,type,
member4921012686109714013tion_f: option3789488934265196358tion_f > set_op6172961469967127676tion_f > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J_J,type,
member1604975567284567801_nat_f: option5916524851906092002_nat_f > set_op1790067033808154392_nat_f > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member7226740684066999833at_nat: produc8199716216217303280at_nat > set_Pr9093778441882193744at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__f_J_Mt__Ground____Terms__Ogterm_Itf__f_J_J,type,
member848276444142703440term_f: produc7245736746747425831term_f > set_Pr989862937836626183term_f > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
member4041562125048333488rm_f_v: set_Pr8827868859434726151rm_f_v > set_se4583834864486174823rm_f_v > $o ).
thf(sy_c_member_001t__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J,type,
member6626405664465942413tion_f: tree_r733329426570293750tion_f > set_Tr6476182622925392812tion_f > $o ).
thf(sy_c_member_001t__Tree____Automata__Oreg_It__Nat__Onat_Mtf__f_J,type,
member2049775546588854825_nat_f: tree_reg_nat_f > set_Tree_reg_nat_f > $o ).
thf(sy_v_Rs,type,
rs: list_f1824981274722084755rm_f_v ).
thf(sy_v__092_060F_062,type,
f: fset_P6228066233360383026_f_nat ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_r____,type,
r: fOR_rr2_rel_FOR_ftrs ).
% Relevant facts (1263)
thf(fact_0_R1Proj,axiom,
! [Ta2: tree_r733329426570293750tion_f] :
( ( ( fOR_rr7226795042121552277el_f_v @ f @ rs @ r )
= ( some_T4055341017772447441tion_f @ Ta2 ) )
=> ( rRn_RR2_spec_nat_f_f @ Ta2 @ ( fOR_eval_rr2_rel_f_v @ ( fset_P3576968334923099475_f_nat @ f ) @ ( map_fs8602507653405230974rm_f_v @ fset_P4617584883882644886rm_f_v @ rs ) @ r ) ) ) ).
% R1Proj
thf(fact_1__092_060open_062RR2__spec_A_Ithe_A_Irr2__of__rr2__rel_A_092_060F_062_ARs_Ar_J_J_A_Ieval__rr2__rel_A_Ifset_A_092_060F_062_J_A_Imap_Afset_ARs_J_Ar_J_A_092_060Longrightarrow_062_ARR1__spec_A_Iproj__1__reg_A_Ithe_A_Irr2__of__rr2__rel_A_092_060F_062_ARs_Ar_J_J_J_A_Ifst_A_096_Aeval__rr2__rel_A_Ifset_A_092_060F_062_J_A_Imap_Afset_ARs_J_Ar_J_092_060close_062,axiom,
( ( rRn_RR2_spec_nat_f_f @ ( the_Tr3488611493235794018tion_f @ ( fOR_rr7226795042121552277el_f_v @ f @ rs @ r ) ) @ ( fOR_eval_rr2_rel_f_v @ ( fset_P3576968334923099475_f_nat @ f ) @ ( map_fs8602507653405230974rm_f_v @ fset_P4617584883882644886rm_f_v @ rs ) @ r ) )
=> ( rRn_RR1_spec_nat_f @ ( rRn_pr8053562776630354921tion_f @ ( the_Tr3488611493235794018tion_f @ ( fOR_rr7226795042121552277el_f_v @ f @ rs @ r ) ) ) @ ( image_6328483948524962770term_f @ produc1239122367099457539term_f @ ( fOR_eval_rr2_rel_f_v @ ( fset_P3576968334923099475_f_nat @ f ) @ ( map_fs8602507653405230974rm_f_v @ fset_P4617584883882644886rm_f_v @ rs ) @ r ) ) ) ) ).
% \<open>RR2_spec (the (rr2_of_rr2_rel \<F> Rs r)) (eval_rr2_rel (fset \<F>) (map fset Rs) r) \<Longrightarrow> RR1_spec (proj_1_reg (the (rr2_of_rr2_rel \<F> Rs r))) (fst ` eval_rr2_rel (fset \<F>) (map fset Rs) r)\<close>
thf(fact_2_RR1__spec__simplify__reg,axiom,
! [A: tree_reg_nat_f,R: set_Ground_gterm_f] :
( ( rRn_RR1_spec_nat_f @ ( fOR_si5663490455393327601_nat_f @ A ) @ R )
= ( rRn_RR1_spec_nat_f @ A @ R ) ) ).
% RR1_spec_simplify_reg
thf(fact_3_rr1__rel_Oinject_I3_J,axiom,
! [X41: nat,X42: fOR_rr2_rel_FOR_ftrs,Y41: nat,Y42: fOR_rr2_rel_FOR_ftrs] :
( ( ( fOR_rr305201390985280028R_ftrs @ X41 @ X42 )
= ( fOR_rr305201390985280028R_ftrs @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% rr1_rel.inject(3)
thf(fact_4_RR1__spec__eps__free__reg,axiom,
! [A: tree_reg_nat_f,R: set_Ground_gterm_f] :
( ( rRn_RR1_spec_nat_f @ ( tree_e3620564068972927063_nat_f @ A ) @ R )
= ( rRn_RR1_spec_nat_f @ A @ R ) ) ).
% RR1_spec_eps_free_reg
thf(fact_5_option_Oinject,axiom,
! [X2: tree_reg_nat_f,Y2: tree_reg_nat_f] :
( ( ( some_Tree_reg_nat_f @ X2 )
= ( some_Tree_reg_nat_f @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_6_option_Oinject,axiom,
! [X2: tree_r733329426570293750tion_f,Y2: tree_r733329426570293750tion_f] :
( ( ( some_T4055341017772447441tion_f @ X2 )
= ( some_T4055341017772447441tion_f @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_7__C0_C,axiom,
i = zero_zero_nat ).
% "0"
thf(fact_8_relabel__RR1__spec,axiom,
! [A2: tree_reg_nat_f,T: set_Ground_gterm_f] :
( ( rRn_RR1_spec_nat_f @ ( tree_r7387757160296218660_nat_f @ A2 ) @ T )
= ( rRn_RR1_spec_nat_f @ A2 @ T ) ) ).
% relabel_RR1_spec
thf(fact_9_fset__cong,axiom,
! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
( ( ( fset_P3576968334923099475_f_nat @ X )
= ( fset_P3576968334923099475_f_nat @ Y ) )
= ( X = Y ) ) ).
% fset_cong
thf(fact_10_fset__cong,axiom,
! [X: fset_P8018961893305114765rm_f_v,Y: fset_P8018961893305114765rm_f_v] :
( ( ( fset_P4617584883882644886rm_f_v @ X )
= ( fset_P4617584883882644886rm_f_v @ Y ) )
= ( X = Y ) ) ).
% fset_cong
thf(fact_11_liftO2__Some,axiom,
! [F: tree_reg_nat_f > tree_reg_nat_f > tree_reg_nat_f,X: option5916524851906092002_nat_f,Y: option5916524851906092002_nat_f,Z: tree_reg_nat_f] :
( ( ( fOR_li3434644546498476466_nat_f @ F @ X @ Y )
= ( some_Tree_reg_nat_f @ Z ) )
= ( ? [X3: tree_reg_nat_f,Y3: tree_reg_nat_f] :
( ( X
= ( some_Tree_reg_nat_f @ X3 ) )
& ( Y
= ( some_Tree_reg_nat_f @ Y3 ) ) )
& ( Z
= ( F @ ( the_Tree_reg_nat_f @ X ) @ ( the_Tree_reg_nat_f @ Y ) ) ) ) ) ).
% liftO2_Some
thf(fact_12_liftO2__Some,axiom,
! [F: tree_reg_nat_f > tree_r733329426570293750tion_f > tree_reg_nat_f,X: option5916524851906092002_nat_f,Y: option3789488934265196358tion_f,Z: tree_reg_nat_f] :
( ( ( fOR_li6346090120295541526_nat_f @ F @ X @ Y )
= ( some_Tree_reg_nat_f @ Z ) )
= ( ? [X3: tree_reg_nat_f,Y3: tree_r733329426570293750tion_f] :
( ( X
= ( some_Tree_reg_nat_f @ X3 ) )
& ( Y
= ( some_T4055341017772447441tion_f @ Y3 ) ) )
& ( Z
= ( F @ ( the_Tree_reg_nat_f @ X ) @ ( the_Tr3488611493235794018tion_f @ Y ) ) ) ) ) ).
% liftO2_Some
thf(fact_13_liftO2__Some,axiom,
! [F: tree_r733329426570293750tion_f > tree_reg_nat_f > tree_reg_nat_f,X: option3789488934265196358tion_f,Y: option5916524851906092002_nat_f,Z: tree_reg_nat_f] :
( ( ( fOR_li2310238201417421590_nat_f @ F @ X @ Y )
= ( some_Tree_reg_nat_f @ Z ) )
= ( ? [X3: tree_r733329426570293750tion_f,Y3: tree_reg_nat_f] :
( ( X
= ( some_T4055341017772447441tion_f @ X3 ) )
& ( Y
= ( some_Tree_reg_nat_f @ Y3 ) ) )
& ( Z
= ( F @ ( the_Tr3488611493235794018tion_f @ X ) @ ( the_Tree_reg_nat_f @ Y ) ) ) ) ) ).
% liftO2_Some
thf(fact_14_liftO2__Some,axiom,
! [F: tree_r733329426570293750tion_f > tree_r733329426570293750tion_f > tree_reg_nat_f,X: option3789488934265196358tion_f,Y: option3789488934265196358tion_f,Z: tree_reg_nat_f] :
( ( ( fOR_li3014203053008925818_nat_f @ F @ X @ Y )
= ( some_Tree_reg_nat_f @ Z ) )
= ( ? [X3: tree_r733329426570293750tion_f,Y3: tree_r733329426570293750tion_f] :
( ( X
= ( some_T4055341017772447441tion_f @ X3 ) )
& ( Y
= ( some_T4055341017772447441tion_f @ Y3 ) ) )
& ( Z
= ( F @ ( the_Tr3488611493235794018tion_f @ X ) @ ( the_Tr3488611493235794018tion_f @ Y ) ) ) ) ) ).
% liftO2_Some
thf(fact_15_liftO2__Some,axiom,
! [F: tree_reg_nat_f > tree_reg_nat_f > tree_r733329426570293750tion_f,X: option5916524851906092002_nat_f,Y: option5916524851906092002_nat_f,Z: tree_r733329426570293750tion_f] :
( ( ( fOR_li5638306150758482710tion_f @ F @ X @ Y )
= ( some_T4055341017772447441tion_f @ Z ) )
= ( ? [X3: tree_reg_nat_f,Y3: tree_reg_nat_f] :
( ( X
= ( some_Tree_reg_nat_f @ X3 ) )
& ( Y
= ( some_Tree_reg_nat_f @ Y3 ) ) )
& ( Z
= ( F @ ( the_Tree_reg_nat_f @ X ) @ ( the_Tree_reg_nat_f @ Y ) ) ) ) ) ).
% liftO2_Some
thf(fact_16_liftO2__Some,axiom,
! [F: tree_reg_nat_f > tree_r733329426570293750tion_f > tree_r733329426570293750tion_f,X: option5916524851906092002_nat_f,Y: option3789488934265196358tion_f,Z: tree_r733329426570293750tion_f] :
( ( ( fOR_li5786706929538884730tion_f @ F @ X @ Y )
= ( some_T4055341017772447441tion_f @ Z ) )
= ( ? [X3: tree_reg_nat_f,Y3: tree_r733329426570293750tion_f] :
( ( X
= ( some_Tree_reg_nat_f @ X3 ) )
& ( Y
= ( some_T4055341017772447441tion_f @ Y3 ) ) )
& ( Z
= ( F @ ( the_Tree_reg_nat_f @ X ) @ ( the_Tr3488611493235794018tion_f @ Y ) ) ) ) ) ).
% liftO2_Some
thf(fact_17_liftO2__Some,axiom,
! [F: tree_r733329426570293750tion_f > tree_reg_nat_f > tree_r733329426570293750tion_f,X: option3789488934265196358tion_f,Y: option5916524851906092002_nat_f,Z: tree_r733329426570293750tion_f] :
( ( ( fOR_li2306419083471867002tion_f @ F @ X @ Y )
= ( some_T4055341017772447441tion_f @ Z ) )
= ( ? [X3: tree_r733329426570293750tion_f,Y3: tree_reg_nat_f] :
( ( X
= ( some_T4055341017772447441tion_f @ X3 ) )
& ( Y
= ( some_Tree_reg_nat_f @ Y3 ) ) )
& ( Z
= ( F @ ( the_Tr3488611493235794018tion_f @ X ) @ ( the_Tree_reg_nat_f @ Y ) ) ) ) ) ).
% liftO2_Some
thf(fact_18_liftO2__Some,axiom,
! [F: tree_r733329426570293750tion_f > tree_r733329426570293750tion_f > tree_r733329426570293750tion_f,X: option3789488934265196358tion_f,Y: option3789488934265196358tion_f,Z: tree_r733329426570293750tion_f] :
( ( ( fOR_li3518057619300867550tion_f @ F @ X @ Y )
= ( some_T4055341017772447441tion_f @ Z ) )
= ( ? [X3: tree_r733329426570293750tion_f,Y3: tree_r733329426570293750tion_f] :
( ( X
= ( some_T4055341017772447441tion_f @ X3 ) )
& ( Y
= ( some_T4055341017772447441tion_f @ Y3 ) ) )
& ( Z
= ( F @ ( the_Tr3488611493235794018tion_f @ X ) @ ( the_Tr3488611493235794018tion_f @ Y ) ) ) ) ) ).
% liftO2_Some
thf(fact_19_assms,axiom,
! [X4: fset_P8018961893305114765rm_f_v] :
( ( member6790519936504491446rm_f_v @ X4 @ ( set_fs7270820277574336546rm_f_v @ rs ) )
=> ( ( lV_to_lv_trs_f_v @ ( fset_P4617584883882644886rm_f_v @ X4 ) )
& ( ord_le1552505484586773650_f_nat @ ( lV_to_ffunas_trs_f_v @ X4 ) @ f ) ) ) ).
% assms
thf(fact_20_liftO1__Some,axiom,
! [F: tree_reg_nat_f > tree_reg_nat_f,X: option5916524851906092002_nat_f,Y: tree_reg_nat_f] :
( ( ( fOR_li1500713569213856382_nat_f @ F @ X )
= ( some_Tree_reg_nat_f @ Y ) )
= ( ? [X3: tree_reg_nat_f] :
( X
= ( some_Tree_reg_nat_f @ X3 ) )
& ( Y
= ( F @ ( the_Tree_reg_nat_f @ X ) ) ) ) ) ).
% liftO1_Some
thf(fact_21_liftO1__Some,axiom,
! [F: tree_r733329426570293750tion_f > tree_reg_nat_f,X: option3789488934265196358tion_f,Y: tree_reg_nat_f] :
( ( ( fOR_li967653975282842594_nat_f @ F @ X )
= ( some_Tree_reg_nat_f @ Y ) )
= ( ? [X3: tree_r733329426570293750tion_f] :
( X
= ( some_T4055341017772447441tion_f @ X3 ) )
& ( Y
= ( F @ ( the_Tr3488611493235794018tion_f @ X ) ) ) ) ) ).
% liftO1_Some
thf(fact_22_liftO1__Some,axiom,
! [F: tree_reg_nat_f > tree_r733329426570293750tion_f,X: option5916524851906092002_nat_f,Y: tree_r733329426570293750tion_f] :
( ( ( fOR_li259870005745783778tion_f @ F @ X )
= ( some_T4055341017772447441tion_f @ Y ) )
= ( ? [X3: tree_reg_nat_f] :
( X
= ( some_Tree_reg_nat_f @ X3 ) )
& ( Y
= ( F @ ( the_Tree_reg_nat_f @ X ) ) ) ) ) ).
% liftO1_Some
thf(fact_23_liftO1__Some,axiom,
! [F: tree_r733329426570293750tion_f > tree_r733329426570293750tion_f,X: option3789488934265196358tion_f,Y: tree_r733329426570293750tion_f] :
( ( ( fOR_li2611159955866944838tion_f @ F @ X )
= ( some_T4055341017772447441tion_f @ Y ) )
= ( ? [X3: tree_r733329426570293750tion_f] :
( X
= ( some_T4055341017772447441tion_f @ X3 ) )
& ( Y
= ( F @ ( the_Tr3488611493235794018tion_f @ X ) ) ) ) ) ).
% liftO1_Some
thf(fact_24_fsubset__antisym,axiom,
! [A2: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ A2 @ B )
=> ( ( ord_le1552505484586773650_f_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% fsubset_antisym
thf(fact_25_RR2__spec__eps__free__reg,axiom,
! [A: tree_r733329426570293750tion_f,R: set_Pr989862937836626183term_f] :
( ( rRn_RR2_spec_nat_f_f @ ( tree_e1535168946329719739tion_f @ A ) @ R )
= ( rRn_RR2_spec_nat_f_f @ A @ R ) ) ).
% RR2_spec_eps_free_reg
thf(fact_26_map__eq__conv,axiom,
! [F: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v,Xs: list_f1824981274722084755rm_f_v,G: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v] :
( ( ( map_fs8602507653405230974rm_f_v @ F @ Xs )
= ( map_fs8602507653405230974rm_f_v @ G @ Xs ) )
= ( ! [X5: fset_P8018961893305114765rm_f_v] :
( ( member6790519936504491446rm_f_v @ X5 @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
=> ( ( F @ X5 )
= ( G @ X5 ) ) ) ) ) ).
% map_eq_conv
thf(fact_27_relabel__RR2__spec,axiom,
! [A2: tree_r733329426570293750tion_f,T: set_Pr989862937836626183term_f] :
( ( rRn_RR2_spec_nat_f_f @ ( tree_r1168672525086783368tion_f @ A2 ) @ T )
= ( rRn_RR2_spec_nat_f_f @ A2 @ T ) ) ).
% relabel_RR2_spec
thf(fact_28_RR2__spec__simplify__reg,axiom,
! [A: tree_r733329426570293750tion_f,R: set_Pr989862937836626183term_f] :
( ( rRn_RR2_spec_nat_f_f @ ( fOR_si5451137711280541013tion_f @ A ) @ R )
= ( rRn_RR2_spec_nat_f_f @ A @ R ) ) ).
% RR2_spec_simplify_reg
thf(fact_29_list_Oset__map,axiom,
! [F: produc7245736746747425831term_f > ground_gterm_f,V: list_P4363786793477243949term_f] :
( ( set_Ground_gterm_f2 @ ( map_Pr4088600317852009361term_f @ F @ V ) )
= ( image_6328483948524962770term_f @ F @ ( set_Pr1901606489578307004term_f @ V ) ) ) ).
% list.set_map
thf(fact_30_list_Oset__map,axiom,
! [F: nat > nat,V: list_nat] :
( ( set_nat2 @ ( map_nat_nat @ F @ V ) )
= ( image_nat_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_31_list_Oset__map,axiom,
! [F: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v,V: list_f1824981274722084755rm_f_v] :
( ( set_se722422988665441564rm_f_v @ ( map_fs8602507653405230974rm_f_v @ F @ V ) )
= ( image_1525103038981075903rm_f_v @ F @ ( set_fs7270820277574336546rm_f_v @ V ) ) ) ).
% list.set_map
thf(fact_32_list_Oset__map,axiom,
! [F: fset_P8018961893305114765rm_f_v > fset_P8018961893305114765rm_f_v,V: list_f1824981274722084755rm_f_v] :
( ( set_fs7270820277574336546rm_f_v @ ( map_fs3095252337778551172rm_f_v @ F @ V ) )
= ( image_1909965058460517701rm_f_v @ F @ ( set_fs7270820277574336546rm_f_v @ V ) ) ) ).
% list.set_map
thf(fact_33_fequalityE,axiom,
! [A2: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
( ( A2 = B )
=> ~ ( ( ord_le1552505484586773650_f_nat @ A2 @ B )
=> ~ ( ord_le1552505484586773650_f_nat @ B @ A2 ) ) ) ).
% fequalityE
thf(fact_34_fequalityD1,axiom,
! [A2: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
( ( A2 = B )
=> ( ord_le1552505484586773650_f_nat @ A2 @ B ) ) ).
% fequalityD1
thf(fact_35_fequalityD2,axiom,
! [A2: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat] :
( ( A2 = B )
=> ( ord_le1552505484586773650_f_nat @ B @ A2 ) ) ).
% fequalityD2
thf(fact_36_fsubset__refl,axiom,
! [A2: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ A2 @ A2 ) ).
% fsubset_refl
thf(fact_37_fsubset__trans,axiom,
! [A2: fset_P6228066233360383026_f_nat,B: fset_P6228066233360383026_f_nat,C: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ A2 @ B )
=> ( ( ord_le1552505484586773650_f_nat @ B @ C )
=> ( ord_le1552505484586773650_f_nat @ A2 @ C ) ) ) ).
% fsubset_trans
thf(fact_38_fset__eq__fsubset,axiom,
( ( ^ [Y4: fset_P6228066233360383026_f_nat,Z2: fset_P6228066233360383026_f_nat] : ( Y4 = Z2 ) )
= ( ^ [A3: fset_P6228066233360383026_f_nat,B2: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ A3 @ B2 )
& ( ord_le1552505484586773650_f_nat @ B2 @ A3 ) ) ) ) ).
% fset_eq_fsubset
thf(fact_39_image__set,axiom,
! [F: produc7245736746747425831term_f > ground_gterm_f,Xs: list_P4363786793477243949term_f] :
( ( image_6328483948524962770term_f @ F @ ( set_Pr1901606489578307004term_f @ Xs ) )
= ( set_Ground_gterm_f2 @ ( map_Pr4088600317852009361term_f @ F @ Xs ) ) ) ).
% image_set
thf(fact_40_image__set,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( image_nat_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( map_nat_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_41_image__set,axiom,
! [F: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v,Xs: list_f1824981274722084755rm_f_v] :
( ( image_1525103038981075903rm_f_v @ F @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
= ( set_se722422988665441564rm_f_v @ ( map_fs8602507653405230974rm_f_v @ F @ Xs ) ) ) ).
% image_set
thf(fact_42_image__set,axiom,
! [F: fset_P8018961893305114765rm_f_v > fset_P8018961893305114765rm_f_v,Xs: list_f1824981274722084755rm_f_v] :
( ( image_1909965058460517701rm_f_v @ F @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
= ( set_fs7270820277574336546rm_f_v @ ( map_fs3095252337778551172rm_f_v @ F @ Xs ) ) ) ).
% image_set
thf(fact_43_ex__map__conv,axiom,
! [Ys: list_s8746099396510718605rm_f_v,F: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v] :
( ( ? [Xs2: list_f1824981274722084755rm_f_v] :
( Ys
= ( map_fs8602507653405230974rm_f_v @ F @ Xs2 ) ) )
= ( ! [X5: set_Pr8827868859434726151rm_f_v] :
( ( member4041562125048333488rm_f_v @ X5 @ ( set_se722422988665441564rm_f_v @ Ys ) )
=> ? [Y5: fset_P8018961893305114765rm_f_v] :
( X5
= ( F @ Y5 ) ) ) ) ) ).
% ex_map_conv
thf(fact_44_map__cong,axiom,
! [Xs: list_f1824981274722084755rm_f_v,Ys: list_f1824981274722084755rm_f_v,F: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v,G: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v] :
( ( Xs = Ys )
=> ( ! [X6: fset_P8018961893305114765rm_f_v] :
( ( member6790519936504491446rm_f_v @ X6 @ ( set_fs7270820277574336546rm_f_v @ Ys ) )
=> ( ( F @ X6 )
= ( G @ X6 ) ) )
=> ( ( map_fs8602507653405230974rm_f_v @ F @ Xs )
= ( map_fs8602507653405230974rm_f_v @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_45_map__idI,axiom,
! [Xs: list_real,F: real > real] :
( ! [X6: real] :
( ( member_real @ X6 @ ( set_real2 @ Xs ) )
=> ( ( F @ X6 )
= X6 ) )
=> ( ( map_real_real @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_46_map__idI,axiom,
! [Xs: list_o,F: $o > $o] :
( ! [X6: $o] :
( ( member_o @ X6 @ ( set_o2 @ Xs ) )
=> ( ( F @ X6 )
= X6 ) )
=> ( ( map_o_o @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_47_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X6 )
= X6 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_48_map__idI,axiom,
! [Xs: list_f1824981274722084755rm_f_v,F: fset_P8018961893305114765rm_f_v > fset_P8018961893305114765rm_f_v] :
( ! [X6: fset_P8018961893305114765rm_f_v] :
( ( member6790519936504491446rm_f_v @ X6 @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
=> ( ( F @ X6 )
= X6 ) )
=> ( ( map_fs3095252337778551172rm_f_v @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_49_map__ext,axiom,
! [Xs: list_f1824981274722084755rm_f_v,F: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v,G: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v] :
( ! [X6: fset_P8018961893305114765rm_f_v] :
( ( member6790519936504491446rm_f_v @ X6 @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
=> ( ( F @ X6 )
= ( G @ X6 ) ) )
=> ( ( map_fs8602507653405230974rm_f_v @ F @ Xs )
= ( map_fs8602507653405230974rm_f_v @ G @ Xs ) ) ) ).
