TPTP Problem File: SLH0085^1.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 : VYDRA_MDL/0003_Metric_Point_Structure/prob_00068_002585__16157224_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1511 ( 636 unt; 231 typ; 0 def)
% Number of atoms : 3670 (1262 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 10471 ( 397 ~; 53 |; 258 &;8281 @)
% ( 0 <=>;1482 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 1146 (1146 >; 0 *; 0 +; 0 <<)
% Number of symbols : 207 ( 204 usr; 33 con; 0-4 aty)
% Number of variables : 3522 ( 181 ^;3158 !; 183 ?;3522 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:52:20.939
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
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__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Sum_sum_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
set_Pr4934435412358123699_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
set_Pr4193341848836149977_nat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
set_Sum_sum_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mtf__a_J_J,type,
set_Sum_sum_nat_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
list_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Product_prod_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mtf__a_J_J,type,
set_Sum_sum_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
set_option_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
set_option_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Nat__Onat_J_J,type,
set_a_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
set_nat_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
set_a_o: $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__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (204)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
comple8317665133742190828_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_Itf__a_M_Eo_J,type,
complete_Sup_Sup_a_o: set_a_o > a > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
condit2214826472909112428ve_nat: set_nat > $o ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
finite_card_set_nat: set_set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_Itf__a_J,type,
finite_card_set_a: set_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite2115694454571419734at_nat: set_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mtf__a_J,type,
finite_finite_nat_a: set_nat_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_Itf__a_Mt__Nat__Onat_J,type,
finite_finite_a_nat: set_a_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_Itf__a_Mtf__a_J,type,
finite_finite_a_a: set_a_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__Nat__Onat_J,type,
finite5523153139673422903on_nat: set_option_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_Itf__a_J,type,
finite1674126218327898605tion_a: set_option_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
finite659689790015031866_nat_a: set_Pr4193341848836149977_nat_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
finite6644898363146130708_a_nat: set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
finite6544458595007987280od_a_a: set_Product_prod_a_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6187706683773761046at_nat: set_Sum_sum_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mtf__a_J,type,
finite3740268481367103950_nat_a: set_Sum_sum_nat_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
finite502105017643426984_a_nat: set_Sum_sum_a_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_Itf__a_Mtf__a_J,type,
finite51705147264084924um_a_a: set_Sum_sum_a_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
monotone_on_nat_nat: set_nat > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
monoto723715495973462885_set_a: set_nat > ( nat > nat > $o ) > ( set_a > set_a > $o ) > ( nat > set_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
monoto1748750089227133045et_nat: set_set_nat > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > ( set_nat > set_nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
monoto4790297507788910087_a_nat: set_set_a > ( set_a > set_a > $o ) > ( nat > nat > $o ) > ( set_a > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
monoto7172710143293369831_set_a: set_set_a > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > ( set_a > set_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__a_001t__Nat__Onat,type,
monotone_on_a_nat: set_a > ( a > a > $o ) > ( nat > nat > $o ) > ( a > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
groups8294997508430121362at_nat: ( set_nat > nat ) > set_set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001t__Nat__Onat,type,
groups6334556678337121940_a_nat: ( a > nat ) > set_a > nat ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001t__Set__Oset_It__Nat__Onat_J,type,
comple1596078789208929544et_nat: ( set_nat > set_nat ) > set_nat ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001t__Set__Oset_Itf__a_J,type,
comple3341859861669737308_set_a: ( set_a > set_a ) > set_a ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp_001t__Set__Oset_Itf__a_J,type,
comple1558298011288954135_set_a: ( set_a > set_a ) > set_a ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_Eo_Mt__Nat__Onat_J,type,
inf_inf_o_nat: ( $o > nat ) > ( $o > nat ) > $o > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
inf_inf_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
inf_inf_set_nat_a: set_nat_a > set_nat_a > set_nat_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_Itf__a_Mt__Nat__Onat_J_J,type,
inf_inf_set_a_nat: set_a_nat > set_a_nat > set_a_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
inf_inf_set_a_a: set_a_a > set_a_a > set_a_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_It__Nat__Onat_J,type,
semila1667268886620078168et_nat: ( set_nat > set_nat > set_nat ) > set_nat > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_Itf__a_J,type,
semila2496817875450240012_set_a: ( set_a > set_a > set_a ) > set_a > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
lattic8721135487736765967in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Nat__Onat,type,
lattic6340287419671400565_a_nat: ( a > nat ) > set_a > a ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Nat__Onat,type,
lattic6009151579333465974et_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Set__Oset_Itf__a_J,type,
lattic8986249270076014136_set_a: ( set_a > set_a > set_a ) > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_001t__Nat__Onat,type,
lattic1029310888574255042et_nat: ( nat > nat > nat ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_001t__Set__Oset_Itf__a_J,type,
lattic1258622339881844972_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF_001t__Nat__Onat,type,
lattic7742739596368939638_F_nat: ( nat > nat > nat ) > set_nat > nat ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
append_list_a: list_list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Obind_001tf__a_001tf__a,type,
bind_a_a: list_a > ( a > list_a ) > list_a ).
thf(sy_c_List_Obutlast_001tf__a,type,
butlast_a: list_a > list_a ).
thf(sy_c_List_Oconcat_001tf__a,type,
concat_a: list_list_a > list_a ).
thf(sy_c_List_Ogen__length_001tf__a,type,
gen_length_a: nat > list_a > nat ).
thf(sy_c_List_Oinsert_001tf__a,type,
insert_a: a > list_a > list_a ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
cons_list_list_a: list_list_a > list_list_list_a > list_list_list_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
nil_list_list_a: list_list_list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist__ex1_001tf__a,type,
list_ex1_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Omaps_001tf__a_001tf__a,type,
maps_a_a: ( a > list_a ) > list_a > list_a ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Oord_Olexordp_001t__List__Olist_Itf__a_J,type,
lexordp_list_a: ( list_a > list_a > $o ) > list_list_a > list_list_a > $o ).
thf(sy_c_List_Oord_Olexordp_001t__Nat__Onat,type,
lexordp_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
thf(sy_c_List_Oord_Olexordp_001tf__a,type,
lexordp_a: ( a > a > $o ) > list_a > list_a > $o ).
thf(sy_c_List_Oord_Olexordp__eq_001t__List__Olist_Itf__a_J,type,
lexordp_eq_list_a: ( list_a > list_a > $o ) > list_list_a > list_list_a > $o ).
thf(sy_c_List_Oord_Olexordp__eq_001t__Nat__Onat,type,
lexordp_eq_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
thf(sy_c_List_Oord_Olexordp__eq_001tf__a,type,
lexordp_eq_a: ( a > a > $o ) > list_a > list_a > $o ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
compow8708494347934031032et_nat: nat > ( set_nat > set_nat ) > set_nat > set_nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
compow_set_a_set_a: nat > ( set_a > set_a ) > set_a > set_a ).
thf(sy_c_Nat_Ocompow_001_062_Itf__a_Mtf__a_J,type,
compow_a_a: nat > ( a > a ) > a > a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord_Omax_001tf__a,type,
max_a: ( a > a > $o ) > a > a > a ).
thf(sy_c_Orderings_Oord_Omin_001tf__a,type,
min_a: ( a > a > $o ) > a > a > a ).
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__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $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__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__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Nat__Onat_J,type,
ordering_top_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_Itf__a_J,type,
ordering_top_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Nat__Onat_Mtf__a_J_M_Eo_J,type,
top_top_nat_a_o: ( nat > a ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_Itf__a_Mt__Nat__Onat_J_M_Eo_J,type,
top_top_a_nat_o: ( a > nat ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_Itf__a_Mtf__a_J_M_Eo_J,type,
top_top_a_a_o: ( a > a ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
top_top_a_o: a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_top_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
top_top_set_nat_a: set_nat_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mt__Nat__Onat_J_J,type,
top_top_set_a_nat: set_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
top_top_set_a_a: set_a_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
top_to8920198386146353926on_nat: set_option_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
top_top_set_option_a: set_option_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
top_to2612598781856825737_nat_a: set_Pr4193341848836149977_nat_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
top_to3353692345378799459_a_nat: set_Pr4934435412358123699_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
top_to8063371432257647191od_a_a: set_Product_prod_a_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
top_top_set_set_a: set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to6661820994512907621at_nat: set_Sum_sum_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mtf__a_J_J,type,
top_to54524901450547413_nat_a: set_Sum_sum_nat_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
top_to795618464972521135_a_nat: set_Sum_sum_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mtf__a_J_J,type,
top_to8848906000605539851um_a_a: set_Sum_sum_a_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mt__Nat__Onat_J,type,
collect_a_nat: ( ( a > nat ) > $o ) > set_a_nat ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mtf__a_J,type,
collect_a_a: ( ( a > a ) > $o ) > set_a_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_Itf__a_M_Eo_J_001t__Set__Oset_Itf__a_J,type,
image_a_o_set_a: ( ( a > $o ) > set_a ) > set_a_o > set_set_a ).
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_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set__Interval_Oord_OatLeastAtMost_001t__Nat__Onat,type,
set_at8086275982456994279st_nat: ( nat > nat > $o ) > nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord_OatLeastAtMost_001tf__a,type,
set_atLeastAtMost_a: ( a > a > $o ) > a > a > set_a ).
thf(sy_c_Set__Interval_Oord_OatLeastLessThan_001t__Nat__Onat,type,
set_at5771681743635938059an_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord_OatLeastLessThan_001tf__a,type,
set_at7181854700642260099Than_a: ( a > a > $o ) > ( a > a > $o ) > a > a > set_a ).
thf(sy_c_Set__Interval_Oord_OatLeast_001t__Nat__Onat,type,
set_atLeast_nat: ( nat > nat > $o ) > nat > set_nat ).
thf(sy_c_Set__Interval_Oord_OatLeast_001tf__a,type,
set_atLeast_a: ( a > a > $o ) > a > set_a ).
thf(sy_c_Set__Interval_Oord_OatMost_001t__Nat__Onat,type,
set_atMost_nat: ( nat > nat > $o ) > nat > set_nat ).
thf(sy_c_Set__Interval_Oord_OatMost_001tf__a,type,
set_atMost_a: ( a > a > $o ) > a > set_a ).
thf(sy_c_Set__Interval_Oord_OgreaterThanAtMost_001t__Nat__Onat,type,
set_gr2512719926906771852st_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord_OgreaterThanAtMost_001tf__a,type,
set_gr7381147536065975362Most_a: ( a > a > $o ) > ( a > a > $o ) > a > a > set_a ).
thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan_001t__Nat__Onat,type,
set_gr7355061613916526640an_nat: ( nat > nat > $o ) > nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_gr8761010429814653926et_nat: ( set_nat > set_nat > $o ) > set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan_001t__Set__Oset_Itf__a_J,type,
set_gr890836759843348478_set_a: ( set_a > set_a > $o ) > set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan_001tf__a,type,
set_gr5113148517155960478Than_a: ( a > a > $o ) > a > a > set_a ).
thf(sy_c_Set__Interval_Oord_OgreaterThan_001t__Nat__Onat,type,
set_greaterThan_nat: ( nat > nat > $o ) > nat > set_nat ).
thf(sy_c_Set__Interval_Oord_OgreaterThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_gr7331712898236694572et_nat: ( set_nat > set_nat > $o ) > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord_OgreaterThan_001t__Set__Oset_Itf__a_J,type,
set_gr7079423240508057912_set_a: ( set_a > set_a > $o ) > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord_OgreaterThan_001tf__a,type,
set_greaterThan_a: ( a > a > $o ) > a > set_a ).
thf(sy_c_Set__Interval_Oord_OlessThan_001t__Nat__Onat,type,
set_lessThan_nat: ( nat > nat > $o ) > nat > set_nat ).
thf(sy_c_Set__Interval_Oord_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_lessThan_set_nat: ( set_nat > set_nat > $o ) > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord_OlessThan_001t__Set__Oset_Itf__a_J,type,
set_lessThan_set_a: ( set_a > set_a > $o ) > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord_OlessThan_001tf__a,type,
set_lessThan_a: ( a > a > $o ) > a > set_a ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Nat__Onat_J,type,
member_a_nat: ( a > nat ) > set_a_nat > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v__092_060iota_062,type,
iota: nat > a ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_c,type,
c: a ).
thf(sy_v_less,type,
less: a > a > $o ).
thf(sy_v_less__eq,type,
less_eq: a > a > $o ).
thf(sy_v_plus,type,
plus: a > a > a ).
thf(sy_v_sup,type,
sup: a > a > a ).
thf(sy_v_tfin,type,
tfin: set_a ).
thf(sy_v_zero,type,
zero: a ).
% Relevant facts (1274)
thf(fact_0_local_O_092_060Delta_0621,axiom,
! [X: a,X2: a] :
( ( plus @ X @ X2 )
= ( plus @ X2 @ X ) ) ).
% local.\<Delta>1
thf(fact_1_local_O_092_060Delta_0623,axiom,
! [X: a] :
( ( plus @ X @ zero )
= X ) ).
% local.\<Delta>3
thf(fact_2_local_Oord__eq__less__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( less @ B @ C )
=> ( less @ A @ C ) ) ) ).
% local.ord_eq_less_trans
thf(fact_3_local_Oord__less__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( less @ A @ B )
=> ( ( B = C )
=> ( less @ A @ C ) ) ) ).
% local.ord_less_eq_trans
thf(fact_4_local_O_092_060Delta_0622,axiom,
! [X: a,X2: a,X3: a] :
( ( plus @ ( plus @ X @ X2 ) @ X3 )
= ( plus @ X @ ( plus @ X2 @ X3 ) ) ) ).
% local.\<Delta>2
thf(fact_5_local_O_092_060Delta_0624,axiom,
! [X: a,X2: a,X3: a] :
( ( ( plus @ X @ X2 )
= ( plus @ X @ X3 ) )
=> ( X2 = X3 ) ) ).
% local.\<Delta>4
thf(fact_6_local_O_092_060Delta_0624_H,axiom,
! [X: a,X3: a,X2: a] :
( ( ( plus @ X @ X3 )
= ( plus @ X2 @ X3 ) )
=> ( X = X2 ) ) ).
% local.\<Delta>4'
thf(fact_7_local_O_092_060Delta_0626,axiom,
! [X: a,X2: a] :
? [X4: a] :
( ( X
= ( plus @ X2 @ X4 ) )
| ( X2
= ( plus @ X @ X4 ) ) ) ).
% local.\<Delta>6
thf(fact_8_local_O_092_060Delta_0623_H,axiom,
! [X: a] :
( X
= ( plus @ zero @ X ) ) ).
% local.\<Delta>3'
thf(fact_9_local_O_092_060Delta_0625,axiom,
! [X: a,X2: a] :
( ( ( plus @ X @ X2 )
= zero )
=> ( X = zero ) ) ).
% local.\<Delta>5
thf(fact_10_local_O_092_060Delta_0625_H,axiom,
! [X: a,X2: a] :
( ( ( plus @ X @ X2 )
= zero )
=> ( X2 = zero ) ) ).
% local.\<Delta>5'
thf(fact_11_local_Ometric__domain__lt__def,axiom,
! [X: a,X2: a] :
( ( less @ X @ X2 )
= ( ? [X5: a] :
( ( X5 != zero )
& ( X2
= ( plus @ X @ X5 ) ) ) ) ) ).
% local.metric_domain_lt_def
thf(fact_12_local_Ometric__domain__tfin__def,axiom,
tfin = top_top_set_a ).
% local.metric_domain_tfin_def
thf(fact_13_local_Olexordp__eq__refl,axiom,
! [Xs: list_a] : ( lexordp_eq_a @ less @ Xs @ Xs ) ).
% local.lexordp_eq_refl
thf(fact_14_local_Olexordp__irreflexive,axiom,
! [Xs: list_a] :
( ! [X6: a] :
~ ( less @ X6 @ X6 )
=> ~ ( lexordp_a @ less @ Xs @ Xs ) ) ).
% local.lexordp_irreflexive
thf(fact_15_local_OgreaterThanLessThan__iff,axiom,
! [I: a,L: a,U: a] :
( ( member_a @ I @ ( set_gr5113148517155960478Than_a @ less @ L @ U ) )
= ( ( less @ L @ I )
& ( less @ I @ U ) ) ) ).
% local.greaterThanLessThan_iff
thf(fact_16_local_OgreaterThan__iff,axiom,
! [I: a,K: a] :
( ( member_a @ I @ ( set_greaterThan_a @ less @ K ) )
= ( less @ K @ I ) ) ).
% local.greaterThan_iff
thf(fact_17_local_OlessThan__iff,axiom,
! [I: a,K: a] :
( ( member_a @ I @ ( set_lessThan_a @ less @ K ) )
= ( less @ I @ K ) ) ).
% local.lessThan_iff
thf(fact_18_local_Ometric__domain__pos,axiom,
! [X: a] : ( less_eq @ zero @ X ) ).
% local.metric_domain_pos
thf(fact_19_local_Ometric__domain__le__def,axiom,
! [X: a,X2: a] :
( ( less_eq @ X @ X2 )
= ( ? [X5: a] :
( X2
= ( plus @ X @ X5 ) ) ) ) ).
% local.metric_domain_le_def
thf(fact_20__092_060open_062_092_060And_062y_Ax_O_A_Ix_A_060_Ay_J_A_061_A_Ix_A_092_060le_062_Ay_A_092_060and_062_A_092_060not_062_Ay_A_092_060le_062_Ax_J_092_060close_062,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
= ( ( less_eq @ X @ Y )
& ~ ( less_eq @ Y @ X ) ) ) ).
% \<open>\<And>y x. (x < y) = (x \<le> y \<and> \<not> y \<le> x)\<close>
thf(fact_21_local_Oless__eq__le__neq,axiom,
! [X: a,X2: a] :
( ( less @ X @ X2 )
= ( ( less_eq @ X @ X2 )
& ( X != X2 ) ) ) ).
% local.less_eq_le_neq
thf(fact_22_local_Ometric__domain___092_060iota_062__progressing,axiom,
! [I: nat,X: a] :
? [J: nat] :
~ ( less_eq @ ( iota @ J ) @ ( plus @ ( iota @ I ) @ X ) ) ).
% local.metric_domain_\<iota>_progressing
thf(fact_23_local_Olexordp__eq_ONil,axiom,
! [Ys: list_a] : ( lexordp_eq_a @ less @ nil_a @ Ys ) ).
% local.lexordp_eq.Nil
thf(fact_24__092_060open_062_092_060And_062y_Ax_O_A_092_060lbrakk_062x_A_092_060le_062_Ay_059_Ay_A_092_060le_062_Ax_092_060rbrakk_062_A_092_060Longrightarrow_062_Ax_A_061_Ay_092_060close_062,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
=> ( ( less_eq @ Y @ X )
=> ( X = Y ) ) ) ).
% \<open>\<And>y x. \<lbrakk>x \<le> y; y \<le> x\<rbrakk> \<Longrightarrow> x = y\<close>
thf(fact_25_local_Oord__eq__le__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( less_eq @ B @ C )
=> ( less_eq @ A @ C ) ) ) ).
% local.ord_eq_le_trans
thf(fact_26_local_Oord__le__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( less_eq @ A @ B )
=> ( ( B = C )
=> ( less_eq @ A @ C ) ) ) ).
% local.ord_le_eq_trans
thf(fact_27_local_Ometric__domain__sup__def,axiom,
! [X: a,X2: a] :
( ( ( less_eq @ X @ X2 )
=> ( ( sup @ X @ X2 )
= X2 ) )
& ( ~ ( less_eq @ X @ X2 )
=> ( ( sup @ X @ X2 )
= X ) ) ) ).
% local.metric_domain_sup_def
thf(fact_28_local_Olexordp__into__lexordp__eq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( lexordp_a @ less @ Xs @ Ys )
=> ( lexordp_eq_a @ less @ Xs @ Ys ) ) ).
% local.lexordp_into_lexordp_eq
thf(fact_29_local_Omax__def,axiom,
! [A: a,B: a] :
( ( ( less_eq @ A @ B )
=> ( ( max_a @ less_eq @ A @ B )
= B ) )
& ( ~ ( less_eq @ A @ B )
=> ( ( max_a @ less_eq @ A @ B )
= A ) ) ) ).
% local.max_def
thf(fact_30_local_Omin__def,axiom,
! [A: a,B: a] :
( ( ( less_eq @ A @ B )
=> ( ( min_a @ less_eq @ A @ B )
= A ) )
& ( ~ ( less_eq @ A @ B )
=> ( ( min_a @ less_eq @ A @ B )
= B ) ) ) ).
% local.min_def
thf(fact_31_local_Olexordp__simps_I1_J,axiom,
! [Ys: list_a] :
( ( lexordp_a @ less @ nil_a @ Ys )
= ( Ys != nil_a ) ) ).
% local.lexordp_simps(1)
thf(fact_32_local_Olexordp__simps_I2_J,axiom,
! [Xs: list_a] :
~ ( lexordp_a @ less @ Xs @ nil_a ) ).
% local.lexordp_simps(2)
thf(fact_33_local_Olexordp__eq__simps_I1_J,axiom,
! [Ys: list_a] : ( lexordp_eq_a @ less @ nil_a @ Ys ) ).
% local.lexordp_eq_simps(1)
thf(fact_34_local_Olexordp__eq__simps_I2_J,axiom,
! [Xs: list_a] :
( ( lexordp_eq_a @ less @ Xs @ nil_a )
= ( Xs = nil_a ) ) ).
% local.lexordp_eq_simps(2)
thf(fact_35_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: list_a > list_a > $o,Xs: list_list_a] :
( ( lexordp_eq_list_a @ Less @ Xs @ nil_list_a )
= ( Xs = nil_list_a ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_36_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: nat > nat > $o,Xs: list_nat] :
( ( lexordp_eq_nat @ Less @ Xs @ nil_nat )
= ( Xs = nil_nat ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_37_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: a > a > $o,Xs: list_a] :
( ( lexordp_eq_a @ Less @ Xs @ nil_a )
= ( Xs = nil_a ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_38_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: list_a > list_a > $o,Ys: list_list_a] : ( lexordp_eq_list_a @ Less @ nil_list_a @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_39_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: nat > nat > $o,Ys: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_40_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: a > a > $o,Ys: list_a] : ( lexordp_eq_a @ Less @ nil_a @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_41_ord_Olexordp__simps_I2_J,axiom,
! [Less: list_a > list_a > $o,Xs: list_list_a] :
~ ( lexordp_list_a @ Less @ Xs @ nil_list_a ) ).
% ord.lexordp_simps(2)
thf(fact_42_ord_Olexordp__simps_I2_J,axiom,
! [Less: nat > nat > $o,Xs: list_nat] :
~ ( lexordp_nat @ Less @ Xs @ nil_nat ) ).
% ord.lexordp_simps(2)
thf(fact_43_ord_Olexordp__simps_I2_J,axiom,
! [Less: a > a > $o,Xs: list_a] :
~ ( lexordp_a @ Less @ Xs @ nil_a ) ).
% ord.lexordp_simps(2)
thf(fact_44_ord_Olexordp__simps_I1_J,axiom,
! [Less: list_a > list_a > $o,Ys: list_list_a] :
( ( lexordp_list_a @ Less @ nil_list_a @ Ys )
= ( Ys != nil_list_a ) ) ).
% ord.lexordp_simps(1)
thf(fact_45_ord_Olexordp__simps_I1_J,axiom,
! [Less: nat > nat > $o,Ys: list_nat] :
( ( lexordp_nat @ Less @ nil_nat @ Ys )
= ( Ys != nil_nat ) ) ).
% ord.lexordp_simps(1)
thf(fact_46_ord_Olexordp__simps_I1_J,axiom,
! [Less: a > a > $o,Ys: list_a] :
( ( lexordp_a @ Less @ nil_a @ Ys )
= ( Ys != nil_a ) ) ).