% map_ext
thf(fact_50_list_Omap__ident__strong,axiom,
! [T2: list_real,F: real > real] :
( ! [Z3: real] :
( ( member_real @ Z3 @ ( set_real2 @ T2 ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_real_real @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_51_list_Omap__ident__strong,axiom,
! [T2: list_o,F: $o > $o] :
( ! [Z3: $o] :
( ( member_o @ Z3 @ ( set_o2 @ T2 ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_o_o @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_52_list_Omap__ident__strong,axiom,
! [T2: list_nat,F: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ T2 ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_nat_nat @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_53_list_Omap__ident__strong,axiom,
! [T2: list_f1824981274722084755rm_f_v,F: fset_P8018961893305114765rm_f_v > fset_P8018961893305114765rm_f_v] :
( ! [Z3: fset_P8018961893305114765rm_f_v] :
( ( member6790519936504491446rm_f_v @ Z3 @ ( set_fs7270820277574336546rm_f_v @ T2 ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_fs3095252337778551172rm_f_v @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_54_list_Oinj__map__strong,axiom,
! [X: list_f1824981274722084755rm_f_v,Xa: list_f1824981274722084755rm_f_v,F: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v,Fa: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v] :
( ! [Z3: fset_P8018961893305114765rm_f_v,Za: fset_P8018961893305114765rm_f_v] :
( ( member6790519936504491446rm_f_v @ Z3 @ ( set_fs7270820277574336546rm_f_v @ X ) )
=> ( ( member6790519936504491446rm_f_v @ Za @ ( set_fs7270820277574336546rm_f_v @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_fs8602507653405230974rm_f_v @ F @ X )
= ( map_fs8602507653405230974rm_f_v @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_55_list_Omap__cong0,axiom,
! [X: list_f1824981274722084755rm_f_v,F: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v,G: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v] :
( ! [Z3: fset_P8018961893305114765rm_f_v] :
( ( member6790519936504491446rm_f_v @ Z3 @ ( set_fs7270820277574336546rm_f_v @ X ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_fs8602507653405230974rm_f_v @ F @ X )
= ( map_fs8602507653405230974rm_f_v @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_56_list_Omap__cong,axiom,
! [X: list_f1824981274722084755rm_f_v,Ya: list_f1824981274722084755rm_f_v,F: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v,G: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v] :
( ( X = Ya )
=> ( ! [Z3: fset_P8018961893305114765rm_f_v] :
( ( member6790519936504491446rm_f_v @ Z3 @ ( set_fs7270820277574336546rm_f_v @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_fs8602507653405230974rm_f_v @ F @ X )
= ( map_fs8602507653405230974rm_f_v @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_57_proj__1,axiom,
! [A2: tree_r733329426570293750tion_f,R: set_Pr989862937836626183term_f] :
( ( rRn_RR2_spec_nat_f_f @ A2 @ R )
=> ( rRn_RR1_spec_nat_f @ ( rRn_pr8053562776630354921tion_f @ A2 ) @ ( image_6328483948524962770term_f @ produc1239122367099457539term_f @ R ) ) ) ).
% proj_1
thf(fact_58_option_Osel,axiom,
! [X2: tree_reg_nat_f] :
( ( the_Tree_reg_nat_f @ ( some_Tree_reg_nat_f @ X2 ) )
= X2 ) ).
% option.sel
thf(fact_59_option_Osel,axiom,
! [X2: tree_r733329426570293750tion_f] :
( ( the_Tr3488611493235794018tion_f @ ( some_T4055341017772447441tion_f @ X2 ) )
= X2 ) ).
% option.sel
thf(fact_60_less__eq__option__Some,axiom,
! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
( ( ord_le1231092944882968930_f_nat @ ( some_f8740016242142974861_f_nat @ X ) @ ( some_f8740016242142974861_f_nat @ Y ) )
= ( ord_le1552505484586773650_f_nat @ X @ Y ) ) ).
% less_eq_option_Some
thf(fact_61_less__eq__option__Some,axiom,
! [X: nat,Y: nat] :
( ( ord_le5914376470875661696on_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% less_eq_option_Some
thf(fact_62_less__eq__option__Some,axiom,
! [X: int,Y: int] :
( ( ord_le1736525451366464988on_int @ ( some_int @ X ) @ ( some_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% less_eq_option_Some
thf(fact_63_less__eq__option__Some,axiom,
! [X: real,Y: real] :
( ( ord_le8614940839814719452n_real @ ( some_real @ X ) @ ( some_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% less_eq_option_Some
thf(fact_64_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_65_is__to__trs_H__props_I2_J,axiom,
! [Rs: list_f1824981274722084755rm_f_v,F2: fset_P6228066233360383026_f_nat,Is: list_FOR_ftrs,S: fset_P8018961893305114765rm_f_v] :
( ! [X6: fset_P8018961893305114765rm_f_v] :
( ( member6790519936504491446rm_f_v @ X6 @ ( set_fs7270820277574336546rm_f_v @ Rs ) )
=> ( ( lV_to_lv_trs_f_v @ ( fset_P4617584883882644886rm_f_v @ X6 ) )
& ( ord_le1552505484586773650_f_nat @ ( lV_to_ffunas_trs_f_v @ X6 ) @ F2 ) ) )
=> ( ( ( fOR_is_to_trs_f_v @ Rs @ Is )
= ( some_f4918923380007242610rm_f_v @ S ) )
=> ( lV_to_lv_trs_f_v @ ( fset_P4617584883882644886rm_f_v @ S ) ) ) ) ).
% is_to_trs'_props(2)
thf(fact_66_is__to__trs_H__props_I1_J,axiom,
! [Rs: list_f1824981274722084755rm_f_v,F2: fset_P6228066233360383026_f_nat,Is: list_FOR_ftrs,S: fset_P8018961893305114765rm_f_v] :
( ! [X6: fset_P8018961893305114765rm_f_v] :
( ( member6790519936504491446rm_f_v @ X6 @ ( set_fs7270820277574336546rm_f_v @ Rs ) )
=> ( ( lV_to_lv_trs_f_v @ ( fset_P4617584883882644886rm_f_v @ X6 ) )
& ( ord_le1552505484586773650_f_nat @ ( lV_to_ffunas_trs_f_v @ X6 ) @ F2 ) ) )
=> ( ( ( fOR_is_to_trs_f_v @ Rs @ Is )
= ( some_f4918923380007242610rm_f_v @ S ) )
=> ( ord_le1552505484586773650_f_nat @ ( lV_to_ffunas_trs_f_v @ S ) @ F2 ) ) ) ).
% is_to_trs'_props(1)
thf(fact_67_image__ident,axiom,
! [Y6: set_nat] :
( ( image_nat_nat
@ ^ [X5: nat] : X5
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_68_image__eqI,axiom,
! [B3: ground_gterm_f,F: produc7245736746747425831term_f > ground_gterm_f,X: produc7245736746747425831term_f,A2: set_Pr989862937836626183term_f] :
( ( B3
= ( F @ X ) )
=> ( ( member848276444142703440term_f @ X @ A2 )
=> ( member5261315044688711901term_f @ B3 @ ( image_6328483948524962770term_f @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_69_image__eqI,axiom,
! [B3: real,F: real > real,X: real,A2: set_real] :
( ( B3
= ( F @ X ) )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ B3 @ ( image_real_real @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_70_image__eqI,axiom,
! [B3: $o,F: real > $o,X: real,A2: set_real] :
( ( B3
= ( F @ X ) )
=> ( ( member_real @ X @ A2 )
=> ( member_o @ B3 @ ( image_real_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_71_image__eqI,axiom,
! [B3: nat,F: real > nat,X: real,A2: set_real] :
( ( B3
= ( F @ X ) )
=> ( ( member_real @ X @ A2 )
=> ( member_nat @ B3 @ ( image_real_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_72_image__eqI,axiom,
! [B3: real,F: $o > real,X: $o,A2: set_o] :
( ( B3
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member_real @ B3 @ ( image_o_real @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_73_image__eqI,axiom,
! [B3: $o,F: $o > $o,X: $o,A2: set_o] :
( ( B3
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member_o @ B3 @ ( image_o_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_74_image__eqI,axiom,
! [B3: nat,F: $o > nat,X: $o,A2: set_o] :
( ( B3
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member_nat @ B3 @ ( image_o_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_75_image__eqI,axiom,
! [B3: real,F: nat > real,X: nat,A2: set_nat] :
( ( B3
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_real @ B3 @ ( image_nat_real @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_76_image__eqI,axiom,
! [B3: $o,F: nat > $o,X: nat,A2: set_nat] :
( ( B3
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_o @ B3 @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_77_image__eqI,axiom,
! [B3: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B3
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B3 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_78_order__refl,axiom,
! [X: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ X @ X ) ).
% order_refl
thf(fact_79_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_80_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_81_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_82_mem__Collect__eq,axiom,
! [A4: real,P: real > $o] :
( ( member_real @ A4 @ ( collect_real @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_83_mem__Collect__eq,axiom,
! [A4: $o,P: $o > $o] :
( ( member_o @ A4 @ ( collect_o @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_84_mem__Collect__eq,axiom,
! [A4: nat,P: nat > $o] :
( ( member_nat @ A4 @ ( collect_nat @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_85_mem__Collect__eq,axiom,
! [A4: product_prod_int_int,P: product_prod_int_int > $o] :
( ( member5262025264175285858nt_int @ A4 @ ( collec213857154873943460nt_int @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_86_mem__Collect__eq,axiom,
! [A4: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A4 @ ( collec3392354462482085612at_nat @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_87_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X5: real] : ( member_real @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A2: set_o] :
( ( collect_o
@ ^ [X5: $o] : ( member_o @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_89_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X5: nat] : ( member_nat @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_90_Collect__mem__eq,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( collec213857154873943460nt_int
@ ^ [X5: product_prod_int_int] : ( member5262025264175285858nt_int @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_91_Collect__mem__eq,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_92_Collect__cong,axiom,
! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ! [X6: product_prod_int_int] :
( ( P @ X6 )
= ( Q @ X6 ) )
=> ( ( collec213857154873943460nt_int @ P )
= ( collec213857154873943460nt_int @ Q ) ) ) ).
% Collect_cong
thf(fact_93_Collect__cong,axiom,
! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ! [X6: product_prod_nat_nat] :
( ( P @ X6 )
= ( Q @ X6 ) )
=> ( ( collec3392354462482085612at_nat @ P )
= ( collec3392354462482085612at_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_94_dual__order_Orefl,axiom,
! [A4: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_95_dual__order_Orefl,axiom,
! [A4: nat] : ( ord_less_eq_nat @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_96_dual__order_Orefl,axiom,
! [A4: int] : ( ord_less_eq_int @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_97_dual__order_Orefl,axiom,
! [A4: real] : ( ord_less_eq_real @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_98_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_99_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_100_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_101_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_102_map__eq__map__tailrec,axiom,
map_fs8602507653405230974rm_f_v = map_ta9055086340571775688rm_f_v ).
% map_eq_map_tailrec
thf(fact_103_subset__image__iff,axiom,
! [B: set_Ground_gterm_f,F: produc7245736746747425831term_f > ground_gterm_f,A2: set_Pr989862937836626183term_f] :
( ( ord_le2735537439747282356term_f @ B @ ( image_6328483948524962770term_f @ F @ A2 ) )
= ( ? [AA: set_Pr989862937836626183term_f] :
( ( ord_le263819222746101927term_f @ AA @ A2 )
& ( B
= ( image_6328483948524962770term_f @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_104_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_105_image__subset__iff,axiom,
! [F: produc7245736746747425831term_f > ground_gterm_f,A2: set_Pr989862937836626183term_f,B: set_Ground_gterm_f] :
( ( ord_le2735537439747282356term_f @ ( image_6328483948524962770term_f @ F @ A2 ) @ B )
= ( ! [X5: produc7245736746747425831term_f] :
( ( member848276444142703440term_f @ X5 @ A2 )
=> ( member5261315044688711901term_f @ ( F @ X5 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_106_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( member_nat @ ( F @ X5 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_107_subset__imageE,axiom,
! [B: set_Ground_gterm_f,F: produc7245736746747425831term_f > ground_gterm_f,A2: set_Pr989862937836626183term_f] :
( ( ord_le2735537439747282356term_f @ B @ ( image_6328483948524962770term_f @ F @ A2 ) )
=> ~ ! [C2: set_Pr989862937836626183term_f] :
( ( ord_le263819222746101927term_f @ C2 @ A2 )
=> ( B
!= ( image_6328483948524962770term_f @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_108_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( B
!= ( image_nat_nat @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_109_image__subsetI,axiom,
! [A2: set_Pr989862937836626183term_f,F: produc7245736746747425831term_f > ground_gterm_f,B: set_Ground_gterm_f] :
( ! [X6: produc7245736746747425831term_f] :
( ( member848276444142703440term_f @ X6 @ A2 )
=> ( member5261315044688711901term_f @ ( F @ X6 ) @ B ) )
=> ( ord_le2735537439747282356term_f @ ( image_6328483948524962770term_f @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_110_image__subsetI,axiom,
! [A2: set_real,F: real > real,B: set_real] :
( ! [X6: real] :
( ( member_real @ X6 @ A2 )
=> ( member_real @ ( F @ X6 ) @ B ) )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_111_image__subsetI,axiom,
! [A2: set_real,F: real > $o,B: set_o] :
( ! [X6: real] :
( ( member_real @ X6 @ A2 )
=> ( member_o @ ( F @ X6 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_real_o @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_112_image__subsetI,axiom,
! [A2: set_real,F: real > nat,B: set_nat] :
( ! [X6: real] :
( ( member_real @ X6 @ A2 )
=> ( member_nat @ ( F @ X6 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_real_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_113_image__subsetI,axiom,
! [A2: set_o,F: $o > real,B: set_real] :
( ! [X6: $o] :
( ( member_o @ X6 @ A2 )
=> ( member_real @ ( F @ X6 ) @ B ) )
=> ( ord_less_eq_set_real @ ( image_o_real @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_114_image__subsetI,axiom,
! [A2: set_o,F: $o > $o,B: set_o] :
( ! [X6: $o] :
( ( member_o @ X6 @ A2 )
=> ( member_o @ ( F @ X6 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_115_image__subsetI,axiom,
! [A2: set_o,F: $o > nat,B: set_nat] :
( ! [X6: $o] :
( ( member_o @ X6 @ A2 )
=> ( member_nat @ ( F @ X6 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_116_image__subsetI,axiom,
! [A2: set_nat,F: nat > real,B: set_real] :
( ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( member_real @ ( F @ X6 ) @ B ) )
=> ( ord_less_eq_set_real @ ( image_nat_real @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_117_image__subsetI,axiom,
! [A2: set_nat,F: nat > $o,B: set_o] :
( ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( member_o @ ( F @ X6 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_118_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B: set_nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( member_nat @ ( F @ X6 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_119_image__mono,axiom,
! [A2: set_Pr989862937836626183term_f,B: set_Pr989862937836626183term_f,F: produc7245736746747425831term_f > ground_gterm_f] :
( ( ord_le263819222746101927term_f @ A2 @ B )
=> ( ord_le2735537439747282356term_f @ ( image_6328483948524962770term_f @ F @ A2 ) @ ( image_6328483948524962770term_f @ F @ B ) ) ) ).
% image_mono
thf(fact_120_image__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_121_subset__code_I1_J,axiom,
! [Xs: list_real,B: set_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B )
= ( ! [X5: real] :
( ( member_real @ X5 @ ( set_real2 @ Xs ) )
=> ( member_real @ X5 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_122_subset__code_I1_J,axiom,
! [Xs: list_o,B: set_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ B )
= ( ! [X5: $o] :
( ( member_o @ X5 @ ( set_o2 @ Xs ) )
=> ( member_o @ X5 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_123_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X5 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_124_subset__code_I1_J,axiom,
! [Xs: list_f1824981274722084755rm_f_v,B: set_fs7307227306443116653rm_f_v] :
( ( ord_le2293696477246793741rm_f_v @ ( set_fs7270820277574336546rm_f_v @ Xs ) @ B )
= ( ! [X5: fset_P8018961893305114765rm_f_v] :
( ( member6790519936504491446rm_f_v @ X5 @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
=> ( member6790519936504491446rm_f_v @ X5 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_125_less__eq__fset_Orep__eq,axiom,
( ord_le4587745213494032429rm_f_v
= ( ^ [X5: fset_P8018961893305114765rm_f_v,Xa2: fset_P8018961893305114765rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( fset_P4617584883882644886rm_f_v @ X5 ) @ ( fset_P4617584883882644886rm_f_v @ Xa2 ) ) ) ) ).
% less_eq_fset.rep_eq
thf(fact_126_less__eq__fset_Orep__eq,axiom,
( ord_le1552505484586773650_f_nat
= ( ^ [X5: fset_P6228066233360383026_f_nat,Xa2: fset_P6228066233360383026_f_nat] : ( ord_le8976984241387448984_f_nat @ ( fset_P3576968334923099475_f_nat @ X5 ) @ ( fset_P3576968334923099475_f_nat @ Xa2 ) ) ) ) ).
% less_eq_fset.rep_eq
thf(fact_127_order__antisym__conv,axiom,
! [Y: fset_P6228066233360383026_f_nat,X: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ Y @ X )
=> ( ( ord_le1552505484586773650_f_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_128_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_129_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_130_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_131_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_132_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_133_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_134_ord__le__eq__subst,axiom,
! [A4: nat,B3: nat,F: nat > nat,C3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_135_ord__le__eq__subst,axiom,
! [A4: nat,B3: nat,F: nat > int,C3: int] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_136_ord__le__eq__subst,axiom,
! [A4: nat,B3: nat,F: nat > real,C3: real] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_137_ord__le__eq__subst,axiom,
! [A4: int,B3: int,F: int > nat,C3: nat] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_138_ord__le__eq__subst,axiom,
! [A4: int,B3: int,F: int > int,C3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_139_ord__le__eq__subst,axiom,
! [A4: int,B3: int,F: int > real,C3: real] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_140_ord__le__eq__subst,axiom,
! [A4: real,B3: real,F: real > nat,C3: nat] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_141_ord__le__eq__subst,axiom,
! [A4: real,B3: real,F: real > int,C3: int] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_142_ord__le__eq__subst,axiom,
! [A4: real,B3: real,F: real > real,C3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_143_ord__le__eq__subst,axiom,
! [A4: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat,F: fset_P6228066233360383026_f_nat > nat,C3: nat] :
( ( ord_le1552505484586773650_f_nat @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: fset_P6228066233360383026_f_nat,Y7: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_144_ord__eq__le__subst,axiom,
! [A4: nat,F: nat > nat,B3: nat,C3: nat] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_145_ord__eq__le__subst,axiom,
! [A4: int,F: nat > int,B3: nat,C3: nat] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_146_ord__eq__le__subst,axiom,
! [A4: real,F: nat > real,B3: nat,C3: nat] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_147_ord__eq__le__subst,axiom,
! [A4: nat,F: int > nat,B3: int,C3: int] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_148_ord__eq__le__subst,axiom,
! [A4: int,F: int > int,B3: int,C3: int] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_149_ord__eq__le__subst,axiom,
! [A4: real,F: int > real,B3: int,C3: int] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_150_ord__eq__le__subst,axiom,
! [A4: nat,F: real > nat,B3: real,C3: real] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_151_ord__eq__le__subst,axiom,
! [A4: int,F: real > int,B3: real,C3: real] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_152_ord__eq__le__subst,axiom,
! [A4: real,F: real > real,B3: real,C3: real] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_153_ord__eq__le__subst,axiom,
! [A4: nat,F: fset_P6228066233360383026_f_nat > nat,B3: fset_P6228066233360383026_f_nat,C3: fset_P6228066233360383026_f_nat] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_le1552505484586773650_f_nat @ B3 @ C3 )
=> ( ! [X6: fset_P6228066233360383026_f_nat,Y7: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_154_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_155_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_156_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_157_order__eq__refl,axiom,
! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
( ( X = Y )
=> ( ord_le1552505484586773650_f_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_158_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_159_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_160_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_161_order__subst2,axiom,
! [A4: nat,B3: nat,F: nat > nat,C3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_162_order__subst2,axiom,
! [A4: nat,B3: nat,F: nat > int,C3: int] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_163_order__subst2,axiom,
! [A4: nat,B3: nat,F: nat > real,C3: real] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_164_order__subst2,axiom,
! [A4: int,B3: int,F: int > nat,C3: nat] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_165_order__subst2,axiom,
! [A4: int,B3: int,F: int > int,C3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_166_order__subst2,axiom,
! [A4: int,B3: int,F: int > real,C3: real] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_167_order__subst2,axiom,
! [A4: real,B3: real,F: real > nat,C3: nat] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_168_order__subst2,axiom,
! [A4: real,B3: real,F: real > int,C3: int] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_169_order__subst2,axiom,
! [A4: real,B3: real,F: real > real,C3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_170_order__subst2,axiom,
! [A4: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat,F: fset_P6228066233360383026_f_nat > nat,C3: nat] :
( ( ord_le1552505484586773650_f_nat @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C3 )
=> ( ! [X6: fset_P6228066233360383026_f_nat,Y7: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_171_order__subst1,axiom,
! [A4: nat,F: nat > nat,B3: nat,C3: nat] :
( ( ord_less_eq_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_172_order__subst1,axiom,
! [A4: nat,F: int > nat,B3: int,C3: int] :
( ( ord_less_eq_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_173_order__subst1,axiom,
! [A4: nat,F: real > nat,B3: real,C3: real] :
( ( ord_less_eq_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_174_order__subst1,axiom,
! [A4: int,F: nat > int,B3: nat,C3: nat] :
( ( ord_less_eq_int @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_175_order__subst1,axiom,
! [A4: int,F: int > int,B3: int,C3: int] :
( ( ord_less_eq_int @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_176_order__subst1,axiom,
! [A4: int,F: real > int,B3: real,C3: real] :
( ( ord_less_eq_int @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_177_order__subst1,axiom,
! [A4: real,F: nat > real,B3: nat,C3: nat] :
( ( ord_less_eq_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_178_order__subst1,axiom,
! [A4: real,F: int > real,B3: int,C3: int] :
( ( ord_less_eq_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_179_order__subst1,axiom,
! [A4: real,F: real > real,B3: real,C3: real] :
( ( ord_less_eq_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_180_order__subst1,axiom,
! [A4: fset_P6228066233360383026_f_nat,F: nat > fset_P6228066233360383026_f_nat,B3: nat,C3: nat] :
( ( ord_le1552505484586773650_f_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_le1552505484586773650_f_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_le1552505484586773650_f_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_181_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: fset_P6228066233360383026_f_nat,Z2: fset_P6228066233360383026_f_nat] : ( Y4 = Z2 ) )
= ( ^ [A5: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ A5 @ B4 )
& ( ord_le1552505484586773650_f_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_182_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_183_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_184_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_185_antisym,axiom,
! [A4: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ A4 @ B3 )
=> ( ( ord_le1552505484586773650_f_nat @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% antisym
thf(fact_186_antisym,axiom,
! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% antisym
thf(fact_187_antisym,axiom,
! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% antisym
thf(fact_188_antisym,axiom,
! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% antisym
thf(fact_189_dual__order_Otrans,axiom,
! [B3: fset_P6228066233360383026_f_nat,A4: fset_P6228066233360383026_f_nat,C3: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ B3 @ A4 )
=> ( ( ord_le1552505484586773650_f_nat @ C3 @ B3 )
=> ( ord_le1552505484586773650_f_nat @ C3 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_190_dual__order_Otrans,axiom,
! [B3: nat,A4: nat,C3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
=> ( ( ord_less_eq_nat @ C3 @ B3 )
=> ( ord_less_eq_nat @ C3 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_191_dual__order_Otrans,axiom,
! [B3: int,A4: int,C3: int] :
( ( ord_less_eq_int @ B3 @ A4 )
=> ( ( ord_less_eq_int @ C3 @ B3 )
=> ( ord_less_eq_int @ C3 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_192_dual__order_Otrans,axiom,
! [B3: real,A4: real,C3: real] :
( ( ord_less_eq_real @ B3 @ A4 )
=> ( ( ord_less_eq_real @ C3 @ B3 )
=> ( ord_less_eq_real @ C3 @ A4 ) ) ) ).
% dual_order.trans
thf(fact_193_dual__order_Oantisym,axiom,
! [B3: fset_P6228066233360383026_f_nat,A4: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ B3 @ A4 )
=> ( ( ord_le1552505484586773650_f_nat @ A4 @ B3 )
=> ( A4 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_194_dual__order_Oantisym,axiom,
! [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
=> ( ( ord_less_eq_nat @ A4 @ B3 )
=> ( A4 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_195_dual__order_Oantisym,axiom,
! [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
=> ( ( ord_less_eq_int @ A4 @ B3 )
=> ( A4 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_196_dual__order_Oantisym,axiom,
! [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
=> ( ( ord_less_eq_real @ A4 @ B3 )
=> ( A4 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_197_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: fset_P6228066233360383026_f_nat,Z2: fset_P6228066233360383026_f_nat] : ( Y4 = Z2 ) )
= ( ^ [A5: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ B4 @ A5 )
& ( ord_le1552505484586773650_f_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_198_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_199_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_200_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_201_linorder__wlog,axiom,
! [P: nat > nat > $o,A4: nat,B3: nat] :
( ! [A6: nat,B5: nat] :
( ( ord_less_eq_nat @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: nat,B5: nat] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A4 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_202_linorder__wlog,axiom,
! [P: int > int > $o,A4: int,B3: int] :
( ! [A6: int,B5: int] :
( ( ord_less_eq_int @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: int,B5: int] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A4 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_203_linorder__wlog,axiom,
! [P: real > real > $o,A4: real,B3: real] :
( ! [A6: real,B5: real] :
( ( ord_less_eq_real @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: real,B5: real] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A4 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_204_order__trans,axiom,
! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat,Z: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ X @ Y )
=> ( ( ord_le1552505484586773650_f_nat @ Y @ Z )
=> ( ord_le1552505484586773650_f_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_205_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_206_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_207_order__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X @ Z ) ) ) ).
% order_trans
thf(fact_208_order_Otrans,axiom,
! [A4: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat,C3: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ A4 @ B3 )
=> ( ( ord_le1552505484586773650_f_nat @ B3 @ C3 )
=> ( ord_le1552505484586773650_f_nat @ A4 @ C3 ) ) ) ).
% order.trans
thf(fact_209_order_Otrans,axiom,
! [A4: nat,B3: nat,C3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C3 )
=> ( ord_less_eq_nat @ A4 @ C3 ) ) ) ).
% order.trans
thf(fact_210_order_Otrans,axiom,
! [A4: int,B3: int,C3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C3 )
=> ( ord_less_eq_int @ A4 @ C3 ) ) ) ).
% order.trans
thf(fact_211_order_Otrans,axiom,
! [A4: real,B3: real,C3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( ord_less_eq_real @ A4 @ C3 ) ) ) ).
% order.trans
thf(fact_212_order__antisym,axiom,
! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ X @ Y )
=> ( ( ord_le1552505484586773650_f_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_213_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_214_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_215_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_216_ord__le__eq__trans,axiom,
! [A4: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat,C3: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ A4 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_le1552505484586773650_f_nat @ A4 @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_217_ord__le__eq__trans,axiom,
! [A4: nat,B3: nat,C3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_eq_nat @ A4 @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_218_ord__le__eq__trans,axiom,
! [A4: int,B3: int,C3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_eq_int @ A4 @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_219_ord__le__eq__trans,axiom,
! [A4: real,B3: real,C3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_eq_real @ A4 @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_220_ord__eq__le__trans,axiom,
! [A4: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat,C3: fset_P6228066233360383026_f_nat] :
( ( A4 = B3 )
=> ( ( ord_le1552505484586773650_f_nat @ B3 @ C3 )
=> ( ord_le1552505484586773650_f_nat @ A4 @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_221_ord__eq__le__trans,axiom,
! [A4: nat,B3: nat,C3: nat] :
( ( A4 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C3 )
=> ( ord_less_eq_nat @ A4 @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_222_ord__eq__le__trans,axiom,
! [A4: int,B3: int,C3: int] :
( ( A4 = B3 )
=> ( ( ord_less_eq_int @ B3 @ C3 )
=> ( ord_less_eq_int @ A4 @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_223_ord__eq__le__trans,axiom,
! [A4: real,B3: real,C3: real] :
( ( A4 = B3 )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( ord_less_eq_real @ A4 @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_224_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: fset_P6228066233360383026_f_nat,Z2: fset_P6228066233360383026_f_nat] : ( Y4 = Z2 ) )
= ( ^ [X5: fset_P6228066233360383026_f_nat,Y5: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ X5 @ Y5 )
& ( ord_le1552505484586773650_f_nat @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_225_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_226_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_227_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
& ( ord_less_eq_real @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_228_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_229_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_230_le__cases3,axiom,
! [X: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z ) )
=> ( ( ( ord_less_eq_real @ X @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X ) )
=> ~ ( ( ord_less_eq_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_231_nle__le,axiom,
! [A4: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A4 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A4 )
& ( B3 != A4 ) ) ) ).
% nle_le
thf(fact_232_nle__le,axiom,
! [A4: int,B3: int] :
( ( ~ ( ord_less_eq_int @ A4 @ B3 ) )
= ( ( ord_less_eq_int @ B3 @ A4 )
& ( B3 != A4 ) ) ) ).
% nle_le
thf(fact_233_nle__le,axiom,
! [A4: real,B3: real] :
( ( ~ ( ord_less_eq_real @ A4 @ B3 ) )
= ( ( ord_less_eq_real @ B3 @ A4 )
& ( B3 != A4 ) ) ) ).
% nle_le
thf(fact_234_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_235_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_236_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_237_rev__image__eqI,axiom,
! [X: produc7245736746747425831term_f,A2: set_Pr989862937836626183term_f,B3: ground_gterm_f,F: produc7245736746747425831term_f > ground_gterm_f] :
( ( member848276444142703440term_f @ X @ A2 )
=> ( ( B3
= ( F @ X ) )
=> ( member5261315044688711901term_f @ B3 @ ( image_6328483948524962770term_f @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_238_rev__image__eqI,axiom,
! [X: real,A2: set_real,B3: real,F: real > real] :
( ( member_real @ X @ A2 )
=> ( ( B3
= ( F @ X ) )
=> ( member_real @ B3 @ ( image_real_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_239_rev__image__eqI,axiom,
! [X: real,A2: set_real,B3: $o,F: real > $o] :
( ( member_real @ X @ A2 )
=> ( ( B3
= ( F @ X ) )
=> ( member_o @ B3 @ ( image_real_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_240_rev__image__eqI,axiom,
! [X: real,A2: set_real,B3: nat,F: real > nat] :
( ( member_real @ X @ A2 )
=> ( ( B3
= ( F @ X ) )
=> ( member_nat @ B3 @ ( image_real_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_241_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B3: real,F: $o > real] :
( ( member_o @ X @ A2 )
=> ( ( B3
= ( F @ X ) )
=> ( member_real @ B3 @ ( image_o_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_242_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B3: $o,F: $o > $o] :
( ( member_o @ X @ A2 )
=> ( ( B3
= ( F @ X ) )
=> ( member_o @ B3 @ ( image_o_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_243_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B3: nat,F: $o > nat] :
( ( member_o @ X @ A2 )
=> ( ( B3
= ( F @ X ) )
=> ( member_nat @ B3 @ ( image_o_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_244_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B3: real,F: nat > real] :
( ( member_nat @ X @ A2 )
=> ( ( B3
= ( F @ X ) )
=> ( member_real @ B3 @ ( image_nat_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_245_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B3: $o,F: nat > $o] :
( ( member_nat @ X @ A2 )
=> ( ( B3
= ( F @ X ) )
=> ( member_o @ B3 @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_246_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B3: nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B3
= ( F @ X ) )
=> ( member_nat @ B3 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_247_ball__imageD,axiom,
! [F: produc7245736746747425831term_f > ground_gterm_f,A2: set_Pr989862937836626183term_f,P: ground_gterm_f > $o] :
( ! [X6: ground_gterm_f] :
( ( member5261315044688711901term_f @ X6 @ ( image_6328483948524962770term_f @ F @ A2 ) )
=> ( P @ X6 ) )
=> ! [X4: produc7245736746747425831term_f] :
( ( member848276444142703440term_f @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_248_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X6: nat] :
( ( member_nat @ X6 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X6 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_249_image__cong,axiom,
! [M: set_Pr989862937836626183term_f,N2: set_Pr989862937836626183term_f,F: produc7245736746747425831term_f > ground_gterm_f,G: produc7245736746747425831term_f > ground_gterm_f] :
( ( M = N2 )
=> ( ! [X6: produc7245736746747425831term_f] :
( ( member848276444142703440term_f @ X6 @ N2 )
=> ( ( F @ X6 )
= ( G @ X6 ) ) )
=> ( ( image_6328483948524962770term_f @ F @ M )
= ( image_6328483948524962770term_f @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_250_image__cong,axiom,
! [M: set_nat,N2: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N2 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ N2 )
=> ( ( F @ X6 )
= ( G @ X6 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_251_bex__imageD,axiom,
! [F: produc7245736746747425831term_f > ground_gterm_f,A2: set_Pr989862937836626183term_f,P: ground_gterm_f > $o] :
( ? [X4: ground_gterm_f] :
( ( member5261315044688711901term_f @ X4 @ ( image_6328483948524962770term_f @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X6: produc7245736746747425831term_f] :
( ( member848276444142703440term_f @ X6 @ A2 )
& ( P @ ( F @ X6 ) ) ) ) ).
% bex_imageD
thf(fact_252_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A2 )
& ( P @ ( F @ X6 ) ) ) ) ).
% bex_imageD
thf(fact_253_image__iff,axiom,
! [Z: ground_gterm_f,F: produc7245736746747425831term_f > ground_gterm_f,A2: set_Pr989862937836626183term_f] :
( ( member5261315044688711901term_f @ Z @ ( image_6328483948524962770term_f @ F @ A2 ) )
= ( ? [X5: produc7245736746747425831term_f] :
( ( member848276444142703440term_f @ X5 @ A2 )
& ( Z
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_254_image__iff,axiom,
! [Z: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( Z
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_255_imageI,axiom,
! [X: produc7245736746747425831term_f,A2: set_Pr989862937836626183term_f,F: produc7245736746747425831term_f > ground_gterm_f] :
( ( member848276444142703440term_f @ X @ A2 )
=> ( member5261315044688711901term_f @ ( F @ X ) @ ( image_6328483948524962770term_f @ F @ A2 ) ) ) ).
% imageI
thf(fact_256_imageI,axiom,
! [X: real,A2: set_real,F: real > real] :
( ( member_real @ X @ A2 )
=> ( member_real @ ( F @ X ) @ ( image_real_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_257_imageI,axiom,
! [X: real,A2: set_real,F: real > $o] :
( ( member_real @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_real_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_258_imageI,axiom,
! [X: real,A2: set_real,F: real > nat] :
( ( member_real @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_real_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_259_imageI,axiom,
! [X: $o,A2: set_o,F: $o > real] :
( ( member_o @ X @ A2 )
=> ( member_real @ ( F @ X ) @ ( image_o_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_260_imageI,axiom,
! [X: $o,A2: set_o,F: $o > $o] :
( ( member_o @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_o_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_261_imageI,axiom,
! [X: $o,A2: set_o,F: $o > nat] :
( ( member_o @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_o_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_262_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > real] :
( ( member_nat @ X @ A2 )
=> ( member_real @ ( F @ X ) @ ( image_nat_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_263_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > $o] :
( ( member_nat @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_nat_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_264_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_265_imageE,axiom,
! [B3: ground_gterm_f,F: produc7245736746747425831term_f > ground_gterm_f,A2: set_Pr989862937836626183term_f] :
( ( member5261315044688711901term_f @ B3 @ ( image_6328483948524962770term_f @ F @ A2 ) )
=> ~ ! [X6: produc7245736746747425831term_f] :
( ( B3
= ( F @ X6 ) )
=> ~ ( member848276444142703440term_f @ X6 @ A2 ) ) ) ).
% imageE
thf(fact_266_imageE,axiom,
! [B3: real,F: real > real,A2: set_real] :
( ( member_real @ B3 @ ( image_real_real @ F @ A2 ) )
=> ~ ! [X6: real] :
( ( B3
= ( F @ X6 ) )
=> ~ ( member_real @ X6 @ A2 ) ) ) ).
% imageE
thf(fact_267_imageE,axiom,
! [B3: real,F: $o > real,A2: set_o] :
( ( member_real @ B3 @ ( image_o_real @ F @ A2 ) )
=> ~ ! [X6: $o] :
( ( B3
= ( F @ X6 ) )
=> ~ ( member_o @ X6 @ A2 ) ) ) ).
% imageE
thf(fact_268_imageE,axiom,
! [B3: real,F: nat > real,A2: set_nat] :
( ( member_real @ B3 @ ( image_nat_real @ F @ A2 ) )
=> ~ ! [X6: nat] :
( ( B3
= ( F @ X6 ) )
=> ~ ( member_nat @ X6 @ A2 ) ) ) ).
% imageE
thf(fact_269_imageE,axiom,
! [B3: $o,F: real > $o,A2: set_real] :
( ( member_o @ B3 @ ( image_real_o @ F @ A2 ) )
=> ~ ! [X6: real] :
( ( B3
= ( F @ X6 ) )
=> ~ ( member_real @ X6 @ A2 ) ) ) ).
% imageE
thf(fact_270_imageE,axiom,
! [B3: $o,F: $o > $o,A2: set_o] :
( ( member_o @ B3 @ ( image_o_o @ F @ A2 ) )
=> ~ ! [X6: $o] :
( ( B3
= ( F @ X6 ) )
=> ~ ( member_o @ X6 @ A2 ) ) ) ).
% imageE
thf(fact_271_imageE,axiom,
! [B3: $o,F: nat > $o,A2: set_nat] :
( ( member_o @ B3 @ ( image_nat_o @ F @ A2 ) )
=> ~ ! [X6: nat] :
( ( B3
= ( F @ X6 ) )
=> ~ ( member_nat @ X6 @ A2 ) ) ) ).
% imageE
thf(fact_272_imageE,axiom,
! [B3: nat,F: real > nat,A2: set_real] :
( ( member_nat @ B3 @ ( image_real_nat @ F @ A2 ) )
=> ~ ! [X6: real] :
( ( B3
= ( F @ X6 ) )
=> ~ ( member_real @ X6 @ A2 ) ) ) ).
% imageE
thf(fact_273_imageE,axiom,
! [B3: nat,F: $o > nat,A2: set_o] :
( ( member_nat @ B3 @ ( image_o_nat @ F @ A2 ) )
=> ~ ! [X6: $o] :
( ( B3
= ( F @ X6 ) )
=> ~ ( member_o @ X6 @ A2 ) ) ) ).
% imageE
thf(fact_274_imageE,axiom,
! [B3: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X6: nat] :
( ( B3
= ( F @ X6 ) )
=> ~ ( member_nat @ X6 @ A2 ) ) ) ).
% imageE
thf(fact_275_image__image,axiom,
! [F: ground_gterm_f > ground_gterm_f,G: produc7245736746747425831term_f > ground_gterm_f,A2: set_Pr989862937836626183term_f] :
( ( image_8650110728916761733term_f @ F @ ( image_6328483948524962770term_f @ G @ A2 ) )
= ( image_6328483948524962770term_f
@ ^ [X5: produc7245736746747425831term_f] : ( F @ ( G @ X5 ) )
@ A2 ) ) ).
% image_image
thf(fact_276_image__image,axiom,
! [F: produc7245736746747425831term_f > ground_gterm_f,G: produc7245736746747425831term_f > produc7245736746747425831term_f,A2: set_Pr989862937836626183term_f] :
( ( image_6328483948524962770term_f @ F @ ( image_2537294313875852229term_f @ G @ A2 ) )
= ( image_6328483948524962770term_f
@ ^ [X5: produc7245736746747425831term_f] : ( F @ ( G @ X5 ) )
@ A2 ) ) ).
% image_image
thf(fact_277_image__image,axiom,
! [F: nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X5: nat] : ( F @ ( G @ X5 ) )
@ A2 ) ) ).
% image_image
thf(fact_278_Compr__image__eq,axiom,
! [F: real > real,A2: set_real,P: real > $o] :
( ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ ( image_real_real @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_real_real @ F
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_279_Compr__image__eq,axiom,
! [F: $o > real,A2: set_o,P: real > $o] :
( ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ ( image_o_real @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_o_real @ F
@ ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_280_Compr__image__eq,axiom,
! [F: nat > real,A2: set_nat,P: real > $o] :
( ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ ( image_nat_real @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_nat_real @ F
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_281_Compr__image__eq,axiom,
! [F: real > $o,A2: set_real,P: $o > $o] :
( ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ ( image_real_o @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_real_o @ F
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_282_Compr__image__eq,axiom,
! [F: $o > $o,A2: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ ( image_o_o @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_283_Compr__image__eq,axiom,
! [F: nat > $o,A2: set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ ( image_nat_o @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_nat_o @ F
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_284_Compr__image__eq,axiom,
! [F: real > nat,A2: set_real,P: nat > $o] :
( ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ ( image_real_nat @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_real_nat @ F
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_285_Compr__image__eq,axiom,
! [F: $o > nat,A2: set_o,P: nat > $o] :
( ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ ( image_o_nat @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_o_nat @ F
@ ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_286_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_287_Compr__image__eq,axiom,
! [F: product_prod_int_int > real,A2: set_Pr958786334691620121nt_int,P: real > $o] :
( ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ ( image_2737096548053899296t_real @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_2737096548053899296t_real @ F
@ ( collec213857154873943460nt_int
@ ^ [X5: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_288_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_289_bot__nat__0_Oextremum,axiom,
! [A4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A4 ) ).
% bot_nat_0.extremum
thf(fact_290_is__to__trs__conv,axiom,
! [Rs: list_f1824981274722084755rm_f_v,Is: list_FOR_ftrs,S: fset_P8018961893305114765rm_f_v] :
( ( ( fOR_is_to_trs_f_v @ Rs @ Is )
= ( some_f4918923380007242610rm_f_v @ S ) )
=> ( ( fOR_is_to_trs_f_v2 @ ( map_fs8602507653405230974rm_f_v @ fset_P4617584883882644886rm_f_v @ Rs ) @ Is )
= ( fset_P4617584883882644886rm_f_v @ S ) ) ) ).
% is_to_trs_conv
thf(fact_291_image__Collect__subsetI,axiom,
! [P: produc7245736746747425831term_f > $o,F: produc7245736746747425831term_f > ground_gterm_f,B: set_Ground_gterm_f] :
( ! [X6: produc7245736746747425831term_f] :
( ( P @ X6 )
=> ( member5261315044688711901term_f @ ( F @ X6 ) @ B ) )
=> ( ord_le2735537439747282356term_f @ ( image_6328483948524962770term_f @ F @ ( collec3209569126566747026term_f @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_292_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B: set_nat] :
( ! [X6: nat] :
( ( P @ X6 )
=> ( member_nat @ ( F @ X6 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_293_image__Collect__subsetI,axiom,
! [P: product_prod_int_int > $o,F: product_prod_int_int > real,B: set_real] :
( ! [X6: product_prod_int_int] :
( ( P @ X6 )
=> ( member_real @ ( F @ X6 ) @ B ) )
=> ( ord_less_eq_set_real @ ( image_2737096548053899296t_real @ F @ ( collec213857154873943460nt_int @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_294_image__Collect__subsetI,axiom,
! [P: product_prod_int_int > $o,F: product_prod_int_int > $o,B: set_o] :
( ! [X6: product_prod_int_int] :
( ( P @ X6 )
=> ( member_o @ ( F @ X6 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_2135063354759101220_int_o @ F @ ( collec213857154873943460nt_int @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_295_image__Collect__subsetI,axiom,
! [P: product_prod_int_int > $o,F: product_prod_int_int > nat,B: set_nat] :
( ! [X6: product_prod_int_int] :
( ( P @ X6 )
=> ( member_nat @ ( F @ X6 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_5044651549707136836nt_nat @ F @ ( collec213857154873943460nt_int @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_296_image__Collect__subsetI,axiom,
! [P: product_prod_nat_nat > $o,F: product_prod_nat_nat > real,B: set_real] :
( ! [X6: product_prod_nat_nat] :
( ( P @ X6 )
=> ( member_real @ ( F @ X6 ) @ B ) )
=> ( ord_less_eq_set_real @ ( image_2732955339748730216t_real @ F @ ( collec3392354462482085612at_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_297_image__Collect__subsetI,axiom,
! [P: product_prod_nat_nat > $o,F: product_prod_nat_nat > $o,B: set_o] :
( ! [X6: product_prod_nat_nat] :
( ( P @ X6 )
=> ( member_o @ ( F @ X6 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_3693632289388996572_nat_o @ F @ ( collec3392354462482085612at_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_298_image__Collect__subsetI,axiom,
! [P: product_prod_nat_nat > $o,F: product_prod_nat_nat > nat,B: set_nat] :
( ! [X6: product_prod_nat_nat] :
( ( P @ X6 )
=> ( member_nat @ ( F @ X6 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ F @ ( collec3392354462482085612at_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_299_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A2: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X5: nat] : X5
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_300_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A2: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X5: nat] : X5
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_301_simplify__reg__def,axiom,
( fOR_si5663490455393327601_nat_f
= ( ^ [A7: tree_reg_nat_f] : ( tree_r7387757160296218660_nat_f @ ( tree_trim_reg_nat_f @ A7 ) ) ) ) ).
% simplify_reg_def
thf(fact_302_count__notin,axiom,
! [X: real,Xs: list_real] :
( ~ ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( ( count_list_real @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_303_count__notin,axiom,
! [X: $o,Xs: list_o] :
( ~ ( member_o @ X @ ( set_o2 @ Xs ) )
=> ( ( count_list_o @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_304_count__notin,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( count_list_nat @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_305_count__notin,axiom,
! [X: fset_P8018961893305114765rm_f_v,Xs: list_f1824981274722084755rm_f_v] :
( ~ ( member6790519936504491446rm_f_v @ X @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
=> ( ( count_1704377189486916105rm_f_v @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_306_insort__insert__key__triv,axiom,
! [F: nat > nat,X: nat,Xs: list_nat] :
( ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ ( set_nat2 @ Xs ) ) )
=> ( ( linord1921536354676448932at_nat @ F @ X @ Xs )
= Xs ) ) ).
% insort_insert_key_triv
thf(fact_307_insort__insert__key__triv,axiom,
! [F: fset_P8018961893305114765rm_f_v > real,X: fset_P8018961893305114765rm_f_v,Xs: list_f1824981274722084755rm_f_v] :
( ( member_real @ ( F @ X ) @ ( image_7709105692213329908v_real @ F @ ( set_fs7270820277574336546rm_f_v @ Xs ) ) )
=> ( ( linord405245991147262135v_real @ F @ X @ Xs )
= Xs ) ) ).
% insort_insert_key_triv
thf(fact_308_insort__insert__key__triv,axiom,
! [F: fset_P8018961893305114765rm_f_v > $o,X: fset_P8018961893305114765rm_f_v,Xs: list_f1824981274722084755rm_f_v] :
( ( member_o @ ( F @ X ) @ ( image_7665057491163335760_f_v_o @ F @ ( set_fs7270820277574336546rm_f_v @ Xs ) ) )
=> ( ( linord8610820856414126925_f_v_o @ F @ X @ Xs )
= Xs ) ) ).
% insort_insert_key_triv
thf(fact_309_insort__insert__key__triv,axiom,
! [F: fset_P8018961893305114765rm_f_v > nat,X: fset_P8018961893305114765rm_f_v,Xs: list_f1824981274722084755rm_f_v] :
( ( member_nat @ ( F @ X ) @ ( image_3972722278727975192_v_nat @ F @ ( set_fs7270820277574336546rm_f_v @ Xs ) ) )
=> ( ( linord7013379258955989851_v_nat @ F @ X @ Xs )
= Xs ) ) ).
% insort_insert_key_triv
thf(fact_310_trim__RR1__spec,axiom,
! [A2: tree_reg_nat_f,T: set_Ground_gterm_f] :
( ( rRn_RR1_spec_nat_f @ ( tree_trim_reg_nat_f @ A2 ) @ T )
= ( rRn_RR1_spec_nat_f @ A2 @ T ) ) ).
% trim_RR1_spec
thf(fact_311_subsetI,axiom,
! [A2: set_real,B: set_real] :
( ! [X6: real] :
( ( member_real @ X6 @ A2 )
=> ( member_real @ X6 @ B ) )
=> ( ord_less_eq_set_real @ A2 @ B ) ) ).
% subsetI
thf(fact_312_subsetI,axiom,
! [A2: set_o,B: set_o] :
( ! [X6: $o] :
( ( member_o @ X6 @ A2 )
=> ( member_o @ X6 @ B ) )
=> ( ord_less_eq_set_o @ A2 @ B ) ) ).
% subsetI
thf(fact_313_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( member_nat @ X6 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_314_trim__RR2__spec,axiom,
! [A2: tree_r733329426570293750tion_f,T: set_Pr989862937836626183term_f] :
( ( rRn_RR2_spec_nat_f_f @ ( tree_t6100411961293590077tion_f @ A2 ) @ T )
= ( rRn_RR2_spec_nat_f_f @ A2 @ T ) ) ).
% trim_RR2_spec
thf(fact_315_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_316_in__mono,axiom,
! [A2: set_real,B: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ X @ B ) ) ) ).
% in_mono
thf(fact_317_in__mono,axiom,
! [A2: set_o,B: set_o,X: $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ( member_o @ X @ A2 )
=> ( member_o @ X @ B ) ) ) ).
% in_mono
thf(fact_318_in__mono,axiom,
! [A2: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_319_subsetD,axiom,
! [A2: set_real,B: set_real,C3: real] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( member_real @ C3 @ A2 )
=> ( member_real @ C3 @ B ) ) ) ).
% subsetD
thf(fact_320_subsetD,axiom,
! [A2: set_o,B: set_o,C3: $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ( member_o @ C3 @ A2 )
=> ( member_o @ C3 @ B ) ) ) ).
% subsetD
thf(fact_321_subsetD,axiom,
! [A2: set_nat,B: set_nat,C3: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ C3 @ A2 )
=> ( member_nat @ C3 @ B ) ) ) ).
% subsetD
thf(fact_322_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_323_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_324_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B2: set_real] :
! [X5: real] :
( ( member_real @ X5 @ A3 )
=> ( member_real @ X5 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_325_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B2: set_o] :
! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( member_o @ X5 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_326_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [X5: nat] :
( ( member_nat @ X5 @ A3 )
=> ( member_nat @ X5 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_327_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_328_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B2: set_real] :
! [T3: real] :
( ( member_real @ T3 @ A3 )
=> ( member_real @ T3 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_329_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B2: set_o] :
! [T3: $o] :
( ( member_o @ T3 @ A3 )
=> ( member_o @ T3 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_330_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A3 )
=> ( member_nat @ T3 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_331_Collect__mono,axiom,
! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ! [X6: product_prod_int_int] :
( ( P @ X6 )
=> ( Q @ X6 ) )
=> ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).