% ord.lexordp_simps(1)
thf(fact_47_local_OgreaterThanLessThan__eq,axiom,
! [A: a,B: a] :
( ( set_gr5113148517155960478Than_a @ less @ A @ B )
= ( inf_inf_set_a @ ( set_greaterThan_a @ less @ A ) @ ( set_lessThan_a @ less @ B ) ) ) ).
% local.greaterThanLessThan_eq
thf(fact_48_local_OgreaterThanLessThan__def,axiom,
! [L: a,U: a] :
( ( set_gr5113148517155960478Than_a @ less @ L @ U )
= ( inf_inf_set_a @ ( set_greaterThan_a @ less @ L ) @ ( set_lessThan_a @ less @ U ) ) ) ).
% local.greaterThanLessThan_def
thf(fact_49_local_Ometric__domain___092_060iota_062__mono,axiom,
! [I: nat,J2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( less_eq @ ( iota @ I ) @ ( iota @ J2 ) ) ) ).
% local.metric_domain_\<iota>_mono
thf(fact_50_local_Olexordp__eq_Osimps,axiom,
! [A1: list_a,A2: list_a] :
( ( lexordp_eq_a @ less @ A1 @ A2 )
= ( ? [Ys2: list_a] :
( ( A1 = nil_a )
& ( A2 = Ys2 ) )
| ? [X7: a,Y2: a,Xs2: list_a,Ys2: list_a] :
( ( A1
= ( cons_a @ X7 @ Xs2 ) )
& ( A2
= ( cons_a @ Y2 @ Ys2 ) )
& ( less @ X7 @ Y2 ) )
| ? [X7: a,Y2: a,Xs2: list_a,Ys2: list_a] :
( ( A1
= ( cons_a @ X7 @ Xs2 ) )
& ( A2
= ( cons_a @ Y2 @ Ys2 ) )
& ~ ( less @ X7 @ Y2 )
& ~ ( less @ Y2 @ X7 )
& ( lexordp_eq_a @ less @ Xs2 @ Ys2 ) ) ) ) ).
% local.lexordp_eq.simps
thf(fact_51_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A3: set_set_a] :
( ( collect_set_a
@ ^ [X7: set_a] : ( member_set_a @ X7 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A3: set_set_nat] :
( ( collect_set_nat
@ ^ [X7: set_nat] : ( member_set_nat @ X7 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X7: a] : ( member_a @ X7 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X7: nat] : ( member_nat @ X7 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_59_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X6: nat] :
( ( P @ X6 )
= ( Q @ X6 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_60_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X6: a] :
( ( P @ X6 )
= ( Q @ X6 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_61_local_Olexordp__eq_Ocases,axiom,
! [A1: list_a,A2: list_a] :
( ( lexordp_eq_a @ less @ A1 @ A2 )
=> ( ( A1 != nil_a )
=> ( ! [X6: a] :
( ? [Xs3: list_a] :
( A1
= ( cons_a @ X6 @ Xs3 ) )
=> ! [Y3: a] :
( ? [Ys3: list_a] :
( A2
= ( cons_a @ Y3 @ Ys3 ) )
=> ~ ( less @ X6 @ Y3 ) ) )
=> ~ ! [X6: a,Y3: a,Xs3: list_a] :
( ( A1
= ( cons_a @ X6 @ Xs3 ) )
=> ! [Ys3: list_a] :
( ( A2
= ( cons_a @ Y3 @ Ys3 ) )
=> ( ~ ( less @ X6 @ Y3 )
=> ( ~ ( less @ Y3 @ X6 )
=> ~ ( lexordp_eq_a @ less @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% local.lexordp_eq.cases
thf(fact_62_local_Olexordp_Osimps,axiom,
! [A1: list_a,A2: list_a] :
( ( lexordp_a @ less @ A1 @ A2 )
= ( ? [Y2: a,Ys2: list_a] :
( ( A1 = nil_a )
& ( A2
= ( cons_a @ Y2 @ Ys2 ) ) )
| ? [X7: a,Y2: a,Xs2: list_a,Ys2: list_a] :
( ( A1
= ( cons_a @ X7 @ Xs2 ) )
& ( A2
= ( cons_a @ Y2 @ Ys2 ) )
& ( less @ X7 @ Y2 ) )
| ? [X7: a,Y2: a,Xs2: list_a,Ys2: list_a] :
( ( A1
= ( cons_a @ X7 @ Xs2 ) )
& ( A2
= ( cons_a @ Y2 @ Ys2 ) )
& ~ ( less @ X7 @ Y2 )
& ~ ( less @ Y2 @ X7 )
& ( lexordp_a @ less @ Xs2 @ Ys2 ) ) ) ) ).
% local.lexordp.simps
thf(fact_63_local_Olexordp_OCons,axiom,
! [X: a,Y: a,Xs: list_a,Ys: list_a] :
( ( less @ X @ Y )
=> ( lexordp_a @ less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ).
% local.lexordp.Cons
thf(fact_64_local_Olexordp_OCons__eq,axiom,
! [X: a,Y: a,Xs: list_a,Ys: list_a] :
( ~ ( less @ X @ Y )
=> ( ~ ( less @ Y @ X )
=> ( ( lexordp_a @ less @ Xs @ Ys )
=> ( lexordp_a @ less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ) ) ).
% local.lexordp.Cons_eq
thf(fact_65_local_Olexordp__eq_OCons,axiom,
! [X: a,Y: a,Xs: list_a,Ys: list_a] :
( ( less @ X @ Y )
=> ( lexordp_eq_a @ less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ).
% local.lexordp_eq.Cons
thf(fact_66_local_Olexordp__eq_OCons__eq,axiom,
! [X: a,Y: a,Xs: list_a,Ys: list_a] :
( ~ ( less @ X @ Y )
=> ( ~ ( less @ Y @ X )
=> ( ( lexordp_eq_a @ less @ Xs @ Ys )
=> ( lexordp_eq_a @ less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ) ) ).
% local.lexordp_eq.Cons_eq
thf(fact_67_list_Oinject,axiom,
! [X21: list_a,X22: list_list_a,Y21: list_a,Y22: list_list_a] :
( ( ( cons_list_a @ X21 @ X22 )
= ( cons_list_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_68_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_69_local_Olexordp_ONil,axiom,
! [Y: a,Ys: list_a] : ( lexordp_a @ less @ nil_a @ ( cons_a @ Y @ Ys ) ) ).
% local.lexordp.Nil
thf(fact_70_local_Olexordp_Ocases,axiom,
! [A1: list_a,A2: list_a] :
( ( lexordp_a @ less @ A1 @ A2 )
=> ( ( ( A1 = nil_a )
=> ! [Y3: a,Ys3: list_a] :
( A2
!= ( cons_a @ Y3 @ Ys3 ) ) )
=> ( ! [X6: a] :
( ? [Xs3: list_a] :
( A1
= ( cons_a @ X6 @ Xs3 ) )
=> ! [Y3: a] :
( ? [Ys3: list_a] :
( A2
= ( cons_a @ Y3 @ Ys3 ) )
=> ~ ( less @ X6 @ Y3 ) ) )
=> ~ ! [X6: a,Y3: a,Xs3: list_a] :
( ( A1
= ( cons_a @ X6 @ Xs3 ) )
=> ! [Ys3: list_a] :
( ( A2
= ( cons_a @ Y3 @ Ys3 ) )
=> ( ~ ( less @ X6 @ Y3 )
=> ( ~ ( less @ Y3 @ X6 )
=> ~ ( lexordp_a @ less @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% local.lexordp.cases
thf(fact_71_ord_Olexordp__simps_I3_J,axiom,
! [Less: list_a > list_a > $o,X: list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a] :
( ( lexordp_list_a @ Less @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_list_a @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_simps(3)
thf(fact_72_ord_Olexordp__simps_I3_J,axiom,
! [Less: a > a > $o,X: a,Xs: list_a,Y: a,Ys: list_a] :
( ( lexordp_a @ Less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_a @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_simps(3)
thf(fact_73_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: list_a > list_a > $o,X: list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a] :
( ( lexordp_eq_list_a @ Less @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_eq_list_a @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_74_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: a > a > $o,X: a,Xs: list_a,Y: a,Ys: list_a] :
( ( lexordp_eq_a @ Less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_eq_a @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_75_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: nat > nat > $o,X: nat,Xs: list_nat] :
~ ( lexordp_eq_nat @ Less @ ( cons_nat @ X @ Xs ) @ nil_nat ) ).
% ord.lexordp_eq_simps(3)
thf(fact_76_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: list_a > list_a > $o,X: list_a,Xs: list_list_a] :
~ ( lexordp_eq_list_a @ Less @ ( cons_list_a @ X @ Xs ) @ nil_list_a ) ).
% ord.lexordp_eq_simps(3)
thf(fact_77_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: a > a > $o,X: a,Xs: list_a] :
~ ( lexordp_eq_a @ Less @ ( cons_a @ X @ Xs ) @ nil_a ) ).
% ord.lexordp_eq_simps(3)
thf(fact_78_local_Olexordp__simps_I3_J,axiom,
! [X: a,Xs: list_a,Y: a,Ys: list_a] :
( ( lexordp_a @ less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( ( less @ X @ Y )
| ( ~ ( less @ Y @ X )
& ( lexordp_a @ less @ Xs @ Ys ) ) ) ) ).
% local.lexordp_simps(3)
thf(fact_79_local_Olexordp__eq__simps_I4_J,axiom,
! [X: a,Xs: list_a,Y: a,Ys: list_a] :
( ( lexordp_eq_a @ less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( ( less @ X @ Y )
| ( ~ ( less @ Y @ X )
& ( lexordp_eq_a @ less @ Xs @ Ys ) ) ) ) ).
% local.lexordp_eq_simps(4)
thf(fact_80_local_Olexordp__eq__simps_I3_J,axiom,
! [X: a,Xs: list_a] :
~ ( lexordp_eq_a @ less @ ( cons_a @ X @ Xs ) @ nil_a ) ).
% local.lexordp_eq_simps(3)
thf(fact_81_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X6: nat,Xs3: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X6 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_82_transpose_Ocases,axiom,
! [X: list_list_list_a] :
( ( X != nil_list_list_a )
=> ( ! [Xss: list_list_list_a] :
( X
!= ( cons_list_list_a @ nil_list_a @ Xss ) )
=> ~ ! [X6: list_a,Xs3: list_list_a,Xss: list_list_list_a] :
( X
!= ( cons_list_list_a @ ( cons_list_a @ X6 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_83_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X6: a,Xs3: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X6 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_84_not__Cons__self2,axiom,
! [X: list_a,Xs: list_list_a] :
( ( cons_list_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_85_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_86_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X6: nat] : ( P @ ( cons_nat @ X6 @ nil_nat ) )
=> ( ! [X6: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X6 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_87_list__nonempty__induct,axiom,
! [Xs: list_list_a,P: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X6: list_a] : ( P @ ( cons_list_a @ X6 @ nil_list_a ) )
=> ( ! [X6: list_a,Xs3: list_list_a] :
( ( Xs3 != nil_list_a )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_list_a @ X6 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_88_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X6: a] : ( P @ ( cons_a @ X6 @ nil_a ) )
=> ( ! [X6: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_89_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X6: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X6 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys3: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys3 ) )
=> ( ! [X6: nat,Xs3: list_nat,Y3: nat,Ys3: list_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_nat @ X6 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_90_list__induct2_H,axiom,
! [P: list_nat > list_a > $o,Xs: list_nat,Ys: list_a] :
( ( P @ nil_nat @ nil_a )
=> ( ! [X6: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X6 @ Xs3 ) @ nil_a )
=> ( ! [Y3: a,Ys3: list_a] : ( P @ nil_nat @ ( cons_a @ Y3 @ Ys3 ) )
=> ( ! [X6: nat,Xs3: list_nat,Y3: a,Ys3: list_a] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_nat @ X6 @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_91_list__induct2_H,axiom,
! [P: list_nat > list_list_a > $o,Xs: list_nat,Ys: list_list_a] :
( ( P @ nil_nat @ nil_list_a )
=> ( ! [X6: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X6 @ Xs3 ) @ nil_list_a )
=> ( ! [Y3: list_a,Ys3: list_list_a] : ( P @ nil_nat @ ( cons_list_a @ Y3 @ Ys3 ) )
=> ( ! [X6: nat,Xs3: list_nat,Y3: list_a,Ys3: list_list_a] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_nat @ X6 @ Xs3 ) @ ( cons_list_a @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_92_list__induct2_H,axiom,
! [P: list_a > list_nat > $o,Xs: list_a,Ys: list_nat] :
( ( P @ nil_a @ nil_nat )
=> ( ! [X6: a,Xs3: list_a] : ( P @ ( cons_a @ X6 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys3: list_nat] : ( P @ nil_a @ ( cons_nat @ Y3 @ Ys3 ) )
=> ( ! [X6: a,Xs3: list_a,Y3: nat,Ys3: list_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_93_list__induct2_H,axiom,
! [P: list_a > list_list_a > $o,Xs: list_a,Ys: list_list_a] :
( ( P @ nil_a @ nil_list_a )
=> ( ! [X6: a,Xs3: list_a] : ( P @ ( cons_a @ X6 @ Xs3 ) @ nil_list_a )
=> ( ! [Y3: list_a,Ys3: list_list_a] : ( P @ nil_a @ ( cons_list_a @ Y3 @ Ys3 ) )
=> ( ! [X6: a,Xs3: list_a,Y3: list_a,Ys3: list_list_a] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_list_a @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_94_list__induct2_H,axiom,
! [P: list_list_a > list_nat > $o,Xs: list_list_a,Ys: list_nat] :
( ( P @ nil_list_a @ nil_nat )
=> ( ! [X6: list_a,Xs3: list_list_a] : ( P @ ( cons_list_a @ X6 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys3: list_nat] : ( P @ nil_list_a @ ( cons_nat @ Y3 @ Ys3 ) )
=> ( ! [X6: list_a,Xs3: list_list_a,Y3: nat,Ys3: list_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_list_a @ X6 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_95_list__induct2_H,axiom,
! [P: list_list_a > list_a > $o,Xs: list_list_a,Ys: list_a] :
( ( P @ nil_list_a @ nil_a )
=> ( ! [X6: list_a,Xs3: list_list_a] : ( P @ ( cons_list_a @ X6 @ Xs3 ) @ nil_a )
=> ( ! [Y3: a,Ys3: list_a] : ( P @ nil_list_a @ ( cons_a @ Y3 @ Ys3 ) )
=> ( ! [X6: list_a,Xs3: list_list_a,Y3: a,Ys3: list_a] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_list_a @ X6 @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_96_list__induct2_H,axiom,
! [P: list_list_a > list_list_a > $o,Xs: list_list_a,Ys: list_list_a] :
( ( P @ nil_list_a @ nil_list_a )
=> ( ! [X6: list_a,Xs3: list_list_a] : ( P @ ( cons_list_a @ X6 @ Xs3 ) @ nil_list_a )
=> ( ! [Y3: list_a,Ys3: list_list_a] : ( P @ nil_list_a @ ( cons_list_a @ Y3 @ Ys3 ) )
=> ( ! [X6: list_a,Xs3: list_list_a,Y3: list_a,Ys3: list_list_a] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_list_a @ X6 @ Xs3 ) @ ( cons_list_a @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_97_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X6: a,Xs3: list_a] : ( P @ ( cons_a @ X6 @ Xs3 ) @ nil_a )
=> ( ! [Y3: a,Ys3: list_a] : ( P @ nil_a @ ( cons_a @ Y3 @ Ys3 ) )
=> ( ! [X6: a,Xs3: list_a,Y3: a,Ys3: list_a] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_a @ X6 @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_98_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y2: nat,Ys2: list_nat] :
( Xs
= ( cons_nat @ Y2 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_99_neq__Nil__conv,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
= ( ? [Y2: list_a,Ys2: list_list_a] :
( Xs
= ( cons_list_a @ Y2 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_100_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y2: a,Ys2: list_a] :
( Xs
= ( cons_a @ Y2 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_101_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X6: nat] :
( X
!= ( cons_nat @ X6 @ nil_nat ) )
=> ~ ! [X6: nat,Y3: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X6 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_102_remdups__adj_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [X6: list_a] :
( X
!= ( cons_list_a @ X6 @ nil_list_a ) )
=> ~ ! [X6: list_a,Y3: list_a,Xs3: list_list_a] :
( X
!= ( cons_list_a @ X6 @ ( cons_list_a @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_103_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X6: a] :
( X
!= ( cons_a @ X6 @ nil_a ) )
=> ~ ! [X6: a,Y3: a,Xs3: list_a] :
( X
!= ( cons_a @ X6 @ ( cons_a @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_104_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X6: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X6 @ Xs3 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_105_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_106_list_Oexhaust,axiom,
! [Y: list_list_a] :
( ( Y != nil_list_a )
=> ~ ! [X212: list_a,X222: list_list_a] :
( Y
!= ( cons_list_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_107_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_108_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_109_list_OdiscI,axiom,
! [List: list_list_a,X21: list_a,X22: list_list_a] :
( ( List
= ( cons_list_a @ X21 @ X22 ) )
=> ( List != nil_list_a ) ) ).
% list.discI
thf(fact_110_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_111_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_112_list_Odistinct_I1_J,axiom,
! [X21: list_a,X22: list_list_a] :
( nil_list_a
!= ( cons_list_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_113_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_114_ord_Olexordp_OCons__eq,axiom,
! [Less: list_a > list_a > $o,X: list_a,Y: list_a,Xs: list_list_a,Ys: list_list_a] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_list_a @ Less @ Xs @ Ys )
=> ( lexordp_list_a @ Less @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp.Cons_eq
thf(fact_115_ord_Olexordp_OCons__eq,axiom,
! [Less: a > a > $o,X: a,Y: a,Xs: list_a,Ys: list_a] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_a @ Less @ Xs @ Ys )
=> ( lexordp_a @ Less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp.Cons_eq
thf(fact_116_ord_Olexordp_OCons,axiom,
! [Less: list_a > list_a > $o,X: list_a,Y: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( Less @ X @ Y )
=> ( lexordp_list_a @ Less @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) ) ).
% ord.lexordp.Cons
thf(fact_117_ord_Olexordp_OCons,axiom,
! [Less: a > a > $o,X: a,Y: a,Xs: list_a,Ys: list_a] :
( ( Less @ X @ Y )
=> ( lexordp_a @ Less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ).
% ord.lexordp.Cons
thf(fact_118_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: list_a > list_a > $o,X: list_a,Y: list_a,Xs: list_list_a,Ys: list_list_a] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_eq_list_a @ Less @ Xs @ Ys )
=> ( lexordp_eq_list_a @ Less @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_119_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: a > a > $o,X: a,Y: a,Xs: list_a,Ys: list_a] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_eq_a @ Less @ Xs @ Ys )
=> ( lexordp_eq_a @ Less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_120_ord_Olexordp__eq_OCons,axiom,
! [Less: list_a > list_a > $o,X: list_a,Y: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( Less @ X @ Y )
=> ( lexordp_eq_list_a @ Less @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_121_ord_Olexordp__eq_OCons,axiom,
! [Less: a > a > $o,X: a,Y: a,Xs: list_a,Ys: list_a] :
( ( Less @ X @ Y )
=> ( lexordp_eq_a @ Less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_122_ord_Olexordp_Osimps,axiom,
( lexordp_nat
= ( ^ [Less2: nat > nat > $o,A12: list_nat,A22: list_nat] :
( ? [Y2: nat,Ys2: list_nat] :
( ( A12 = nil_nat )
& ( A22
= ( cons_nat @ Y2 @ Ys2 ) ) )
| ? [X7: nat,Y2: nat,Xs2: list_nat,Ys2: list_nat] :
( ( A12
= ( cons_nat @ X7 @ Xs2 ) )
& ( A22
= ( cons_nat @ Y2 @ Ys2 ) )
& ( Less2 @ X7 @ Y2 ) )
| ? [X7: nat,Y2: nat,Xs2: list_nat,Ys2: list_nat] :
( ( A12
= ( cons_nat @ X7 @ Xs2 ) )
& ( A22
= ( cons_nat @ Y2 @ Ys2 ) )
& ~ ( Less2 @ X7 @ Y2 )
& ~ ( Less2 @ Y2 @ X7 )
& ( lexordp_nat @ Less2 @ Xs2 @ Ys2 ) ) ) ) ) ).
% ord.lexordp.simps
thf(fact_123_ord_Olexordp_Osimps,axiom,
( lexordp_list_a
= ( ^ [Less2: list_a > list_a > $o,A12: list_list_a,A22: list_list_a] :
( ? [Y2: list_a,Ys2: list_list_a] :
( ( A12 = nil_list_a )
& ( A22
= ( cons_list_a @ Y2 @ Ys2 ) ) )
| ? [X7: list_a,Y2: list_a,Xs2: list_list_a,Ys2: list_list_a] :
( ( A12
= ( cons_list_a @ X7 @ Xs2 ) )
& ( A22
= ( cons_list_a @ Y2 @ Ys2 ) )
& ( Less2 @ X7 @ Y2 ) )
| ? [X7: list_a,Y2: list_a,Xs2: list_list_a,Ys2: list_list_a] :
( ( A12
= ( cons_list_a @ X7 @ Xs2 ) )
& ( A22
= ( cons_list_a @ Y2 @ Ys2 ) )
& ~ ( Less2 @ X7 @ Y2 )
& ~ ( Less2 @ Y2 @ X7 )
& ( lexordp_list_a @ Less2 @ Xs2 @ Ys2 ) ) ) ) ) ).
% ord.lexordp.simps
thf(fact_124_ord_Olexordp_Osimps,axiom,
( lexordp_a
= ( ^ [Less2: a > a > $o,A12: list_a,A22: list_a] :
( ? [Y2: a,Ys2: list_a] :
( ( A12 = nil_a )
& ( A22
= ( cons_a @ Y2 @ Ys2 ) ) )
| ? [X7: a,Y2: a,Xs2: list_a,Ys2: list_a] :
( ( A12
= ( cons_a @ X7 @ Xs2 ) )
& ( A22
= ( cons_a @ Y2 @ Ys2 ) )
& ( Less2 @ X7 @ Y2 ) )
| ? [X7: a,Y2: a,Xs2: list_a,Ys2: list_a] :
( ( A12
= ( cons_a @ X7 @ Xs2 ) )
& ( A22
= ( cons_a @ Y2 @ Ys2 ) )
& ~ ( Less2 @ X7 @ Y2 )
& ~ ( Less2 @ Y2 @ X7 )
& ( lexordp_a @ Less2 @ Xs2 @ Ys2 ) ) ) ) ) ).