% Collect_mono
thf(fact_332_Collect__mono,axiom,
! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ! [X6: product_prod_nat_nat] :
( ( P @ X6 )
=> ( Q @ X6 ) )
=> ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_333_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_334_Collect__subset,axiom,
! [A2: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A2 )
& ( P @ X5 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_335_Collect__subset,axiom,
! [A2: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( P @ X5 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_336_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( P @ X5 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_337_Collect__subset,axiom,
! [A2: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
( ord_le2843351958646193337nt_int
@ ( collec213857154873943460nt_int
@ ^ [X5: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X5 @ A2 )
& ( P @ X5 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_338_Collect__subset,axiom,
! [A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
( ord_le3146513528884898305at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A2 )
& ( P @ X5 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_339_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B2: set_real] :
( ord_less_eq_real_o
@ ^ [X5: real] : ( member_real @ X5 @ A3 )
@ ^ [X5: real] : ( member_real @ X5 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_340_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B2: set_o] :
( ord_less_eq_o_o
@ ^ [X5: $o] : ( member_o @ X5 @ A3 )
@ ^ [X5: $o] : ( member_o @ X5 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_341_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ord_less_eq_nat_o
@ ^ [X5: nat] : ( member_nat @ X5 @ A3 )
@ ^ [X5: nat] : ( member_nat @ X5 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_342_Collect__mono__iff,axiom,
! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) )
= ( ! [X5: product_prod_int_int] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_343_Collect__mono__iff,axiom,
! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) )
= ( ! [X5: product_prod_nat_nat] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_344_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ B3 ) )
=> ? [X6: nat] :
( ( P @ X6 )
& ! [Y8: nat] :
( ( P @ Y8 )
=> ( ord_less_eq_nat @ Y8 @ X6 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_345_prop__restrict,axiom,
! [X: real,Z4: set_real,X7: set_real,P: real > $o] :
( ( member_real @ X @ Z4 )
=> ( ( ord_less_eq_set_real @ Z4
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ X7 )
& ( P @ X5 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_346_prop__restrict,axiom,
! [X: $o,Z4: set_o,X7: set_o,P: $o > $o] :
( ( member_o @ X @ Z4 )
=> ( ( ord_less_eq_set_o @ Z4
@ ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ X7 )
& ( P @ X5 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_347_prop__restrict,axiom,
! [X: nat,Z4: set_nat,X7: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ X7 )
& ( P @ X5 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_348_prop__restrict,axiom,
! [X: product_prod_int_int,Z4: set_Pr958786334691620121nt_int,X7: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
( ( member5262025264175285858nt_int @ X @ Z4 )
=> ( ( ord_le2843351958646193337nt_int @ Z4
@ ( collec213857154873943460nt_int
@ ^ [X5: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X5 @ X7 )
& ( P @ X5 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_349_prop__restrict,axiom,
! [X: product_prod_nat_nat,Z4: set_Pr1261947904930325089at_nat,X7: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ X @ Z4 )
=> ( ( ord_le3146513528884898305at_nat @ Z4
@ ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ X7 )
& ( P @ X5 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_350_Collect__restrict,axiom,
! [X7: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ X7 )
& ( P @ X5 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_351_Collect__restrict,axiom,
! [X7: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ X7 )
& ( P @ X5 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_352_Collect__restrict,axiom,
! [X7: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ X7 )
& ( P @ X5 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_353_Collect__restrict,axiom,
! [X7: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
( ord_le2843351958646193337nt_int
@ ( collec213857154873943460nt_int
@ ^ [X5: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X5 @ X7 )
& ( P @ X5 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_354_Collect__restrict,axiom,
! [X7: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
( ord_le3146513528884898305at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ X7 )
& ( P @ X5 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_355_count__list__map__ge,axiom,
! [Xs: list_f1824981274722084755rm_f_v,X: fset_P8018961893305114765rm_f_v,F: fset_P8018961893305114765rm_f_v > set_Pr8827868859434726151rm_f_v] : ( ord_less_eq_nat @ ( count_1704377189486916105rm_f_v @ Xs @ X ) @ ( count_4977700719192441475rm_f_v @ ( map_fs8602507653405230974rm_f_v @ F @ Xs ) @ ( F @ X ) ) ) ).
% count_list_map_ge
thf(fact_356_insort__insert__triv,axiom,
! [X: real,Xs: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( ( linord1891625487229344476l_real
@ ^ [X5: real] : X5
@ X
@ Xs )
= Xs ) ) ).
% insort_insert_triv
thf(fact_357_insort__insert__triv,axiom,
! [X: $o,Xs: list_o] :
( ( member_o @ X @ ( set_o2 @ Xs ) )
=> ( ( linord6472470733373143810ey_o_o
@ ^ [X5: $o] : X5
@ X
@ Xs )
= Xs ) ) ).
% insort_insert_triv
thf(fact_358_insort__insert__triv,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( linord1921536354676448932at_nat
@ ^ [X5: nat] : X5
@ X
@ Xs )
= Xs ) ) ).
% insort_insert_triv
thf(fact_359_count__list__0__iff,axiom,
! [Xs: list_real,X: real] :
( ( ( count_list_real @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_real @ X @ ( set_real2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_360_count__list__0__iff,axiom,
! [Xs: list_o,X: $o] :
( ( ( count_list_o @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_o @ X @ ( set_o2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_361_count__list__0__iff,axiom,
! [Xs: list_nat,X: nat] :
( ( ( count_list_nat @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_362_count__list__0__iff,axiom,
! [Xs: list_f1824981274722084755rm_f_v,X: fset_P8018961893305114765rm_f_v] :
( ( ( count_1704377189486916105rm_f_v @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member6790519936504491446rm_f_v @ X @ ( set_fs7270820277574336546rm_f_v @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_363_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_364_bot__nat__0_Oextremum__unique,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
= ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_365_bot__nat__0_Oextremum__uniqueI,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
=> ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_366_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_367_Inf_OINF__cong,axiom,
! [A2: set_Pr989862937836626183term_f,B: set_Pr989862937836626183term_f,C: produc7245736746747425831term_f > ground_gterm_f,D: produc7245736746747425831term_f > ground_gterm_f,Inf: set_Ground_gterm_f > ground_gterm_f] :
( ( A2 = B )
=> ( ! [X6: produc7245736746747425831term_f] :
( ( member848276444142703440term_f @ X6 @ B )
=> ( ( C @ X6 )
= ( D @ X6 ) ) )
=> ( ( Inf @ ( image_6328483948524962770term_f @ C @ A2 ) )
= ( Inf @ ( image_6328483948524962770term_f @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_368_Inf_OINF__cong,axiom,
! [A2: set_nat,B: set_nat,C: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A2 = B )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ B )
=> ( ( C @ X6 )
= ( D @ X6 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C @ A2 ) )
= ( Inf @ ( image_nat_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_369_Sup_OSUP__cong,axiom,
! [A2: set_Pr989862937836626183term_f,B: set_Pr989862937836626183term_f,C: produc7245736746747425831term_f > ground_gterm_f,D: produc7245736746747425831term_f > ground_gterm_f,Sup: set_Ground_gterm_f > ground_gterm_f] :
( ( A2 = B )
=> ( ! [X6: produc7245736746747425831term_f] :
( ( member848276444142703440term_f @ X6 @ B )
=> ( ( C @ X6 )
= ( D @ X6 ) ) )
=> ( ( Sup @ ( image_6328483948524962770term_f @ C @ A2 ) )
= ( Sup @ ( image_6328483948524962770term_f @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_370_Sup_OSUP__cong,axiom,
! [A2: set_nat,B: set_nat,C: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A2 = B )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ B )
=> ( ( C @ X6 )
= ( D @ X6 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C @ A2 ) )
= ( Sup @ ( image_nat_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_371_all__subset__image,axiom,
! [F: produc7245736746747425831term_f > ground_gterm_f,A2: set_Pr989862937836626183term_f,P: set_Ground_gterm_f > $o] :
( ( ! [B2: set_Ground_gterm_f] :
( ( ord_le2735537439747282356term_f @ B2 @ ( image_6328483948524962770term_f @ F @ A2 ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_Pr989862937836626183term_f] :
( ( ord_le263819222746101927term_f @ B2 @ A2 )
=> ( P @ ( image_6328483948524962770term_f @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_372_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_373_pred__subset__eq,axiom,
! [R: set_real,S: set_real] :
( ( ord_less_eq_real_o
@ ^ [X5: real] : ( member_real @ X5 @ R )
@ ^ [X5: real] : ( member_real @ X5 @ S ) )
= ( ord_less_eq_set_real @ R @ S ) ) ).
% pred_subset_eq
thf(fact_374_pred__subset__eq,axiom,
! [R: set_o,S: set_o] :
( ( ord_less_eq_o_o
@ ^ [X5: $o] : ( member_o @ X5 @ R )
@ ^ [X5: $o] : ( member_o @ X5 @ S ) )
= ( ord_less_eq_set_o @ R @ S ) ) ).
% pred_subset_eq
thf(fact_375_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X5: nat] : ( member_nat @ X5 @ R )
@ ^ [X5: nat] : ( member_nat @ X5 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_376_conj__subset__def,axiom,
! [A2: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ( ord_le2843351958646193337nt_int @ A2
@ ( collec213857154873943460nt_int
@ ^ [X5: product_prod_int_int] :
( ( P @ X5 )
& ( Q @ X5 ) ) ) )
= ( ( ord_le2843351958646193337nt_int @ A2 @ ( collec213857154873943460nt_int @ P ) )
& ( ord_le2843351958646193337nt_int @ A2 @ ( collec213857154873943460nt_int @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_377_conj__subset__def,axiom,
! [A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ A2
@ ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] :
( ( P @ X5 )
& ( Q @ X5 ) ) ) )
= ( ( ord_le3146513528884898305at_nat @ A2 @ ( collec3392354462482085612at_nat @ P ) )
& ( ord_le3146513528884898305at_nat @ A2 @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_378_subset__CollectI,axiom,
! [B: set_real,A2: set_real,Q: real > $o,P: real > $o] :
( ( ord_less_eq_set_real @ B @ A2 )
=> ( ! [X6: real] :
( ( member_real @ X6 @ B )
=> ( ( Q @ X6 )
=> ( P @ X6 ) ) )
=> ( ord_less_eq_set_real
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ B )
& ( Q @ X5 ) ) )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A2 )
& ( P @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_379_subset__CollectI,axiom,
! [B: set_o,A2: set_o,Q: $o > $o,P: $o > $o] :
( ( ord_less_eq_set_o @ B @ A2 )
=> ( ! [X6: $o] :
( ( member_o @ X6 @ B )
=> ( ( Q @ X6 )
=> ( P @ X6 ) ) )
=> ( ord_less_eq_set_o
@ ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ B )
& ( Q @ X5 ) ) )
@ ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( P @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_380_subset__CollectI,axiom,
! [B: set_nat,A2: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ B )
=> ( ( Q @ X6 )
=> ( P @ X6 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ B )
& ( Q @ X5 ) ) )
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( P @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_381_subset__CollectI,axiom,
! [B: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int,Q: product_prod_int_int > $o,P: product_prod_int_int > $o] :
( ( ord_le2843351958646193337nt_int @ B @ A2 )
=> ( ! [X6: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X6 @ B )
=> ( ( Q @ X6 )
=> ( P @ X6 ) ) )
=> ( ord_le2843351958646193337nt_int
@ ( collec213857154873943460nt_int
@ ^ [X5: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X5 @ B )
& ( Q @ X5 ) ) )
@ ( collec213857154873943460nt_int
@ ^ [X5: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X5 @ A2 )
& ( P @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_382_subset__CollectI,axiom,
! [B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,Q: product_prod_nat_nat > $o,P: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ B @ A2 )
=> ( ! [X6: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X6 @ B )
=> ( ( Q @ X6 )
=> ( P @ X6 ) ) )
=> ( ord_le3146513528884898305at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ B )
& ( Q @ X5 ) ) )
@ ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A2 )
& ( P @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_383_subset__Collect__iff,axiom,
! [B: set_real,A2: set_real,P: real > $o] :
( ( ord_less_eq_set_real @ B @ A2 )
=> ( ( ord_less_eq_set_real @ B
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A2 )
& ( P @ X5 ) ) ) )
= ( ! [X5: real] :
( ( member_real @ X5 @ B )
=> ( P @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_384_subset__Collect__iff,axiom,
! [B: set_o,A2: set_o,P: $o > $o] :
( ( ord_less_eq_set_o @ B @ A2 )
=> ( ( ord_less_eq_set_o @ B
@ ( collect_o
@ ^ [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( P @ X5 ) ) ) )
= ( ! [X5: $o] :
( ( member_o @ X5 @ B )
=> ( P @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_385_subset__Collect__iff,axiom,
! [B: set_nat,A2: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( P @ X5 ) ) ) )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ B )
=> ( P @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_386_subset__Collect__iff,axiom,
! [B: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
( ( ord_le2843351958646193337nt_int @ B @ A2 )
=> ( ( ord_le2843351958646193337nt_int @ B
@ ( collec213857154873943460nt_int
@ ^ [X5: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X5 @ A2 )
& ( P @ X5 ) ) ) )
= ( ! [X5: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X5 @ B )
=> ( P @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_387_subset__Collect__iff,axiom,
! [B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ B @ A2 )
=> ( ( ord_le3146513528884898305at_nat @ B
@ ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A2 )
& ( P @ X5 ) ) ) )
= ( ! [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ B )
=> ( P @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_388_fset__of__list__subset,axiom,
! [Xs: list_f1824981274722084755rm_f_v,Ys: list_f1824981274722084755rm_f_v] :
( ( ord_le2293696477246793741rm_f_v @ ( set_fs7270820277574336546rm_f_v @ Xs ) @ ( set_fs7270820277574336546rm_f_v @ Ys ) )
=> ( ord_le7606603380504677011rm_f_v @ ( fset_o5782745410740424019rm_f_v @ Xs ) @ ( fset_o5782745410740424019rm_f_v @ Ys ) ) ) ).
% fset_of_list_subset
thf(fact_389_fset__of__list__subset,axiom,
! [Xs: list_P3903862279629787026_f_nat,Ys: list_P3903862279629787026_f_nat] :
( ( ord_le8976984241387448984_f_nat @ ( set_Pr7102205019285007021_f_nat @ Xs ) @ ( set_Pr7102205019285007021_f_nat @ Ys ) )
=> ( ord_le1552505484586773650_f_nat @ ( fset_o8009517685352940092_f_nat @ Xs ) @ ( fset_o8009517685352940092_f_nat @ Ys ) ) ) ).
% fset_of_list_subset
thf(fact_390_ffunas__trs_Orep__eq,axiom,
! [X: fset_P8018961893305114765rm_f_v] :
( ( fset_P3576968334923099475_f_nat @ ( lV_to_ffunas_trs_f_v @ X ) )
= ( funas_trs_f_v_v @ ( fset_P4617584883882644886rm_f_v @ X ) ) ) ).
% ffunas_trs.rep_eq
thf(fact_391_GreatestI2__order,axiom,
! [P: fset_P6228066233360383026_f_nat > $o,X: fset_P6228066233360383026_f_nat,Q: fset_P6228066233360383026_f_nat > $o] :
( ( P @ X )
=> ( ! [Y7: fset_P6228066233360383026_f_nat] :
( ( P @ Y7 )
=> ( ord_le1552505484586773650_f_nat @ Y7 @ X ) )
=> ( ! [X6: fset_P6228066233360383026_f_nat] :
( ( P @ X6 )
=> ( ! [Y8: fset_P6228066233360383026_f_nat] :
( ( P @ Y8 )
=> ( ord_le1552505484586773650_f_nat @ Y8 @ X6 ) )
=> ( Q @ X6 ) ) )
=> ( Q @ ( order_5949861133752320921_f_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_392_GreatestI2__order,axiom,
! [P: int > $o,X: int,Q: int > $o] :
( ( P @ X )
=> ( ! [Y7: int] :
( ( P @ Y7 )
=> ( ord_less_eq_int @ Y7 @ X ) )
=> ( ! [X6: int] :
( ( P @ X6 )
=> ( ! [Y8: int] :
( ( P @ Y8 )
=> ( ord_less_eq_int @ Y8 @ X6 ) )
=> ( Q @ X6 ) ) )
=> ( Q @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_393_GreatestI2__order,axiom,
! [P: real > $o,X: real,Q: real > $o] :
( ( P @ X )
=> ( ! [Y7: real] :
( ( P @ Y7 )
=> ( ord_less_eq_real @ Y7 @ X ) )
=> ( ! [X6: real] :
( ( P @ X6 )
=> ( ! [Y8: real] :
( ( P @ Y8 )
=> ( ord_less_eq_real @ Y8 @ X6 ) )
=> ( Q @ X6 ) ) )
=> ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_394_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X ) )
=> ( ! [X6: nat] :
( ( P @ X6 )
=> ( ! [Y8: nat] :
( ( P @ Y8 )
=> ( ord_less_eq_nat @ Y8 @ X6 ) )
=> ( Q @ X6 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_395_Greatest__equality,axiom,
! [P: fset_P6228066233360383026_f_nat > $o,X: fset_P6228066233360383026_f_nat] :
( ( P @ X )
=> ( ! [Y7: fset_P6228066233360383026_f_nat] :
( ( P @ Y7 )
=> ( ord_le1552505484586773650_f_nat @ Y7 @ X ) )
=> ( ( order_5949861133752320921_f_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_396_Greatest__equality,axiom,
! [P: int > $o,X: int] :
( ( P @ X )
=> ( ! [Y7: int] :
( ( P @ Y7 )
=> ( ord_less_eq_int @ Y7 @ X ) )
=> ( ( order_Greatest_int @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_397_Greatest__equality,axiom,
! [P: real > $o,X: real] :
( ( P @ X )
=> ( ! [Y7: real] :
( ( P @ Y7 )
=> ( ord_less_eq_real @ Y7 @ X ) )
=> ( ( order_Greatest_real @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_398_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_399_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_400_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_401_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_402_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_403_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_404_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_405_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_406_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_407_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_408_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_409_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_410_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_411_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_412_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_413_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_414_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_415_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_416_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_417_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_418_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_419_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_420_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_421_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_422_fset__of__list_Orep__eq,axiom,
! [X: list_f1824981274722084755rm_f_v] :
( ( fset_f1410452810862158076rm_f_v @ ( fset_o5782745410740424019rm_f_v @ X ) )
= ( set_fs7270820277574336546rm_f_v @ X ) ) ).
% fset_of_list.rep_eq
thf(fact_423_fset__of__list_Orep__eq,axiom,
! [X: list_P3903862279629787026_f_nat] :
( ( fset_P3576968334923099475_f_nat @ ( fset_o8009517685352940092_f_nat @ X ) )
= ( set_Pr7102205019285007021_f_nat @ X ) ) ).
% fset_of_list.rep_eq
thf(fact_424_fset__of__list_Orep__eq,axiom,
! [X: list_P4093298276913796397rm_f_v] :
( ( fset_P4617584883882644886rm_f_v @ ( fset_o4970844032613833069rm_f_v @ X ) )
= ( set_Pr817814403484925884rm_f_v @ X ) ) ).
% fset_of_list.rep_eq
thf(fact_425_GreatestI__ex__nat,axiom,
! [P: nat > $o,B3: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ B3 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_426_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ B3 ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_427_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ B3 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_428_real__arch__simple,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% real_arch_simple
thf(fact_429_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_430_insort__insert__insort__key,axiom,
! [F: nat > nat,X: nat,Xs: list_nat] :
( ~ ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ ( set_nat2 @ Xs ) ) )
=> ( ( linord1921536354676448932at_nat @ F @ X @ Xs )
= ( linord8961336180081300637at_nat @ F @ X @ Xs ) ) ) ).
% insort_insert_insort_key
thf(fact_431_insort__insert__insort__key,axiom,
! [F: fset_P8018961893305114765rm_f_v > real,X: fset_P8018961893305114765rm_f_v,Xs: list_f1824981274722084755rm_f_v] :
( ~ ( member_real @ ( F @ X ) @ ( image_7709105692213329908v_real @ F @ ( set_fs7270820277574336546rm_f_v @ Xs ) ) )
=> ( ( linord405245991147262135v_real @ F @ X @ Xs )
= ( linord3420383945269706110v_real @ F @ X @ Xs ) ) ) ).
% insort_insert_insort_key
thf(fact_432_insort__insert__insort__key,axiom,
! [F: fset_P8018961893305114765rm_f_v > $o,X: fset_P8018961893305114765rm_f_v,Xs: list_f1824981274722084755rm_f_v] :
( ~ ( member_o @ ( F @ X ) @ ( image_7665057491163335760_f_v_o @ F @ ( set_fs7270820277574336546rm_f_v @ Xs ) ) )
=> ( ( linord8610820856414126925_f_v_o @ F @ X @ Xs )
= ( linord9188204533028590406_f_v_o @ F @ X @ Xs ) ) ) ).
% insort_insert_insort_key
thf(fact_433_insort__insert__insort__key,axiom,
! [F: fset_P8018961893305114765rm_f_v > nat,X: fset_P8018961893305114765rm_f_v,Xs: list_f1824981274722084755rm_f_v] :
( ~ ( member_nat @ ( F @ X ) @ ( image_3972722278727975192_v_nat @ F @ ( set_fs7270820277574336546rm_f_v @ Xs ) ) )
=> ( ( linord7013379258955989851_v_nat @ F @ X @ Xs )
= ( linord45516334601593762_v_nat @ F @ X @ Xs ) ) ) ).
% insort_insert_insort_key
thf(fact_434_insort__insert__key__def,axiom,
( linord1921536354676448932at_nat
= ( ^ [F3: nat > nat,X5: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat @ ( F3 @ X5 ) @ ( image_nat_nat @ F3 @ ( set_nat2 @ Xs2 ) ) ) @ Xs2 @ ( linord8961336180081300637at_nat @ F3 @ X5 @ Xs2 ) ) ) ) ).
% insort_insert_key_def
thf(fact_435_insort__insert__key__def,axiom,
( linord405245991147262135v_real
= ( ^ [F3: fset_P8018961893305114765rm_f_v > real,X5: fset_P8018961893305114765rm_f_v,Xs2: list_f1824981274722084755rm_f_v] : ( if_lis137525736101459289rm_f_v @ ( member_real @ ( F3 @ X5 ) @ ( image_7709105692213329908v_real @ F3 @ ( set_fs7270820277574336546rm_f_v @ Xs2 ) ) ) @ Xs2 @ ( linord3420383945269706110v_real @ F3 @ X5 @ Xs2 ) ) ) ) ).
% insort_insert_key_def
thf(fact_436_insort__insert__key__def,axiom,
( linord8610820856414126925_f_v_o
= ( ^ [F3: fset_P8018961893305114765rm_f_v > $o,X5: fset_P8018961893305114765rm_f_v,Xs2: list_f1824981274722084755rm_f_v] : ( if_lis137525736101459289rm_f_v @ ( member_o @ ( F3 @ X5 ) @ ( image_7665057491163335760_f_v_o @ F3 @ ( set_fs7270820277574336546rm_f_v @ Xs2 ) ) ) @ Xs2 @ ( linord9188204533028590406_f_v_o @ F3 @ X5 @ Xs2 ) ) ) ) ).
% insort_insert_key_def
thf(fact_437_insort__insert__key__def,axiom,
( linord7013379258955989851_v_nat
= ( ^ [F3: fset_P8018961893305114765rm_f_v > nat,X5: fset_P8018961893305114765rm_f_v,Xs2: list_f1824981274722084755rm_f_v] : ( if_lis137525736101459289rm_f_v @ ( member_nat @ ( F3 @ X5 ) @ ( image_3972722278727975192_v_nat @ F3 @ ( set_fs7270820277574336546rm_f_v @ Xs2 ) ) ) @ Xs2 @ ( linord45516334601593762_v_nat @ F3 @ X5 @ Xs2 ) ) ) ) ).
% insort_insert_key_def
thf(fact_438_subset__code_I2_J,axiom,
! [A2: set_real,Ys: list_real] :
( ( ord_less_eq_set_real @ A2 @ ( coset_real @ Ys ) )
= ( ! [X5: real] :
( ( member_real @ X5 @ ( set_real2 @ Ys ) )
=> ~ ( member_real @ X5 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_439_subset__code_I2_J,axiom,
! [A2: set_o,Ys: list_o] :
( ( ord_less_eq_set_o @ A2 @ ( coset_o @ Ys ) )
= ( ! [X5: $o] :
( ( member_o @ X5 @ ( set_o2 @ Ys ) )
=> ~ ( member_o @ X5 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_440_subset__code_I2_J,axiom,
! [A2: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( coset_nat @ Ys ) )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat @ X5 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_441_subset__code_I2_J,axiom,
! [A2: set_fs7307227306443116653rm_f_v,Ys: list_f1824981274722084755rm_f_v] :
( ( ord_le2293696477246793741rm_f_v @ A2 @ ( coset_7382170490471766784rm_f_v @ Ys ) )
= ( ! [X5: fset_P8018961893305114765rm_f_v] :
( ( member6790519936504491446rm_f_v @ X5 @ ( set_fs7270820277574336546rm_f_v @ Ys ) )
=> ~ ( member6790519936504491446rm_f_v @ X5 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_442_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% zle_int
thf(fact_443_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_444_not__Some__eq,axiom,
! [X: option5916524851906092002_nat_f] :
( ( ! [Y5: tree_reg_nat_f] :
( X
!= ( some_Tree_reg_nat_f @ Y5 ) ) )
= ( X = none_Tree_reg_nat_f ) ) ).
% not_Some_eq
thf(fact_445_not__Some__eq,axiom,
! [X: option3789488934265196358tion_f] :
( ( ! [Y5: tree_r733329426570293750tion_f] :
( X
!= ( some_T4055341017772447441tion_f @ Y5 ) ) )
= ( X = none_T5277256714714431317tion_f ) ) ).
% not_Some_eq
thf(fact_446_not__None__eq,axiom,
! [X: option5916524851906092002_nat_f] :
( ( X != none_Tree_reg_nat_f )
= ( ? [Y5: tree_reg_nat_f] :
( X
= ( some_Tree_reg_nat_f @ Y5 ) ) ) ) ).
% not_None_eq
thf(fact_447_not__None__eq,axiom,
! [X: option3789488934265196358tion_f] :
( ( X != none_T5277256714714431317tion_f )
= ( ? [Y5: tree_r733329426570293750tion_f] :
( X
= ( some_T4055341017772447441tion_f @ Y5 ) ) ) ) ).
% not_None_eq
thf(fact_448_option_Ocollapse,axiom,
! [Option: option5916524851906092002_nat_f] :
( ( Option != none_Tree_reg_nat_f )
=> ( ( some_Tree_reg_nat_f @ ( the_Tree_reg_nat_f @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_449_option_Ocollapse,axiom,
! [Option: option3789488934265196358tion_f] :
( ( Option != none_T5277256714714431317tion_f )
=> ( ( some_T4055341017772447441tion_f @ ( the_Tr3488611493235794018tion_f @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_450_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_451_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_452_verit__la__generic,axiom,
! [A4: int,X: int] :
( ( ord_less_eq_int @ A4 @ X )
| ( A4 = X )
| ( ord_less_eq_int @ X @ A4 ) ) ).
% verit_la_generic
thf(fact_453_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_454_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_455_int__if,axiom,
! [P: $o,A4: nat,B3: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A4 @ B3 ) )
= ( semiri1314217659103216013at_int @ A4 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A4 @ B3 ) )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% int_if
thf(fact_456_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A5 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_457_combine__options__cases,axiom,
! [X: option5916524851906092002_nat_f,P: option5916524851906092002_nat_f > option5916524851906092002_nat_f > $o,Y: option5916524851906092002_nat_f] :
( ( ( X = none_Tree_reg_nat_f )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_Tree_reg_nat_f )
=> ( P @ X @ Y ) )
=> ( ! [A6: tree_reg_nat_f,B5: tree_reg_nat_f] :
( ( X
= ( some_Tree_reg_nat_f @ A6 ) )
=> ( ( Y
= ( some_Tree_reg_nat_f @ B5 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_458_combine__options__cases,axiom,
! [X: option5916524851906092002_nat_f,P: option5916524851906092002_nat_f > option3789488934265196358tion_f > $o,Y: option3789488934265196358tion_f] :
( ( ( X = none_Tree_reg_nat_f )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_T5277256714714431317tion_f )
=> ( P @ X @ Y ) )
=> ( ! [A6: tree_reg_nat_f,B5: tree_r733329426570293750tion_f] :
( ( X
= ( some_Tree_reg_nat_f @ A6 ) )
=> ( ( Y
= ( some_T4055341017772447441tion_f @ B5 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_459_combine__options__cases,axiom,
! [X: option3789488934265196358tion_f,P: option3789488934265196358tion_f > option5916524851906092002_nat_f > $o,Y: option5916524851906092002_nat_f] :
( ( ( X = none_T5277256714714431317tion_f )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_Tree_reg_nat_f )
=> ( P @ X @ Y ) )
=> ( ! [A6: tree_r733329426570293750tion_f,B5: tree_reg_nat_f] :
( ( X
= ( some_T4055341017772447441tion_f @ A6 ) )
=> ( ( Y
= ( some_Tree_reg_nat_f @ B5 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_460_combine__options__cases,axiom,
! [X: option3789488934265196358tion_f,P: option3789488934265196358tion_f > option3789488934265196358tion_f > $o,Y: option3789488934265196358tion_f] :
( ( ( X = none_T5277256714714431317tion_f )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_T5277256714714431317tion_f )
=> ( P @ X @ Y ) )
=> ( ! [A6: tree_r733329426570293750tion_f,B5: tree_r733329426570293750tion_f] :
( ( X
= ( some_T4055341017772447441tion_f @ A6 ) )
=> ( ( Y
= ( some_T4055341017772447441tion_f @ B5 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_461_split__option__all,axiom,
( ( ^ [P2: option5916524851906092002_nat_f > $o] :
! [X8: option5916524851906092002_nat_f] : ( P2 @ X8 ) )
= ( ^ [P3: option5916524851906092002_nat_f > $o] :
( ( P3 @ none_Tree_reg_nat_f )
& ! [X5: tree_reg_nat_f] : ( P3 @ ( some_Tree_reg_nat_f @ X5 ) ) ) ) ) ).
% split_option_all
thf(fact_462_split__option__all,axiom,
( ( ^ [P2: option3789488934265196358tion_f > $o] :
! [X8: option3789488934265196358tion_f] : ( P2 @ X8 ) )
= ( ^ [P3: option3789488934265196358tion_f > $o] :
( ( P3 @ none_T5277256714714431317tion_f )
& ! [X5: tree_r733329426570293750tion_f] : ( P3 @ ( some_T4055341017772447441tion_f @ X5 ) ) ) ) ) ).
% split_option_all
thf(fact_463_split__option__ex,axiom,
( ( ^ [P2: option5916524851906092002_nat_f > $o] :
? [X8: option5916524851906092002_nat_f] : ( P2 @ X8 ) )
= ( ^ [P3: option5916524851906092002_nat_f > $o] :
( ( P3 @ none_Tree_reg_nat_f )
| ? [X5: tree_reg_nat_f] : ( P3 @ ( some_Tree_reg_nat_f @ X5 ) ) ) ) ) ).
% split_option_ex
thf(fact_464_split__option__ex,axiom,
( ( ^ [P2: option3789488934265196358tion_f > $o] :
? [X8: option3789488934265196358tion_f] : ( P2 @ X8 ) )
= ( ^ [P3: option3789488934265196358tion_f > $o] :
( ( P3 @ none_T5277256714714431317tion_f )
| ? [X5: tree_r733329426570293750tion_f] : ( P3 @ ( some_T4055341017772447441tion_f @ X5 ) ) ) ) ) ).
% split_option_ex
thf(fact_465_option_Oexhaust,axiom,
! [Y: option5916524851906092002_nat_f] :
( ( Y != none_Tree_reg_nat_f )
=> ~ ! [X22: tree_reg_nat_f] :
( Y
!= ( some_Tree_reg_nat_f @ X22 ) ) ) ).
% option.exhaust
thf(fact_466_option_Oexhaust,axiom,
! [Y: option3789488934265196358tion_f] :
( ( Y != none_T5277256714714431317tion_f )
=> ~ ! [X22: tree_r733329426570293750tion_f] :
( Y
!= ( some_T4055341017772447441tion_f @ X22 ) ) ) ).
% option.exhaust
thf(fact_467_option_OdiscI,axiom,
! [Option: option5916524851906092002_nat_f,X2: tree_reg_nat_f] :
( ( Option
= ( some_Tree_reg_nat_f @ X2 ) )
=> ( Option != none_Tree_reg_nat_f ) ) ).
% option.discI
thf(fact_468_option_OdiscI,axiom,
! [Option: option3789488934265196358tion_f,X2: tree_r733329426570293750tion_f] :
( ( Option
= ( some_T4055341017772447441tion_f @ X2 ) )
=> ( Option != none_T5277256714714431317tion_f ) ) ).
% option.discI
thf(fact_469_option_Odistinct_I1_J,axiom,
! [X2: tree_reg_nat_f] :
( none_Tree_reg_nat_f
!= ( some_Tree_reg_nat_f @ X2 ) ) ).
% option.distinct(1)
thf(fact_470_option_Odistinct_I1_J,axiom,
! [X2: tree_r733329426570293750tion_f] :
( none_T5277256714714431317tion_f
!= ( some_T4055341017772447441tion_f @ X2 ) ) ).
% option.distinct(1)
thf(fact_471_None__notin__image__Some,axiom,
! [A2: set_Tree_reg_nat_f] :
~ ( member1604975567284567801_nat_f @ none_Tree_reg_nat_f @ ( image_868921236233539067_nat_f @ some_Tree_reg_nat_f @ A2 ) ) ).
% None_notin_image_Some
thf(fact_472_None__notin__image__Some,axiom,
! [A2: set_Tr6476182622925392812tion_f] :
~ ( member4921012686109714013tion_f @ none_T5277256714714431317tion_f @ ( image_8199543339563177155tion_f @ some_T4055341017772447441tion_f @ A2 ) ) ).
% None_notin_image_Some
thf(fact_473_option_Oexpand,axiom,
! [Option: option3789488934265196358tion_f,Option2: option3789488934265196358tion_f] :
( ( ( Option = none_T5277256714714431317tion_f )
= ( Option2 = none_T5277256714714431317tion_f ) )
=> ( ( ( Option != none_T5277256714714431317tion_f )
=> ( ( Option2 != none_T5277256714714431317tion_f )
=> ( ( the_Tr3488611493235794018tion_f @ Option )
= ( the_Tr3488611493235794018tion_f @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_474_option_Oexhaust__sel,axiom,
! [Option: option5916524851906092002_nat_f] :
( ( Option != none_Tree_reg_nat_f )
=> ( Option
= ( some_Tree_reg_nat_f @ ( the_Tree_reg_nat_f @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_475_option_Oexhaust__sel,axiom,
! [Option: option3789488934265196358tion_f] :
( ( Option != none_T5277256714714431317tion_f )
=> ( Option
= ( some_T4055341017772447441tion_f @ ( the_Tr3488611493235794018tion_f @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_476_insort__insert__insort,axiom,
! [X: real,Xs: list_real] :
( ~ ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( ( linord1891625487229344476l_real
@ ^ [X5: real] : X5
@ X
@ Xs )
= ( linord1674302359176591317l_real
@ ^ [X5: real] : X5
@ X
@ Xs ) ) ) ).
% insort_insert_insort
thf(fact_477_insort__insert__insort,axiom,
! [X: $o,Xs: list_o] :
( ~ ( member_o @ X @ ( set_o2 @ Xs ) )
=> ( ( linord6472470733373143810ey_o_o
@ ^ [X5: $o] : X5
@ X
@ Xs )
= ( linord5141348845282165115ey_o_o
@ ^ [X5: $o] : X5
@ X
@ Xs ) ) ) ).
% insort_insert_insort
thf(fact_478_insort__insert__insort,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( linord1921536354676448932at_nat
@ ^ [X5: nat] : X5
@ X
@ Xs )
= ( linord8961336180081300637at_nat
@ ^ [X5: nat] : X5
@ X
@ Xs ) ) ) ).
% insort_insert_insort
thf(fact_479_verit__comp__simplify1_I2_J,axiom,
! [A4: fset_P6228066233360383026_f_nat] : ( ord_le1552505484586773650_f_nat @ A4 @ A4 ) ).
% verit_comp_simplify1(2)
thf(fact_480_verit__comp__simplify1_I2_J,axiom,
! [A4: nat] : ( ord_less_eq_nat @ A4 @ A4 ) ).
% verit_comp_simplify1(2)
thf(fact_481_verit__comp__simplify1_I2_J,axiom,
! [A4: int] : ( ord_less_eq_int @ A4 @ A4 ) ).
% verit_comp_simplify1(2)
thf(fact_482_verit__comp__simplify1_I2_J,axiom,
! [A4: real] : ( ord_less_eq_real @ A4 @ A4 ) ).
% verit_comp_simplify1(2)
thf(fact_483_verit__la__disequality,axiom,
! [A4: nat,B3: nat] :
( ( A4 = B3 )
| ~ ( ord_less_eq_nat @ A4 @ B3 )
| ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ).
% verit_la_disequality
thf(fact_484_verit__la__disequality,axiom,
! [A4: int,B3: int] :
( ( A4 = B3 )
| ~ ( ord_less_eq_int @ A4 @ B3 )
| ~ ( ord_less_eq_int @ B3 @ A4 ) ) ).
% verit_la_disequality
thf(fact_485_verit__la__disequality,axiom,
! [A4: real,B3: real] :
( ( A4 = B3 )
| ~ ( ord_less_eq_real @ A4 @ B3 )
| ~ ( ord_less_eq_real @ B3 @ A4 ) ) ).
% verit_la_disequality
thf(fact_486_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_487_mapM__Some,axiom,
! [F: fset_P8018961893305114765rm_f_v > option3296083141436081229rm_f_v,Xs: list_f1824981274722084755rm_f_v,Ys: list_s8746099396510718605rm_f_v] :
( ( ( option5007661016126702777rm_f_v @ F @ Xs )
= ( some_l530080162245194098rm_f_v @ Ys ) )
=> ( ( Ys
= ( map_fs8602507653405230974rm_f_v
@ ^ [X5: fset_P8018961893305114765rm_f_v] : ( the_se7542970557029289371rm_f_v @ ( F @ X5 ) )
@ Xs ) )
& ! [X4: fset_P8018961893305114765rm_f_v] :
( ( member6790519936504491446rm_f_v @ X4 @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
=> ( ( F @ X4 )
!= none_s1889411087592137448rm_f_v ) ) ) ) ).
% mapM_Some
thf(fact_488_mapM__Some,axiom,
! [F: fset_P8018961893305114765rm_f_v > option3789488934265196358tion_f,Xs: list_f1824981274722084755rm_f_v,Ys: list_T5268601877343193350tion_f] :
( ( ( option5852021067674027716tion_f @ F @ Xs )
= ( some_l8507658222393836641tion_f @ Ys ) )
=> ( ( Ys
= ( map_fs7137784065334668863tion_f
@ ^ [X5: fset_P8018961893305114765rm_f_v] : ( the_Tr3488611493235794018tion_f @ ( F @ X5 ) )
@ Xs ) )
& ! [X4: fset_P8018961893305114765rm_f_v] :
( ( member6790519936504491446rm_f_v @ X4 @ ( set_fs7270820277574336546rm_f_v @ Xs ) )
=> ( ( F @ X4 )
!= none_T5277256714714431317tion_f ) ) ) ) ).
% mapM_Some
thf(fact_489_round__of__nat,axiom,
! [N: nat] :
( ( archim8280529875227126926d_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% round_of_nat
thf(fact_490_Option_Othese__def,axiom,
( these_4693336772737898105tion_f
= ( ^ [A3: set_op6172961469967127676tion_f] :
( image_7469968060601971523tion_f @ the_Tr3488611493235794018tion_f
@ ( collec5662084080931014299tion_f
@ ^ [X5: option3789488934265196358tion_f] :
( ( member4921012686109714013tion_f @ X5 @ A3 )
& ( X5 != none_T5277256714714431317tion_f ) ) ) ) ) ) ).
% Option.these_def
thf(fact_491_Some__image__these__eq,axiom,
! [A2: set_op1790067033808154392_nat_f] :
( ( image_868921236233539067_nat_f @ some_Tree_reg_nat_f @ ( these_Tree_reg_nat_f @ A2 ) )
= ( collec6098388481013462839_nat_f
@ ^ [X5: option5916524851906092002_nat_f] :
( ( member1604975567284567801_nat_f @ X5 @ A2 )
& ( X5 != none_Tree_reg_nat_f ) ) ) ) ).
% Some_image_these_eq
thf(fact_492_Some__image__these__eq,axiom,
! [A2: set_op6172961469967127676tion_f] :
( ( image_8199543339563177155tion_f @ some_T4055341017772447441tion_f @ ( these_4693336772737898105tion_f @ A2 ) )
= ( collec5662084080931014299tion_f
@ ^ [X5: option3789488934265196358tion_f] :
( ( member4921012686109714013tion_f @ X5 @ A2 )
& ( X5 != none_T5277256714714431317tion_f ) ) ) ) ).
% Some_image_these_eq
thf(fact_493_round__0,axiom,
( ( archim8280529875227126926d_real @ zero_zero_real )
= zero_zero_int ) ).
% round_0
thf(fact_494_these__image__Some__eq,axiom,
! [A2: set_Tree_reg_nat_f] :
( ( these_Tree_reg_nat_f @ ( image_868921236233539067_nat_f @ some_Tree_reg_nat_f @ A2 ) )
= A2 ) ).
% these_image_Some_eq
thf(fact_495_these__image__Some__eq,axiom,
! [A2: set_Tr6476182622925392812tion_f] :
( ( these_4693336772737898105tion_f @ ( image_8199543339563177155tion_f @ some_T4055341017772447441tion_f @ A2 ) )
= A2 ) ).
% these_image_Some_eq
thf(fact_496_in__these__eq,axiom,
! [X: real,A2: set_option_real] :
( ( member_real @ X @ ( these_real @ A2 ) )
= ( member_option_real @ ( some_real @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_497_in__these__eq,axiom,
! [X: $o,A2: set_option_o] :
( ( member_o @ X @ ( these_o @ A2 ) )
= ( member_option_o @ ( some_o @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_498_in__these__eq,axiom,
! [X: nat,A2: set_option_nat] :
( ( member_nat @ X @ ( these_nat @ A2 ) )
= ( member_option_nat @ ( some_nat @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_499_in__these__eq,axiom,
! [X: tree_reg_nat_f,A2: set_op1790067033808154392_nat_f] :
( ( member2049775546588854825_nat_f @ X @ ( these_Tree_reg_nat_f @ A2 ) )
= ( member1604975567284567801_nat_f @ ( some_Tree_reg_nat_f @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_500_in__these__eq,axiom,
! [X: tree_r733329426570293750tion_f,A2: set_op6172961469967127676tion_f] :
( ( member6626405664465942413tion_f @ X @ ( these_4693336772737898105tion_f @ A2 ) )
= ( member4921012686109714013tion_f @ ( some_T4055341017772447441tion_f @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_501_round__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim8280529875227126926d_real @ Y ) ) ) ).
% round_mono
thf(fact_502_conj__le__cong,axiom,
! [X: int,X9: int,P: $o,P4: $o] :
( ( X = X9 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X9 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X9 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_503_imp__le__cong,axiom,
! [X: int,X9: int,P: $o,P4: $o] :
( ( X = X9 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X9 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X9 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_504_mapM__map,axiom,
( option5007661016126702777rm_f_v
= ( ^ [F3: fset_P8018961893305114765rm_f_v > option3296083141436081229rm_f_v,Xs2: list_f1824981274722084755rm_f_v] :
( if_opt1743852810818088729rm_f_v
@ ! [X5: fset_P8018961893305114765rm_f_v] :
( ( member6790519936504491446rm_f_v @ X5 @ ( set_fs7270820277574336546rm_f_v @ Xs2 ) )
=> ( ( F3 @ X5 )
!= none_s1889411087592137448rm_f_v ) )
@ ( some_l530080162245194098rm_f_v
@ ( map_fs8602507653405230974rm_f_v
@ ^ [X5: fset_P8018961893305114765rm_f_v] : ( the_se7542970557029289371rm_f_v @ ( F3 @ X5 ) )
@ Xs2 ) )
@ none_l3004429212090885614rm_f_v ) ) ) ).
% mapM_map
thf(fact_505_mapM__map,axiom,
( option5852021067674027716tion_f
= ( ^ [F3: fset_P8018961893305114765rm_f_v > option3789488934265196358tion_f,Xs2: list_f1824981274722084755rm_f_v] :
( if_opt7165310518185895824tion_f
@ ! [X5: fset_P8018961893305114765rm_f_v] :
( ( member6790519936504491446rm_f_v @ X5 @ ( set_fs7270820277574336546rm_f_v @ Xs2 ) )
=> ( ( F3 @ X5 )
!= none_T5277256714714431317tion_f ) )
@ ( some_l8507658222393836641tion_f
@ ( map_fs7137784065334668863tion_f
@ ^ [X5: fset_P8018961893305114765rm_f_v] : ( the_Tr3488611493235794018tion_f @ ( F3 @ X5 ) )
@ Xs2 ) )
@ none_l7671671938703071973tion_f ) ) ) ).
% mapM_map
thf(fact_506_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_507_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_508_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_509_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_510_ceiling__le__zero,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% ceiling_le_zero
thf(fact_511_zero__le__floor,axiom,
! [X: real] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% zero_le_floor
thf(fact_512_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( ( ring_1_of_int_real @ W )
= ( ring_1_of_int_real @ Z ) )
= ( W = Z ) ) ).
% of_int_eq_iff
thf(fact_513_floor__of__int,axiom,
! [Z: int] :
( ( archim6058952711729229775r_real @ ( ring_1_of_int_real @ Z ) )
= Z ) ).
% floor_of_int
thf(fact_514_ceiling__of__int,axiom,
! [Z: int] :
( ( archim7802044766580827645g_real @ ( ring_1_of_int_real @ Z ) )
= Z ) ).
% ceiling_of_int
thf(fact_515_round__of__int,axiom,
! [N: int] :
( ( archim8280529875227126926d_real @ ( ring_1_of_int_real @ N ) )
= N ) ).
% round_of_int
thf(fact_516_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= zero_zero_int )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_517_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= zero_zero_real )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_518_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_519_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_520_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_521_of__int__0,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_0
thf(fact_522_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_523_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_524_floor__zero,axiom,
( ( archim6058952711729229775r_real @ zero_zero_real )
= zero_zero_int ) ).
% floor_zero
thf(fact_525_ceiling__zero,axiom,
( ( archim7802044766580827645g_real @ zero_zero_real )
= zero_zero_int ) ).
% ceiling_zero
thf(fact_526_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% of_int_of_nat_eq
thf(fact_527_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% of_int_of_nat_eq
thf(fact_528_floor__of__nat,axiom,
! [N: nat] :
( ( archim6058952711729229775r_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% floor_of_nat
thf(fact_529_ceiling__of__nat,axiom,
! [N: nat] :
( ( archim7802044766580827645g_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% ceiling_of_nat
thf(fact_530_cproper__interval__prod_Ocases,axiom,
! [X: produc2858277421346412199at_nat] :
( ( X
!= ( produc254078196866515351at_nat @ none_P3005240565499088977at_nat @ none_P3005240565499088977at_nat ) )
=> ( ! [Y1: nat > nat,Y22: nat] :
( X
!= ( produc254078196866515351at_nat @ none_P3005240565499088977at_nat @ ( some_P4370590140854848469at_nat @ ( produc72220940542539688at_nat @ Y1 @ Y22 ) ) ) )
=> ( ! [X1: nat > nat,X22: nat] :
( X
!= ( produc254078196866515351at_nat @ ( some_P4370590140854848469at_nat @ ( produc72220940542539688at_nat @ X1 @ X22 ) ) @ none_P3005240565499088977at_nat ) )
=> ~ ! [X1: nat > nat,X22: nat,Y1: nat > nat,Y22: nat] :
( X
!= ( produc254078196866515351at_nat @ ( some_P4370590140854848469at_nat @ ( produc72220940542539688at_nat @ X1 @ X22 ) ) @ ( some_P4370590140854848469at_nat @ ( produc72220940542539688at_nat @ Y1 @ Y22 ) ) ) ) ) ) ) ).
% cproper_interval_prod.cases
thf(fact_531_cproper__interval__prod_Ocases,axiom,
! [X: produc6121120109295599847at_nat] :
( ( X
!= ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ none_P5556105721700978146at_nat ) )
=> ( ! [Y1: nat,Y22: nat] :
( X
!= ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Y1 @ Y22 ) ) ) )
=> ( ! [X1: nat,X22: nat] :
( X
!= ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) ) @ none_P5556105721700978146at_nat ) )
=> ~ ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
( X
!= ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) ) @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Y1 @ Y22 ) ) ) ) ) ) ) ).
% cproper_interval_prod.cases
thf(fact_532_pred__equals__eq2,axiom,
! [R: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
( ( ( ^ [X5: nat > nat,Y5: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X5 @ Y5 ) @ R ) )
= ( ^ [X5: nat > nat,Y5: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X5 @ Y5 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_533_pred__equals__eq2,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ( ^ [X5: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ R ) )
= ( ^ [X5: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_534_Ball__def,axiom,
( ball_real
= ( ^ [A3: set_real,P3: real > $o] :
! [X5: real] :
( ( member_real @ X5 @ A3 )
=> ( P3 @ X5 ) ) ) ) ).
% Ball_def
thf(fact_535_Ball__def,axiom,
( ball_o
= ( ^ [A3: set_o,P3: $o > $o] :
! [X5: $o] :
( ( member_o @ X5 @ A3 )
=> ( P3 @ X5 ) ) ) ) ).
% Ball_def
thf(fact_536_Ball__def,axiom,
( ball_nat
= ( ^ [A3: set_nat,P3: nat > $o] :
! [X5: nat] :
( ( member_nat @ X5 @ A3 )
=> ( P3 @ X5 ) ) ) ) ).
% Ball_def
thf(fact_537_ssubst__Pair__rhs,axiom,
! [R2: nat > nat,S2: nat,R: set_Pr9093778441882193744at_nat,S3: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ R2 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ R2 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_538_ssubst__Pair__rhs,axiom,
! [R2: nat,S2: nat,R: set_Pr1261947904930325089at_nat,S3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_539_le__of__int__ceiling,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).
% le_of_int_ceiling
thf(fact_540_floor__le__ceiling,axiom,
! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% floor_le_ceiling
thf(fact_541_of__int__floor__le,axiom,
! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X ) ).
% of_int_floor_le
thf(fact_542_pred__subset__eq2,axiom,
! [R: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
( ( ord_le5974136086019949911_nat_o
@ ^ [X5: nat > nat,Y5: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X5 @ Y5 ) @ R )
@ ^ [X5: nat > nat,Y5: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X5 @ Y5 ) @ S ) )
= ( ord_le3678578370064672496at_nat @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_543_pred__subset__eq2,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le2646555220125990790_nat_o
@ ^ [X5: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ R )
@ ^ [X5: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ S ) )
= ( ord_le3146513528884898305at_nat @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_544_subrelI,axiom,
! [R2: set_Pr9093778441882193744at_nat,S2: set_Pr9093778441882193744at_nat] :
( ! [X6: nat > nat,Y7: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X6 @ Y7 ) @ R2 )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X6 @ Y7 ) @ S2 ) )
=> ( ord_le3678578370064672496at_nat @ R2 @ S2 ) ) ).
% subrelI
thf(fact_545_subrelI,axiom,
! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ! [X6: nat,Y7: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y7 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y7 ) @ S2 ) )
=> ( ord_le3146513528884898305at_nat @ R2 @ S2 ) ) ).
% subrelI
thf(fact_546_le__floor__iff,axiom,
! [Z: int,X: real] :
( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X ) )
= ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% le_floor_iff
thf(fact_547_ceiling__le__iff,axiom,
! [X: real,Z: int] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z )
= ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% ceiling_le_iff
thf(fact_548_ex__le__of__int,axiom,
! [X: real] :
? [Z3: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% ex_le_of_int
thf(fact_549_floor__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ).
% floor_mono
thf(fact_550_ceiling__mono,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X ) ) ) ).
% ceiling_mono
thf(fact_551_floor__le__round,axiom,
! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim8280529875227126926d_real @ X ) ) ).
% floor_le_round
thf(fact_552_ceiling__ge__round,axiom,
! [X: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).
% ceiling_ge_round
thf(fact_553_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_nonneg
thf(fact_554_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_nonneg
thf(fact_555_ceiling__le,axiom,
! [X: real,A4: int] :
( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A4 ) )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A4 ) ) ).
% ceiling_le
thf(fact_556_Ball__Collect,axiom,
( ball_P4917565930384053347nt_int
= ( ^ [A3: set_Pr958786334691620121nt_int,P3: product_prod_int_int > $o] : ( ord_le2843351958646193337nt_int @ A3 @ ( collec213857154873943460nt_int @ P3 ) ) ) ) ).
% Ball_Collect
thf(fact_557_Ball__Collect,axiom,
( ball_P8096063237992195499at_nat
= ( ^ [A3: set_Pr1261947904930325089at_nat,P3: product_prod_nat_nat > $o] : ( ord_le3146513528884898305at_nat @ A3 @ ( collec3392354462482085612at_nat @ P3 ) ) ) ) ).