% ord.lexordp.simps
thf(fact_125_ord_Olexordp_Ocases,axiom,
! [Less: nat > nat > $o,A1: list_nat,A2: list_nat] :
( ( lexordp_nat @ Less @ A1 @ A2 )
=> ( ( ( A1 = nil_nat )
=> ! [Y3: nat,Ys3: list_nat] :
( A2
!= ( cons_nat @ Y3 @ Ys3 ) ) )
=> ( ! [X6: nat] :
( ? [Xs3: list_nat] :
( A1
= ( cons_nat @ X6 @ Xs3 ) )
=> ! [Y3: nat] :
( ? [Ys3: list_nat] :
( A2
= ( cons_nat @ Y3 @ Ys3 ) )
=> ~ ( Less @ X6 @ Y3 ) ) )
=> ~ ! [X6: nat,Y3: nat,Xs3: list_nat] :
( ( A1
= ( cons_nat @ X6 @ Xs3 ) )
=> ! [Ys3: list_nat] :
( ( A2
= ( cons_nat @ Y3 @ Ys3 ) )
=> ( ~ ( Less @ X6 @ Y3 )
=> ( ~ ( Less @ Y3 @ X6 )
=> ~ ( lexordp_nat @ Less @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% ord.lexordp.cases
thf(fact_126_ord_Olexordp_Ocases,axiom,
! [Less: list_a > list_a > $o,A1: list_list_a,A2: list_list_a] :
( ( lexordp_list_a @ Less @ A1 @ A2 )
=> ( ( ( A1 = nil_list_a )
=> ! [Y3: list_a,Ys3: list_list_a] :
( A2
!= ( cons_list_a @ Y3 @ Ys3 ) ) )
=> ( ! [X6: list_a] :
( ? [Xs3: list_list_a] :
( A1
= ( cons_list_a @ X6 @ Xs3 ) )
=> ! [Y3: list_a] :
( ? [Ys3: list_list_a] :
( A2
= ( cons_list_a @ Y3 @ Ys3 ) )
=> ~ ( Less @ X6 @ Y3 ) ) )
=> ~ ! [X6: list_a,Y3: list_a,Xs3: list_list_a] :
( ( A1
= ( cons_list_a @ X6 @ Xs3 ) )
=> ! [Ys3: list_list_a] :
( ( A2
= ( cons_list_a @ Y3 @ Ys3 ) )
=> ( ~ ( Less @ X6 @ Y3 )
=> ( ~ ( Less @ Y3 @ X6 )
=> ~ ( lexordp_list_a @ Less @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% ord.lexordp.cases
thf(fact_127_ord_Olexordp_Ocases,axiom,
! [Less: a > a > $o,A1: list_a,A2: list_a] :
( ( lexordp_a @ Less @ A1 @ A2 )
=> ( ( ( A1 = nil_a )
=> ! [Y3: a,Ys3: list_a] :
( A2
!= ( cons_a @ Y3 @ Ys3 ) ) )
=> ( ! [X6: a] :
( ? [Xs3: list_a] :
( A1
= ( cons_a @ X6 @ Xs3 ) )
=> ! [Y3: a] :
( ? [Ys3: list_a] :
( A2
= ( cons_a @ Y3 @ Ys3 ) )
=> ~ ( Less @ X6 @ Y3 ) ) )
=> ~ ! [X6: a,Y3: a,Xs3: list_a] :
( ( A1
= ( cons_a @ X6 @ Xs3 ) )
=> ! [Ys3: list_a] :
( ( A2
= ( cons_a @ Y3 @ Ys3 ) )
=> ( ~ ( Less @ X6 @ Y3 )
=> ( ~ ( Less @ Y3 @ X6 )
=> ~ ( lexordp_a @ Less @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% ord.lexordp.cases
thf(fact_128_ord_Olexordp_ONil,axiom,
! [Less: nat > nat > $o,Y: nat,Ys: list_nat] : ( lexordp_nat @ Less @ nil_nat @ ( cons_nat @ Y @ Ys ) ) ).
% ord.lexordp.Nil
thf(fact_129_ord_Olexordp_ONil,axiom,
! [Less: list_a > list_a > $o,Y: list_a,Ys: list_list_a] : ( lexordp_list_a @ Less @ nil_list_a @ ( cons_list_a @ Y @ Ys ) ) ).
% ord.lexordp.Nil
thf(fact_130_ord_Olexordp_ONil,axiom,
! [Less: a > a > $o,Y: a,Ys: list_a] : ( lexordp_a @ Less @ nil_a @ ( cons_a @ Y @ Ys ) ) ).
% ord.lexordp.Nil
thf(fact_131_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_nat
= ( ^ [Less2: nat > nat > $o,A12: list_nat,A22: list_nat] :
( ? [Ys2: list_nat] :
( ( A12 = nil_nat )
& ( A22 = Ys2 ) )
| ? [X7: nat,Y2: nat,Xs2: list_nat,Ys2: list_nat] :
( ( A12
= ( cons_nat @ X7 @ Xs2 ) )
& ( A22
= ( cons_nat @ Y2 @ Ys2 ) )
& ( Less2 @ X7 @ Y2 ) )
| ? [X7: nat,Y2: nat,Xs2: list_nat,Ys2: list_nat] :
( ( A12
= ( cons_nat @ X7 @ Xs2 ) )
& ( A22
= ( cons_nat @ Y2 @ Ys2 ) )
& ~ ( Less2 @ X7 @ Y2 )
& ~ ( Less2 @ Y2 @ X7 )
& ( lexordp_eq_nat @ Less2 @ Xs2 @ Ys2 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_132_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_list_a
= ( ^ [Less2: list_a > list_a > $o,A12: list_list_a,A22: list_list_a] :
( ? [Ys2: list_list_a] :
( ( A12 = nil_list_a )
& ( A22 = Ys2 ) )
| ? [X7: list_a,Y2: list_a,Xs2: list_list_a,Ys2: list_list_a] :
( ( A12
= ( cons_list_a @ X7 @ Xs2 ) )
& ( A22
= ( cons_list_a @ Y2 @ Ys2 ) )
& ( Less2 @ X7 @ Y2 ) )
| ? [X7: list_a,Y2: list_a,Xs2: list_list_a,Ys2: list_list_a] :
( ( A12
= ( cons_list_a @ X7 @ Xs2 ) )
& ( A22
= ( cons_list_a @ Y2 @ Ys2 ) )
& ~ ( Less2 @ X7 @ Y2 )
& ~ ( Less2 @ Y2 @ X7 )
& ( lexordp_eq_list_a @ Less2 @ Xs2 @ Ys2 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_133_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_a
= ( ^ [Less2: a > a > $o,A12: list_a,A22: list_a] :
( ? [Ys2: list_a] :
( ( A12 = nil_a )
& ( A22 = Ys2 ) )
| ? [X7: a,Y2: a,Xs2: list_a,Ys2: list_a] :
( ( A12
= ( cons_a @ X7 @ Xs2 ) )
& ( A22
= ( cons_a @ Y2 @ Ys2 ) )
& ( Less2 @ X7 @ Y2 ) )
| ? [X7: a,Y2: a,Xs2: list_a,Ys2: list_a] :
( ( A12
= ( cons_a @ X7 @ Xs2 ) )
& ( A22
= ( cons_a @ Y2 @ Ys2 ) )
& ~ ( Less2 @ X7 @ Y2 )
& ~ ( Less2 @ Y2 @ X7 )
& ( lexordp_eq_a @ Less2 @ Xs2 @ Ys2 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_134_ord_Olexordp__eq_Ocases,axiom,
! [Less: nat > nat > $o,A1: list_nat,A2: list_nat] :
( ( lexordp_eq_nat @ Less @ A1 @ A2 )
=> ( ( A1 != nil_nat )
=> ( ! [X6: nat] :
( ? [Xs3: list_nat] :
( A1
= ( cons_nat @ X6 @ Xs3 ) )
=> ! [Y3: nat] :
( ? [Ys3: list_nat] :
( A2
= ( cons_nat @ Y3 @ Ys3 ) )
=> ~ ( Less @ X6 @ Y3 ) ) )
=> ~ ! [X6: nat,Y3: nat,Xs3: list_nat] :
( ( A1
= ( cons_nat @ X6 @ Xs3 ) )
=> ! [Ys3: list_nat] :
( ( A2
= ( cons_nat @ Y3 @ Ys3 ) )
=> ( ~ ( Less @ X6 @ Y3 )
=> ( ~ ( Less @ Y3 @ X6 )
=> ~ ( lexordp_eq_nat @ Less @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_135_ord_Olexordp__eq_Ocases,axiom,
! [Less: list_a > list_a > $o,A1: list_list_a,A2: list_list_a] :
( ( lexordp_eq_list_a @ Less @ A1 @ A2 )
=> ( ( A1 != nil_list_a )
=> ( ! [X6: list_a] :
( ? [Xs3: list_list_a] :
( A1
= ( cons_list_a @ X6 @ Xs3 ) )
=> ! [Y3: list_a] :
( ? [Ys3: list_list_a] :
( A2
= ( cons_list_a @ Y3 @ Ys3 ) )
=> ~ ( Less @ X6 @ Y3 ) ) )
=> ~ ! [X6: list_a,Y3: list_a,Xs3: list_list_a] :
( ( A1
= ( cons_list_a @ X6 @ Xs3 ) )
=> ! [Ys3: list_list_a] :
( ( A2
= ( cons_list_a @ Y3 @ Ys3 ) )
=> ( ~ ( Less @ X6 @ Y3 )
=> ( ~ ( Less @ Y3 @ X6 )
=> ~ ( lexordp_eq_list_a @ Less @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_136_ord_Olexordp__eq_Ocases,axiom,
! [Less: a > a > $o,A1: list_a,A2: list_a] :
( ( lexordp_eq_a @ Less @ A1 @ A2 )
=> ( ( A1 != nil_a )
=> ( ! [X6: a] :
( ? [Xs3: list_a] :
( A1
= ( cons_a @ X6 @ Xs3 ) )
=> ! [Y3: a] :
( ? [Ys3: list_a] :
( A2
= ( cons_a @ Y3 @ Ys3 ) )
=> ~ ( Less @ X6 @ Y3 ) ) )
=> ~ ! [X6: a,Y3: a,Xs3: list_a] :
( ( A1
= ( cons_a @ X6 @ Xs3 ) )
=> ! [Ys3: list_a] :
( ( A2
= ( cons_a @ Y3 @ Ys3 ) )
=> ( ~ ( Less @ X6 @ Y3 )
=> ( ~ ( Less @ Y3 @ X6 )
=> ~ ( lexordp_eq_a @ Less @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_137_ord_Olexordp__irreflexive,axiom,
! [Less: a > a > $o,Xs: list_a] :
( ! [X6: a] :
~ ( Less @ X6 @ X6 )
=> ~ ( lexordp_a @ Less @ Xs @ Xs ) ) ).
% ord.lexordp_irreflexive
thf(fact_138_ord_Olexordp_Ocong,axiom,
lexordp_a = lexordp_a ).
% ord.lexordp.cong
thf(fact_139_ord_Olexordp__eq__refl,axiom,
! [Less: a > a > $o,Xs: list_a] : ( lexordp_eq_a @ Less @ Xs @ Xs ) ).
% ord.lexordp_eq_refl
thf(fact_140_ord_Olexordp__eq_Ocong,axiom,
lexordp_eq_a = lexordp_eq_a ).
% ord.lexordp_eq.cong
thf(fact_141_ord_Olexordp__eq_ONil,axiom,
! [Less: list_a > list_a > $o,Ys: list_list_a] : ( lexordp_eq_list_a @ Less @ nil_list_a @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_142_ord_Olexordp__eq_ONil,axiom,
! [Less: nat > nat > $o,Ys: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_143_ord_Olexordp__eq_ONil,axiom,
! [Less: a > a > $o,Ys: list_a] : ( lexordp_eq_a @ Less @ nil_a @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_144_ord_Olexordp__into__lexordp__eq,axiom,
! [Less: a > a > $o,Xs: list_a,Ys: list_a] :
( ( lexordp_a @ Less @ Xs @ Ys )
=> ( lexordp_eq_a @ Less @ Xs @ Ys ) ) ).
% ord.lexordp_into_lexordp_eq
thf(fact_145_ord_OgreaterThanLessThan__def,axiom,
( set_gr890836759843348478_set_a
= ( ^ [Less2: set_a > set_a > $o,L2: set_a,U2: set_a] : ( inf_inf_set_set_a @ ( set_gr7079423240508057912_set_a @ Less2 @ L2 ) @ ( set_lessThan_set_a @ Less2 @ U2 ) ) ) ) ).
% ord.greaterThanLessThan_def
thf(fact_146_ord_OgreaterThanLessThan__def,axiom,
( set_gr7355061613916526640an_nat
= ( ^ [Less2: nat > nat > $o,L2: nat,U2: nat] : ( inf_inf_set_nat @ ( set_greaterThan_nat @ Less2 @ L2 ) @ ( set_lessThan_nat @ Less2 @ U2 ) ) ) ) ).
% ord.greaterThanLessThan_def
thf(fact_147_ord_OgreaterThanLessThan__def,axiom,
( set_gr5113148517155960478Than_a
= ( ^ [Less2: a > a > $o,L2: a,U2: a] : ( inf_inf_set_a @ ( set_greaterThan_a @ Less2 @ L2 ) @ ( set_lessThan_a @ Less2 @ U2 ) ) ) ) ).
% ord.greaterThanLessThan_def
thf(fact_148_ord_OgreaterThanLessThan__eq,axiom,
( set_gr890836759843348478_set_a
= ( ^ [Less2: set_a > set_a > $o,A4: set_a,B2: set_a] : ( inf_inf_set_set_a @ ( set_gr7079423240508057912_set_a @ Less2 @ A4 ) @ ( set_lessThan_set_a @ Less2 @ B2 ) ) ) ) ).
% ord.greaterThanLessThan_eq
thf(fact_149_ord_OgreaterThanLessThan__eq,axiom,
( set_gr7355061613916526640an_nat
= ( ^ [Less2: nat > nat > $o,A4: nat,B2: nat] : ( inf_inf_set_nat @ ( set_greaterThan_nat @ Less2 @ A4 ) @ ( set_lessThan_nat @ Less2 @ B2 ) ) ) ) ).
% ord.greaterThanLessThan_eq
thf(fact_150_ord_OgreaterThanLessThan__eq,axiom,
( set_gr5113148517155960478Than_a
= ( ^ [Less2: a > a > $o,A4: a,B2: a] : ( inf_inf_set_a @ ( set_greaterThan_a @ Less2 @ A4 ) @ ( set_lessThan_a @ Less2 @ B2 ) ) ) ) ).
% ord.greaterThanLessThan_eq
thf(fact_151_Int__UNIV,axiom,
! [A3: set_set_a,B3: set_set_a] :
( ( ( inf_inf_set_set_a @ A3 @ B3 )
= top_top_set_set_a )
= ( ( A3 = top_top_set_set_a )
& ( B3 = top_top_set_set_a ) ) ) ).
% Int_UNIV
thf(fact_152_Int__UNIV,axiom,
! [A3: set_nat_a,B3: set_nat_a] :
( ( ( inf_inf_set_nat_a @ A3 @ B3 )
= top_top_set_nat_a )
= ( ( A3 = top_top_set_nat_a )
& ( B3 = top_top_set_nat_a ) ) ) ).
% Int_UNIV
thf(fact_153_Int__UNIV,axiom,
! [A3: set_a_nat,B3: set_a_nat] :
( ( ( inf_inf_set_a_nat @ A3 @ B3 )
= top_top_set_a_nat )
= ( ( A3 = top_top_set_a_nat )
& ( B3 = top_top_set_a_nat ) ) ) ).
% Int_UNIV
thf(fact_154_Int__UNIV,axiom,
! [A3: set_a_a,B3: set_a_a] :
( ( ( inf_inf_set_a_a @ A3 @ B3 )
= top_top_set_a_a )
= ( ( A3 = top_top_set_a_a )
& ( B3 = top_top_set_a_a ) ) ) ).
% Int_UNIV
thf(fact_155_Int__UNIV,axiom,
! [A3: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A3 @ B3 )
= top_top_set_a )
= ( ( A3 = top_top_set_a )
& ( B3 = top_top_set_a ) ) ) ).
% Int_UNIV
thf(fact_156_Int__UNIV,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B3 )
= top_top_set_nat )
= ( ( A3 = top_top_set_nat )
& ( B3 = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_157_inf__top_Oright__neutral,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ top_top_set_set_a )
= A ) ).
% inf_top.right_neutral
thf(fact_158_inf__top_Oright__neutral,axiom,
! [A: set_nat_a] :
( ( inf_inf_set_nat_a @ A @ top_top_set_nat_a )
= A ) ).
% inf_top.right_neutral
thf(fact_159_inf__top_Oright__neutral,axiom,
! [A: set_a_nat] :
( ( inf_inf_set_a_nat @ A @ top_top_set_a_nat )
= A ) ).
% inf_top.right_neutral
thf(fact_160_inf__top_Oright__neutral,axiom,
! [A: set_a_a] :
( ( inf_inf_set_a_a @ A @ top_top_set_a_a )
= A ) ).
% inf_top.right_neutral
thf(fact_161_inf__top_Oright__neutral,axiom,
! [A: nat > $o] :
( ( inf_inf_nat_o @ A @ top_top_nat_o )
= A ) ).
% inf_top.right_neutral
thf(fact_162_inf__top_Oright__neutral,axiom,
! [A: a > $o] :
( ( inf_inf_a_o @ A @ top_top_a_o )
= A ) ).
% inf_top.right_neutral
thf(fact_163_inf__top_Oright__neutral,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ top_top_set_a )
= A ) ).
% inf_top.right_neutral
thf(fact_164_inf__top_Oright__neutral,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% inf_top.right_neutral
thf(fact_165_inf__top_Oneutr__eq__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( top_top_set_set_a
= ( inf_inf_set_set_a @ A @ B ) )
= ( ( A = top_top_set_set_a )
& ( B = top_top_set_set_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_166_inf__top_Oneutr__eq__iff,axiom,
! [A: set_nat_a,B: set_nat_a] :
( ( top_top_set_nat_a
= ( inf_inf_set_nat_a @ A @ B ) )
= ( ( A = top_top_set_nat_a )
& ( B = top_top_set_nat_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_167_inf__top_Oneutr__eq__iff,axiom,
! [A: set_a_nat,B: set_a_nat] :
( ( top_top_set_a_nat
= ( inf_inf_set_a_nat @ A @ B ) )
= ( ( A = top_top_set_a_nat )
& ( B = top_top_set_a_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_168_inf__top_Oneutr__eq__iff,axiom,
! [A: set_a_a,B: set_a_a] :
( ( top_top_set_a_a
= ( inf_inf_set_a_a @ A @ B ) )
= ( ( A = top_top_set_a_a )
& ( B = top_top_set_a_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_169_inf__top_Oneutr__eq__iff,axiom,
! [A: nat > $o,B: nat > $o] :
( ( top_top_nat_o
= ( inf_inf_nat_o @ A @ B ) )
= ( ( A = top_top_nat_o )
& ( B = top_top_nat_o ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_170_inf__top_Oneutr__eq__iff,axiom,
! [A: a > $o,B: a > $o] :
( ( top_top_a_o
= ( inf_inf_a_o @ A @ B ) )
= ( ( A = top_top_a_o )
& ( B = top_top_a_o ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_171_inf__top_Oneutr__eq__iff,axiom,
! [A: set_a,B: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ A @ B ) )
= ( ( A = top_top_set_a )
& ( B = top_top_set_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_172_inf__top_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A @ B ) )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_173_inf__top_Oleft__neutral,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ top_top_set_set_a @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_174_inf__top_Oleft__neutral,axiom,
! [A: set_nat_a] :
( ( inf_inf_set_nat_a @ top_top_set_nat_a @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_175_inf__top_Oleft__neutral,axiom,
! [A: set_a_nat] :
( ( inf_inf_set_a_nat @ top_top_set_a_nat @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_176_inf__top_Oleft__neutral,axiom,
! [A: set_a_a] :
( ( inf_inf_set_a_a @ top_top_set_a_a @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_177_inf__top_Oleft__neutral,axiom,
! [A: nat > $o] :
( ( inf_inf_nat_o @ top_top_nat_o @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_178_inf__top_Oleft__neutral,axiom,
! [A: a > $o] :
( ( inf_inf_a_o @ top_top_a_o @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_179_inf__top_Oleft__neutral,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_180_inf__top_Oleft__neutral,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_181_inf__top_Oeq__neutr__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= top_top_set_set_a )
= ( ( A = top_top_set_set_a )
& ( B = top_top_set_set_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_182_inf__top_Oeq__neutr__iff,axiom,
! [A: set_nat_a,B: set_nat_a] :
( ( ( inf_inf_set_nat_a @ A @ B )
= top_top_set_nat_a )
= ( ( A = top_top_set_nat_a )
& ( B = top_top_set_nat_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_183_inf__top_Oeq__neutr__iff,axiom,
! [A: set_a_nat,B: set_a_nat] :
( ( ( inf_inf_set_a_nat @ A @ B )
= top_top_set_a_nat )
= ( ( A = top_top_set_a_nat )
& ( B = top_top_set_a_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_184_inf__top_Oeq__neutr__iff,axiom,
! [A: set_a_a,B: set_a_a] :
( ( ( inf_inf_set_a_a @ A @ B )
= top_top_set_a_a )
= ( ( A = top_top_set_a_a )
& ( B = top_top_set_a_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_185_inf__top_Oeq__neutr__iff,axiom,
! [A: nat > $o,B: nat > $o] :
( ( ( inf_inf_nat_o @ A @ B )
= top_top_nat_o )
= ( ( A = top_top_nat_o )
& ( B = top_top_nat_o ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_186_inf__top_Oeq__neutr__iff,axiom,
! [A: a > $o,B: a > $o] :
( ( ( inf_inf_a_o @ A @ B )
= top_top_a_o )
= ( ( A = top_top_a_o )
& ( B = top_top_a_o ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_187_inf__top_Oeq__neutr__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= top_top_set_a )
= ( ( A = top_top_set_a )
& ( B = top_top_set_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_188_inf__top_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_189_top__eq__inf__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( top_top_set_set_a
= ( inf_inf_set_set_a @ X @ Y ) )
= ( ( X = top_top_set_set_a )
& ( Y = top_top_set_set_a ) ) ) ).
% top_eq_inf_iff
thf(fact_190_top__eq__inf__iff,axiom,
! [X: set_nat_a,Y: set_nat_a] :
( ( top_top_set_nat_a
= ( inf_inf_set_nat_a @ X @ Y ) )
= ( ( X = top_top_set_nat_a )
& ( Y = top_top_set_nat_a ) ) ) ).
% top_eq_inf_iff
thf(fact_191_top__eq__inf__iff,axiom,
! [X: set_a_nat,Y: set_a_nat] :
( ( top_top_set_a_nat
= ( inf_inf_set_a_nat @ X @ Y ) )
= ( ( X = top_top_set_a_nat )
& ( Y = top_top_set_a_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_192_top__eq__inf__iff,axiom,
! [X: set_a_a,Y: set_a_a] :
( ( top_top_set_a_a
= ( inf_inf_set_a_a @ X @ Y ) )
= ( ( X = top_top_set_a_a )
& ( Y = top_top_set_a_a ) ) ) ).
% top_eq_inf_iff
thf(fact_193_top__eq__inf__iff,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( top_top_nat_o
= ( inf_inf_nat_o @ X @ Y ) )
= ( ( X = top_top_nat_o )
& ( Y = top_top_nat_o ) ) ) ).
% top_eq_inf_iff
thf(fact_194_top__eq__inf__iff,axiom,
! [X: a > $o,Y: a > $o] :
( ( top_top_a_o
= ( inf_inf_a_o @ X @ Y ) )
= ( ( X = top_top_a_o )
& ( Y = top_top_a_o ) ) ) ).
% top_eq_inf_iff
thf(fact_195_top__eq__inf__iff,axiom,
! [X: set_a,Y: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ X @ Y ) )
= ( ( X = top_top_set_a )
& ( Y = top_top_set_a ) ) ) ).
% top_eq_inf_iff
thf(fact_196_top__eq__inf__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X @ Y ) )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_197_inf__eq__top__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( inf_inf_set_set_a @ X @ Y )
= top_top_set_set_a )
= ( ( X = top_top_set_set_a )
& ( Y = top_top_set_set_a ) ) ) ).
% inf_eq_top_iff
thf(fact_198_inf__eq__top__iff,axiom,
! [X: set_nat_a,Y: set_nat_a] :
( ( ( inf_inf_set_nat_a @ X @ Y )
= top_top_set_nat_a )
= ( ( X = top_top_set_nat_a )
& ( Y = top_top_set_nat_a ) ) ) ).