% Ball_Collect
thf(fact_558_eq__fst__iff,axiom,
! [A4: nat > nat,P5: produc8199716216217303280at_nat] :
( ( A4
= ( produc6156676138143019412at_nat @ P5 ) )
= ( ? [B4: nat] :
( P5
= ( produc72220940542539688at_nat @ A4 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_559_eq__fst__iff,axiom,
! [A4: ground_gterm_f,P5: produc7245736746747425831term_f] :
( ( A4
= ( produc1239122367099457539term_f @ P5 ) )
= ( ? [B4: ground_gterm_f] :
( P5
= ( produc3560254623552331287term_f @ A4 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_560_eq__fst__iff,axiom,
! [A4: nat,P5: product_prod_nat_nat] :
( ( A4
= ( product_fst_nat_nat @ P5 ) )
= ( ? [B4: nat] :
( P5
= ( product_Pair_nat_nat @ A4 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_561_fst__conv,axiom,
! [X12: nat > nat,X2: nat] :
( ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X12 @ X2 ) )
= X12 ) ).
% fst_conv
thf(fact_562_fst__conv,axiom,
! [X12: ground_gterm_f,X2: ground_gterm_f] :
( ( produc1239122367099457539term_f @ ( produc3560254623552331287term_f @ X12 @ X2 ) )
= X12 ) ).
% fst_conv
thf(fact_563_fst__conv,axiom,
! [X12: nat,X2: nat] :
( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X12 @ X2 ) )
= X12 ) ).
% fst_conv
thf(fact_564_fst__eqD,axiom,
! [X: nat > nat,Y: nat,A4: nat > nat] :
( ( ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X @ Y ) )
= A4 )
=> ( X = A4 ) ) ).
% fst_eqD
thf(fact_565_fst__eqD,axiom,
! [X: ground_gterm_f,Y: ground_gterm_f,A4: ground_gterm_f] :
( ( ( produc1239122367099457539term_f @ ( produc3560254623552331287term_f @ X @ Y ) )
= A4 )
=> ( X = A4 ) ) ).
% fst_eqD
thf(fact_566_fst__eqD,axiom,
! [X: nat,Y: nat,A4: nat] :
( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
= A4 )
=> ( X = A4 ) ) ).
% fst_eqD
thf(fact_567_fstI,axiom,
! [X: produc8199716216217303280at_nat,Y: nat > nat,Z: nat] :
( ( X
= ( produc72220940542539688at_nat @ Y @ Z ) )
=> ( ( produc6156676138143019412at_nat @ X )
= Y ) ) ).
% fstI
thf(fact_568_fstI,axiom,
! [X: produc7245736746747425831term_f,Y: ground_gterm_f,Z: ground_gterm_f] :
( ( X
= ( produc3560254623552331287term_f @ Y @ Z ) )
=> ( ( produc1239122367099457539term_f @ X )
= Y ) ) ).
% fstI
thf(fact_569_fstI,axiom,
! [X: product_prod_nat_nat,Y: nat,Z: nat] :
( ( X
= ( product_Pair_nat_nat @ Y @ Z ) )
=> ( ( product_fst_nat_nat @ X )
= Y ) ) ).
% fstI
thf(fact_570_of__nat__nat,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= ( ring_1_of_int_int @ Z ) ) ) ).
% of_nat_nat
thf(fact_571_of__nat__nat,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri5074537144036343181t_real @ ( nat2 @ Z ) )
= ( ring_1_of_int_real @ Z ) ) ) ).
% of_nat_nat
thf(fact_572_nat__ceiling__le__eq,axiom,
! [X: real,A4: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A4 )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A4 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_573_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_574_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_575_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_576_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_577_nat__floor__neg,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= zero_zero_nat ) ) ).
% nat_floor_neg
thf(fact_578_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_579_le__nat__floor,axiom,
! [X: nat,A4: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A4 )
=> ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A4 ) ) ) ) ).
% le_nat_floor
thf(fact_580_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_581_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_582_eq__nat__nat__iff,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z5 ) )
= ( Z = Z5 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_583_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X8: nat] : ( P2 @ X8 ) )
= ( ^ [P3: nat > $o] :
! [X5: int] :
( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P3 @ ( nat2 @ X5 ) ) ) ) ) ).
% all_nat
thf(fact_584_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X8: nat] : ( P2 @ X8 ) )
= ( ^ [P3: nat > $o] :
? [X5: int] :
( ( ord_less_eq_int @ zero_zero_int @ X5 )
& ( P3 @ ( nat2 @ X5 ) ) ) ) ) ).
% ex_nat
thf(fact_585_of__nat__ceiling,axiom,
! [R2: real] : ( ord_less_eq_real @ R2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R2 ) ) ) ) ).
% of_nat_ceiling
thf(fact_586_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_587_int__eq__iff,axiom,
! [M2: nat,Z: int] :
( ( ( semiri1314217659103216013at_int @ M2 )
= Z )
= ( ( M2
= ( nat2 @ Z ) )
& ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% int_eq_iff
thf(fact_588_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_589_lift__less__eq__total_Ocases,axiom,
! [X: produc8574009693863894890_nat_f] :
( ! [F4: tree_reg_nat_f > tree_reg_nat_f > $o,Uu: option5916524851906092002_nat_f] :
( X
!= ( produc822877007188873434_nat_f @ F4 @ ( produc8844834930725908189_nat_f @ Uu @ none_Tree_reg_nat_f ) ) )
=> ( ! [F4: tree_reg_nat_f > tree_reg_nat_f > $o,V2: tree_reg_nat_f] :
( X
!= ( produc822877007188873434_nat_f @ F4 @ ( produc8844834930725908189_nat_f @ none_Tree_reg_nat_f @ ( some_Tree_reg_nat_f @ V2 ) ) ) )
=> ~ ! [F4: tree_reg_nat_f > tree_reg_nat_f > $o,S4: tree_reg_nat_f,T4: tree_reg_nat_f] :
( X
!= ( produc822877007188873434_nat_f @ F4 @ ( produc8844834930725908189_nat_f @ ( some_Tree_reg_nat_f @ S4 ) @ ( some_Tree_reg_nat_f @ T4 ) ) ) ) ) ) ).
% lift_less_eq_total.cases
thf(fact_590_lift__less__eq__total_Ocases,axiom,
! [X: produc7762418835577779250_nat_f] :
( ! [F4: tree_r733329426570293750tion_f > tree_reg_nat_f > $o,Uu: option3789488934265196358tion_f] :
( X
!= ( produc1791491351022178210_nat_f @ F4 @ ( produc7608536704123013185_nat_f @ Uu @ none_Tree_reg_nat_f ) ) )
=> ( ! [F4: tree_r733329426570293750tion_f > tree_reg_nat_f > $o,V2: tree_reg_nat_f] :
( X
!= ( produc1791491351022178210_nat_f @ F4 @ ( produc7608536704123013185_nat_f @ none_T5277256714714431317tion_f @ ( some_Tree_reg_nat_f @ V2 ) ) ) )
=> ~ ! [F4: tree_r733329426570293750tion_f > tree_reg_nat_f > $o,S4: tree_r733329426570293750tion_f,T4: tree_reg_nat_f] :
( X
!= ( produc1791491351022178210_nat_f @ F4 @ ( produc7608536704123013185_nat_f @ ( some_T4055341017772447441tion_f @ S4 ) @ ( some_Tree_reg_nat_f @ T4 ) ) ) ) ) ) ).
% lift_less_eq_total.cases
thf(fact_591_lift__less__eq__total_Ocases,axiom,
! [X: produc3878888619416440882tion_f] :
( ! [F4: tree_reg_nat_f > tree_r733329426570293750tion_f > $o,Uu: option5916524851906092002_nat_f] :
( X
!= ( produc230842789923892130tion_f @ F4 @ ( produc751580840900675137tion_f @ Uu @ none_T5277256714714431317tion_f ) ) )
=> ( ! [F4: tree_reg_nat_f > tree_r733329426570293750tion_f > $o,V2: tree_r733329426570293750tion_f] :
( X
!= ( produc230842789923892130tion_f @ F4 @ ( produc751580840900675137tion_f @ none_Tree_reg_nat_f @ ( some_T4055341017772447441tion_f @ V2 ) ) ) )
=> ~ ! [F4: tree_reg_nat_f > tree_r733329426570293750tion_f > $o,S4: tree_reg_nat_f,T4: tree_r733329426570293750tion_f] :
( X
!= ( produc230842789923892130tion_f @ F4 @ ( produc751580840900675137tion_f @ ( some_Tree_reg_nat_f @ S4 ) @ ( some_T4055341017772447441tion_f @ T4 ) ) ) ) ) ) ).
% lift_less_eq_total.cases
thf(fact_592_lift__less__eq__total_Ocases,axiom,
! [X: produc2070832938513523962tion_f] :
( ! [F4: tree_r733329426570293750tion_f > tree_r733329426570293750tion_f > $o,Uu: option3789488934265196358tion_f] :
( X
!= ( produc7984526239635384938tion_f @ F4 @ ( produc6849691629296390053tion_f @ Uu @ none_T5277256714714431317tion_f ) ) )
=> ( ! [F4: tree_r733329426570293750tion_f > tree_r733329426570293750tion_f > $o,V2: tree_r733329426570293750tion_f] :
( X
!= ( produc7984526239635384938tion_f @ F4 @ ( produc6849691629296390053tion_f @ none_T5277256714714431317tion_f @ ( some_T4055341017772447441tion_f @ V2 ) ) ) )
=> ~ ! [F4: tree_r733329426570293750tion_f > tree_r733329426570293750tion_f > $o,S4: tree_r733329426570293750tion_f,T4: tree_r733329426570293750tion_f] :
( X
!= ( produc7984526239635384938tion_f @ F4 @ ( produc6849691629296390053tion_f @ ( some_T4055341017772447441tion_f @ S4 ) @ ( some_T4055341017772447441tion_f @ T4 ) ) ) ) ) ) ).
% lift_less_eq_total.cases
thf(fact_593_of__nat__floor,axiom,
! [R2: real] :
( ( ord_less_eq_real @ zero_zero_real @ R2 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R2 ) ) ) @ R2 ) ) ).
% of_nat_floor
thf(fact_594_nat__eq__iff2,axiom,
! [M2: nat,W: int] :
( ( M2
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_595_nat__eq__iff,axiom,
! [W: int,M2: nat] :
( ( ( nat2 @ W )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_596_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_597_nat__le__eq__zle,axiom,
! [W: int,Z: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_598_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N4: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N4 ) )
=> ( P @ N4 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_599_distinct__insort__key,axiom,
! [F: nat > nat,X: nat,Xs: list_nat] :
( ( distinct_nat @ ( map_nat_nat @ F @ ( linord8961336180081300637at_nat @ F @ X @ Xs ) ) )
= ( ~ ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ ( set_nat2 @ Xs ) ) )
& ( distinct_nat @ ( map_nat_nat @ F @ Xs ) ) ) ) ).
% distinct_insort_key
thf(fact_600_distinct__insort__key,axiom,
! [F: fset_P8018961893305114765rm_f_v > real,X: fset_P8018961893305114765rm_f_v,Xs: list_f1824981274722084755rm_f_v] :
( ( distinct_real @ ( map_fs3787659953648829301v_real @ F @ ( linord3420383945269706110v_real @ F @ X @ Xs ) ) )
= ( ~ ( member_real @ ( F @ X ) @ ( image_7709105692213329908v_real @ F @ ( set_fs7270820277574336546rm_f_v @ Xs ) ) )
& ( distinct_real @ ( map_fs3787659953648829301v_real @ F @ Xs ) ) ) ) ).
% distinct_insort_key
thf(fact_601_distinct__insort__key,axiom,
! [F: fset_P8018961893305114765rm_f_v > $o,X: fset_P8018961893305114765rm_f_v,Xs: list_f1824981274722084755rm_f_v] :
( ( distinct_o @ ( map_fs1183804113021257231_f_v_o @ F @ ( linord9188204533028590406_f_v_o @ F @ X @ Xs ) ) )
= ( ~ ( member_o @ ( F @ X ) @ ( image_7665057491163335760_f_v_o @ F @ ( set_fs7270820277574336546rm_f_v @ Xs ) ) )
& ( distinct_o @ ( map_fs1183804113021257231_f_v_o @ F @ Xs ) ) ) ) ).
% distinct_insort_key
thf(fact_602_distinct__insort__key,axiom,
! [F: fset_P8018961893305114765rm_f_v > nat,X: fset_P8018961893305114765rm_f_v,Xs: list_f1824981274722084755rm_f_v] :
( ( distinct_nat @ ( map_fs2439897442395436313_v_nat @ F @ ( linord45516334601593762_v_nat @ F @ X @ Xs ) ) )
= ( ~ ( member_nat @ ( F @ X ) @ ( image_3972722278727975192_v_nat @ F @ ( set_fs7270820277574336546rm_f_v @ Xs ) ) )
& ( distinct_nat @ ( map_fs2439897442395436313_v_nat @ F @ Xs ) ) ) ) ).
% distinct_insort_key
thf(fact_603_divides__aux__eq,axiom,
! [Q2: nat,R2: nat] :
( ( unique5332122412489317741ux_nat @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
= ( R2 = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_604_divides__aux__eq,axiom,
! [Q2: int,R2: int] :
( ( unique5329631941980267465ux_int @ ( product_Pair_int_int @ Q2 @ R2 ) )
= ( R2 = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_605_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_606_relChain__def,axiom,
( bNF_Ca4696266947080128040_f_nat
= ( ^ [R3: set_Pr1261947904930325089at_nat,As: nat > fset_P6228066233360383026_f_nat] :
! [I2: nat,J2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R3 )
=> ( ord_le1552505484586773650_f_nat @ ( As @ I2 ) @ ( As @ J2 ) ) ) ) ) ).
% relChain_def
thf(fact_607_relChain__def,axiom,
( bNF_Ca968750328013420230at_nat
= ( ^ [R3: set_Pr1261947904930325089at_nat,As: nat > nat] :
! [I2: nat,J2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R3 )
=> ( ord_less_eq_nat @ ( As @ I2 ) @ ( As @ J2 ) ) ) ) ) ).
% relChain_def
thf(fact_608_relChain__def,axiom,
( bNF_Ca966259857504369954at_int
= ( ^ [R3: set_Pr1261947904930325089at_nat,As: nat > int] :
! [I2: nat,J2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R3 )
=> ( ord_less_eq_int @ ( As @ I2 ) @ ( As @ J2 ) ) ) ) ) ).
% relChain_def
thf(fact_609_relChain__def,axiom,
( bNF_Ca9191250440166129314t_real
= ( ^ [R3: set_Pr1261947904930325089at_nat,As: nat > real] :
! [I2: nat,J2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R3 )
=> ( ord_less_eq_real @ ( As @ I2 ) @ ( As @ J2 ) ) ) ) ) ).
% relChain_def
thf(fact_610_neg__equal__iff__equal,axiom,
! [A4: int,B3: int] :
( ( ( uminus_uminus_int @ A4 )
= ( uminus_uminus_int @ B3 ) )
= ( A4 = B3 ) ) ).
% neg_equal_iff_equal
thf(fact_611_neg__equal__iff__equal,axiom,
! [A4: real,B3: real] :
( ( ( uminus_uminus_real @ A4 )
= ( uminus_uminus_real @ B3 ) )
= ( A4 = B3 ) ) ).
% neg_equal_iff_equal
thf(fact_612_add_Oinverse__inverse,axiom,
! [A4: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A4 ) )
= A4 ) ).
% add.inverse_inverse
thf(fact_613_add_Oinverse__inverse,axiom,
! [A4: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A4 ) )
= A4 ) ).
% add.inverse_inverse
thf(fact_614_verit__minus__simplify_I4_J,axiom,
! [B3: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B3 ) )
= B3 ) ).
% verit_minus_simplify(4)
thf(fact_615_verit__minus__simplify_I4_J,axiom,
! [B3: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B3 ) )
= B3 ) ).
% verit_minus_simplify(4)
thf(fact_616_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_617_less__option__Some,axiom,
! [X: int,Y: int] :
( ( ord_less_option_int @ ( some_int @ X ) @ ( some_int @ Y ) )
= ( ord_less_int @ X @ Y ) ) ).
% less_option_Some
thf(fact_618_less__option__Some,axiom,
! [X: nat,Y: nat] :
( ( ord_less_option_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ).
% less_option_Some
thf(fact_619_less__option__Some,axiom,
! [X: real,Y: real] :
( ( ord_less_option_real @ ( some_real @ X ) @ ( some_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% less_option_Some
thf(fact_620_neg__le__iff__le,axiom,
! [B3: int,A4: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_eq_int @ A4 @ B3 ) ) ).
% neg_le_iff_le
thf(fact_621_neg__le__iff__le,axiom,
! [B3: real,A4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A4 ) )
= ( ord_less_eq_real @ A4 @ B3 ) ) ).
% neg_le_iff_le
thf(fact_622_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_623_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_624_neg__0__equal__iff__equal,axiom,
! [A4: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A4 ) )
= ( zero_zero_int = A4 ) ) ).
% neg_0_equal_iff_equal
thf(fact_625_neg__0__equal__iff__equal,axiom,
! [A4: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A4 ) )
= ( zero_zero_real = A4 ) ) ).
% neg_0_equal_iff_equal
thf(fact_626_neg__equal__0__iff__equal,axiom,
! [A4: int] :
( ( ( uminus_uminus_int @ A4 )
= zero_zero_int )
= ( A4 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_627_neg__equal__0__iff__equal,axiom,
! [A4: real] :
( ( ( uminus_uminus_real @ A4 )
= zero_zero_real )
= ( A4 = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_628_equal__neg__zero,axiom,
! [A4: int] :
( ( A4
= ( uminus_uminus_int @ A4 ) )
= ( A4 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_629_equal__neg__zero,axiom,
! [A4: real] :
( ( A4
= ( uminus_uminus_real @ A4 ) )
= ( A4 = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_630_neg__equal__zero,axiom,
! [A4: int] :
( ( ( uminus_uminus_int @ A4 )
= A4 )
= ( A4 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_631_neg__equal__zero,axiom,
! [A4: real] :
( ( ( uminus_uminus_real @ A4 )
= A4 )
= ( A4 = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_632_neg__less__iff__less,axiom,
! [B3: int,A4: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_int @ A4 @ B3 ) ) ).
% neg_less_iff_less
thf(fact_633_neg__less__iff__less,axiom,
! [B3: real,A4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A4 ) )
= ( ord_less_real @ A4 @ B3 ) ) ).
% neg_less_iff_less
thf(fact_634_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_635_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_636_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_637_INF__identity__eq,axiom,
! [A2: set_nat] :
( ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X5: nat] : X5
@ A2 ) )
= ( complete_Inf_Inf_nat @ A2 ) ) ).
% INF_identity_eq
thf(fact_638_INF__identity__eq,axiom,
! [A2: set_o] :
( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [X5: $o] : X5
@ A2 ) )
= ( complete_Inf_Inf_o @ A2 ) ) ).
% INF_identity_eq
thf(fact_639_neg__0__le__iff__le,axiom,
! [A4: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_eq_int @ A4 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_640_neg__0__le__iff__le,axiom,
! [A4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A4 ) )
= ( ord_less_eq_real @ A4 @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_641_neg__le__0__iff__le,axiom,
! [A4: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A4 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A4 ) ) ).
% neg_le_0_iff_le
thf(fact_642_neg__le__0__iff__le,axiom,
! [A4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A4 ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A4 ) ) ).
% neg_le_0_iff_le
thf(fact_643_less__eq__neg__nonpos,axiom,
! [A4: int] :
( ( ord_less_eq_int @ A4 @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_eq_int @ A4 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_644_less__eq__neg__nonpos,axiom,
! [A4: real] :
( ( ord_less_eq_real @ A4 @ ( uminus_uminus_real @ A4 ) )
= ( ord_less_eq_real @ A4 @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_645_neg__less__eq__nonneg,axiom,
! [A4: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A4 ) @ A4 )
= ( ord_less_eq_int @ zero_zero_int @ A4 ) ) ).
% neg_less_eq_nonneg
thf(fact_646_neg__less__eq__nonneg,axiom,
! [A4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A4 ) @ A4 )
= ( ord_less_eq_real @ zero_zero_real @ A4 ) ) ).
% neg_less_eq_nonneg
thf(fact_647_neg__less__0__iff__less,axiom,
! [A4: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A4 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A4 ) ) ).
% neg_less_0_iff_less
thf(fact_648_neg__less__0__iff__less,axiom,
! [A4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A4 ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A4 ) ) ).
% neg_less_0_iff_less
thf(fact_649_neg__0__less__iff__less,axiom,
! [A4: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_int @ A4 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_650_neg__0__less__iff__less,axiom,
! [A4: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A4 ) )
= ( ord_less_real @ A4 @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_651_neg__less__pos,axiom,
! [A4: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A4 ) @ A4 )
= ( ord_less_int @ zero_zero_int @ A4 ) ) ).
% neg_less_pos
thf(fact_652_neg__less__pos,axiom,
! [A4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A4 ) @ A4 )
= ( ord_less_real @ zero_zero_real @ A4 ) ) ).
% neg_less_pos
thf(fact_653_less__neg__neg,axiom,
! [A4: int] :
( ( ord_less_int @ A4 @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_int @ A4 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_654_less__neg__neg,axiom,
! [A4: real] :
( ( ord_less_real @ A4 @ ( uminus_uminus_real @ A4 ) )
= ( ord_less_real @ A4 @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_655_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_656_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_657_of__int__minus,axiom,
! [Z: int] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z ) )
= ( uminus_uminus_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_minus
thf(fact_658_of__int__minus,axiom,
! [Z: int] :
( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z ) )
= ( uminus_uminus_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_minus
thf(fact_659_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_660_negative__zle,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zle
thf(fact_661_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_662_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_663_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_664_floor__uminus__of__int,axiom,
! [Z: int] :
( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( ring_1_of_int_real @ Z ) ) )
= ( uminus_uminus_int @ Z ) ) ).
% floor_uminus_of_int
thf(fact_665_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_666_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_667_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_668_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_669_floor__less__zero,axiom,
! [X: real] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% floor_less_zero
thf(fact_670_zero__less__ceiling,axiom,
! [X: real] :
( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% zero_less_ceiling
thf(fact_671_complete__real,axiom,
! [S: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S )
=> ( ? [Z6: real] :
! [X6: real] :
( ( member_real @ X6 @ S )
=> ( ord_less_eq_real @ X6 @ Z6 ) )
=> ? [Y7: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Y7 ) )
& ! [Z6: real] :
( ! [X6: real] :
( ( member_real @ X6 @ S )
=> ( ord_less_eq_real @ X6 @ Z6 ) )
=> ( ord_less_eq_real @ Y7 @ Z6 ) ) ) ) ) ).
% complete_real
thf(fact_672_folding__Map__graph_Odistinct__if__distinct__map,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( distinct_nat @ ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ Xs ) )
=> ( distin6923225563576452346at_nat @ Xs ) ) ).
% folding_Map_graph.distinct_if_distinct_map
thf(fact_673_less__INF__D,axiom,
! [Y: $o,F: real > $o,A2: set_real,I: real] :
( ( ord_less_o @ Y @ ( complete_Inf_Inf_o @ ( image_real_o @ F @ A2 ) ) )
=> ( ( member_real @ I @ A2 )
=> ( ord_less_o @ Y @ ( F @ I ) ) ) ) ).
% less_INF_D
thf(fact_674_less__INF__D,axiom,
! [Y: $o,F: $o > $o,A2: set_o,I: $o] :
( ( ord_less_o @ Y @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) )
=> ( ( member_o @ I @ A2 )
=> ( ord_less_o @ Y @ ( F @ I ) ) ) ) ).
% less_INF_D
thf(fact_675_less__INF__D,axiom,
! [Y: $o,F: nat > $o,A2: set_nat,I: nat] :
( ( ord_less_o @ Y @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A2 ) ) )
=> ( ( member_nat @ I @ A2 )
=> ( ord_less_o @ Y @ ( F @ I ) ) ) ) ).