% inf_eq_top_iff
thf(fact_199_inf__eq__top__iff,axiom,
! [X: set_a_nat,Y: set_a_nat] :
( ( ( inf_inf_set_a_nat @ X @ Y )
= top_top_set_a_nat )
= ( ( X = top_top_set_a_nat )
& ( Y = top_top_set_a_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_200_inf__eq__top__iff,axiom,
! [X: set_a_a,Y: set_a_a] :
( ( ( inf_inf_set_a_a @ X @ Y )
= top_top_set_a_a )
= ( ( X = top_top_set_a_a )
& ( Y = top_top_set_a_a ) ) ) ).
% inf_eq_top_iff
thf(fact_201_inf__eq__top__iff,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( ( inf_inf_nat_o @ X @ Y )
= top_top_nat_o )
= ( ( X = top_top_nat_o )
& ( Y = top_top_nat_o ) ) ) ).
% inf_eq_top_iff
thf(fact_202_inf__eq__top__iff,axiom,
! [X: a > $o,Y: a > $o] :
( ( ( inf_inf_a_o @ X @ Y )
= top_top_a_o )
= ( ( X = top_top_a_o )
& ( Y = top_top_a_o ) ) ) ).
% inf_eq_top_iff
thf(fact_203_inf__eq__top__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ X @ Y )
= top_top_set_a )
= ( ( X = top_top_set_a )
& ( Y = top_top_set_a ) ) ) ).
% inf_eq_top_iff
thf(fact_204_inf__eq__top__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= top_top_set_nat )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_205_inf__top__right,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ top_top_set_set_a )
= X ) ).
% inf_top_right
thf(fact_206_inf__top__right,axiom,
! [X: set_nat_a] :
( ( inf_inf_set_nat_a @ X @ top_top_set_nat_a )
= X ) ).
% inf_top_right
thf(fact_207_inf__top__right,axiom,
! [X: set_a_nat] :
( ( inf_inf_set_a_nat @ X @ top_top_set_a_nat )
= X ) ).
% inf_top_right
thf(fact_208_inf__top__right,axiom,
! [X: set_a_a] :
( ( inf_inf_set_a_a @ X @ top_top_set_a_a )
= X ) ).
% inf_top_right
thf(fact_209_inf__top__right,axiom,
! [X: nat > $o] :
( ( inf_inf_nat_o @ X @ top_top_nat_o )
= X ) ).
% inf_top_right
thf(fact_210_inf__top__right,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ X @ top_top_a_o )
= X ) ).
% inf_top_right
thf(fact_211_inf__top__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ top_top_set_a )
= X ) ).
% inf_top_right
thf(fact_212_inf__top__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% inf_top_right
thf(fact_213_inf__top__left,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ top_top_set_set_a @ X )
= X ) ).
% inf_top_left
thf(fact_214_inf__top__left,axiom,
! [X: set_nat_a] :
( ( inf_inf_set_nat_a @ top_top_set_nat_a @ X )
= X ) ).
% inf_top_left
thf(fact_215_inf__top__left,axiom,
! [X: set_a_nat] :
( ( inf_inf_set_a_nat @ top_top_set_a_nat @ X )
= X ) ).
% inf_top_left
thf(fact_216_inf__top__left,axiom,
! [X: set_a_a] :
( ( inf_inf_set_a_a @ top_top_set_a_a @ X )
= X ) ).
% inf_top_left
thf(fact_217_inf__top__left,axiom,
! [X: nat > $o] :
( ( inf_inf_nat_o @ top_top_nat_o @ X )
= X ) ).
% inf_top_left
thf(fact_218_inf__top__left,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ top_top_a_o @ X )
= X ) ).
% inf_top_left
thf(fact_219_inf__top__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ X )
= X ) ).
% inf_top_left
thf(fact_220_inf__top__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X )
= X ) ).
% inf_top_left
thf(fact_221_UNIV__I,axiom,
! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_222_UNIV__I,axiom,
! [X: set_nat] : ( member_set_nat @ X @ top_top_set_set_nat ) ).
% UNIV_I
thf(fact_223_UNIV__I,axiom,
! [X: nat > a] : ( member_nat_a @ X @ top_top_set_nat_a ) ).
% UNIV_I
thf(fact_224_UNIV__I,axiom,
! [X: a > nat] : ( member_a_nat @ X @ top_top_set_a_nat ) ).
% UNIV_I
thf(fact_225_UNIV__I,axiom,
! [X: a > a] : ( member_a_a @ X @ top_top_set_a_a ) ).
% UNIV_I
thf(fact_226_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_227_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_228_inf_Oidem,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_229_inf_Oidem,axiom,
! [A: nat] :
( ( inf_inf_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_230_inf_Oidem,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_231_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_232_inf__idem,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_233_inf__idem,axiom,
! [X: nat] :
( ( inf_inf_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_234_inf__idem,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_235_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_236_inf_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% inf.left_idem
thf(fact_237_inf_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( inf_inf_nat @ A @ ( inf_inf_nat @ A @ B ) )
= ( inf_inf_nat @ A @ B ) ) ).
% inf.left_idem
thf(fact_238_inf_Oleft__idem,axiom,
! [A: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ).
% inf.left_idem
thf(fact_239_inf_Oleft__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.left_idem
thf(fact_240_inf__left__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_241_inf__left__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_242_inf__left__idem,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y ) )
= ( inf_inf_set_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_243_inf__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_244_inf_Oright__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B )
= ( inf_inf_set_nat @ A @ B ) ) ).
% inf.right_idem
thf(fact_245_inf_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ B )
= ( inf_inf_nat @ A @ B ) ) ).
% inf.right_idem
thf(fact_246_inf_Oright__idem,axiom,
! [A: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ B )
= ( inf_inf_set_set_a @ A @ B ) ) ).
% inf.right_idem
thf(fact_247_inf_Oright__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ B )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.right_idem
thf(fact_248_inf__right__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_249_inf__right__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Y )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_250_inf__right__idem,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_251_inf__right__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_252_IntI,axiom,
! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ A3 )
=> ( ( member_set_nat @ C @ B3 )
=> ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_253_IntI,axiom,
! [C: set_a,A3: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ A3 )
=> ( ( member_set_a @ C @ B3 )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_254_IntI,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ A3 )
=> ( ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_255_IntI,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ A3 )
=> ( ( member_a @ C @ B3 )
=> ( member_a @ C @ ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_256_Int__iff,axiom,
! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B3 ) )
= ( ( member_set_nat @ C @ A3 )
& ( member_set_nat @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_257_Int__iff,axiom,
! [C: set_a,A3: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A3 @ B3 ) )
= ( ( member_set_a @ C @ A3 )
& ( member_set_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_258_Int__iff,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
= ( ( member_nat @ C @ A3 )
& ( member_nat @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_259_Int__iff,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B3 ) )
= ( ( member_a @ C @ A3 )
& ( member_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_260_le__inf__iff,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( ( ord_le3724670747650509150_set_a @ X @ Y )
& ( ord_le3724670747650509150_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_261_le__inf__iff,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( ( ord_less_eq_set_nat @ X @ Y )
& ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_262_le__inf__iff,axiom,
! [X: $o > nat,Y: $o > nat,Z: $o > nat] :
( ( ord_less_eq_o_nat @ X @ ( inf_inf_o_nat @ Y @ Z ) )
= ( ( ord_less_eq_o_nat @ X @ Y )
& ( ord_less_eq_o_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_263_le__inf__iff,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z ) )
= ( ( ord_le6893508408891458716et_nat @ X @ Y )
& ( ord_le6893508408891458716et_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_264_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_265_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_266_inf_Obounded__iff,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B @ C ) )
= ( ( ord_le3724670747650509150_set_a @ A @ B )
& ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_267_inf_Obounded__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
= ( ( ord_less_eq_set_nat @ A @ B )
& ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_268_inf_Obounded__iff,axiom,
! [A: $o > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A @ ( inf_inf_o_nat @ B @ C ) )
= ( ( ord_less_eq_o_nat @ A @ B )
& ( ord_less_eq_o_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_269_inf_Obounded__iff,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( inf_inf_set_set_nat @ B @ C ) )
= ( ( ord_le6893508408891458716et_nat @ A @ B )
& ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_270_inf_Obounded__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_271_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_272_top__set__def,axiom,
( top_top_set_nat_a
= ( collect_nat_a @ top_top_nat_a_o ) ) ).
% top_set_def
thf(fact_273_top__set__def,axiom,
( top_top_set_a_nat
= ( collect_a_nat @ top_top_a_nat_o ) ) ).
% top_set_def
thf(fact_274_top__set__def,axiom,
( top_top_set_a_a
= ( collect_a_a @ top_top_a_a_o ) ) ).
% top_set_def
thf(fact_275_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_276_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_277_UNIV__eq__I,axiom,
! [A3: set_set_a] :
( ! [X6: set_a] : ( member_set_a @ X6 @ A3 )
=> ( top_top_set_set_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_278_UNIV__eq__I,axiom,
! [A3: set_set_nat] :
( ! [X6: set_nat] : ( member_set_nat @ X6 @ A3 )
=> ( top_top_set_set_nat = A3 ) ) ).
% UNIV_eq_I
thf(fact_279_UNIV__eq__I,axiom,
! [A3: set_nat_a] :
( ! [X6: nat > a] : ( member_nat_a @ X6 @ A3 )
=> ( top_top_set_nat_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_280_UNIV__eq__I,axiom,
! [A3: set_a_nat] :
( ! [X6: a > nat] : ( member_a_nat @ X6 @ A3 )
=> ( top_top_set_a_nat = A3 ) ) ).
% UNIV_eq_I
thf(fact_281_UNIV__eq__I,axiom,
! [A3: set_a_a] :
( ! [X6: a > a] : ( member_a_a @ X6 @ A3 )
=> ( top_top_set_a_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_282_UNIV__eq__I,axiom,
! [A3: set_a] :
( ! [X6: a] : ( member_a @ X6 @ A3 )
=> ( top_top_set_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_283_UNIV__eq__I,axiom,
! [A3: set_nat] :
( ! [X6: nat] : ( member_nat @ X6 @ A3 )
=> ( top_top_set_nat = A3 ) ) ).
% UNIV_eq_I
thf(fact_284_UNIV__witness,axiom,
? [X6: set_a] : ( member_set_a @ X6 @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_285_UNIV__witness,axiom,
? [X6: set_nat] : ( member_set_nat @ X6 @ top_top_set_set_nat ) ).
% UNIV_witness
thf(fact_286_UNIV__witness,axiom,
? [X6: nat > a] : ( member_nat_a @ X6 @ top_top_set_nat_a ) ).
% UNIV_witness
thf(fact_287_UNIV__witness,axiom,
? [X6: a > nat] : ( member_a_nat @ X6 @ top_top_set_a_nat ) ).
% UNIV_witness
thf(fact_288_UNIV__witness,axiom,
? [X6: a > a] : ( member_a_a @ X6 @ top_top_set_a_a ) ).
% UNIV_witness
thf(fact_289_UNIV__witness,axiom,
? [X6: a] : ( member_a @ X6 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_290_UNIV__witness,axiom,
? [X6: nat] : ( member_nat @ X6 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_291_inf__sup__aci_I4_J,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_292_inf__sup__aci_I4_J,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_293_inf__sup__aci_I4_J,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y ) )
= ( inf_inf_set_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_294_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_295_inf__sup__aci_I3_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_296_inf__sup__aci_I3_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_297_inf__sup__aci_I3_J,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( inf_inf_set_set_a @ Y @ ( inf_inf_set_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_298_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_299_inf__sup__aci_I2_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
= ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_300_inf__sup__aci_I2_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_301_inf__sup__aci_I2_J,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_302_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_303_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X7: set_nat,Y2: set_nat] : ( inf_inf_set_nat @ Y2 @ X7 ) ) ) ).
% inf_sup_aci(1)
thf(fact_304_inf__sup__aci_I1_J,axiom,
( inf_inf_nat
= ( ^ [X7: nat,Y2: nat] : ( inf_inf_nat @ Y2 @ X7 ) ) ) ).
% inf_sup_aci(1)
thf(fact_305_inf__sup__aci_I1_J,axiom,
( inf_inf_set_set_a
= ( ^ [X7: set_set_a,Y2: set_set_a] : ( inf_inf_set_set_a @ Y2 @ X7 ) ) ) ).
% inf_sup_aci(1)
thf(fact_306_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X7: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X7 ) ) ) ).
% inf_sup_aci(1)
thf(fact_307_inf_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_308_inf_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ C )
= ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_309_inf_Oassoc,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ C )
= ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_310_inf_Oassoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_311_inf__assoc,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
= ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_312_inf__assoc,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_313_inf__assoc,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_314_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_315_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B2: set_nat] : ( inf_inf_set_nat @ B2 @ A4 ) ) ) ).
% inf.commute
thf(fact_316_inf_Ocommute,axiom,
( inf_inf_nat
= ( ^ [A4: nat,B2: nat] : ( inf_inf_nat @ B2 @ A4 ) ) ) ).
% inf.commute
thf(fact_317_inf_Ocommute,axiom,
( inf_inf_set_set_a
= ( ^ [A4: set_set_a,B2: set_set_a] : ( inf_inf_set_set_a @ B2 @ A4 ) ) ) ).
% inf.commute
thf(fact_318_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B2: set_a] : ( inf_inf_set_a @ B2 @ A4 ) ) ) ).
% inf.commute
thf(fact_319_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X7: set_nat,Y2: set_nat] : ( inf_inf_set_nat @ Y2 @ X7 ) ) ) ).
% inf_commute
thf(fact_320_inf__commute,axiom,
( inf_inf_nat
= ( ^ [X7: nat,Y2: nat] : ( inf_inf_nat @ Y2 @ X7 ) ) ) ).
% inf_commute
thf(fact_321_inf__commute,axiom,
( inf_inf_set_set_a
= ( ^ [X7: set_set_a,Y2: set_set_a] : ( inf_inf_set_set_a @ Y2 @ X7 ) ) ) ).
% inf_commute
thf(fact_322_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X7: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X7 ) ) ) ).
% inf_commute
thf(fact_323_inf_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A @ C ) )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_324_inf_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( inf_inf_nat @ B @ ( inf_inf_nat @ A @ C ) )
= ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_325_inf_Oleft__commute,axiom,
! [B: set_set_a,A: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ B @ ( inf_inf_set_set_a @ A @ C ) )
= ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_326_inf_Oleft__commute,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_327_inf__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_328_inf__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_329_inf__left__commute,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( inf_inf_set_set_a @ Y @ ( inf_inf_set_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_330_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_331_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X8: nat] :
( ( P @ X8 )
=> ( ord_less_eq_nat @ X8 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_332_IntE,axiom,
! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B3 ) )
=> ~ ( ( member_set_nat @ C @ A3 )
=> ~ ( member_set_nat @ C @ B3 ) ) ) ).
% IntE
thf(fact_333_IntE,axiom,
! [C: set_a,A3: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A3 @ B3 ) )
=> ~ ( ( member_set_a @ C @ A3 )
=> ~ ( member_set_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_334_IntE,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
=> ~ ( ( member_nat @ C @ A3 )
=> ~ ( member_nat @ C @ B3 ) ) ) ).
% IntE
thf(fact_335_IntE,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B3 ) )
=> ~ ( ( member_a @ C @ A3 )
=> ~ ( member_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_336_IntD1,axiom,
! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B3 ) )
=> ( member_set_nat @ C @ A3 ) ) ).
% IntD1
thf(fact_337_IntD1,axiom,
! [C: set_a,A3: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A3 @ B3 ) )
=> ( member_set_a @ C @ A3 ) ) ).
% IntD1
thf(fact_338_IntD1,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
=> ( member_nat @ C @ A3 ) ) ).
% IntD1
thf(fact_339_IntD1,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B3 ) )
=> ( member_a @ C @ A3 ) ) ).
% IntD1
thf(fact_340_IntD2,axiom,
! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B3 ) )
=> ( member_set_nat @ C @ B3 ) ) ).
% IntD2
thf(fact_341_IntD2,axiom,
! [C: set_a,A3: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A3 @ B3 ) )
=> ( member_set_a @ C @ B3 ) ) ).
% IntD2
thf(fact_342_IntD2,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
=> ( member_nat @ C @ B3 ) ) ).
% IntD2
thf(fact_343_IntD2,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B3 ) )
=> ( member_a @ C @ B3 ) ) ).
% IntD2
thf(fact_344_Int__assoc,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ C2 )
= ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_345_Int__assoc,axiom,
! [A3: set_set_a,B3: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A3 @ B3 ) @ C2 )
= ( inf_inf_set_set_a @ A3 @ ( inf_inf_set_set_a @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_346_Int__assoc,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ C2 )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_347_Int__absorb,axiom,
! [A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_348_Int__absorb,axiom,
! [A3: set_set_a] :
( ( inf_inf_set_set_a @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_349_Int__absorb,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_350_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A5: set_nat,B4: set_nat] : ( inf_inf_set_nat @ B4 @ A5 ) ) ) ).
% Int_commute
thf(fact_351_Int__commute,axiom,
( inf_inf_set_set_a
= ( ^ [A5: set_set_a,B4: set_set_a] : ( inf_inf_set_set_a @ B4 @ A5 ) ) ) ).
% Int_commute
thf(fact_352_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A5: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A5 ) ) ) ).
% Int_commute
thf(fact_353_Int__left__absorb,axiom,
! [A3: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ A3 @ B3 ) )
= ( inf_inf_set_nat @ A3 @ B3 ) ) ).
% Int_left_absorb
thf(fact_354_Int__left__absorb,axiom,
! [A3: set_set_a,B3: set_set_a] :
( ( inf_inf_set_set_a @ A3 @ ( inf_inf_set_set_a @ A3 @ B3 ) )
= ( inf_inf_set_set_a @ A3 @ B3 ) ) ).
% Int_left_absorb
thf(fact_355_Int__left__absorb,axiom,
! [A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ A3 @ B3 ) )
= ( inf_inf_set_a @ A3 @ B3 ) ) ).
% Int_left_absorb
thf(fact_356_Int__left__commute,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C2 ) )
= ( inf_inf_set_nat @ B3 @ ( inf_inf_set_nat @ A3 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_357_Int__left__commute,axiom,
! [A3: set_set_a,B3: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ A3 @ ( inf_inf_set_set_a @ B3 @ C2 ) )
= ( inf_inf_set_set_a @ B3 @ ( inf_inf_set_set_a @ A3 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_358_Int__left__commute,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C2 ) )
= ( inf_inf_set_a @ B3 @ ( inf_inf_set_a @ A3 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_359_ord_OlessThan__iff,axiom,
! [I: set_a,Less: set_a > set_a > $o,K: set_a] :
( ( member_set_a @ I @ ( set_lessThan_set_a @ Less @ K ) )
= ( Less @ I @ K ) ) ).
% ord.lessThan_iff
thf(fact_360_ord_OlessThan__iff,axiom,
! [I: set_nat,Less: set_nat > set_nat > $o,K: set_nat] :
( ( member_set_nat @ I @ ( set_lessThan_set_nat @ Less @ K ) )
= ( Less @ I @ K ) ) ).
% ord.lessThan_iff
thf(fact_361_ord_OlessThan__iff,axiom,
! [I: nat,Less: nat > nat > $o,K: nat] :
( ( member_nat @ I @ ( set_lessThan_nat @ Less @ K ) )
= ( Less @ I @ K ) ) ).
% ord.lessThan_iff
thf(fact_362_ord_OlessThan__iff,axiom,
! [I: a,Less: a > a > $o,K: a] :
( ( member_a @ I @ ( set_lessThan_a @ Less @ K ) )
= ( Less @ I @ K ) ) ).
% ord.lessThan_iff
thf(fact_363_ord_OlessThan_Ocong,axiom,
set_lessThan_nat = set_lessThan_nat ).
% ord.lessThan.cong
thf(fact_364_ord_OlessThan_Ocong,axiom,
set_lessThan_a = set_lessThan_a ).
% ord.lessThan.cong
thf(fact_365_ord_OgreaterThan__iff,axiom,
! [I: set_a,Less: set_a > set_a > $o,K: set_a] :
( ( member_set_a @ I @ ( set_gr7079423240508057912_set_a @ Less @ K ) )
= ( Less @ K @ I ) ) ).
% ord.greaterThan_iff
thf(fact_366_ord_OgreaterThan__iff,axiom,
! [I: set_nat,Less: set_nat > set_nat > $o,K: set_nat] :
( ( member_set_nat @ I @ ( set_gr7331712898236694572et_nat @ Less @ K ) )
= ( Less @ K @ I ) ) ).
% ord.greaterThan_iff
thf(fact_367_ord_OgreaterThan__iff,axiom,
! [I: nat,Less: nat > nat > $o,K: nat] :
( ( member_nat @ I @ ( set_greaterThan_nat @ Less @ K ) )
= ( Less @ K @ I ) ) ).
% ord.greaterThan_iff
thf(fact_368_ord_OgreaterThan__iff,axiom,
! [I: a,Less: a > a > $o,K: a] :
( ( member_a @ I @ ( set_greaterThan_a @ Less @ K ) )
= ( Less @ K @ I ) ) ).
% ord.greaterThan_iff
thf(fact_369_ord_OgreaterThan_Ocong,axiom,
set_greaterThan_nat = set_greaterThan_nat ).
% ord.greaterThan.cong
thf(fact_370_ord_OgreaterThan_Ocong,axiom,
set_greaterThan_a = set_greaterThan_a ).
% ord.greaterThan.cong
thf(fact_371_ord_OgreaterThanLessThan__iff,axiom,
! [I: set_a,Less: set_a > set_a > $o,L: set_a,U: set_a] :
( ( member_set_a @ I @ ( set_gr890836759843348478_set_a @ Less @ L @ U ) )
= ( ( Less @ L @ I )
& ( Less @ I @ U ) ) ) ).
% ord.greaterThanLessThan_iff
thf(fact_372_ord_OgreaterThanLessThan__iff,axiom,
! [I: set_nat,Less: set_nat > set_nat > $o,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_gr8761010429814653926et_nat @ Less @ L @ U ) )
= ( ( Less @ L @ I )
& ( Less @ I @ U ) ) ) ).
% ord.greaterThanLessThan_iff
thf(fact_373_ord_OgreaterThanLessThan__iff,axiom,
! [I: nat,Less: nat > nat > $o,L: nat,U: nat] :
( ( member_nat @ I @ ( set_gr7355061613916526640an_nat @ Less @ L @ U ) )
= ( ( Less @ L @ I )
& ( Less @ I @ U ) ) ) ).
% ord.greaterThanLessThan_iff
thf(fact_374_ord_OgreaterThanLessThan__iff,axiom,
! [I: a,Less: a > a > $o,L: a,U: a] :
( ( member_a @ I @ ( set_gr5113148517155960478Than_a @ Less @ L @ U ) )
= ( ( Less @ L @ I )
& ( Less @ I @ U ) ) ) ).
% ord.greaterThanLessThan_iff
thf(fact_375_ord_OgreaterThanLessThan_Ocong,axiom,
set_gr7355061613916526640an_nat = set_gr7355061613916526640an_nat ).
% ord.greaterThanLessThan.cong
thf(fact_376_ord_OgreaterThanLessThan_Ocong,axiom,
set_gr5113148517155960478Than_a = set_gr5113148517155960478Than_a ).
% ord.greaterThanLessThan.cong
thf(fact_377_inf__sup__ord_I2_J,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_378_inf__sup__ord_I2_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_379_inf__sup__ord_I2_J,axiom,
! [X: $o > nat,Y: $o > nat] : ( ord_less_eq_o_nat @ ( inf_inf_o_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_380_inf__sup__ord_I2_J,axiom,
! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_381_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_382_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_383_inf__sup__ord_I1_J,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_384_inf__sup__ord_I1_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_385_inf__sup__ord_I1_J,axiom,
! [X: $o > nat,Y: $o > nat] : ( ord_less_eq_o_nat @ ( inf_inf_o_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_386_inf__sup__ord_I1_J,axiom,
! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_387_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_388_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_389_inf__le1,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_390_inf__le1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_391_inf__le1,axiom,
! [X: $o > nat,Y: $o > nat] : ( ord_less_eq_o_nat @ ( inf_inf_o_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_392_inf__le1,axiom,
! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_393_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_394_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_395_inf__le2,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_396_inf__le2,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_397_inf__le2,axiom,
! [X: $o > nat,Y: $o > nat] : ( ord_less_eq_o_nat @ ( inf_inf_o_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_398_inf__le2,axiom,
! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_399_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_400_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_401_le__infE,axiom,
! [X: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A @ B ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ X @ A )
=> ~ ( ord_le3724670747650509150_set_a @ X @ B ) ) ) ).
% le_infE
thf(fact_402_le__infE,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_set_nat @ X @ A )
=> ~ ( ord_less_eq_set_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_403_le__infE,axiom,
! [X: $o > nat,A: $o > nat,B: $o > nat] :
( ( ord_less_eq_o_nat @ X @ ( inf_inf_o_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_o_nat @ X @ A )
=> ~ ( ord_less_eq_o_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_404_le__infE,axiom,
! [X: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ A @ B ) )
=> ~ ( ( ord_le6893508408891458716et_nat @ X @ A )
=> ~ ( ord_le6893508408891458716et_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_405_le__infE,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B ) ) ) ).
% le_infE
thf(fact_406_le__infE,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_407_le__infI,axiom,
! [X: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ A )
=> ( ( ord_le3724670747650509150_set_a @ X @ B )
=> ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_408_le__infI,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ X @ B )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_409_le__infI,axiom,
! [X: $o > nat,A: $o > nat,B: $o > nat] :
( ( ord_less_eq_o_nat @ X @ A )
=> ( ( ord_less_eq_o_nat @ X @ B )
=> ( ord_less_eq_o_nat @ X @ ( inf_inf_o_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_410_le__infI,axiom,
! [X: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ A )
=> ( ( ord_le6893508408891458716et_nat @ X @ B )
=> ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_411_le__infI,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_412_le__infI,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_413_inf__mono,axiom,
! [A: set_set_a,C: set_set_a,B: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ( ord_le3724670747650509150_set_a @ B @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( inf_inf_set_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_414_inf__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_415_inf__mono,axiom,
! [A: $o > nat,C: $o > nat,B: $o > nat,D: $o > nat] :
( ( ord_less_eq_o_nat @ A @ C )
=> ( ( ord_less_eq_o_nat @ B @ D )
=> ( ord_less_eq_o_nat @ ( inf_inf_o_nat @ A @ B ) @ ( inf_inf_o_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_416_inf__mono,axiom,
! [A: set_set_nat,C: set_set_nat,B: set_set_nat,D: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C )
=> ( ( ord_le6893508408891458716et_nat @ B @ D )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( inf_inf_set_set_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_417_inf__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_418_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_419_le__infI1,axiom,
! [A: set_set_a,X: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ X )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_420_le__infI1,axiom,
! [A: set_nat,X: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_421_le__infI1,axiom,
! [A: $o > nat,X: $o > nat,B: $o > nat] :
( ( ord_less_eq_o_nat @ A @ X )
=> ( ord_less_eq_o_nat @ ( inf_inf_o_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_422_le__infI1,axiom,
! [A: set_set_nat,X: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ X )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_423_le__infI1,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_424_le__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_425_le__infI2,axiom,
! [B: set_set_a,X: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ X )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_426_le__infI2,axiom,
! [B: set_nat,X: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_427_le__infI2,axiom,
! [B: $o > nat,X: $o > nat,A: $o > nat] :
( ( ord_less_eq_o_nat @ B @ X )
=> ( ord_less_eq_o_nat @ ( inf_inf_o_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_428_le__infI2,axiom,
! [B: set_set_nat,X: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ X )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_429_le__infI2,axiom,
! [B: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_430_le__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_431_inf_OorderE,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( A
= ( inf_inf_set_set_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_432_inf_OorderE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( A
= ( inf_inf_set_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_433_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_434_inf_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% inf.orderI
thf(fact_435_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_436_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X6: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X6 @ Y3 ) @ X6 )
=> ( ! [X6: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X6 @ Y3 ) @ Y3 )
=> ( ! [X6: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X6 @ Y3 )
=> ( ( ord_less_eq_set_a @ X6 @ Z2 )
=> ( ord_less_eq_set_a @ X6 @ ( F @ Y3 @ Z2 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_437_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X6: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X6 @ Y3 ) @ X6 )
=> ( ! [X6: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X6 @ Y3 ) @ Y3 )
=> ( ! [X6: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X6 @ Y3 )
=> ( ( ord_less_eq_nat @ X6 @ Z2 )
=> ( ord_less_eq_nat @ X6 @ ( F @ Y3 @ Z2 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_438_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X7: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X7 @ Y2 )
= X7 ) ) ) ).
% le_iff_inf
thf(fact_439_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X7: nat,Y2: nat] :
( ( inf_inf_nat @ X7 @ Y2 )
= X7 ) ) ) ).
% le_iff_inf
thf(fact_440_inf_Oabsorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_441_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_442_inf_Oabsorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_443_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_444_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_445_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_446_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_447_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_448_inf_OboundedE,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_449_inf_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_450_inf_OboundedI,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_451_inf_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_452_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_453_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_454_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B2: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_455_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] :
( A4
= ( inf_inf_nat @ A4 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_456_inf_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_457_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_458_inf_Ocobounded2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_459_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_460_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B2: set_a] :
( ( inf_inf_set_a @ A4 @ B2 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_461_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] :
( ( inf_inf_nat @ A4 @ B2 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_462_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_463_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_464_inf_OcoboundedI1,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_465_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_466_inf_OcoboundedI2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_467_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_468_Int__UNIV__left,axiom,
! [B3: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ B3 )
= B3 ) ).
% Int_UNIV_left
thf(fact_469_Int__UNIV__left,axiom,
! [B3: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B3 )
= B3 ) ).
% Int_UNIV_left
thf(fact_470_Int__UNIV__right,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ top_top_set_a )
= A3 ) ).
% Int_UNIV_right
thf(fact_471_Int__UNIV__right,axiom,
! [A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ top_top_set_nat )
= A3 ) ).
% Int_UNIV_right
thf(fact_472_local_OatLeastLessThan__def,axiom,
! [L: a,U: a] :
( ( set_at7181854700642260099Than_a @ less_eq @ less @ L @ U )
= ( inf_inf_set_a @ ( set_atLeast_a @ less_eq @ L ) @ ( set_lessThan_a @ less @ U ) ) ) ).
% local.atLeastLessThan_def
thf(fact_473_local_OgreaterThanAtMost__def,axiom,
! [L: a,U: a] :
( ( set_gr7381147536065975362Most_a @ less_eq @ less @ L @ U )
= ( inf_inf_set_a @ ( set_greaterThan_a @ less @ L ) @ ( set_atMost_a @ less_eq @ U ) ) ) ).
% local.greaterThanAtMost_def
thf(fact_474_local_OgreaterThanAtMost__iff,axiom,
! [I: a,L: a,U: a] :
( ( member_a @ I @ ( set_gr7381147536065975362Most_a @ less_eq @ less @ L @ U ) )
= ( ( less @ L @ I )
& ( less_eq @ I @ U ) ) ) ).
% local.greaterThanAtMost_iff
thf(fact_475_local_OatLeastLessThan__iff,axiom,
! [I: a,L: a,U: a] :
( ( member_a @ I @ ( set_at7181854700642260099Than_a @ less_eq @ less @ L @ U ) )
= ( ( less_eq @ L @ I )
& ( less @ I @ U ) ) ) ).
% local.atLeastLessThan_iff
thf(fact_476_local_Olexordp__append__rightI,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys != nil_a )
=> ( lexordp_a @ less @ Xs @ ( append_a @ Xs @ Ys ) ) ) ).
% local.lexordp_append_rightI
thf(fact_477_local_Olexordp__append__left__rightI,axiom,
! [X: a,Y: a,Us: list_a,Xs: list_a,Ys: list_a] :
( ( less @ X @ Y )
=> ( lexordp_a @ less @ ( append_a @ Us @ ( cons_a @ X @ Xs ) ) @ ( append_a @ Us @ ( cons_a @ Y @ Ys ) ) ) ) ).
% local.lexordp_append_left_rightI
thf(fact_478_iso__tuple__UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_479_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_480_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_481_local_Olexordp__append__leftD,axiom,
! [Xs: list_a,Us: list_a,Vs: list_a] :
( ( lexordp_a @ less @ ( append_a @ Xs @ Us ) @ ( append_a @ Xs @ Vs ) )
=> ( ! [A6: a] :
~ ( less @ A6 @ A6 )
=> ( lexordp_a @ less @ Us @ Vs ) ) ) ).
% local.lexordp_append_leftD
thf(fact_482_local_Olexordp__append__leftI,axiom,
! [Us: list_a,Vs: list_a,Xs: list_a] :
( ( lexordp_a @ less @ Us @ Vs )
=> ( lexordp_a @ less @ ( append_a @ Xs @ Us ) @ ( append_a @ Xs @ Vs ) ) ) ).
% local.lexordp_append_leftI
thf(fact_483_local_Olexordp__eq__pref,axiom,
! [U: list_a,V: list_a] : ( lexordp_eq_a @ less @ U @ ( append_a @ U @ V ) ) ).
% local.lexordp_eq_pref
thf(fact_484_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_485_Int__subset__iff,axiom,
! [C2: set_a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A3 @ B3 ) )
= ( ( ord_less_eq_set_a @ C2 @ A3 )
& ( ord_less_eq_set_a @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_486_append_Oassoc,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( append_a @ ( append_a @ A @ B ) @ C )
= ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_487_append__assoc,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
= ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_488_append__same__eq,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs )
= ( append_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_489_same__append__eq,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_490_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_491_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_492_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_493_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_494_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_495_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_496_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_497_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_498_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_499_local_OatMost__iff,axiom,
! [I: a,K: a] :
( ( member_a @ I @ ( set_atMost_a @ less_eq @ K ) )
= ( less_eq @ I @ K ) ) ).
% local.atMost_iff
thf(fact_500_local_OatLeast__iff,axiom,
! [I: a,K: a] :
( ( member_a @ I @ ( set_atLeast_a @ less_eq @ K ) )
= ( less_eq @ K @ I ) ) ).
% local.atLeast_iff
thf(fact_501_local_OatLeastAtMost__def,axiom,
! [L: a,U: a] :
( ( set_atLeastAtMost_a @ less_eq @ L @ U )
= ( inf_inf_set_a @ ( set_atLeast_a @ less_eq @ L ) @ ( set_atMost_a @ less_eq @ U ) ) ) ).
% local.atLeastAtMost_def
thf(fact_502_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_503_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us2: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us2 ) )
& ( ( append_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_504_ord_OatLeast_Ocong,axiom,
set_atLeast_a = set_atLeast_a ).
% ord.atLeast.cong
thf(fact_505_ord_OatMost_Ocong,axiom,
set_atMost_a = set_atMost_a ).
% ord.atMost.cong
thf(fact_506_ord_OatLeast__iff,axiom,
! [I: nat,Less_eq: nat > nat > $o,K: nat] :
( ( member_nat @ I @ ( set_atLeast_nat @ Less_eq @ K ) )
= ( Less_eq @ K @ I ) ) ).
% ord.atLeast_iff
thf(fact_507_ord_OatLeast__iff,axiom,
! [I: a,Less_eq: a > a > $o,K: a] :
( ( member_a @ I @ ( set_atLeast_a @ Less_eq @ K ) )
= ( Less_eq @ K @ I ) ) ).
% ord.atLeast_iff
thf(fact_508_ord_OatMost__iff,axiom,
! [I: nat,Less_eq: nat > nat > $o,K: nat] :
( ( member_nat @ I @ ( set_atMost_nat @ Less_eq @ K ) )
= ( Less_eq @ I @ K ) ) ).
% ord.atMost_iff
thf(fact_509_ord_OatMost__iff,axiom,
! [I: a,Less_eq: a > a > $o,K: a] :
( ( member_a @ I @ ( set_atMost_a @ Less_eq @ K ) )
= ( Less_eq @ I @ K ) ) ).
% ord.atMost_iff
thf(fact_510_ord_OgreaterThanAtMost_Ocong,axiom,
set_gr7381147536065975362Most_a = set_gr7381147536065975362Most_a ).
% ord.greaterThanAtMost.cong
thf(fact_511_ord_OgreaterThanAtMost__iff,axiom,
! [I: nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,L: nat,U: nat] :
( ( member_nat @ I @ ( set_gr2512719926906771852st_nat @ Less_eq @ Less @ L @ U ) )
= ( ( Less @ L @ I )
& ( Less_eq @ I @ U ) ) ) ).
% ord.greaterThanAtMost_iff
thf(fact_512_ord_OgreaterThanAtMost__iff,axiom,
! [I: a,Less_eq: a > a > $o,Less: a > a > $o,L: a,U: a] :
( ( member_a @ I @ ( set_gr7381147536065975362Most_a @ Less_eq @ Less @ L @ U ) )
= ( ( Less @ L @ I )
& ( Less_eq @ I @ U ) ) ) ).
% ord.greaterThanAtMost_iff
thf(fact_513_ord_OatLeastLessThan_Ocong,axiom,
set_at7181854700642260099Than_a = set_at7181854700642260099Than_a ).
% ord.atLeastLessThan.cong
thf(fact_514_ord_OatLeastLessThan__iff,axiom,
! [I: nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,L: nat,U: nat] :
( ( member_nat @ I @ ( set_at5771681743635938059an_nat @ Less_eq @ Less @ L @ U ) )
= ( ( Less_eq @ L @ I )
& ( Less @ I @ U ) ) ) ).
% ord.atLeastLessThan_iff
thf(fact_515_ord_OatLeastLessThan__iff,axiom,
! [I: a,Less_eq: a > a > $o,Less: a > a > $o,L: a,U: a] :
( ( member_a @ I @ ( set_at7181854700642260099Than_a @ Less_eq @ Less @ L @ U ) )
= ( ( Less_eq @ L @ I )
& ( Less @ I @ U ) ) ) ).
% ord.atLeastLessThan_iff
thf(fact_516_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_517_append__Cons,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X @ Xs ) @ Ys )
= ( cons_a @ X @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_518_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_519_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_520_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_521_subset__UNIV,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ top_top_set_a ) ).
% subset_UNIV
thf(fact_522_subset__UNIV,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_523_Int__Collect__mono,axiom,
! [A3: set_nat,B3: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ( ( P @ X6 )
=> ( Q @ X6 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B3 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_524_Int__Collect__mono,axiom,
! [A3: set_a,B3: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A3 )
=> ( ( P @ X6 )
=> ( Q @ X6 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B3 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_525_Int__greatest,axiom,
! [C2: set_a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ A3 )
=> ( ( ord_less_eq_set_a @ C2 @ B3 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_526_Int__absorb2,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= A3 ) ) ).
% Int_absorb2
thf(fact_527_Int__absorb1,axiom,
! [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_528_Int__lower2,axiom,
! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_529_Int__lower1,axiom,
! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ A3 ) ).
% Int_lower1
thf(fact_530_Int__mono,axiom,
! [A3: set_a,C2: set_a,B3: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_531_ord_Olexordp__append__leftI,axiom,
! [Less: a > a > $o,Us: list_a,Vs: list_a,Xs: list_a] :
( ( lexordp_a @ Less @ Us @ Vs )
=> ( lexordp_a @ Less @ ( append_a @ Xs @ Us ) @ ( append_a @ Xs @ Vs ) ) ) ).
% ord.lexordp_append_leftI
thf(fact_532_ord_Olexordp__append__leftD,axiom,
! [Less: a > a > $o,Xs: list_a,Us: list_a,Vs: list_a] :
( ( lexordp_a @ Less @ ( append_a @ Xs @ Us ) @ ( append_a @ Xs @ Vs ) )
=> ( ! [A6: a] :
~ ( Less @ A6 @ A6 )
=> ( lexordp_a @ Less @ Us @ Vs ) ) ) ).
% ord.lexordp_append_leftD
thf(fact_533_ord_Olexordp__eq__pref,axiom,
! [Less: a > a > $o,U: list_a,V: list_a] : ( lexordp_eq_a @ Less @ U @ ( append_a @ U @ V ) ) ).
% ord.lexordp_eq_pref
thf(fact_534_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X6: a,Xs3: list_a] :
( ( P @ Xs3 )
=> ( P @ ( append_a @ Xs3 @ ( cons_a @ X6 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_535_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys3: list_a,Y3: a] :
( Xs
!= ( append_a @ Ys3 @ ( cons_a @ Y3 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_536_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs ) )
| ? [Ys4: list_a] :
( ( ( cons_a @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_537_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X @ Xs ) ) )
| ? [Ys4: list_a] :
( ( Ys
= ( cons_a @ X @ Ys4 ) )
& ( ( append_a @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_538_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X6: a] : ( P @ ( cons_a @ X6 @ nil_a ) )
=> ( ! [X6: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P @ Xs3 )
=> ( P @ ( append_a @ Xs3 @ ( cons_a @ X6 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_539_ord_Olexordp__append__left__rightI,axiom,
! [Less: a > a > $o,X: a,Y: a,Us: list_a,Xs: list_a,Ys: list_a] :
( ( Less @ X @ Y )
=> ( lexordp_a @ Less @ ( append_a @ Us @ ( cons_a @ X @ Xs ) ) @ ( append_a @ Us @ ( cons_a @ Y @ Ys ) ) ) ) ).
% ord.lexordp_append_left_rightI
thf(fact_540_ord_Olexordp__append__rightI,axiom,
! [Ys: list_a,Less: a > a > $o,Xs: list_a] :
( ( Ys != nil_a )
=> ( lexordp_a @ Less @ Xs @ ( append_a @ Xs @ Ys ) ) ) ).
% ord.lexordp_append_rightI
thf(fact_541_ord_OgreaterThanAtMost__def,axiom,
( set_gr7381147536065975362Most_a
= ( ^ [Less_eq2: a > a > $o,Less2: a > a > $o,L2: a,U2: a] : ( inf_inf_set_a @ ( set_greaterThan_a @ Less2 @ L2 ) @ ( set_atMost_a @ Less_eq2 @ U2 ) ) ) ) ).
% ord.greaterThanAtMost_def
thf(fact_542_ord_OatLeastLessThan__def,axiom,
( set_at7181854700642260099Than_a
= ( ^ [Less_eq2: a > a > $o,Less2: a > a > $o,L2: a,U2: a] : ( inf_inf_set_a @ ( set_atLeast_a @ Less_eq2 @ L2 ) @ ( set_lessThan_a @ Less2 @ U2 ) ) ) ) ).
% ord.atLeastLessThan_def
thf(fact_543_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_544_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_545_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y3: nat] :
( ( ord_less_eq_nat @ X6 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_546_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y3: nat] :
( ( ord_less_eq_nat @ X6 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_547_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_548_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_549_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y3: nat] :
( ( ord_less_eq_nat @ X6 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_550_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y3: nat] :
( ( ord_less_eq_nat @ X6 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_551_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
& ( ord_less_eq_nat @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_552_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_553_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_554_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_555_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
& ( ord_less_eq_nat @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_556_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: 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 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_557_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_558_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_559_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_560_ord__class_Oord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_class.ord_le_eq_trans
thf(fact_561_ord__class_Oord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_class.ord_eq_le_trans
thf(fact_562_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [X7: nat,Y2: nat] :
( ( ord_less_eq_nat @ X7 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X7 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_563_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_564_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_565_ord_Omin_Ocong,axiom,
min_a = min_a ).
% ord.min.cong
thf(fact_566_ord_Omax_Ocong,axiom,
max_a = max_a ).
% ord.max.cong
thf(fact_567_ord_Omin__def,axiom,
( min_a
= ( ^ [Less_eq2: a > a > $o,A4: a,B2: a] : ( if_a @ ( Less_eq2 @ A4 @ B2 ) @ A4 @ B2 ) ) ) ).
% ord.min_def
thf(fact_568_ord_Omax__def,axiom,
( max_a
= ( ^ [Less_eq2: a > a > $o,A4: a,B2: a] : ( if_a @ ( Less_eq2 @ A4 @ B2 ) @ B2 @ A4 ) ) ) ).
% ord.max_def
thf(fact_569_top_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
=> ( A = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_570_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_571_top_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
= ( A = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_572_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_573_top__greatest,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% top_greatest
thf(fact_574_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_575_ord_OatLeastAtMost__def,axiom,
( set_atLeastAtMost_a
= ( ^ [Less_eq2: a > a > $o,L2: a,U2: a] : ( inf_inf_set_a @ ( set_atLeast_a @ Less_eq2 @ L2 ) @ ( set_atMost_a @ Less_eq2 @ U2 ) ) ) ) ).
% ord.atLeastAtMost_def
thf(fact_576_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X7: a] : ( member_a @ X7 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_577_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X7: nat] : ( member_nat @ X7 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_578_boolean__algebra_Oconj__one__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ top_top_set_a )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_579_boolean__algebra_Oconj__one__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_580_product__lists_Osimps_I1_J,axiom,
( ( product_lists_a @ nil_list_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% product_lists.simps(1)
thf(fact_581_bind__simps_I2_J,axiom,
! [X: a,Xs: list_a,F: a > list_a] :
( ( bind_a_a @ ( cons_a @ X @ Xs ) @ F )
= ( append_a @ ( F @ X ) @ ( bind_a_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_582_local_OatLeastAtMost__iff,axiom,
! [I: a,L: a,U: a] :
( ( member_a @ I @ ( set_atLeastAtMost_a @ less_eq @ L @ U ) )
= ( ( less_eq @ L @ I )
& ( less_eq @ I @ U ) ) ) ).
% local.atLeastAtMost_iff
thf(fact_583_local_Omono__on__subset,axiom,
! [A3: set_a,F: a > nat,B3: set_a] :
( ( monotone_on_a_nat @ A3 @ less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( monotone_on_a_nat @ B3 @ less_eq @ ord_less_eq_nat @ F ) ) ) ).
% local.mono_on_subset
thf(fact_584_local_Omono__onI,axiom,
! [A3: set_a,F: a > nat] :
( ! [R: a,S: a] :
( ( member_a @ R @ A3 )
=> ( ( member_a @ S @ A3 )
=> ( ( less_eq @ R @ S )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monotone_on_a_nat @ A3 @ less_eq @ ord_less_eq_nat @ F ) ) ).
% local.mono_onI
thf(fact_585_local_Omono__onD,axiom,
! [A3: set_a,F: a > nat,R2: a,S2: a] :
( ( monotone_on_a_nat @ A3 @ less_eq @ ord_less_eq_nat @ F )
=> ( ( member_a @ R2 @ A3 )
=> ( ( member_a @ S2 @ A3 )
=> ( ( less_eq @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% local.mono_onD
thf(fact_586_subsetI,axiom,
! [A3: set_a,B3: set_a] :
( ! [X6: a] :
( ( member_a @ X6 @ A3 )
=> ( member_a @ X6 @ B3 ) )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_587_subsetI,axiom,
! [A3: set_nat,B3: set_nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ( member_nat @ X6 @ B3 ) )
=> ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).