% less_INF_D
thf(fact_676_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_677_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_678_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_679_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_680_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_681_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_682_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_683_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_684_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_685_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_686_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_687_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_688_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_689_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_690_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_691_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_692_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_693_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_694_order__less__subst2,axiom,
! [A4: int,B3: int,F: int > int,C3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_695_order__less__subst2,axiom,
! [A4: int,B3: int,F: int > nat,C3: nat] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_696_order__less__subst2,axiom,
! [A4: int,B3: int,F: int > real,C3: real] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_697_order__less__subst2,axiom,
! [A4: nat,B3: nat,F: nat > int,C3: int] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_698_order__less__subst2,axiom,
! [A4: nat,B3: nat,F: nat > nat,C3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_699_order__less__subst2,axiom,
! [A4: nat,B3: nat,F: nat > real,C3: real] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_700_order__less__subst2,axiom,
! [A4: real,B3: real,F: real > int,C3: int] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_701_order__less__subst2,axiom,
! [A4: real,B3: real,F: real > nat,C3: nat] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_702_order__less__subst2,axiom,
! [A4: real,B3: real,F: real > real,C3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_703_order__less__subst1,axiom,
! [A4: int,F: int > int,B3: int,C3: int] :
( ( ord_less_int @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_704_order__less__subst1,axiom,
! [A4: int,F: nat > int,B3: nat,C3: nat] :
( ( ord_less_int @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_705_order__less__subst1,axiom,
! [A4: int,F: real > int,B3: real,C3: real] :
( ( ord_less_int @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_706_order__less__subst1,axiom,
! [A4: nat,F: int > nat,B3: int,C3: int] :
( ( ord_less_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_707_order__less__subst1,axiom,
! [A4: nat,F: nat > nat,B3: nat,C3: nat] :
( ( ord_less_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_708_order__less__subst1,axiom,
! [A4: nat,F: real > nat,B3: real,C3: real] :
( ( ord_less_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_709_order__less__subst1,axiom,
! [A4: real,F: int > real,B3: int,C3: int] :
( ( ord_less_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_710_order__less__subst1,axiom,
! [A4: real,F: nat > real,B3: nat,C3: nat] :
( ( ord_less_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_711_order__less__subst1,axiom,
! [A4: real,F: real > real,B3: real,C3: real] :
( ( ord_less_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_712_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_713_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_714_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_715_ord__less__eq__subst,axiom,
! [A4: int,B3: int,F: int > int,C3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_716_ord__less__eq__subst,axiom,
! [A4: int,B3: int,F: int > nat,C3: nat] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_717_ord__less__eq__subst,axiom,
! [A4: int,B3: int,F: int > real,C3: real] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_718_ord__less__eq__subst,axiom,
! [A4: nat,B3: nat,F: nat > int,C3: int] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_719_ord__less__eq__subst,axiom,
! [A4: nat,B3: nat,F: nat > nat,C3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_720_ord__less__eq__subst,axiom,
! [A4: nat,B3: nat,F: nat > real,C3: real] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_721_ord__less__eq__subst,axiom,
! [A4: real,B3: real,F: real > int,C3: int] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_722_ord__less__eq__subst,axiom,
! [A4: real,B3: real,F: real > nat,C3: nat] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_723_ord__less__eq__subst,axiom,
! [A4: real,B3: real,F: real > real,C3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_724_ord__eq__less__subst,axiom,
! [A4: int,F: int > int,B3: int,C3: int] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_725_ord__eq__less__subst,axiom,
! [A4: nat,F: int > nat,B3: int,C3: int] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_726_ord__eq__less__subst,axiom,
! [A4: real,F: int > real,B3: int,C3: int] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_727_ord__eq__less__subst,axiom,
! [A4: int,F: nat > int,B3: nat,C3: nat] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_728_ord__eq__less__subst,axiom,
! [A4: nat,F: nat > nat,B3: nat,C3: nat] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_729_ord__eq__less__subst,axiom,
! [A4: real,F: nat > real,B3: nat,C3: nat] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_730_ord__eq__less__subst,axiom,
! [A4: int,F: real > int,B3: real,C3: real] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_731_ord__eq__less__subst,axiom,
! [A4: nat,F: real > nat,B3: real,C3: real] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_732_ord__eq__less__subst,axiom,
! [A4: real,F: real > real,B3: real,C3: real] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_733_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_734_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_735_order__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_736_order__less__asym_H,axiom,
! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ~ ( ord_less_int @ B3 @ A4 ) ) ).
% order_less_asym'
thf(fact_737_order__less__asym_H,axiom,
! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A4 ) ) ).
% order_less_asym'
thf(fact_738_order__less__asym_H,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ~ ( ord_less_real @ B3 @ A4 ) ) ).
% order_less_asym'
thf(fact_739_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_740_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_741_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_742_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_743_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_744_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_745_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_746_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_747_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_748_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: int,A4: int] :
( ( ord_less_int @ B3 @ A4 )
=> ( A4 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_749_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
=> ( A4 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_750_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: real,A4: real] :
( ( ord_less_real @ B3 @ A4 )
=> ( A4 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_751_order_Ostrict__implies__not__eq,axiom,
! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( A4 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_752_order_Ostrict__implies__not__eq,axiom,
! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( A4 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_753_order_Ostrict__implies__not__eq,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( A4 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_754_dual__order_Ostrict__trans,axiom,
! [B3: int,A4: int,C3: int] :
( ( ord_less_int @ B3 @ A4 )
=> ( ( ord_less_int @ C3 @ B3 )
=> ( ord_less_int @ C3 @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_755_dual__order_Ostrict__trans,axiom,
! [B3: nat,A4: nat,C3: nat] :
( ( ord_less_nat @ B3 @ A4 )
=> ( ( ord_less_nat @ C3 @ B3 )
=> ( ord_less_nat @ C3 @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_756_dual__order_Ostrict__trans,axiom,
! [B3: real,A4: real,C3: real] :
( ( ord_less_real @ B3 @ A4 )
=> ( ( ord_less_real @ C3 @ B3 )
=> ( ord_less_real @ C3 @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_757_minus__less__iff,axiom,
! [A4: int,B3: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A4 ) @ B3 )
= ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ A4 ) ) ).
% minus_less_iff
thf(fact_758_minus__less__iff,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A4 ) @ B3 )
= ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ A4 ) ) ).
% minus_less_iff
thf(fact_759_less__minus__iff,axiom,
! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ ( uminus_uminus_int @ B3 ) )
= ( ord_less_int @ B3 @ ( uminus_uminus_int @ A4 ) ) ) ).
% less_minus_iff
thf(fact_760_less__minus__iff,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ ( uminus_uminus_real @ B3 ) )
= ( ord_less_real @ B3 @ ( uminus_uminus_real @ A4 ) ) ) ).
% less_minus_iff
thf(fact_761_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_762_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_763_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_764_order_Ostrict__trans,axiom,
! [A4: int,B3: int,C3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( ord_less_int @ B3 @ C3 )
=> ( ord_less_int @ A4 @ C3 ) ) ) ).
% order.strict_trans
thf(fact_765_order_Ostrict__trans,axiom,
! [A4: nat,B3: nat,C3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_less_nat @ B3 @ C3 )
=> ( ord_less_nat @ A4 @ C3 ) ) ) ).
% order.strict_trans
thf(fact_766_order_Ostrict__trans,axiom,
! [A4: real,B3: real,C3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ord_less_real @ B3 @ C3 )
=> ( ord_less_real @ A4 @ C3 ) ) ) ).
% order.strict_trans
thf(fact_767_linorder__less__wlog,axiom,
! [P: int > int > $o,A4: int,B3: int] :
( ! [A6: int,B5: int] :
( ( ord_less_int @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: int] : ( P @ A6 @ A6 )
=> ( ! [A6: int,B5: int] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A4 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_768_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A4: nat,B3: nat] :
( ! [A6: nat,B5: nat] :
( ( ord_less_nat @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: nat] : ( P @ A6 @ A6 )
=> ( ! [A6: nat,B5: nat] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A4 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_769_linorder__less__wlog,axiom,
! [P: real > real > $o,A4: real,B3: real] :
( ! [A6: real,B5: real] :
( ( ord_less_real @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: real] : ( P @ A6 @ A6 )
=> ( ! [A6: real,B5: real] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A4 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_770_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X8: nat] : ( P2 @ X8 ) )
= ( ^ [P3: nat > $o] :
? [N4: nat] :
( ( P3 @ N4 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ~ ( P3 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_771_dual__order_Oirrefl,axiom,
! [A4: int] :
~ ( ord_less_int @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_772_dual__order_Oirrefl,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_773_dual__order_Oirrefl,axiom,
! [A4: real] :
~ ( ord_less_real @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_774_minus__equation__iff,axiom,
! [A4: int,B3: int] :
( ( ( uminus_uminus_int @ A4 )
= B3 )
= ( ( uminus_uminus_int @ B3 )
= A4 ) ) ).
% minus_equation_iff
thf(fact_775_minus__equation__iff,axiom,
! [A4: real,B3: real] :
( ( ( uminus_uminus_real @ A4 )
= B3 )
= ( ( uminus_uminus_real @ B3 )
= A4 ) ) ).
% minus_equation_iff
thf(fact_776_equation__minus__iff,axiom,
! [A4: int,B3: int] :
( ( A4
= ( uminus_uminus_int @ B3 ) )
= ( B3
= ( uminus_uminus_int @ A4 ) ) ) ).
% equation_minus_iff
thf(fact_777_equation__minus__iff,axiom,
! [A4: real,B3: real] :
( ( A4
= ( uminus_uminus_real @ B3 ) )
= ( B3
= ( uminus_uminus_real @ A4 ) ) ) ).
% equation_minus_iff
thf(fact_778_dual__order_Oasym,axiom,
! [B3: int,A4: int] :
( ( ord_less_int @ B3 @ A4 )
=> ~ ( ord_less_int @ A4 @ B3 ) ) ).
% dual_order.asym
thf(fact_779_dual__order_Oasym,axiom,
! [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
=> ~ ( ord_less_nat @ A4 @ B3 ) ) ).
% dual_order.asym
thf(fact_780_dual__order_Oasym,axiom,
! [B3: real,A4: real] :
( ( ord_less_real @ B3 @ A4 )
=> ~ ( ord_less_real @ A4 @ B3 ) ) ).
% dual_order.asym
thf(fact_781_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_782_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_783_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_784_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_785_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_786_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_787_less__induct,axiom,
! [P: nat > $o,A4: nat] :
( ! [X6: nat] :
( ! [Y8: nat] :
( ( ord_less_nat @ Y8 @ X6 )
=> ( P @ Y8 ) )
=> ( P @ X6 ) )
=> ( P @ A4 ) ) ).
% less_induct
thf(fact_788_ord__less__eq__trans,axiom,
! [A4: int,B3: int,C3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_int @ A4 @ C3 ) ) ) ).
% ord_less_eq_trans
thf(fact_789_ord__less__eq__trans,axiom,
! [A4: nat,B3: nat,C3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_nat @ A4 @ C3 ) ) ) ).
% ord_less_eq_trans
thf(fact_790_ord__less__eq__trans,axiom,
! [A4: real,B3: real,C3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_real @ A4 @ C3 ) ) ) ).
% ord_less_eq_trans
thf(fact_791_ord__eq__less__trans,axiom,
! [A4: int,B3: int,C3: int] :
( ( A4 = B3 )
=> ( ( ord_less_int @ B3 @ C3 )
=> ( ord_less_int @ A4 @ C3 ) ) ) ).
% ord_eq_less_trans
thf(fact_792_ord__eq__less__trans,axiom,
! [A4: nat,B3: nat,C3: nat] :
( ( A4 = B3 )
=> ( ( ord_less_nat @ B3 @ C3 )
=> ( ord_less_nat @ A4 @ C3 ) ) ) ).
% ord_eq_less_trans
thf(fact_793_ord__eq__less__trans,axiom,
! [A4: real,B3: real,C3: real] :
( ( A4 = B3 )
=> ( ( ord_less_real @ B3 @ C3 )
=> ( ord_less_real @ A4 @ C3 ) ) ) ).
% ord_eq_less_trans
thf(fact_794_order_Oasym,axiom,
! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ~ ( ord_less_int @ B3 @ A4 ) ) ).
% order.asym
thf(fact_795_order_Oasym,axiom,
! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A4 ) ) ).
% order.asym
thf(fact_796_order_Oasym,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ~ ( ord_less_real @ B3 @ A4 ) ) ).
% order.asym
thf(fact_797_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_798_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_799_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_800_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y ) ) ) ).
% dense
thf(fact_801_gt__ex,axiom,
! [X: int] :
? [X_12: int] : ( ord_less_int @ X @ X_12 ) ).
% gt_ex
thf(fact_802_gt__ex,axiom,
! [X: nat] :
? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).
% gt_ex
thf(fact_803_gt__ex,axiom,
! [X: real] :
? [X_12: real] : ( ord_less_real @ X @ X_12 ) ).
% gt_ex
thf(fact_804_lt__ex,axiom,
! [X: int] :
? [Y7: int] : ( ord_less_int @ Y7 @ X ) ).
% lt_ex
thf(fact_805_lt__ex,axiom,
! [X: real] :
? [Y7: real] : ( ord_less_real @ Y7 @ X ) ).
% lt_ex
thf(fact_806_Inf__bool__def,axiom,
( complete_Inf_Inf_o
= ( ^ [A3: set_o] :
~ ( member_o @ $false @ A3 ) ) ) ).
% Inf_bool_def
thf(fact_807_verit__negate__coefficient_I2_J,axiom,
! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A4 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_808_verit__negate__coefficient_I2_J,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A4 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_809_verit__negate__coefficient_I3_J,axiom,
! [A4: int,B3: int] :
( ( A4 = B3 )
=> ( ( uminus_uminus_int @ A4 )
= ( uminus_uminus_int @ B3 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_810_verit__negate__coefficient_I3_J,axiom,
! [A4: real,B3: real] :
( ( A4 = B3 )
=> ( ( uminus_uminus_real @ A4 )
= ( uminus_uminus_real @ B3 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_811_verit__comp__simplify1_I1_J,axiom,
! [A4: int] :
~ ( ord_less_int @ A4 @ A4 ) ).
% verit_comp_simplify1(1)
thf(fact_812_verit__comp__simplify1_I1_J,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ A4 ) ).
% verit_comp_simplify1(1)
thf(fact_813_verit__comp__simplify1_I1_J,axiom,
! [A4: real] :
~ ( ord_less_real @ A4 @ A4 ) ).
% verit_comp_simplify1(1)
thf(fact_814_not__int__zless__negative,axiom,
! [N: nat,M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_815_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_816_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_817_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_818_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_819_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_820_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_821_floor__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
% floor_less_cancel
thf(fact_822_ceiling__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
% ceiling_less_cancel
thf(fact_823_Some__Inf,axiom,
! [A2: set_o] :
( ( some_o @ ( complete_Inf_Inf_o @ A2 ) )
= ( comple2387459607550929125tion_o @ ( image_o_option_o @ some_o @ A2 ) ) ) ).
% Some_Inf
thf(fact_824_Inf__eqI,axiom,
! [A2: set_o,X: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ( ord_less_eq_o @ X @ I3 ) )
=> ( ! [Y7: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_o @ Y7 @ I4 ) )
=> ( ord_less_eq_o @ Y7 @ X ) )
=> ( ( complete_Inf_Inf_o @ A2 )
= X ) ) ) ).
% Inf_eqI
thf(fact_825_Inf__mono,axiom,
! [B: set_o,A2: set_o] :
( ! [B5: $o] :
( ( member_o @ B5 @ B )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ X4 @ B5 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Inf_Inf_o @ B ) ) ) ).
% Inf_mono
thf(fact_826_Inf__lower,axiom,
! [X: $o,A2: set_o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ X ) ) ).
% Inf_lower
thf(fact_827_Inf__lower2,axiom,
! [U: $o,A2: set_o,V: $o] :
( ( member_o @ U @ A2 )
=> ( ( ord_less_eq_o @ U @ V )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_828_le__Inf__iff,axiom,
! [B3: $o,A2: set_o] :
( ( ord_less_eq_o @ B3 @ ( complete_Inf_Inf_o @ A2 ) )
= ( ! [X5: $o] :
( ( member_o @ X5 @ A2 )
=> ( ord_less_eq_o @ B3 @ X5 ) ) ) ) ).
% le_Inf_iff
thf(fact_829_Inf__greatest,axiom,
! [A2: set_o,Z: $o] :
( ! [X6: $o] :
( ( member_o @ X6 @ A2 )
=> ( ord_less_eq_o @ Z @ X6 ) )
=> ( ord_less_eq_o @ Z @ ( complete_Inf_Inf_o @ A2 ) ) ) ).
% Inf_greatest
thf(fact_830_floor__less__iff,axiom,
! [X: real,Z: int] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ Z )
= ( ord_less_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% floor_less_iff
thf(fact_831_INF__cong,axiom,
! [A2: set_nat,B: set_nat,C: nat > nat,D: nat > nat] :
( ( A2 = B )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ B )
=> ( ( C @ X6 )
= ( D @ X6 ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ C @ A2 ) )
= ( complete_Inf_Inf_nat @ ( image_nat_nat @ D @ B ) ) ) ) ) ).
% INF_cong
thf(fact_832_INF__cong,axiom,
! [A2: set_real,B: set_real,C: real > $o,D: real > $o] :
( ( A2 = B )
=> ( ! [X6: real] :
( ( member_real @ X6 @ B )
=> ( ( C @ X6 )
= ( D @ X6 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_real_o @ C @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_real_o @ D @ B ) ) ) ) ) ).
% INF_cong
thf(fact_833_INF__cong,axiom,
! [A2: set_o,B: set_o,C: $o > $o,D: $o > $o] :
( ( A2 = B )
=> ( ! [X6: $o] :
( ( member_o @ X6 @ B )
=> ( ( C @ X6 )
= ( D @ X6 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ C @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ D @ B ) ) ) ) ) ).
% INF_cong
thf(fact_834_INF__cong,axiom,
! [A2: set_nat,B: set_nat,C: nat > $o,D: nat > $o] :
( ( A2 = B )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ B )
=> ( ( C @ X6 )
= ( D @ X6 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ C @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ D @ B ) ) ) ) ) ).
% INF_cong
thf(fact_835_ceiling__minus,axiom,
! [X: real] :
( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_int @ ( archim6058952711729229775r_real @ X ) ) ) ).
% ceiling_minus
thf(fact_836_floor__minus,axiom,
! [X: real] :
( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_int @ ( archim7802044766580827645g_real @ X ) ) ) ).
% floor_minus
thf(fact_837_ceiling__def,axiom,
( archim7802044766580827645g_real
= ( ^ [X5: real] : ( uminus_uminus_int @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X5 ) ) ) ) ) ).
% ceiling_def
thf(fact_838_less__ceiling__iff,axiom,
! [Z: int,X: real] :
( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% less_ceiling_iff
thf(fact_839_le__minus__iff,axiom,
! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ ( uminus_uminus_int @ B3 ) )
= ( ord_less_eq_int @ B3 @ ( uminus_uminus_int @ A4 ) ) ) ).
% le_minus_iff
thf(fact_840_le__minus__iff,axiom,
! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ ( uminus_uminus_real @ B3 ) )
= ( ord_less_eq_real @ B3 @ ( uminus_uminus_real @ A4 ) ) ) ).
% le_minus_iff
thf(fact_841_minus__le__iff,axiom,
! [A4: int,B3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A4 ) @ B3 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ A4 ) ) ).
% minus_le_iff
thf(fact_842_minus__le__iff,axiom,
! [A4: real,B3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A4 ) @ B3 )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ A4 ) ) ).
% minus_le_iff
thf(fact_843_le__imp__neg__le,axiom,
! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A4 ) ) ) ).
% le_imp_neg_le
thf(fact_844_le__imp__neg__le,axiom,
! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A4 ) ) ) ).
% le_imp_neg_le
thf(fact_845_minf_I8_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ~ ( ord_less_eq_nat @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_846_minf_I8_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ~ ( ord_less_eq_int @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_847_minf_I8_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z3 )
=> ~ ( ord_less_eq_real @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_848_minf_I6_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_849_minf_I6_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ord_less_eq_int @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_850_minf_I6_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z3 )
=> ( ord_less_eq_real @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_851_pinf_I8_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_852_pinf_I8_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ord_less_eq_int @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_853_pinf_I8_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ Z3 @ X4 )
=> ( ord_less_eq_real @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_854_pinf_I6_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_855_pinf_I6_J,axiom,
! [T2: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_856_pinf_I6_J,axiom,
! [T2: real] :
? [Z3: real] :
! [X4: real] :
( ( ord_less_real @ Z3 @ X4 )
=> ~ ( ord_less_eq_real @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_857_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A8: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A8 ) )
= ( ord_less_nat @ A8 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_858_verit__comp__simplify1_I3_J,axiom,
! [B6: int,A8: int] :
( ( ~ ( ord_less_eq_int @ B6 @ A8 ) )
= ( ord_less_int @ A8 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_859_verit__comp__simplify1_I3_J,axiom,
! [B6: real,A8: real] :
( ( ~ ( ord_less_eq_real @ B6 @ A8 ) )
= ( ord_less_real @ A8 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_860_leD,axiom,
! [Y: fset_P6228066233360383026_f_nat,X: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ Y @ X )
=> ~ ( ord_le7711977203798163590_f_nat @ X @ Y ) ) ).
% leD
thf(fact_861_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_862_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_863_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_864_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_865_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_866_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_867_nless__le,axiom,
! [A4: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat] :
( ( ~ ( ord_le7711977203798163590_f_nat @ A4 @ B3 ) )
= ( ~ ( ord_le1552505484586773650_f_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ).
% nless_le
thf(fact_868_nless__le,axiom,
! [A4: nat,B3: nat] :
( ( ~ ( ord_less_nat @ A4 @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ).
% nless_le
thf(fact_869_nless__le,axiom,
! [A4: int,B3: int] :
( ( ~ ( ord_less_int @ A4 @ B3 ) )
= ( ~ ( ord_less_eq_int @ A4 @ B3 )
| ( A4 = B3 ) ) ) ).
% nless_le
thf(fact_870_nless__le,axiom,
! [A4: real,B3: real] :
( ( ~ ( ord_less_real @ A4 @ B3 ) )
= ( ~ ( ord_less_eq_real @ A4 @ B3 )
| ( A4 = B3 ) ) ) ).
% nless_le
thf(fact_871_antisym__conv1,axiom,
! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
( ~ ( ord_le7711977203798163590_f_nat @ X @ Y )
=> ( ( ord_le1552505484586773650_f_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_872_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_873_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_874_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_875_antisym__conv2,axiom,
! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ X @ Y )
=> ( ( ~ ( ord_le7711977203798163590_f_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_876_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_877_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_878_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_879_dense__ge,axiom,
! [Z: real,Y: real] :
( ! [X6: real] :
( ( ord_less_real @ Z @ X6 )
=> ( ord_less_eq_real @ Y @ X6 ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_ge
thf(fact_880_dense__le,axiom,
! [Y: real,Z: real] :
( ! [X6: real] :
( ( ord_less_real @ X6 @ Y )
=> ( ord_less_eq_real @ X6 @ Z ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_le
thf(fact_881_less__le__not__le,axiom,
( ord_le7711977203798163590_f_nat
= ( ^ [X5: fset_P6228066233360383026_f_nat,Y5: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ X5 @ Y5 )
& ~ ( ord_le1552505484586773650_f_nat @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_882_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_883_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_884_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_885_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_886_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_887_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_888_order_Oorder__iff__strict,axiom,
( ord_le1552505484586773650_f_nat
= ( ^ [A5: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
( ( ord_le7711977203798163590_f_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_889_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_890_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_int @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_891_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_real @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_892_order_Ostrict__iff__order,axiom,
( ord_le7711977203798163590_f_nat
= ( ^ [A5: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_893_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_894_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_895_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_896_order_Ostrict__trans1,axiom,
! [A4: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat,C3: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ A4 @ B3 )
=> ( ( ord_le7711977203798163590_f_nat @ B3 @ C3 )
=> ( ord_le7711977203798163590_f_nat @ A4 @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_897_order_Ostrict__trans1,axiom,
! [A4: nat,B3: nat,C3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_nat @ B3 @ C3 )
=> ( ord_less_nat @ A4 @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_898_order_Ostrict__trans1,axiom,
! [A4: int,B3: int,C3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( ord_less_int @ B3 @ C3 )
=> ( ord_less_int @ A4 @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_899_order_Ostrict__trans1,axiom,
! [A4: real,B3: real,C3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_real @ B3 @ C3 )
=> ( ord_less_real @ A4 @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_900_order_Ostrict__trans2,axiom,
! [A4: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat,C3: fset_P6228066233360383026_f_nat] :
( ( ord_le7711977203798163590_f_nat @ A4 @ B3 )
=> ( ( ord_le1552505484586773650_f_nat @ B3 @ C3 )
=> ( ord_le7711977203798163590_f_nat @ A4 @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_901_order_Ostrict__trans2,axiom,
! [A4: nat,B3: nat,C3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C3 )
=> ( ord_less_nat @ A4 @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_902_order_Ostrict__trans2,axiom,
! [A4: int,B3: int,C3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C3 )
=> ( ord_less_int @ A4 @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_903_order_Ostrict__trans2,axiom,
! [A4: real,B3: real,C3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( ord_less_real @ A4 @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_904_order_Ostrict__iff__not,axiom,
( ord_le7711977203798163590_f_nat
= ( ^ [A5: fset_P6228066233360383026_f_nat,B4: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ A5 @ B4 )
& ~ ( ord_le1552505484586773650_f_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_905_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_906_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_907_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ~ ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_908_dense__ge__bounded,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ Z @ X )
=> ( ! [W2: real] :
( ( ord_less_real @ Z @ W2 )
=> ( ( ord_less_real @ W2 @ X )
=> ( ord_less_eq_real @ Y @ W2 ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_909_dense__le__bounded,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W2: real] :
( ( ord_less_real @ X @ W2 )
=> ( ( ord_less_real @ W2 @ Y )
=> ( ord_less_eq_real @ W2 @ Z ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_910_dual__order_Oorder__iff__strict,axiom,
( ord_le1552505484586773650_f_nat
= ( ^ [B4: fset_P6228066233360383026_f_nat,A5: fset_P6228066233360383026_f_nat] :
( ( ord_le7711977203798163590_f_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_911_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_912_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_int @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_913_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_real @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_914_dual__order_Ostrict__iff__order,axiom,
( ord_le7711977203798163590_f_nat
= ( ^ [B4: fset_P6228066233360383026_f_nat,A5: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_915_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_916_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_917_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_918_dual__order_Ostrict__trans1,axiom,
! [B3: fset_P6228066233360383026_f_nat,A4: fset_P6228066233360383026_f_nat,C3: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ B3 @ A4 )
=> ( ( ord_le7711977203798163590_f_nat @ C3 @ B3 )
=> ( ord_le7711977203798163590_f_nat @ C3 @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_919_dual__order_Ostrict__trans1,axiom,
! [B3: nat,A4: nat,C3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
=> ( ( ord_less_nat @ C3 @ B3 )
=> ( ord_less_nat @ C3 @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_920_dual__order_Ostrict__trans1,axiom,
! [B3: int,A4: int,C3: int] :
( ( ord_less_eq_int @ B3 @ A4 )
=> ( ( ord_less_int @ C3 @ B3 )
=> ( ord_less_int @ C3 @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_921_dual__order_Ostrict__trans1,axiom,
! [B3: real,A4: real,C3: real] :
( ( ord_less_eq_real @ B3 @ A4 )
=> ( ( ord_less_real @ C3 @ B3 )
=> ( ord_less_real @ C3 @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_922_dual__order_Ostrict__trans2,axiom,
! [B3: fset_P6228066233360383026_f_nat,A4: fset_P6228066233360383026_f_nat,C3: fset_P6228066233360383026_f_nat] :
( ( ord_le7711977203798163590_f_nat @ B3 @ A4 )
=> ( ( ord_le1552505484586773650_f_nat @ C3 @ B3 )
=> ( ord_le7711977203798163590_f_nat @ C3 @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_923_dual__order_Ostrict__trans2,axiom,
! [B3: nat,A4: nat,C3: nat] :
( ( ord_less_nat @ B3 @ A4 )
=> ( ( ord_less_eq_nat @ C3 @ B3 )
=> ( ord_less_nat @ C3 @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_924_dual__order_Ostrict__trans2,axiom,
! [B3: int,A4: int,C3: int] :
( ( ord_less_int @ B3 @ A4 )
=> ( ( ord_less_eq_int @ C3 @ B3 )
=> ( ord_less_int @ C3 @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_925_dual__order_Ostrict__trans2,axiom,
! [B3: real,A4: real,C3: real] :
( ( ord_less_real @ B3 @ A4 )
=> ( ( ord_less_eq_real @ C3 @ B3 )
=> ( ord_less_real @ C3 @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_926_dual__order_Ostrict__iff__not,axiom,
( ord_le7711977203798163590_f_nat
= ( ^ [B4: fset_P6228066233360383026_f_nat,A5: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ B4 @ A5 )
& ~ ( ord_le1552505484586773650_f_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_927_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_928_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_929_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ~ ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_930_order_Ostrict__implies__order,axiom,
! [A4: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat] :
( ( ord_le7711977203798163590_f_nat @ A4 @ B3 )
=> ( ord_le1552505484586773650_f_nat @ A4 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_931_order_Ostrict__implies__order,axiom,
! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ord_less_eq_nat @ A4 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_932_order_Ostrict__implies__order,axiom,
! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( ord_less_eq_int @ A4 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_933_order_Ostrict__implies__order,axiom,
! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( ord_less_eq_real @ A4 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_934_dual__order_Ostrict__implies__order,axiom,
! [B3: fset_P6228066233360383026_f_nat,A4: fset_P6228066233360383026_f_nat] :
( ( ord_le7711977203798163590_f_nat @ B3 @ A4 )
=> ( ord_le1552505484586773650_f_nat @ B3 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_935_dual__order_Ostrict__implies__order,axiom,
! [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
=> ( ord_less_eq_nat @ B3 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_936_dual__order_Ostrict__implies__order,axiom,
! [B3: int,A4: int] :
( ( ord_less_int @ B3 @ A4 )
=> ( ord_less_eq_int @ B3 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_937_dual__order_Ostrict__implies__order,axiom,
! [B3: real,A4: real] :
( ( ord_less_real @ B3 @ A4 )
=> ( ord_less_eq_real @ B3 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_938_order__le__less,axiom,
( ord_le1552505484586773650_f_nat
= ( ^ [X5: fset_P6228066233360383026_f_nat,Y5: fset_P6228066233360383026_f_nat] :
( ( ord_le7711977203798163590_f_nat @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_939_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_940_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X5: int,Y5: int] :
( ( ord_less_int @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_941_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_942_order__less__le,axiom,
( ord_le7711977203798163590_f_nat
= ( ^ [X5: fset_P6228066233360383026_f_nat,Y5: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_943_order__less__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_944_order__less__le,axiom,
( ord_less_int
= ( ^ [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_945_order__less__le,axiom,
( ord_less_real
= ( ^ [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_946_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_947_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_948_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_949_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_950_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_951_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_952_order__less__imp__le,axiom,
! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
( ( ord_le7711977203798163590_f_nat @ X @ Y )
=> ( ord_le1552505484586773650_f_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_953_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_954_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_955_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_956_order__le__neq__trans,axiom,
! [A4: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ A4 @ B3 )
=> ( ( A4 != B3 )
=> ( ord_le7711977203798163590_f_nat @ A4 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_957_order__le__neq__trans,axiom,
! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( A4 != B3 )
=> ( ord_less_nat @ A4 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_958_order__le__neq__trans,axiom,
! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( A4 != B3 )
=> ( ord_less_int @ A4 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_959_order__le__neq__trans,axiom,
! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( A4 != B3 )
=> ( ord_less_real @ A4 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_960_order__neq__le__trans,axiom,
! [A4: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat] :
( ( A4 != B3 )
=> ( ( ord_le1552505484586773650_f_nat @ A4 @ B3 )
=> ( ord_le7711977203798163590_f_nat @ A4 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_961_order__neq__le__trans,axiom,
! [A4: nat,B3: nat] :
( ( A4 != B3 )
=> ( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ord_less_nat @ A4 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_962_order__neq__le__trans,axiom,
! [A4: int,B3: int] :
( ( A4 != B3 )
=> ( ( ord_less_eq_int @ A4 @ B3 )
=> ( ord_less_int @ A4 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_963_order__neq__le__trans,axiom,
! [A4: real,B3: real] :
( ( A4 != B3 )
=> ( ( ord_less_eq_real @ A4 @ B3 )
=> ( ord_less_real @ A4 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_964_order__le__less__trans,axiom,
! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat,Z: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ X @ Y )
=> ( ( ord_le7711977203798163590_f_nat @ Y @ Z )
=> ( ord_le7711977203798163590_f_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_965_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_966_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_967_order__le__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_968_order__less__le__trans,axiom,
! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat,Z: fset_P6228066233360383026_f_nat] :
( ( ord_le7711977203798163590_f_nat @ X @ Y )
=> ( ( ord_le1552505484586773650_f_nat @ Y @ Z )
=> ( ord_le7711977203798163590_f_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_969_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_970_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_971_order__less__le__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_972_order__le__less__subst1,axiom,
! [A4: nat,F: int > nat,B3: int,C3: int] :
( ( ord_less_eq_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_973_order__le__less__subst1,axiom,
! [A4: nat,F: nat > nat,B3: nat,C3: nat] :
( ( ord_less_eq_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_974_order__le__less__subst1,axiom,
! [A4: nat,F: real > nat,B3: real,C3: real] :
( ( ord_less_eq_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_975_order__le__less__subst1,axiom,
! [A4: int,F: int > int,B3: int,C3: int] :
( ( ord_less_eq_int @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_976_order__le__less__subst1,axiom,
! [A4: int,F: nat > int,B3: nat,C3: nat] :
( ( ord_less_eq_int @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_977_order__le__less__subst1,axiom,
! [A4: int,F: real > int,B3: real,C3: real] :
( ( ord_less_eq_int @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_978_order__le__less__subst1,axiom,
! [A4: real,F: int > real,B3: int,C3: int] :
( ( ord_less_eq_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_979_order__le__less__subst1,axiom,
! [A4: real,F: nat > real,B3: nat,C3: nat] :
( ( ord_less_eq_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_980_order__le__less__subst1,axiom,
! [A4: real,F: real > real,B3: real,C3: real] :
( ( ord_less_eq_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_981_order__le__less__subst1,axiom,
! [A4: fset_P6228066233360383026_f_nat,F: int > fset_P6228066233360383026_f_nat,B3: int,C3: int] :
( ( ord_le1552505484586773650_f_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_le7711977203798163590_f_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_le7711977203798163590_f_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_982_order__le__less__subst2,axiom,
! [A4: nat,B3: nat,F: nat > nat,C3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_983_order__le__less__subst2,axiom,
! [A4: nat,B3: nat,F: nat > int,C3: int] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_984_order__le__less__subst2,axiom,
! [A4: nat,B3: nat,F: nat > real,C3: real] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_985_order__le__less__subst2,axiom,
! [A4: int,B3: int,F: int > nat,C3: nat] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_986_order__le__less__subst2,axiom,
! [A4: int,B3: int,F: int > int,C3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_987_order__le__less__subst2,axiom,
! [A4: int,B3: int,F: int > real,C3: real] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_988_order__le__less__subst2,axiom,
! [A4: real,B3: real,F: real > nat,C3: nat] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_989_order__le__less__subst2,axiom,
! [A4: real,B3: real,F: real > int,C3: int] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_990_order__le__less__subst2,axiom,
! [A4: real,B3: real,F: real > real,C3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_991_order__le__less__subst2,axiom,
! [A4: fset_P6228066233360383026_f_nat,B3: fset_P6228066233360383026_f_nat,F: fset_P6228066233360383026_f_nat > nat,C3: nat] :
( ( ord_le1552505484586773650_f_nat @ A4 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C3 )
=> ( ! [X6: fset_P6228066233360383026_f_nat,Y7: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_992_order__less__le__subst1,axiom,
! [A4: nat,F: nat > nat,B3: nat,C3: nat] :
( ( ord_less_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_993_order__less__le__subst1,axiom,
! [A4: int,F: nat > int,B3: nat,C3: nat] :
( ( ord_less_int @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_994_order__less__le__subst1,axiom,
! [A4: real,F: nat > real,B3: nat,C3: nat] :
( ( ord_less_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_eq_nat @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_995_order__less__le__subst1,axiom,
! [A4: nat,F: int > nat,B3: int,C3: int] :
( ( ord_less_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_996_order__less__le__subst1,axiom,
! [A4: int,F: int > int,B3: int,C3: int] :
( ( ord_less_int @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_997_order__less__le__subst1,axiom,
! [A4: real,F: int > real,B3: int,C3: int] :
( ( ord_less_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_eq_int @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_998_order__less__le__subst1,axiom,
! [A4: nat,F: real > nat,B3: real,C3: real] :
( ( ord_less_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_999_order__less__le__subst1,axiom,
! [A4: int,F: real > int,B3: real,C3: real] :
( ( ord_less_int @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1000_order__less__le__subst1,axiom,
! [A4: real,F: real > real,B3: real,C3: real] :
( ( ord_less_real @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_eq_real @ X6 @ Y7 )
=> ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1001_order__less__le__subst1,axiom,
! [A4: nat,F: fset_P6228066233360383026_f_nat > nat,B3: fset_P6228066233360383026_f_nat,C3: fset_P6228066233360383026_f_nat] :
( ( ord_less_nat @ A4 @ ( F @ B3 ) )
=> ( ( ord_le1552505484586773650_f_nat @ B3 @ C3 )
=> ( ! [X6: fset_P6228066233360383026_f_nat,Y7: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ X6 @ Y7 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1002_order__less__le__subst2,axiom,
! [A4: int,B3: int,F: int > nat,C3: nat] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1003_order__less__le__subst2,axiom,
! [A4: nat,B3: nat,F: nat > nat,C3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1004_order__less__le__subst2,axiom,
! [A4: real,B3: real,F: real > nat,C3: nat] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1005_order__less__le__subst2,axiom,
! [A4: int,B3: int,F: int > int,C3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1006_order__less__le__subst2,axiom,
! [A4: nat,B3: nat,F: nat > int,C3: int] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1007_order__less__le__subst2,axiom,
! [A4: real,B3: real,F: real > int,C3: int] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_int @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1008_order__less__le__subst2,axiom,
! [A4: int,B3: int,F: int > real,C3: real] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1009_order__less__le__subst2,axiom,
! [A4: nat,B3: nat,F: nat > real,C3: real] :
( ( ord_less_nat @ A4 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C3 )
=> ( ! [X6: nat,Y7: nat] :
( ( ord_less_nat @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1010_order__less__le__subst2,axiom,
! [A4: real,B3: real,F: real > real,C3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C3 )
=> ( ! [X6: real,Y7: real] :
( ( ord_less_real @ X6 @ Y7 )
=> ( ord_less_real @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1011_order__less__le__subst2,axiom,
! [A4: int,B3: int,F: int > fset_P6228066233360383026_f_nat,C3: fset_P6228066233360383026_f_nat] :
( ( ord_less_int @ A4 @ B3 )
=> ( ( ord_le1552505484586773650_f_nat @ ( F @ B3 ) @ C3 )
=> ( ! [X6: int,Y7: int] :
( ( ord_less_int @ X6 @ Y7 )
=> ( ord_le7711977203798163590_f_nat @ ( F @ X6 ) @ ( F @ Y7 ) ) )
=> ( ord_le7711977203798163590_f_nat @ ( F @ A4 ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1012_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1013_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1014_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1015_order__le__imp__less__or__eq,axiom,
! [X: fset_P6228066233360383026_f_nat,Y: fset_P6228066233360383026_f_nat] :
( ( ord_le1552505484586773650_f_nat @ X @ Y )
=> ( ( ord_le7711977203798163590_f_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1016_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1017_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1018_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1019_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_1020_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1021_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_1022_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_1023_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1024_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1025_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1026_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1027_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1028_reals__Archimedean2,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% reals_Archimedean2
thf(fact_1029_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1030_int__cases2,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_1031_ex__of__int__less,axiom,
! [X: real] :
? [Z3: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X ) ).
% ex_of_int_less
thf(fact_1032_ex__less__of__int,axiom,
! [X: real] :
? [Z3: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% ex_less_of_int
thf(fact_1033_of__int__of__nat,axiom,
( ring_1_of_int_int
= ( ^ [K2: int] : ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K2 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_1034_of__int__of__nat,axiom,
( ring_1_of_int_real
= ( ^ [K2: int] : ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K2 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_1035_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_pos
thf(fact_1036_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_pos
thf(fact_1037_Inf__option__def,axiom,
( comple2387459607550929125tion_o
= ( ^ [A3: set_option_o] : ( if_option_o @ ( member_option_o @ none_o @ A3 ) @ none_o @ ( some_o @ ( complete_Inf_Inf_o @ ( these_o @ A3 ) ) ) ) ) ) ).
% Inf_option_def
thf(fact_1038_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_1039_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_1040_INF__eq,axiom,
! [A2: set_real,B: set_o,G: $o > $o,F: real > $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_real_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1041_INF__eq,axiom,
! [A2: set_real,B: set_nat,G: nat > $o,F: real > $o] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_real_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1042_INF__eq,axiom,
! [A2: set_o,B: set_real,G: real > $o,F: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: real] :
( ( member_real @ X4 @ B )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_real_o @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1043_INF__eq,axiom,
! [A2: set_o,B: set_o,G: $o > $o,F: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1044_INF__eq,axiom,
! [A2: set_o,B: set_nat,G: nat > $o,F: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1045_INF__eq,axiom,
! [A2: set_nat,B: set_real,G: real > $o,F: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: real] :
( ( member_real @ X4 @ B )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_real_o @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1046_INF__eq,axiom,
! [A2: set_nat,B: set_o,G: $o > $o,F: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1047_INF__eq,axiom,
! [A2: set_nat,B: set_nat,G: nat > $o,F: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ( ord_less_eq_o @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1048_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1049_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1050_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1051_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1052_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1053_bot__nat__0_Onot__eq__extremum,axiom,
! [A4: nat] :
( ( A4 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A4 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1054_zless__nat__conj,axiom,
! [W: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W @ Z ) ) ) ).
% zless_nat_conj
thf(fact_1055_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% zero_less_nat_eq
thf(fact_1056_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1057_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1058_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1059_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1060_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_1061_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_1062_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1063_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_1064_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1065_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1066_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1067_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1068_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1069_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1070_bot__nat__0_Oextremum__strict,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1071_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_1072_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1073_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1074_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1075_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
| ( M3 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1076_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_1077_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
& ( M3 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1078_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1079_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1080_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_1081_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1082_nat__mono__iff,axiom,
! [Z: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W @ Z ) ) ) ).
% nat_mono_iff
thf(fact_1083_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_1084_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1085_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1086_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_1087_nat__less__eq__zless,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_1088_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_1089_nat__less__iff,axiom,
! [W: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M2 )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_1090_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K3 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1091_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K3: nat] :
? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( F @ K3 @ I4 ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ! [K4: nat] :
? [K5: nat] :
( ( ord_less_eq_nat @ K4 @ K5 )
& ( F @ K5 @ I3 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_1092_forall__finite_I1_J,axiom,
! [P: nat > $o,I4: nat] :
( ( ord_less_nat @ I4 @ zero_zero_nat )
=> ( P @ I4 ) ) ).
% forall_finite(1)
thf(fact_1093_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1094_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1095_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1096_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1097_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D3: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z7: int,Z8: int] :
( ( ord_less_eq_int @ D3 @ Z8 )
& ( ord_less_int @ Z7 @ Z8 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_1098_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D3: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z7: int,Z8: int] :
( ( ord_less_eq_int @ D3 @ Z7 )
& ( ord_less_int @ Z7 @ Z8 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_1099_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).
% natLess_def
thf(fact_1100_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% one_less_nat_eq
thf(fact_1101_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_1102_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1103_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_1104_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1105_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1106_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_1107_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1108_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1109_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1110_negative__zless,axiom,
! [N: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zless
thf(fact_1111_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X6: nat] : ( R @ X6 @ X6 )
=> ( ! [X6: nat,Y7: nat,Z3: nat] :
( ( R @ X6 @ Y7 )
=> ( ( R @ Y7 @ Z3 )
=> ( R @ X6 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1112_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1113_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1114_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_1115_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1116_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1117_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M6: nat] :
( M5
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_1118_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1119_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1120_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_1121_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1122_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1123_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1124_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_1125_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1126_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1127_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1128_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1129_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1130_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X6: nat] : ( P @ X6 @ zero_zero_nat )
=> ( ! [Y7: nat] : ( P @ zero_zero_nat @ ( suc @ Y7 ) )
=> ( ! [X6: nat,Y7: nat] :
( ( P @ X6 @ Y7 )
=> ( P @ ( suc @ X6 ) @ ( suc @ Y7 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_1131_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1132_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1133_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_1134_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_1135_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_1136_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1137_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_1138_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_1139_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1140_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_1141_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1142_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1143_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_1144_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_1145_Nat_OAll__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% Nat.All_less_Suc
thf(fact_1146_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M7: nat] :
( ( M2
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1147_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_1148_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_1149_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1150_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K3 )
=> ( P @ I3 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1151_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1152_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1153_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1154_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_1155_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J2: nat] :
( ( M2
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1156_forall__finite_I3_J,axiom,
! [X: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ ( suc @ X ) ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ X ) )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% forall_finite(3)
thf(fact_1157_forall__finite_I2_J,axiom,
! [P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ zero_zero_nat ) )
=> ( P @ I2 ) ) )
= ( P @ zero_zero_nat ) ) ).
% forall_finite(2)
thf(fact_1158_ex__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_1159_all__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_1160_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_1161_Comparator__Generator_OAll__less__Suc,axiom,
! [X: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ X ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ X )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% Comparator_Generator.All_less_Suc
thf(fact_1162_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_1163_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_1164_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1165_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1166_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_1167_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1168_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_1169_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1170_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1171_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1172_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_1173_int__cases,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_1174_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1175_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1176_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1177_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N3: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_1178_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1179_floor__eq3,axiom,
! [N: nat,X: real] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= N ) ) ) ).
% floor_eq3
thf(fact_1180_floor__eq4,axiom,
! [N: nat,X: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= N ) ) ) ).
% floor_eq4
thf(fact_1181_inf__concat__simple_Ocases,axiom,
! [X: produc8199716216217303280at_nat] :
( ! [F4: nat > nat] :
( X
!= ( produc72220940542539688at_nat @ F4 @ zero_zero_nat ) )
=> ~ ! [F4: nat > nat,N3: nat] :
( X
!= ( produc72220940542539688at_nat @ F4 @ ( suc @ N3 ) ) ) ) ).
% inf_concat_simple.cases
thf(fact_1182_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_1183_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M ) )
=> ~ ! [M6: nat] :
( ( P @ M6 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M6 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1184_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_1185_pred__nat__def,axiom,
( pred_nat
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [M3: nat,N4: nat] :
( N4
= ( suc @ M3 ) ) ) ) ) ).
% pred_nat_def
thf(fact_1186_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X5: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X5 )
@ ^ [X5: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X5 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1187_Suc__nat__eq__nat__zadd1,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( suc @ ( nat2 @ Z ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1188_inf__concat__simple_Oelims,axiom,
! [X: nat > nat,Xa: nat,Y: product_prod_nat_nat] :
( ( ( inf_concat_simple @ X @ Xa )
= Y )
=> ( ( ( Xa = zero_zero_nat )
=> ( Y
!= ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) )
=> ~ ! [N3: nat] :
( ( Xa
= ( suc @ N3 ) )
=> ( Y
!= ( produc2626176000494625587at_nat
@ ^ [I2: nat,J2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_nat @ ( suc @ J2 ) @ ( X @ I2 ) ) @ ( product_Pair_nat_nat @ I2 @ ( suc @ J2 ) ) @ ( product_Pair_nat_nat @ ( suc @ I2 ) @ zero_zero_nat ) )
@ ( inf_concat_simple @ X @ N3 ) ) ) ) ) ) ).
% inf_concat_simple.elims
thf(fact_1189_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1190_inf__concat__simple_Osimps_I1_J,axiom,
! [F: nat > nat] :
( ( inf_concat_simple @ F @ zero_zero_nat )
= ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ).
% inf_concat_simple.simps(1)
thf(fact_1191_inf__concat__simple__surj__zero,axiom,
! [F: nat > nat,I: nat] :
? [K3: nat] :
( ( inf_concat_simple @ F @ K3 )
= ( product_Pair_nat_nat @ I @ zero_zero_nat ) ) ).
% inf_concat_simple_surj_zero
thf(fact_1192_inf__concat__simple__mono,axiom,
! [K: nat,K6: nat,F: nat > nat] :
( ( ord_less_eq_nat @ K @ K6 )
=> ( ord_less_eq_nat @ ( product_fst_nat_nat @ ( inf_concat_simple @ F @ K ) ) @ ( product_fst_nat_nat @ ( inf_concat_simple @ F @ K6 ) ) ) ) ).
% inf_concat_simple_mono
thf(fact_1193_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1194_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1195_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1196_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_1197_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W3: int,Z8: int] :
? [N4: nat] :
( Z8
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1198_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1199_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1200_int__ops_I4_J,axiom,
! [A4: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A4 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1201_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1202_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W3: int,Z8: int] :
? [N4: nat] :
( Z8
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1203_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1204_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_1205_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1206_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1207_Suc__as__int,axiom,
( suc
= ( ^ [A5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_1208_inf__concat__simple_Osimps_I2_J,axiom,
! [F: nat > nat,N: nat] :
( ( inf_concat_simple @ F @ ( suc @ N ) )
= ( produc2626176000494625587at_nat
@ ^ [I2: nat,J2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_nat @ ( suc @ J2 ) @ ( F @ I2 ) ) @ ( product_Pair_nat_nat @ I2 @ ( suc @ J2 ) ) @ ( product_Pair_nat_nat @ ( suc @ I2 ) @ zero_zero_nat ) )
@ ( inf_concat_simple @ F @ N ) ) ) ).
% inf_concat_simple.simps(2)
thf(fact_1209_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1210_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1211_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_1212_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1213_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1214_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1215_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1216_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1217_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1218_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1219_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N4: nat] :
? [K2: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1220_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1221_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1222_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1223_real__add__le__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_1224_real__0__le__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1225_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M6: nat,N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ( ord_less_nat @ ( F @ M6 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1226_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N4: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N4 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_1227_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1228_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1229_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1230_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N4: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N4 ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).
% int_less_real_le
thf(fact_1231_real__of__int__floor__add__one__ge,axiom,
! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_ge
thf(fact_1232_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1233_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1234_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1235_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1236_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1237_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1238_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1239_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_1240_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_1241_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_1242_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1243_int__ops_I5_J,axiom,
! [A4: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A4 @ B3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).
% int_ops(5)
thf(fact_1244_int__plus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_1245_zadd__int__left,axiom,
! [M2: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_1246_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1247_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1248_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1249_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1250_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1251_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1252_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1253_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1254_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1255_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1256_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1257_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_1258_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A4: nat] :
( ( A2
= ( plus_plus_nat @ K @ A4 ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A4 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1259_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N4: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N4 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).
% nat_less_real_le
thf(fact_1260_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N4: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1261_nat__int__add,axiom,
! [A4: nat,B3: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) )
= ( plus_plus_nat @ A4 @ B3 ) ) ).
% nat_int_add
thf(fact_1262_floor__eq2,axiom,
! [N: int,X: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= N ) ) ) ).
% floor_eq2
% Helper facts (19)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Option__Ooption_I_Eo_J_T,axiom,
! [X: option_o,Y: option_o] :
( ( if_option_o @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_I_Eo_J_T,axiom,
! [X: option_o,Y: option_o] :
( ( if_option_o @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J_T,axiom,
! [X: list_f1824981274722084755rm_f_v,Y: list_f1824981274722084755rm_f_v] :
( ( if_lis137525736101459289rm_f_v @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J_T,axiom,
! [X: list_f1824981274722084755rm_f_v,Y: list_f1824981274722084755rm_f_v] :
( ( if_lis137525736101459289rm_f_v @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J_J_T,axiom,
! [X: option6825207169704394579rm_f_v,Y: option6825207169704394579rm_f_v] :
( ( if_opt1743852810818088729rm_f_v @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J_J_T,axiom,
! [X: option6825207169704394579rm_f_v,Y: option6825207169704394579rm_f_v] :
( ( if_opt1743852810818088729rm_f_v @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Option__Ooption_It__List__Olist_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__List__Olist_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_J_T,axiom,
! [X: option2150321469529786326tion_f,Y: option2150321469529786326tion_f] :
( ( if_opt7165310518185895824tion_f @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__List__Olist_It__Tree____Automata__Oreg_It__Nat__Onat_Mt__Product____Type__Oprod_It__Option__Ooption_Itf__f_J_Mt__Option__Ooption_Itf__f_J_J_J_J_J_T,axiom,
! [X: option2150321469529786326tion_f,Y: option2150321469529786326tion_f] :
( ( if_opt7165310518185895824tion_f @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [Ta1: tree_reg_nat_f] :
( ( ( fOR_rr7289491001293628373el_f_v @ f @ rs @ ( fOR_rr305201390985280028R_ftrs @ i @ r ) )
!= ( some_Tree_reg_nat_f @ Ta1 ) )
| ( rRn_RR1_spec_nat_f @ Ta1 @ ( fOR_eval_rr1_rel_f_v @ ( fset_P3576968334923099475_f_nat @ f ) @ ( map_fs8602507653405230974rm_f_v @ fset_P4617584883882644886rm_f_v @ rs ) @ ( fOR_rr305201390985280028R_ftrs @ i @ r ) ) ) ) ).
%------------------------------------------------------------------------------