% subsetI
thf(fact_588_bind__simps_I1_J,axiom,
! [F: a > list_a] :
( ( bind_a_a @ nil_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_589_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_590_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A5 )
=> ( member_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_591_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
! [X7: a] :
( ( member_a @ X7 @ A5 )
=> ( member_a @ X7 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_592_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
! [X7: nat] :
( ( member_nat @ X7 @ A5 )
=> ( member_nat @ X7 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_593_subsetD,axiom,
! [A3: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_594_subsetD,axiom,
! [A3: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_595_in__mono,axiom,
! [A3: set_a,B3: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_596_in__mono,axiom,
! [A3: set_nat,B3: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_597_ord_OatLeastAtMost__iff,axiom,
! [I: nat,Less_eq: nat > nat > $o,L: nat,U: nat] :
( ( member_nat @ I @ ( set_at8086275982456994279st_nat @ Less_eq @ L @ U ) )
= ( ( Less_eq @ L @ I )
& ( Less_eq @ I @ U ) ) ) ).
% ord.atLeastAtMost_iff
thf(fact_598_ord_OatLeastAtMost__iff,axiom,
! [I: a,Less_eq: a > a > $o,L: a,U: a] :
( ( member_a @ I @ ( set_atLeastAtMost_a @ Less_eq @ L @ U ) )
= ( ( Less_eq @ L @ I )
& ( Less_eq @ I @ U ) ) ) ).
% ord.atLeastAtMost_iff
thf(fact_599_ord_OatLeastAtMost_Ocong,axiom,
set_atLeastAtMost_a = set_atLeastAtMost_a ).
% ord.atLeastAtMost.cong
thf(fact_600_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_a,K: set_a,A: set_a,B: set_a] :
( ( A3
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A3 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_601_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_a,K: set_a,B: set_a,A: set_a] :
( ( B3
= ( inf_inf_set_a @ K @ B ) )
=> ( ( inf_inf_set_a @ A @ B3 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_602_mono__inf,axiom,
! [F: set_a > set_a,A3: set_a,B3: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( F @ ( inf_inf_set_a @ A3 @ B3 ) ) @ ( inf_inf_set_a @ ( F @ A3 ) @ ( F @ B3 ) ) ) ) ).
% mono_inf
thf(fact_603_mono__inf,axiom,
! [F: set_a > nat,A3: set_a,B3: set_a] :
( ( monoto4790297507788910087_a_nat @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_nat @ F )
=> ( ord_less_eq_nat @ ( F @ ( inf_inf_set_a @ A3 @ B3 ) ) @ ( inf_inf_nat @ ( F @ A3 ) @ ( F @ B3 ) ) ) ) ).
% mono_inf
thf(fact_604_mono__inf,axiom,
! [F: nat > set_a,A3: nat,B3: nat] :
( ( monoto723715495973462885_set_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( F @ ( inf_inf_nat @ A3 @ B3 ) ) @ ( inf_inf_set_a @ ( F @ A3 ) @ ( F @ B3 ) ) ) ) ).
% mono_inf
thf(fact_605_mono__inf,axiom,
! [F: nat > nat,A3: nat,B3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ord_less_eq_nat @ ( F @ ( inf_inf_nat @ A3 @ B3 ) ) @ ( inf_inf_nat @ ( F @ A3 ) @ ( F @ B3 ) ) ) ) ).
% mono_inf
thf(fact_606_ord_Omono__on__subset,axiom,
! [A3: set_nat,Less_eq: nat > nat > $o,F: nat > nat,B3: set_nat] :
( ( monotone_on_nat_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( monotone_on_nat_nat @ B3 @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_607_ord__class_Omono__on__subset,axiom,
! [A3: set_nat,F: nat > nat,B3: set_nat] :
( ( monotone_on_nat_nat @ A3 @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( monotone_on_nat_nat @ B3 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ) ).
% ord_class.mono_on_subset
thf(fact_608_mono__imp__mono__on,axiom,
! [F: nat > nat,A3: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( monotone_on_nat_nat @ A3 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_609_monoI,axiom,
! [F: nat > nat] :
( ! [X6: nat,Y3: nat] :
( ( ord_less_eq_nat @ X6 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_610_monoE,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_611_mono__Int,axiom,
! [F: set_a > set_a,A3: set_a,B3: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( F @ ( inf_inf_set_a @ A3 @ B3 ) ) @ ( inf_inf_set_a @ ( F @ A3 ) @ ( F @ B3 ) ) ) ) ).
% mono_Int
thf(fact_612_monotone__on__def,axiom,
( monotone_on_nat_nat
= ( ^ [A5: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F2: nat > nat] :
! [X7: nat] :
( ( member_nat @ X7 @ A5 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A5 )
=> ( ( Orda @ X7 @ Y2 )
=> ( Ordb @ ( F2 @ X7 ) @ ( F2 @ Y2 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_613_monotone__onI,axiom,
! [A3: set_nat,Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F: nat > nat] :
( ! [X6: nat,Y3: nat] :
( ( member_nat @ X6 @ A3 )
=> ( ( member_nat @ Y3 @ A3 )
=> ( ( Orda2 @ X6 @ Y3 )
=> ( Ordb2 @ ( F @ X6 ) @ ( F @ Y3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A3 @ Orda2 @ Ordb2 @ F ) ) ).
% monotone_onI
thf(fact_614_monotone__onD,axiom,
! [A3: set_nat,Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A3 @ Orda2 @ Ordb2 @ F )
=> ( ( member_nat @ X @ A3 )
=> ( ( member_nat @ Y @ A3 )
=> ( ( Orda2 @ X @ Y )
=> ( Ordb2 @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% monotone_onD
thf(fact_615_ord__class_Omono__onI,axiom,
! [A3: set_nat,F: nat > nat] :
( ! [R: nat,S: nat] :
( ( member_nat @ R @ A3 )
=> ( ( member_nat @ S @ A3 )
=> ( ( ord_less_eq_nat @ R @ S )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monotone_on_nat_nat @ A3 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% ord_class.mono_onI
thf(fact_616_ord__class_Omono__onD,axiom,
! [A3: set_nat,F: nat > nat,R2: nat,S2: nat] :
( ( monotone_on_nat_nat @ A3 @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R2 @ A3 )
=> ( ( member_nat @ S2 @ A3 )
=> ( ( ord_less_eq_nat @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord_class.mono_onD
thf(fact_617_ord_Omono__on__def,axiom,
! [A3: set_a,Less_eq: a > a > $o,F: a > nat] :
( ( monotone_on_a_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: a,S3: a] :
( ( ( member_a @ R3 @ A3 )
& ( member_a @ S3 @ A3 )
& ( Less_eq @ R3 @ S3 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_618_ord_Omono__on__def,axiom,
! [A3: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: nat,S3: nat] :
( ( ( member_nat @ R3 @ A3 )
& ( member_nat @ S3 @ A3 )
& ( Less_eq @ R3 @ S3 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_619_ord_Omono__onI,axiom,
! [A3: set_a,Less_eq: a > a > $o,F: a > nat] :
( ! [R: a,S: a] :
( ( member_a @ R @ A3 )
=> ( ( member_a @ S @ A3 )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monotone_on_a_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_620_ord_Omono__onI,axiom,
! [A3: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ! [R: nat,S: nat] :
( ( member_nat @ R @ A3 )
=> ( ( member_nat @ S @ A3 )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monotone_on_nat_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_621_ord_Omono__onD,axiom,
! [A3: set_a,Less_eq: a > a > $o,F: a > nat,R2: a,S2: a] :
( ( monotone_on_a_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_a @ R2 @ A3 )
=> ( ( member_a @ S2 @ A3 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_622_ord_Omono__onD,axiom,
! [A3: set_nat,Less_eq: nat > nat > $o,F: nat > nat,R2: nat,S2: nat] :
( ( monotone_on_nat_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R2 @ A3 )
=> ( ( member_nat @ S2 @ A3 )
=> ( ( Less_eq @ R2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_623_monotoneI,axiom,
! [Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F: nat > nat] :
( ! [X6: nat,Y3: nat] :
( ( Orda2 @ X6 @ Y3 )
=> ( Ordb2 @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ Orda2 @ Ordb2 @ F ) ) ).
% monotoneI
thf(fact_624_monotoneD,axiom,
! [Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ Orda2 @ Ordb2 @ F )
=> ( ( Orda2 @ X @ Y )
=> ( Ordb2 @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monotoneD
thf(fact_625_monotone__on__subset,axiom,
! [A3: set_nat,Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F: nat > nat,B3: set_nat] :
( ( monotone_on_nat_nat @ A3 @ Orda2 @ Ordb2 @ F )
=> ( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( monotone_on_nat_nat @ B3 @ Orda2 @ Ordb2 @ F ) ) ) ).
% monotone_on_subset
thf(fact_626_monoD,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_627_local_Ostrict__mono__onD,axiom,
! [A3: set_a,F: a > nat,R2: a,S2: a] :
( ( monotone_on_a_nat @ A3 @ less @ ord_less_nat @ F )
=> ( ( member_a @ R2 @ A3 )
=> ( ( member_a @ S2 @ A3 )
=> ( ( less @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% local.strict_mono_onD
thf(fact_628_local_Ostrict__mono__onI,axiom,
! [A3: set_a,F: a > nat] :
( ! [R: a,S: a] :
( ( member_a @ R @ A3 )
=> ( ( member_a @ S @ A3 )
=> ( ( less @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monotone_on_a_nat @ A3 @ less @ ord_less_nat @ F ) ) ).
% local.strict_mono_onI
thf(fact_629_incseqD,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% incseqD
thf(fact_630_incseq__def,axiom,
! [X9: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ X9 )
= ( ! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( X9 @ M3 ) @ ( X9 @ N ) ) ) ) ) ).
% incseq_def
thf(fact_631_subseqs_Osimps_I1_J,axiom,
( ( subseqs_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% subseqs.simps(1)
thf(fact_632_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_633_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_634_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_635_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_636_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_637_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_638_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y3: nat] :
( ( ord_less_nat @ X6 @ Y3 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_639_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y3: nat] :
( ( ord_less_nat @ X6 @ Y3 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_640_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_641_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y3: nat] :
( ( ord_less_nat @ X6 @ Y3 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_642_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y3: nat] :
( ( ord_less_nat @ X6 @ Y3 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_643_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_644_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_645_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_646_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_647_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_648_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_649_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_650_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_651_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_652_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_653_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: 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 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_654_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X10: nat] : ( P2 @ X10 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N )
=> ~ ( P3 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_655_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_656_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_657_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_658_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_659_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X6: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X6 )
=> ( P @ Y5 ) )
=> ( P @ X6 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_660_ord__class_Oord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_class.ord_less_eq_trans
thf(fact_661_ord__class_Oord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_class.ord_eq_less_trans
thf(fact_662_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_663_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_664_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_665_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_666_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_667_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y3: nat] :
( ( ord_less_nat @ X6 @ Y3 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_668_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y3: nat] :
( ( ord_less_eq_nat @ X6 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_669_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y3: nat] :
( ( ord_less_eq_nat @ X6 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_670_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y3: nat] :
( ( ord_less_nat @ X6 @ Y3 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_671_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_672_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_673_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_674_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_675_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_676_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_677_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_678_order__less__le,axiom,
( ord_less_nat
= ( ^ [X7: nat,Y2: nat] :
( ( ord_less_eq_nat @ X7 @ Y2 )
& ( X7 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_679_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X7: nat,Y2: nat] :
( ( ord_less_nat @ X7 @ Y2 )
| ( X7 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_680_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_681_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_682_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_683_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_684_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_685_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_686_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A4: nat] :
( ( ord_less_nat @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_687_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_688_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_689_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_690_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_691_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_692_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_693_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X7: nat,Y2: nat] :
( ( ord_less_eq_nat @ X7 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X7 ) ) ) ) ).
% less_le_not_le
thf(fact_694_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_695_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_696_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_697_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_698_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_699_top_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ top_top_set_a @ A ) ).
% top.extremum_strict
thf(fact_700_top_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A ) ).
% top.extremum_strict
thf(fact_701_top_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != top_top_set_a )
= ( ord_less_set_a @ A @ top_top_set_a ) ) ).
% top.not_eq_extremum
thf(fact_702_top_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != top_top_set_nat )
= ( ord_less_set_nat @ A @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_703_less__infI1,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_set_a @ A @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_704_less__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_nat @ A @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_705_less__infI2,axiom,
! [B: set_a,X: set_a,A: set_a] :
( ( ord_less_set_a @ B @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_706_less__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_nat @ B @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_707_inf_Oabsorb3,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_708_inf_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_709_inf_Oabsorb4,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_710_inf_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_711_inf_Ostrict__boundedE,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_712_inf_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_713_inf_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B2: set_a] :
( ( A4
= ( inf_inf_set_a @ A4 @ B2 ) )
& ( A4 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_714_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A4: nat,B2: nat] :
( ( A4
= ( inf_inf_nat @ A4 @ B2 ) )
& ( A4 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_715_inf_Ostrict__coboundedI1,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_set_a @ A @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_716_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_717_inf_Ostrict__coboundedI2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_718_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_719_strict__mono__on__eqD,axiom,
! [A3: set_nat,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A3 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A3 )
=> ( ( member_nat @ Y @ A3 )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_720_ord__class_Ostrict__mono__onI,axiom,
! [A3: set_nat,F: nat > nat] :
( ! [R: nat,S: nat] :
( ( member_nat @ R @ A3 )
=> ( ( member_nat @ S @ A3 )
=> ( ( ord_less_nat @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monotone_on_nat_nat @ A3 @ ord_less_nat @ ord_less_nat @ F ) ) ).
% ord_class.strict_mono_onI
thf(fact_721_ord__class_Ostrict__mono__onD,axiom,
! [A3: set_nat,F: nat > nat,R2: nat,S2: nat] :
( ( monotone_on_nat_nat @ A3 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ R2 @ A3 )
=> ( ( member_nat @ S2 @ A3 )
=> ( ( ord_less_nat @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord_class.strict_mono_onD
thf(fact_722_ord_Ostrict__mono__on__def,axiom,
! [A3: set_a,Less: a > a > $o,F: a > nat] :
( ( monotone_on_a_nat @ A3 @ Less @ ord_less_nat @ F )
= ( ! [R3: a,S3: a] :
( ( ( member_a @ R3 @ A3 )
& ( member_a @ S3 @ A3 )
& ( Less @ R3 @ S3 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_723_ord_Ostrict__mono__on__def,axiom,
! [A3: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A3 @ Less @ ord_less_nat @ F )
= ( ! [R3: nat,S3: nat] :
( ( ( member_nat @ R3 @ A3 )
& ( member_nat @ S3 @ A3 )
& ( Less @ R3 @ S3 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S3 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_724_ord_Ostrict__mono__onI,axiom,
! [A3: set_a,Less: a > a > $o,F: a > nat] :
( ! [R: a,S: a] :
( ( member_a @ R @ A3 )
=> ( ( member_a @ S @ A3 )
=> ( ( Less @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monotone_on_a_nat @ A3 @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_725_ord_Ostrict__mono__onI,axiom,
! [A3: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ! [R: nat,S: nat] :
( ( member_nat @ R @ A3 )
=> ( ( member_nat @ S @ A3 )
=> ( ( Less @ R @ S )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monotone_on_nat_nat @ A3 @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_726_ord_Ostrict__mono__onD,axiom,
! [A3: set_a,Less: a > a > $o,F: a > nat,R2: a,S2: a] :
( ( monotone_on_a_nat @ A3 @ Less @ ord_less_nat @ F )
=> ( ( member_a @ R2 @ A3 )
=> ( ( member_a @ S2 @ A3 )
=> ( ( Less @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_727_ord_Ostrict__mono__onD,axiom,
! [A3: set_nat,Less: nat > nat > $o,F: nat > nat,R2: nat,S2: nat] :
( ( monotone_on_nat_nat @ A3 @ Less @ ord_less_nat @ F )
=> ( ( member_nat @ R2 @ A3 )
=> ( ( member_nat @ S2 @ A3 )
=> ( ( Less @ R2 @ S2 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_728_strict__mono__leD,axiom,
! [R2: nat > nat,M4: nat,N2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ R2 )
=> ( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ ( R2 @ M4 ) @ ( R2 @ N2 ) ) ) ) ).
% strict_mono_leD
thf(fact_729_mono__on__greaterD,axiom,
! [A3: set_nat,G: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A3 @ ord_less_eq_nat @ ord_less_eq_nat @ G )
=> ( ( member_nat @ X @ A3 )
=> ( ( member_nat @ Y @ A3 )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_nat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_730_strict__mono__on__leD,axiom,
! [A3: set_nat,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A3 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ X @ A3 )
=> ( ( member_nat @ Y @ A3 )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_731_strict__mono__on__imp__mono__on,axiom,
! [A3: set_nat,F: nat > nat] :
( ( monotone_on_nat_nat @ A3 @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ A3 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_732_strict__mono__less,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_733_strict__mono__eq,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_734_strict__monoI,axiom,
! [F: nat > nat] :
( ! [X6: nat,Y3: nat] :
( ( ord_less_nat @ X6 @ Y3 )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_monoI
thf(fact_735_strict__monoD,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_736_mono__invE,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% mono_invE
thf(fact_737_strict__mono__mono,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_mono
thf(fact_738_mono__strict__invE,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_nat @ X @ Y ) ) ) ).
% mono_strict_invE
thf(fact_739_strict__mono__less__eq,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_740_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ord_less_eq_nat @ N2 @ ( F @ N2 ) ) ) ).
% strict_mono_imp_increasing
thf(fact_741_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X8: nat] :
( ( ( ord_less_eq_nat @ A @ X8 )
& ( ord_less_nat @ X8 @ C3 ) )
=> ( P @ X8 ) )
& ! [D3: nat] :
( ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ D3 ) )
=> ( P @ X6 ) )
=> ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_742_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A7 ) )
= ( ord_less_nat @ A7 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_743_pinf_I6_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z2 @ X8 )
=> ~ ( ord_less_eq_nat @ X8 @ T2 ) ) ).
% pinf(6)
thf(fact_744_pinf_I8_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z2 @ X8 )
=> ( ord_less_eq_nat @ T2 @ X8 ) ) ).
% pinf(8)
thf(fact_745_minf_I6_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z2 )
=> ( ord_less_eq_nat @ X8 @ T2 ) ) ).
% minf(6)
thf(fact_746_psubsetD,axiom,
! [A3: set_a,B3: set_a,C: a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_747_psubsetD,axiom,
! [A3: set_nat,B3: set_nat,C: nat] :
( ( ord_less_set_nat @ A3 @ B3 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_748_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_749_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_750_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X6: nat] :
( ( P @ X6 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X6 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_751_nat__le__linear,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
| ( ord_less_eq_nat @ N2 @ M4 ) ) ).
% nat_le_linear
thf(fact_752_le__antisym,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M4 )
=> ( M4 = N2 ) ) ) ).
% le_antisym
thf(fact_753_eq__imp__le,axiom,
! [M4: nat,N2: nat] :
( ( M4 = N2 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% eq_imp_le
thf(fact_754_le__trans,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_755_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_756_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_757_le__neq__implies__less,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( M4 != N2 )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_758_less__or__eq__imp__le,axiom,
! [M4: nat,N2: nat] :
( ( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_759_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
| ( M3 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_760_less__imp__le__nat,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_761_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
& ( M3 != N ) ) ) ) ).
% nat_less_le
thf(fact_762_minf_I8_J,axiom,
! [T2: nat] :
? [Z2: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z2 )
=> ~ ( ord_less_eq_nat @ T2 @ X8 ) ) ).
% minf(8)
thf(fact_763_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M4: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M4 ) ) ) ).
% nat_descend_induct
thf(fact_764_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila2496817875450240012_set_a @ inf_inf_set_a @ top_top_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_765_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila1667268886620078168et_nat @ inf_inf_set_nat @ top_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_766_incseq__imp__monoseq,axiom,
! [X9: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ X9 )
=> ( topolo4902158794631467389eq_nat @ X9 ) ) ).
% incseq_imp_monoseq
thf(fact_767_maps__simps_I1_J,axiom,
! [F: a > list_a,X: a,Xs: list_a] :
( ( maps_a_a @ F @ ( cons_a @ X @ Xs ) )
= ( append_a @ ( F @ X ) @ ( maps_a_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_768_maps__simps_I2_J,axiom,
! [F: a > list_a] :
( ( maps_a_a @ F @ nil_a )
= nil_a ) ).
% maps_simps(2)
thf(fact_769_monoI1,axiom,
! [X9: nat > nat] :
( ! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ ( X9 @ M2 ) @ ( X9 @ N3 ) ) )
=> ( topolo4902158794631467389eq_nat @ X9 ) ) ).
% monoI1
thf(fact_770_monoI2,axiom,
! [X9: nat > nat] :
( ! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ ( X9 @ N3 ) @ ( X9 @ M2 ) ) )
=> ( topolo4902158794631467389eq_nat @ X9 ) ) ).
% monoI2
thf(fact_771_monoseq__def,axiom,
( topolo4902158794631467389eq_nat
= ( ^ [X11: nat > nat] :
( ! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( X11 @ M3 ) @ ( X11 @ N ) ) )
| ! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( X11 @ N ) @ ( X11 @ M3 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_772_insert__Nil,axiom,
! [X: a] :
( ( insert_a @ X @ nil_a )
= ( cons_a @ X @ nil_a ) ) ).
% insert_Nil
thf(fact_773_top_Oordering__top__axioms,axiom,
ordering_top_set_a @ ord_less_eq_set_a @ ord_less_set_a @ top_top_set_a ).
% top.ordering_top_axioms
thf(fact_774_top_Oordering__top__axioms,axiom,
ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).
% top.ordering_top_axioms
thf(fact_775_concat__eq__append__conv,axiom,
! [Xss2: list_list_a,Ys: list_a,Zs: list_a] :
( ( ( concat_a @ Xss2 )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_a )
=> ( ( Ys = nil_a )
& ( Zs = nil_a ) ) )
& ( ( Xss2 != nil_list_a )
=> ? [Xss1: list_list_a,Xs2: list_a,Xs4: list_a,Xss22: list_list_a] :
( ( Xss2
= ( append_list_a @ Xss1 @ ( cons_list_a @ ( append_a @ Xs2 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append_a @ ( concat_a @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_a @ Xs4 @ ( concat_a @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_776_funpow__increasing,axiom,
! [M4: nat,N2: nat,F: set_a > set_a] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( compow_set_a_set_a @ N2 @ F @ top_top_set_a ) @ ( compow_set_a_set_a @ M4 @ F @ top_top_set_a ) ) ) ) ).
% funpow_increasing
thf(fact_777_funpow__increasing,axiom,
! [M4: nat,N2: nat,F: set_nat > set_nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ord_less_eq_set_nat @ ( compow8708494347934031032et_nat @ N2 @ F @ top_top_set_nat ) @ ( compow8708494347934031032et_nat @ M4 @ F @ top_top_set_nat ) ) ) ) ).
% funpow_increasing
thf(fact_778_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic8986249270076014136_set_a @ inf_inf_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_779_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic6009151579333465974et_nat @ inf_inf_nat @ ord_less_eq_nat @ ord_less_nat ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_780_butlast__snoc,axiom,
! [Xs: list_a,X: a] :
( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_781_concat__append,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( concat_a @ ( append_list_a @ Xs @ Ys ) )
= ( append_a @ ( concat_a @ Xs ) @ ( concat_a @ Ys ) ) ) ).
% concat_append
thf(fact_782_butlast_Osimps_I1_J,axiom,
( ( butlast_a @ nil_a )
= nil_a ) ).
% butlast.simps(1)
thf(fact_783_concat_Osimps_I1_J,axiom,
( ( concat_a @ nil_list_a )
= nil_a ) ).
% concat.simps(1)
thf(fact_784_concat_Osimps_I2_J,axiom,
! [X: list_a,Xs: list_list_a] :
( ( concat_a @ ( cons_list_a @ X @ Xs ) )
= ( append_a @ X @ ( concat_a @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_785_butlast_Osimps_I2_J,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs ) )
= nil_a ) )
& ( ( Xs != nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( butlast_a @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_786_butlast__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( butlast_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ Xs @ ( butlast_a @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_787_funpow__mono,axiom,
! [F: nat > nat,A3: nat,B3: nat,N2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( compow_nat_nat @ N2 @ F @ A3 ) @ ( compow_nat_nat @ N2 @ F @ B3 ) ) ) ) ).
% funpow_mono
thf(fact_788_Kleene__iter__gpfp,axiom,
! [F: set_a > set_a,P4: set_a,K: nat] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ P4 @ ( F @ P4 ) )
=> ( ord_less_eq_set_a @ P4 @ ( compow_set_a_set_a @ K @ F @ top_top_set_a ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_789_Kleene__iter__gpfp,axiom,
! [F: set_nat > set_nat,P4: set_nat,K: nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_set_nat @ P4 @ ( F @ P4 ) )
=> ( ord_less_eq_set_nat @ P4 @ ( compow8708494347934031032et_nat @ K @ F @ top_top_set_nat ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_790_concat__eq__appendD,axiom,
! [Xss2: list_list_a,Ys: list_a,Zs: list_a] :
( ( ( concat_a @ Xss2 )
= ( append_a @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_a )
=> ? [Xss12: list_list_a,Xs3: list_a,Xs5: list_a,Xss23: list_list_a] :
( ( Xss2
= ( append_list_a @ Xss12 @ ( cons_list_a @ ( append_a @ Xs3 @ Xs5 ) @ Xss23 ) ) )
& ( Ys
= ( append_a @ ( concat_a @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append_a @ Xs5 @ ( concat_a @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_791_funpow__mono2,axiom,
! [F: nat > nat,I: nat,J2: nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ ( F @ X ) )
=> ( ord_less_eq_nat @ ( compow_nat_nat @ I @ F @ X ) @ ( compow_nat_nat @ J2 @ F @ Y ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_792_append__butlast__last__id,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( butlast_a @ Xs ) @ ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_793_funpow__decreasing,axiom,
! [M4: nat,N2: nat,F: nat > nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ord_less_eq_nat @ ( compow_nat_nat @ M4 @ F @ bot_bot_nat ) @ ( compow_nat_nat @ N2 @ F @ bot_bot_nat ) ) ) ) ).
% funpow_decreasing
thf(fact_794_snoc__eq__iff__butlast,axiom,
! [Xs: list_a,X: a,Ys: list_a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= Ys )
= ( ( Ys != nil_a )
& ( ( butlast_a @ Ys )
= Xs )
& ( ( last_a @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_795_list__ex1__simps_I1_J,axiom,
! [P: a > $o] :
~ ( list_ex1_a @ P @ nil_a ) ).
% list_ex1_simps(1)
thf(fact_796_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_797_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_798_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_799_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_800_last__appendL,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) ) ).
% last_appendL
thf(fact_801_last__appendR,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ).
% last_appendR
thf(fact_802_last__snoc,axiom,
! [Xs: list_a,X: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% last_snoc
thf(fact_803_monotone__on__empty,axiom,
! [Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F: nat > nat] : ( monotone_on_nat_nat @ bot_bot_set_nat @ Orda2 @ Ordb2 @ F ) ).
% monotone_on_empty
thf(fact_804_Int__emptyI,axiom,
! [A3: set_nat,B3: set_nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ~ ( member_nat @ X6 @ B3 ) )
=> ( ( inf_inf_set_nat @ A3 @ B3 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_805_Int__emptyI,axiom,
! [A3: set_a,B3: set_a] :
( ! [X6: a] :
( ( member_a @ X6 @ A3 )
=> ~ ( member_a @ X6 @ B3 ) )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_806_disjoint__iff,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B3 )
= bot_bot_set_nat )
= ( ! [X7: nat] :
( ( member_nat @ X7 @ A3 )
=> ~ ( member_nat @ X7 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_807_disjoint__iff,axiom,
! [A3: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A3 @ B3 )
= bot_bot_set_a )
= ( ! [X7: a] :
( ( member_a @ X7 @ A3 )
=> ~ ( member_a @ X7 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_808_Int__empty__left,axiom,
! [B3: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B3 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_809_Int__empty__right,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_810_disjoint__iff__not__equal,axiom,
! [A3: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A3 @ B3 )
= bot_bot_set_a )
= ( ! [X7: a] :
( ( member_a @ X7 @ A3 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ( X7 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_811_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_812_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_813_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_814_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_815_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_816_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_817_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_818_last_Osimps,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_819_last__ConsL,axiom,
! [Xs: list_a,X: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_820_last__ConsR,axiom,
! [Xs: list_a,X: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_821_longest__common__suffix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ss: list_a,Xs5: list_a,Ys5: list_a] :
( ( Xs
= ( append_a @ Xs5 @ Ss ) )
& ( Ys
= ( append_a @ Ys5 @ Ss ) )
& ( ( Xs5 = nil_a )
| ( Ys5 = nil_a )
| ( ( last_a @ Xs5 )
!= ( last_a @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_822_last__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ) ).
% last_append
thf(fact_823_Kleene__iter__lpfp,axiom,
! [F: nat > nat,P4: nat,K: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ ( F @ P4 ) @ P4 )
=> ( ord_less_eq_nat @ ( compow_nat_nat @ K @ F @ bot_bot_nat ) @ P4 ) ) ) ).
% Kleene_iter_lpfp
thf(fact_824_strict__mono__Suc__iff,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
= ( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) ) ) ) ).
% strict_mono_Suc_iff
thf(fact_825_incseq__Suc__iff,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
= ( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) ) ) ) ).
% incseq_Suc_iff
thf(fact_826_incseq__SucI,axiom,
! [X9: nat > nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ X9 ) ) ).
% incseq_SucI
thf(fact_827_incseq__SucD,axiom,
! [A3: nat > nat,I: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ A3 )
=> ( ord_less_eq_nat @ ( A3 @ I ) @ ( A3 @ ( suc @ I ) ) ) ) ).
% incseq_SucD
thf(fact_828_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_829_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_830_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X7: a] :
~ ( member_a @ X7 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_831_all__not__in__conv,axiom,
! [A3: set_nat] :
( ( ! [X7: nat] :
~ ( member_nat @ X7 @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_832_Suc__le__mono,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M4 ) )
= ( ord_less_eq_nat @ N2 @ M4 ) ) ).
% Suc_le_mono
thf(fact_833_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X7: a] : ( member_a @ X7 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_834_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X7: nat] : ( member_nat @ X7 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_835_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_836_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_837_equals0D,axiom,
! [A3: set_a,A: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A @ A3 ) ) ).
% equals0D
thf(fact_838_equals0D,axiom,
! [A3: set_nat,A: nat] :
( ( A3 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_839_equals0I,axiom,
! [A3: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_840_equals0I,axiom,
! [A3: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A3 )
=> ( A3 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_841_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X7: a] : ( member_a @ X7 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_842_ex__in__conv,axiom,
! [A3: set_nat] :
( ( ? [X7: nat] : ( member_nat @ X7 @ A3 ) )
= ( A3 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_843_transitive__stepwise__le,axiom,
! [M4: nat,N2: nat,R4: nat > nat > $o] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ! [X6: nat] : ( R4 @ X6 @ X6 )
=> ( ! [X6: nat,Y3: nat,Z2: nat] :
( ( R4 @ X6 @ Y3 )
=> ( ( R4 @ Y3 @ Z2 )
=> ( R4 @ X6 @ Z2 ) ) )
=> ( ! [N3: nat] : ( R4 @ N3 @ ( suc @ N3 ) )
=> ( R4 @ M4 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_844_nat__induct__at__least,axiom,
! [M4: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( P @ M4 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_845_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_846_not__less__eq__eq,axiom,
! [M4: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M4 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M4 ) ) ).
% not_less_eq_eq
thf(fact_847_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_848_le__Suc__eq,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M4 @ N2 )
| ( M4
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_849_Suc__le__D,axiom,
! [N2: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
=> ? [M2: nat] :
( M6
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_850_le__SucI,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_851_le__SucE,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M4 @ N2 )
=> ( M4
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_852_Suc__leD,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% Suc_leD
thf(fact_853_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_854_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_855_Suc__leI,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 ) ) ).
% Suc_leI
thf(fact_856_Suc__le__eq,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% Suc_le_eq
thf(fact_857_dec__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_858_inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ J2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_859_Suc__le__lessD,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( ord_less_nat @ M4 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_860_le__less__Suc__eq,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M4 ) )
= ( N2 = M4 ) ) ) ).
% le_less_Suc_eq
thf(fact_861_less__Suc__eq__le,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_862_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_863_le__imp__less__Suc,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_nat @ M4 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_864_mono__SucI1,axiom,
! [X9: nat > nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
=> ( topolo4902158794631467389eq_nat @ X9 ) ) ).
% mono_SucI1
thf(fact_865_mono__SucI2,axiom,
! [X9: nat > nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
=> ( topolo4902158794631467389eq_nat @ X9 ) ) ).
% mono_SucI2
thf(fact_866_monoseq__Suc,axiom,
( topolo4902158794631467389eq_nat
= ( ^ [X11: nat > nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( X11 @ N ) @ ( X11 @ ( suc @ N ) ) )
| ! [N: nat] : ( ord_less_eq_nat @ ( X11 @ ( suc @ N ) ) @ ( X11 @ N ) ) ) ) ) ).
% monoseq_Suc
thf(fact_867_mono__Suc,axiom,
monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ suc ).
% mono_Suc
thf(fact_868_gfp__Kleene__iter,axiom,
! [F: set_a > set_a,K: nat] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ( compow_set_a_set_a @ ( suc @ K ) @ F @ top_top_set_a )
= ( compow_set_a_set_a @ K @ F @ top_top_set_a ) )
=> ( ( comple3341859861669737308_set_a @ F )
= ( compow_set_a_set_a @ K @ F @ top_top_set_a ) ) ) ) ).
% gfp_Kleene_iter
thf(fact_869_gfp__Kleene__iter,axiom,
! [F: set_nat > set_nat,K: nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ( compow8708494347934031032et_nat @ ( suc @ K ) @ F @ top_top_set_nat )
= ( compow8708494347934031032et_nat @ K @ F @ top_top_set_nat ) )
=> ( ( comple1596078789208929544et_nat @ F )
= ( compow8708494347934031032et_nat @ K @ F @ top_top_set_nat ) ) ) ) ).
% gfp_Kleene_iter
thf(fact_870_subset__emptyI,axiom,
! [A3: set_a] :
( ! [X6: a] :
~ ( member_a @ X6 @ A3 )
=> ( ord_less_eq_set_a @ A3 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_871_subset__emptyI,axiom,
! [A3: set_nat] :
( ! [X6: nat] :
~ ( member_nat @ X6 @ A3 )
=> ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_872_lfp__induct,axiom,
! [F: set_a > set_a,P: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ ( F @ ( inf_inf_set_a @ ( comple1558298011288954135_set_a @ F ) @ P ) ) @ P )
=> ( ord_less_eq_set_a @ ( comple1558298011288954135_set_a @ F ) @ P ) ) ) ).
% lfp_induct
thf(fact_873_def__lfp__induct,axiom,
! [A3: set_a,F: set_a > set_a,P: set_a] :
( ( A3
= ( comple1558298011288954135_set_a @ F ) )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ ( F @ ( inf_inf_set_a @ A3 @ P ) ) @ P )
=> ( ord_less_eq_set_a @ A3 @ P ) ) ) ) ).
% def_lfp_induct
thf(fact_874_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X11: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X11 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X11 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_875_weak__coinduct,axiom,
! [A: a,X9: set_a,F: set_a > set_a] :
( ( member_a @ A @ X9 )
=> ( ( ord_less_eq_set_a @ X9 @ ( F @ X9 ) )
=> ( member_a @ A @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% weak_coinduct
thf(fact_876_weak__coinduct,axiom,
! [A: nat,X9: set_nat,F: set_nat > set_nat] :
( ( member_nat @ A @ X9 )
=> ( ( ord_less_eq_set_nat @ X9 @ ( F @ X9 ) )
=> ( member_nat @ A @ ( comple1596078789208929544et_nat @ F ) ) ) ) ).
% weak_coinduct
thf(fact_877_gen__length__code_I2_J,axiom,
! [N2: nat,X: a,Xs: list_a] :
( ( gen_length_a @ N2 @ ( cons_a @ X @ Xs ) )
= ( gen_length_a @ ( suc @ N2 ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_878_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_length_a @ N2 @ nil_a )
= N2 ) ).
% gen_length_code(1)
thf(fact_879_coinduct__set,axiom,
! [F: set_a > set_a,A: a,X9: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_a @ A @ X9 )
=> ( ( ord_less_eq_set_a @ X9 @ ( F @ ( sup_sup_set_a @ X9 @ ( comple3341859861669737308_set_a @ F ) ) ) )
=> ( member_a @ A @ ( comple3341859861669737308_set_a @ F ) ) ) ) ) ).
% coinduct_set
thf(fact_880_coinduct__set,axiom,
! [F: set_nat > set_nat,A: nat,X9: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( member_nat @ A @ X9 )
=> ( ( ord_less_eq_set_nat @ X9 @ ( F @ ( sup_sup_set_nat @ X9 @ ( comple1596078789208929544et_nat @ F ) ) ) )
=> ( member_nat @ A @ ( comple1596078789208929544et_nat @ F ) ) ) ) ) ).
% coinduct_set
thf(fact_881_gfp__fun__UnI2,axiom,
! [F: set_a > set_a,A: a,X9: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_a @ A @ ( comple3341859861669737308_set_a @ F ) )
=> ( member_a @ A @ ( F @ ( sup_sup_set_a @ X9 @ ( comple3341859861669737308_set_a @ F ) ) ) ) ) ) ).
% gfp_fun_UnI2
thf(fact_882_gfp__fun__UnI2,axiom,
! [F: set_nat > set_nat,A: nat,X9: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( member_nat @ A @ ( comple1596078789208929544et_nat @ F ) )
=> ( member_nat @ A @ ( F @ ( sup_sup_set_nat @ X9 @ ( comple1596078789208929544et_nat @ F ) ) ) ) ) ) ).
% gfp_fun_UnI2
thf(fact_883_Un__iff,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) )
= ( ( member_a @ C @ A3 )
| ( member_a @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_884_Un__iff,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) )
= ( ( member_nat @ C @ A3 )
| ( member_nat @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_885_UnCI,axiom,
! [C: a,B3: set_a,A3: set_a] :
( ( ~ ( member_a @ C @ B3 )
=> ( member_a @ C @ A3 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_886_UnCI,axiom,
! [C: nat,B3: set_nat,A3: set_nat] :
( ( ~ ( member_nat @ C @ B3 )
=> ( member_nat @ C @ A3 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_887_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_888_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_889_sup__top__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ top_top_set_a )
= top_top_set_a ) ).
% sup_top_right
thf(fact_890_sup__top__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% sup_top_right
thf(fact_891_sup__top__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ X )
= top_top_set_a ) ).
% sup_top_left
thf(fact_892_sup__top__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% sup_top_left
thf(fact_893_boolean__algebra_Odisj__one__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ X )
= top_top_set_a ) ).
% boolean_algebra.disj_one_left
thf(fact_894_boolean__algebra_Odisj__one__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_895_boolean__algebra_Odisj__one__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ top_top_set_a )
= top_top_set_a ) ).
% boolean_algebra.disj_one_right
thf(fact_896_boolean__algebra_Odisj__one__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_897_sup__inf__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_898_inf__sup__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_899_Un__Int__eq_I1_J,axiom,
! [S4: set_a,T3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S4 @ T3 ) @ S4 )
= S4 ) ).
% Un_Int_eq(1)
thf(fact_900_Un__Int__eq_I2_J,axiom,
! [S4: set_a,T3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S4 @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_901_Un__Int__eq_I3_J,axiom,
! [S4: set_a,T3: set_a] :
( ( inf_inf_set_a @ S4 @ ( sup_sup_set_a @ S4 @ T3 ) )
= S4 ) ).
% Un_Int_eq(3)
thf(fact_902_Un__Int__eq_I4_J,axiom,
! [T3: set_a,S4: set_a] :
( ( inf_inf_set_a @ T3 @ ( sup_sup_set_a @ S4 @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_903_Int__Un__eq_I1_J,axiom,
! [S4: set_a,T3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S4 @ T3 ) @ S4 )
= S4 ) ).
% Int_Un_eq(1)
thf(fact_904_Int__Un__eq_I2_J,axiom,
! [S4: set_a,T3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S4 @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_905_Int__Un__eq_I3_J,axiom,
! [S4: set_a,T3: set_a] :
( ( sup_sup_set_a @ S4 @ ( inf_inf_set_a @ S4 @ T3 ) )
= S4 ) ).
% Int_Un_eq(3)
thf(fact_906_Int__Un__eq_I4_J,axiom,
! [T3: set_a,S4: set_a] :
( ( sup_sup_set_a @ T3 @ ( inf_inf_set_a @ S4 @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_907_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_908_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_909_le__supE,axiom,
! [A: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_910_le__supI,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_911_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_912_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_913_le__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_914_le__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_915_sup_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_916_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_917_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_918_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X7: nat,Y2: nat] :
( ( sup_sup_nat @ X7 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_919_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_920_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_921_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X6: nat,Y3: nat] : ( ord_less_eq_nat @ X6 @ ( F @ X6 @ Y3 ) )
=> ( ! [X6: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F @ X6 @ Y3 ) )
=> ( ! [X6: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y3 @ X6 )
=> ( ( ord_less_eq_nat @ Z2 @ X6 )
=> ( ord_less_eq_nat @ ( F @ Y3 @ Z2 ) @ X6 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_922_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_923_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_924_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_925_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_926_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_927_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_928_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_929_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_930_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_931_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B2 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_932_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] :
( ( sup_sup_nat @ A4 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_933_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_934_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_935_less__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_nat @ X @ A )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_936_less__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_nat @ X @ B )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_937_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_938_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_939_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_940_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B2: nat,A4: nat] :
( ( A4
= ( sup_sup_nat @ A4 @ B2 ) )
& ( A4 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_941_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_942_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_943_sup__inf__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_944_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_945_inf__sup__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_946_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_947_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X6: set_a,Y3: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X6 @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X6 @ Y3 ) @ ( sup_sup_set_a @ X6 @ Z2 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_948_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X6: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X6 @ ( sup_sup_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X6 @ Y3 ) @ ( inf_inf_set_a @ X6 @ Z2 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_949_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_950_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_951_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_952_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_953_Un__UNIV__right,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ top_top_set_a )
= top_top_set_a ) ).
% Un_UNIV_right
thf(fact_954_Un__UNIV__right,axiom,
! [A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_955_Un__UNIV__left,axiom,
! [B3: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ B3 )
= top_top_set_a ) ).
% Un_UNIV_left
thf(fact_956_Un__UNIV__left,axiom,
! [B3: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B3 )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_957_Un__Int__distrib2,axiom,
! [B3: set_a,C2: set_a,A3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B3 @ C2 ) @ A3 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B3 @ A3 ) @ ( sup_sup_set_a @ C2 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_958_Int__Un__distrib2,axiom,
! [B3: set_a,C2: set_a,A3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B3 @ C2 ) @ A3 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B3 @ A3 ) @ ( inf_inf_set_a @ C2 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_959_Un__Int__distrib,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( sup_sup_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ ( sup_sup_set_a @ A3 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_960_Int__Un__distrib,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ ( inf_inf_set_a @ A3 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_961_Un__Int__crazy,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ ( inf_inf_set_a @ B3 @ C2 ) ) @ ( inf_inf_set_a @ C2 @ A3 ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ ( sup_sup_set_a @ B3 @ C2 ) ) @ ( sup_sup_set_a @ C2 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_962_UnI2,axiom,
! [C: a,B3: set_a,A3: set_a] :
( ( member_a @ C @ B3 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_963_UnI2,axiom,
! [C: nat,B3: set_nat,A3: set_nat] :
( ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_964_UnI1,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ A3 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_965_UnI1,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_966_UnE,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) )
=> ( ~ ( member_a @ C @ A3 )
=> ( member_a @ C @ B3 ) ) ) ).
% UnE
thf(fact_967_UnE,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) )
=> ( ~ ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B3 ) ) ) ).
% UnE
thf(fact_968_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_969_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_970_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_971_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_972_Un__Int__assoc__eq,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ C2 )
= ( inf_inf_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_973_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_a,X: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ A @ X )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ A @ X )
= top_top_set_a )
=> ( ( ( inf_inf_set_a @ A @ Y )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ A @ Y )
= top_top_set_a )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_974_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_nat,X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ A @ X )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A @ X )
= top_top_set_nat )
=> ( ( ( inf_inf_set_nat @ A @ Y )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A @ Y )
= top_top_set_nat )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_975_mono__sup,axiom,
! [F: nat > nat,A3: nat,B3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ ( F @ A3 ) @ ( F @ B3 ) ) @ ( F @ ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% mono_sup
thf(fact_976_def__coinduct__set,axiom,
! [A3: set_a,F: set_a > set_a,A: a,X9: set_a] :
( ( A3
= ( comple3341859861669737308_set_a @ F ) )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_a @ A @ X9 )
=> ( ( ord_less_eq_set_a @ X9 @ ( F @ ( sup_sup_set_a @ X9 @ A3 ) ) )
=> ( member_a @ A @ A3 ) ) ) ) ) ).
% def_coinduct_set
thf(fact_977_def__coinduct__set,axiom,
! [A3: set_nat,F: set_nat > set_nat,A: nat,X9: set_nat] :
( ( A3
= ( comple1596078789208929544et_nat @ F ) )
=> ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( member_nat @ A @ X9 )
=> ( ( ord_less_eq_set_nat @ X9 @ ( F @ ( sup_sup_set_nat @ X9 @ A3 ) ) )
=> ( member_nat @ A @ A3 ) ) ) ) ) ).
% def_coinduct_set
thf(fact_978_rotate1_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_979_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_980_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_981_cSup__eq__maximum,axiom,
! [Z: nat,X9: set_nat] :
( ( member_nat @ Z @ X9 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ X9 )
=> ( ord_less_eq_nat @ X6 @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X9 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_982_cSup__least,axiom,
! [X9: set_nat,Z: nat] :
( ( X9 != bot_bot_set_nat )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ X9 )
=> ( ord_less_eq_nat @ X6 @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X9 ) @ Z ) ) ) ).
% cSup_least
thf(fact_983_cSup__eq__non__empty,axiom,
! [X9: set_nat,A: nat] :
( ( X9 != bot_bot_set_nat )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ X9 )
=> ( ord_less_eq_nat @ X6 @ A ) )
=> ( ! [Y3: nat] :
( ! [X8: nat] :
( ( member_nat @ X8 @ X9 )
=> ( ord_less_eq_nat @ X8 @ Y3 ) )
=> ( ord_less_eq_nat @ A @ Y3 ) )
=> ( ( complete_Sup_Sup_nat @ X9 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_984_Sup__UNIV,axiom,
( ( comple2307003609928055243_set_a @ top_top_set_set_a )
= top_top_set_a ) ).
% Sup_UNIV
thf(fact_985_Sup__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_986_Union__disjoint,axiom,
! [C2: set_set_a,A3: set_a] :
( ( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ C2 ) @ A3 )
= bot_bot_set_a )
= ( ! [X7: set_a] :
( ( member_set_a @ X7 @ C2 )
=> ( ( inf_inf_set_a @ X7 @ A3 )
= bot_bot_set_a ) ) ) ) ).
% Union_disjoint
thf(fact_987_Union__Int__subset,axiom,
! [A3: set_set_a,B3: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A3 @ B3 ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A3 ) @ ( comple2307003609928055243_set_a @ B3 ) ) ) ).
% Union_Int_subset
thf(fact_988_Union__UNIV,axiom,
( ( comple2307003609928055243_set_a @ top_top_set_set_a )
= top_top_set_a ) ).
% Union_UNIV
thf(fact_989_Union__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Union_UNIV
thf(fact_990_Sup__inter__less__eq,axiom,
! [A3: set_set_a,B3: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A3 @ B3 ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A3 ) @ ( comple2307003609928055243_set_a @ B3 ) ) ) ).
% Sup_inter_less_eq
thf(fact_991_Sup__inf__eq__bot__iff,axiom,
! [B3: set_set_a,A: set_a] :
( ( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ B3 ) @ A )
= bot_bot_set_a )
= ( ! [X7: set_a] :
( ( member_set_a @ X7 @ B3 )
=> ( ( inf_inf_set_a @ X7 @ A )
= bot_bot_set_a ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_992_cSup__inter__less__eq,axiom,
! [A3: set_nat,B3: set_nat] :
( ( condit2214826472909112428ve_nat @ A3 )
=> ( ( condit2214826472909112428ve_nat @ B3 )
=> ( ( ( inf_inf_set_nat @ A3 @ B3 )
!= bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( inf_inf_set_nat @ A3 @ B3 ) ) @ ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A3 ) @ ( complete_Sup_Sup_nat @ B3 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_993_bdd__above_OI,axiom,
! [A3: set_nat,M: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ( ord_less_eq_nat @ X6 @ M ) )
=> ( condit2214826472909112428ve_nat @ A3 ) ) ).
% bdd_above.I
thf(fact_994_cSup__upper,axiom,
! [X: nat,X9: set_nat] :
( ( member_nat @ X @ X9 )
=> ( ( condit2214826472909112428ve_nat @ X9 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X9 ) ) ) ) ).
% cSup_upper
thf(fact_995_cSup__upper2,axiom,
! [X: nat,X9: set_nat,Y: nat] :
( ( member_nat @ X @ X9 )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( ( condit2214826472909112428ve_nat @ X9 )
=> ( ord_less_eq_nat @ Y @ ( complete_Sup_Sup_nat @ X9 ) ) ) ) ) ).
% cSup_upper2
thf(fact_996_bdd__above_OE,axiom,
! [A3: set_nat] :
( ( condit2214826472909112428ve_nat @ A3 )
=> ~ ! [M7: nat] :
~ ! [X8: nat] :
( ( member_nat @ X8 @ A3 )
=> ( ord_less_eq_nat @ X8 @ M7 ) ) ) ).
% bdd_above.E
thf(fact_997_bdd__above_Ounfold,axiom,
( condit2214826472909112428ve_nat
= ( ^ [A5: set_nat] :
? [M8: nat] :
! [X7: nat] :
( ( member_nat @ X7 @ A5 )
=> ( ord_less_eq_nat @ X7 @ M8 ) ) ) ) ).
% bdd_above.unfold
thf(fact_998_cSup__mono,axiom,
! [B3: set_nat,A3: set_nat] :
( ( B3 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A3 )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ B3 )
=> ? [X8: nat] :
( ( member_nat @ X8 @ A3 )
& ( ord_less_eq_nat @ B5 @ X8 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ B3 ) @ ( complete_Sup_Sup_nat @ A3 ) ) ) ) ) ).
% cSup_mono
thf(fact_999_cSup__le__iff,axiom,
! [S4: set_nat,A: nat] :
( ( S4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ S4 )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ S4 ) @ A )
= ( ! [X7: nat] :
( ( member_nat @ X7 @ S4 )
=> ( ord_less_eq_nat @ X7 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1000_cSup__subset__mono,axiom,
! [A3: set_nat,B3: set_nat] :
( ( A3 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ A3 ) @ ( complete_Sup_Sup_nat @ B3 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1001_le__cSup__iff,axiom,
! [A3: set_nat,X: nat] :
( ( A3 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A3 )
=> ( ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ A3 ) )
= ( ! [Y2: nat] :
( ( ord_less_nat @ Y2 @ X )
=> ? [X7: nat] :
( ( member_nat @ X7 @ A3 )
& ( ord_less_nat @ Y2 @ X7 ) ) ) ) ) ) ) ).
% le_cSup_iff
thf(fact_1002_cSUP__subset__mono,axiom,
! [A3: set_a,G: a > nat,B3: set_a,F: a > nat] :
( ( A3 != bot_bot_set_a )
=> ( ( condit2214826472909112428ve_nat @ ( image_a_nat @ G @ B3 ) )
=> ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A3 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A3 ) ) @ ( complete_Sup_Sup_nat @ ( image_a_nat @ G @ B3 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1003_cSUP__subset__mono,axiom,
! [A3: set_nat,G: nat > nat,B3: set_nat,F: nat > nat] :
( ( A3 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B3 ) )
=> ( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A3 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ B3 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1004_image__eqI,axiom,
! [B: a,F: a > a,X: a,A3: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ B @ ( image_a_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1005_image__eqI,axiom,
! [B: nat,F: a > nat,X: a,A3: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A3 )
=> ( member_nat @ B @ ( image_a_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1006_image__eqI,axiom,
! [B: a,F: nat > a,X: nat,A3: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A3 )
=> ( member_a @ B @ ( image_nat_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1007_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A3: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A3 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1008_image__is__empty,axiom,
! [F: nat > nat,A3: set_nat] :
( ( ( image_nat_nat @ F @ A3 )
= bot_bot_set_nat )
= ( A3 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1009_empty__is__image,axiom,
! [F: nat > nat,A3: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1010_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1011_surj__fn,axiom,
! [F: a > a,N2: nat] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( image_a_a @ ( compow_a_a @ N2 @ F ) @ top_top_set_a )
= top_top_set_a ) ) ).
% surj_fn
thf(fact_1012_surj__fn,axiom,
! [F: nat > nat,N2: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat @ ( compow_nat_nat @ N2 @ F ) @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surj_fn
thf(fact_1013_bdd__above_OI2,axiom,
! [A3: set_a,F: a > nat,M: nat] :
( ! [X6: a] :
( ( member_a @ X6 @ A3 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ M ) )
=> ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ A3 ) ) ) ).
% bdd_above.I2
thf(fact_1014_bdd__above_OI2,axiom,
! [A3: set_nat,F: nat > nat,M: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ M ) )
=> ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A3 ) ) ) ).
% bdd_above.I2
thf(fact_1015_Sup__SUP__eq,axiom,
( complete_Sup_Sup_a_o
= ( ^ [S5: set_a_o,X7: a] : ( member_a @ X7 @ ( comple2307003609928055243_set_a @ ( image_a_o_set_a @ collect_a @ S5 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1016_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S5: set_nat_o,X7: nat] : ( member_nat @ X7 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S5 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1017_imageI,axiom,
! [X: a,A3: set_a,F: a > a] :
( ( member_a @ X @ A3 )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_1018_imageI,axiom,
! [X: a,A3: set_a,F: a > nat] :
( ( member_a @ X @ A3 )
=> ( member_nat @ ( F @ X ) @ ( image_a_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_1019_imageI,axiom,
! [X: nat,A3: set_nat,F: nat > a] :
( ( member_nat @ X @ A3 )
=> ( member_a @ ( F @ X ) @ ( image_nat_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_1020_imageI,axiom,
! [X: nat,A3: set_nat,F: nat > nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_1021_image__iff,axiom,
! [Z: nat,F: nat > nat,A3: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A3 ) )
= ( ? [X7: nat] :
( ( member_nat @ X7 @ A3 )
& ( Z
= ( F @ X7 ) ) ) ) ) ).
% image_iff
thf(fact_1022_bex__imageD,axiom,
! [F: nat > nat,A3: set_nat,P: nat > $o] :
( ? [X8: nat] :
( ( member_nat @ X8 @ ( image_nat_nat @ F @ A3 ) )
& ( P @ X8 ) )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A3 )
& ( P @ ( F @ X6 ) ) ) ) ).
% bex_imageD
thf(fact_1023_image__cong,axiom,
! [M: set_nat,N5: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N5 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ N5 )
=> ( ( F @ X6 )
= ( G @ X6 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_1024_ball__imageD,axiom,
! [F: nat > nat,A3: set_nat,P: nat > $o] :
( ! [X6: nat] :
( ( member_nat @ X6 @ ( image_nat_nat @ F @ A3 ) )
=> ( P @ X6 ) )
=> ! [X8: nat] :
( ( member_nat @ X8 @ A3 )
=> ( P @ ( F @ X8 ) ) ) ) ).
% ball_imageD
thf(fact_1025_rev__image__eqI,axiom,
! [X: a,A3: set_a,B: a,F: a > a] :
( ( member_a @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_a_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_1026_rev__image__eqI,axiom,
! [X: a,A3: set_a,B: nat,F: a > nat] :
( ( member_a @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_a_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_1027_rev__image__eqI,axiom,
! [X: nat,A3: set_nat,B: a,F: nat > a] :
( ( member_nat @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_nat_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_1028_rev__image__eqI,axiom,
! [X: nat,A3: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_1029_rangeI,axiom,
! [F: a > a,X: a] : ( member_a @ ( F @ X ) @ ( image_a_a @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_1030_rangeI,axiom,
! [F: a > nat,X: a] : ( member_nat @ ( F @ X ) @ ( image_a_nat @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_1031_rangeI,axiom,
! [F: nat > a,X: nat] : ( member_a @ ( F @ X ) @ ( image_nat_a @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1032_rangeI,axiom,
! [F: nat > nat,X: nat] : ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1033_range__eqI,axiom,
! [B: a,F: a > a,X: a] :
( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_a_a @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_1034_range__eqI,axiom,
! [B: nat,F: a > nat,X: a] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_a_nat @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_1035_range__eqI,axiom,
! [B: a,F: nat > a,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_nat_a @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1036_range__eqI,axiom,
! [B: nat,F: nat > nat,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1037_surjD,axiom,
! [F: a > a,Y: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ? [X6: a] :
( Y
= ( F @ X6 ) ) ) ).
% surjD
thf(fact_1038_surjD,axiom,
! [F: a > nat,Y: nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ? [X6: a] :
( Y
= ( F @ X6 ) ) ) ).
% surjD
thf(fact_1039_surjD,axiom,
! [F: nat > a,Y: a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ? [X6: nat] :
( Y
= ( F @ X6 ) ) ) ).
% surjD
thf(fact_1040_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X6: nat] :
( Y
= ( F @ X6 ) ) ) ).
% surjD
thf(fact_1041_surjE,axiom,
! [F: a > a,Y: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ~ ! [X6: a] :
( Y
!= ( F @ X6 ) ) ) ).
% surjE
thf(fact_1042_surjE,axiom,
! [F: a > nat,Y: nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ~ ! [X6: a] :
( Y
!= ( F @ X6 ) ) ) ).
% surjE
thf(fact_1043_surjE,axiom,
! [F: nat > a,Y: a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ~ ! [X6: nat] :
( Y
!= ( F @ X6 ) ) ) ).
% surjE
thf(fact_1044_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X6: nat] :
( Y
!= ( F @ X6 ) ) ) ).
% surjE
thf(fact_1045_surjI,axiom,
! [G: a > a,F: a > a] :
( ! [X6: a] :
( ( G @ ( F @ X6 ) )
= X6 )
=> ( ( image_a_a @ G @ top_top_set_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_1046_surjI,axiom,
! [G: a > nat,F: nat > a] :
( ! [X6: nat] :
( ( G @ ( F @ X6 ) )
= X6 )
=> ( ( image_a_nat @ G @ top_top_set_a )
= top_top_set_nat ) ) ).
% surjI
thf(fact_1047_surjI,axiom,
! [G: nat > a,F: a > nat] :
( ! [X6: a] :
( ( G @ ( F @ X6 ) )
= X6 )
=> ( ( image_nat_a @ G @ top_top_set_nat )
= top_top_set_a ) ) ).
% surjI
thf(fact_1048_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X6: nat] :
( ( G @ ( F @ X6 ) )
= X6 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_1049_surj__def,axiom,
! [F: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
= ( ! [Y2: a] :
? [X7: a] :
( Y2
= ( F @ X7 ) ) ) ) ).
% surj_def
thf(fact_1050_surj__def,axiom,
! [F: a > nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X7: a] :
( Y2
= ( F @ X7 ) ) ) ) ).
% surj_def
thf(fact_1051_surj__def,axiom,
! [F: nat > a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
= ( ! [Y2: a] :
? [X7: nat] :
( Y2
= ( F @ X7 ) ) ) ) ).
% surj_def
thf(fact_1052_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X7: nat] :
( Y2
= ( F @ X7 ) ) ) ) ).
% surj_def
thf(fact_1053_image__Un,axiom,
! [F: nat > nat,A3: set_nat,B3: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A3 @ B3 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A3 ) @ ( image_nat_nat @ F @ B3 ) ) ) ).
% image_Un
thf(fact_1054_weak__coinduct__image,axiom,
! [A: a,X9: set_a,G: a > a,F: set_a > set_a] :
( ( member_a @ A @ X9 )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ G @ X9 ) @ ( F @ ( image_a_a @ G @ X9 ) ) )
=> ( member_a @ ( G @ A ) @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_1055_weak__coinduct__image,axiom,
! [A: a,X9: set_a,G: a > nat,F: set_nat > set_nat] :
( ( member_a @ A @ X9 )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ G @ X9 ) @ ( F @ ( image_a_nat @ G @ X9 ) ) )
=> ( member_nat @ ( G @ A ) @ ( comple1596078789208929544et_nat @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_1056_weak__coinduct__image,axiom,
! [A: nat,X9: set_nat,G: nat > a,F: set_a > set_a] :
( ( member_nat @ A @ X9 )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ G @ X9 ) @ ( F @ ( image_nat_a @ G @ X9 ) ) )
=> ( member_a @ ( G @ A ) @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_1057_weak__coinduct__image,axiom,
! [A: nat,X9: set_nat,G: nat > nat,F: set_nat > set_nat] :
( ( member_nat @ A @ X9 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ X9 ) @ ( F @ ( image_nat_nat @ G @ X9 ) ) )
=> ( member_nat @ ( G @ A ) @ ( comple1596078789208929544et_nat @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_1058_range__subsetD,axiom,
! [F: a > a,B3: set_a,I: a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ top_top_set_a ) @ B3 )
=> ( member_a @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_1059_range__subsetD,axiom,
! [F: a > nat,B3: set_nat,I: a] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ top_top_set_a ) @ B3 )
=> ( member_nat @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_1060_range__subsetD,axiom,
! [F: nat > a,B3: set_a,I: nat] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F @ top_top_set_nat ) @ B3 )
=> ( member_a @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_1061_range__subsetD,axiom,
! [F: nat > nat,B3: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B3 )
=> ( member_nat @ ( F @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_1062_image__Int__subset,axiom,
! [F: nat > nat,A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A3 @ B3 ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A3 ) @ ( image_nat_nat @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1063_image__Int__subset,axiom,
! [F: a > a,A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A3 @ B3 ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A3 ) @ ( image_a_a @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1064_subset__image__iff,axiom,
! [B3: set_nat,F: nat > nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A3 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A3 )
& ( B3
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1065_image__subset__iff,axiom,
! [F: nat > nat,A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ B3 )
= ( ! [X7: nat] :
( ( member_nat @ X7 @ A3 )
=> ( member_nat @ ( F @ X7 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_1066_subset__imageE,axiom,
! [B3: set_nat,F: nat > nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A3 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A3 )
=> ( B3
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1067_image__subsetI,axiom,
! [A3: set_a,F: a > a,B3: set_a] :
( ! [X6: a] :
( ( member_a @ X6 @ A3 )
=> ( member_a @ ( F @ X6 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1068_image__subsetI,axiom,
! [A3: set_a,F: a > nat,B3: set_nat] :
( ! [X6: a] :
( ( member_a @ X6 @ A3 )
=> ( member_nat @ ( F @ X6 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1069_image__subsetI,axiom,
! [A3: set_nat,F: nat > a,B3: set_a] :
( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ( member_a @ ( F @ X6 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1070_image__subsetI,axiom,
! [A3: set_nat,F: nat > nat,B3: set_nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ( member_nat @ ( F @ X6 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1071_image__mono,axiom,
! [A3: set_nat,B3: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ ( image_nat_nat @ F @ B3 ) ) ) ).
% image_mono
thf(fact_1072_inf__Sup,axiom,
! [A: set_a,B3: set_set_a] :
( ( inf_inf_set_a @ A @ ( comple2307003609928055243_set_a @ B3 ) )
= ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ ( inf_inf_set_a @ A ) @ B3 ) ) ) ).
% inf_Sup
thf(fact_1073_cSUP__least,axiom,
! [A3: set_a,F: a > nat,M: nat] :
( ( A3 != bot_bot_set_a )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A3 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A3 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1074_cSUP__least,axiom,
! [A3: set_nat,F: nat > nat,M: nat] :
( ( A3 != bot_bot_set_nat )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A3 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1075_strict__mono__inv,axiom,
! [F: nat > nat,G: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X6: nat] :
( ( G @ ( F @ X6 ) )
= X6 )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_1076_cSUP__upper2,axiom,
! [F: a > nat,A3: set_a,X: a,U: nat] :
( ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ A3 ) )
=> ( ( member_a @ X @ A3 )
=> ( ( ord_less_eq_nat @ U @ ( F @ X ) )
=> ( ord_less_eq_nat @ U @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A3 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1077_cSUP__upper2,axiom,
! [F: nat > nat,A3: set_nat,X: nat,U: nat] :
( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A3 ) )
=> ( ( member_nat @ X @ A3 )
=> ( ( ord_less_eq_nat @ U @ ( F @ X ) )
=> ( ord_less_eq_nat @ U @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A3 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1078_cSUP__upper,axiom,
! [X: a,A3: set_a,F: a > nat] :
( ( member_a @ X @ A3 )
=> ( ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ A3 ) )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A3 ) ) ) ) ) ).
% cSUP_upper
thf(fact_1079_cSUP__upper,axiom,
! [X: nat,A3: set_nat,F: nat > nat] :
( ( member_nat @ X @ A3 )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A3 ) )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A3 ) ) ) ) ) ).
% cSUP_upper
thf(fact_1080_le__cSUP__iff,axiom,
! [A3: set_nat,F: nat > nat,X: nat] :
( ( A3 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A3 ) )
=> ( ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A3 ) ) )
= ( ! [Y2: nat] :
( ( ord_less_nat @ Y2 @ X )
=> ? [X7: nat] :
( ( member_nat @ X7 @ A3 )
& ( ord_less_nat @ Y2 @ ( F @ X7 ) ) ) ) ) ) ) ) ).
% le_cSUP_iff
thf(fact_1081_cSUP__mono,axiom,
! [A3: set_a,G: nat > nat,B3: set_nat,F: a > nat] :
( ( A3 != bot_bot_set_a )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B3 ) )
=> ( ! [N3: a] :
( ( member_a @ N3 @ A3 )
=> ? [X8: nat] :
( ( member_nat @ X8 @ B3 )
& ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X8 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A3 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ B3 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_1082_cSUP__mono,axiom,
! [A3: set_nat,G: nat > nat,B3: set_nat,F: nat > nat] :
( ( A3 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B3 ) )
=> ( ! [N3: nat] :
( ( member_nat @ N3 @ A3 )
=> ? [X8: nat] :
( ( member_nat @ X8 @ B3 )
& ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X8 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A3 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ B3 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_1083_cSUP__le__iff,axiom,
! [A3: set_nat,F: nat > nat,U: nat] :
( ( A3 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A3 ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A3 ) ) @ U )
= ( ! [X7: nat] :
( ( member_nat @ X7 @ A3 )
=> ( ord_less_eq_nat @ ( F @ X7 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1084_bdd__above__image__mono,axiom,
! [F: nat > nat,A3: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( condit2214826472909112428ve_nat @ A3 )
=> ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A3 ) ) ) ) ).
% bdd_above_image_mono
thf(fact_1085_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ! [N3: nat] :
( ( ( F @ N3 )
= ( F @ ( suc @ N3 ) ) )
=> ( ( F @ ( suc @ N3 ) )
= ( F @ ( suc @ ( suc @ N3 ) ) ) ) )
=> ? [N6: nat] :
( ! [N7: nat] :
( ( ord_less_eq_nat @ N7 @ N6 )
=> ! [M5: nat] :
( ( ord_less_eq_nat @ M5 @ N6 )
=> ( ( ord_less_nat @ M5 @ N7 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N7 ) ) ) ) )
& ! [N7: nat] :
( ( ord_less_eq_nat @ N6 @ N7 )
=> ( ( F @ N6 )
= ( F @ N7 ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_1086_le__cSup__finite,axiom,
! [X9: set_nat,X: nat] :
( ( finite_finite_nat @ X9 )
=> ( ( member_nat @ X @ X9 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X9 ) ) ) ) ).
% le_cSup_finite
thf(fact_1087_finite__subset__Union,axiom,
! [A3: set_nat,B7: set_set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ord_less_eq_set_nat @ A3 @ ( comple7399068483239264473et_nat @ B7 ) )
=> ~ ! [F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( ord_le6893508408891458716et_nat @ F3 @ B7 )
=> ~ ( ord_less_eq_set_nat @ A3 @ ( comple7399068483239264473et_nat @ F3 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1088_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N8: set_nat] :
? [M3: nat] :
! [X7: nat] :
( ( member_nat @ X7 @ N8 )
=> ( ord_less_eq_nat @ X7 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1089_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N8: set_nat] :
? [M3: nat] :
! [X7: nat] :
( ( member_nat @ X7 @ N8 )
=> ( ord_less_nat @ X7 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1090_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N2: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ N5 )
=> ( ord_less_nat @ X6 @ N2 ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1091_finite__Int,axiom,
! [F4: set_nat,G2: set_nat] :
( ( ( finite_finite_nat @ F4 )
| ( finite_finite_nat @ G2 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F4 @ G2 ) ) ) ).
% finite_Int
thf(fact_1092_finite__Int,axiom,
! [F4: set_a,G2: set_a] :
( ( ( finite_finite_a @ F4 )
| ( finite_finite_a @ G2 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F4 @ G2 ) ) ) ).
% finite_Int
thf(fact_1093_finite__Plus__UNIV__iff,axiom,
( ( finite51705147264084924um_a_a @ top_to8848906000605539851um_a_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1094_finite__Plus__UNIV__iff,axiom,
( ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1095_finite__Plus__UNIV__iff,axiom,
( ( finite3740268481367103950_nat_a @ top_to54524901450547413_nat_a )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1096_finite__Plus__UNIV__iff,axiom,
( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1097_finite__has__minimal2,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A3 )
& ( ord_less_eq_nat @ X6 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ Xa @ X6 )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1098_finite__has__maximal2,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A3 )
& ( ord_less_eq_nat @ A @ X6 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ X6 @ Xa )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1099_ex__new__if__finite,axiom,
! [A3: set_a] :
( ~ ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ A3 )
=> ? [A6: a] :
~ ( member_a @ A6 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1100_ex__new__if__finite,axiom,
! [A3: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A3 )
=> ? [A6: nat] :
~ ( member_nat @ A6 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1101_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_1102_finite__prod,axiom,
( ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_1103_finite__prod,axiom,
( ( finite6644898363146130708_a_nat @ top_to3353692345378799459_a_nat )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_1104_finite__prod,axiom,
( ( finite659689790015031866_nat_a @ top_to2612598781856825737_nat_a )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_1105_finite__prod,axiom,
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_1106_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_1107_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6644898363146130708_a_nat @ top_to3353692345378799459_a_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_1108_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite659689790015031866_nat_a @ top_to2612598781856825737_nat_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_1109_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_1110_all__subset__image,axiom,
! [F: nat > nat,A3: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A3 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A3 )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1111_rev__finite__subset,axiom,
! [B3: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( finite_finite_nat @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_1112_infinite__super,axiom,
! [S4: set_nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S4 @ T3 )
=> ( ~ ( finite_finite_nat @ S4 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_1113_finite__subset,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite_finite_nat @ A3 ) ) ) ).
% finite_subset
thf(fact_1114_finite__has__maximal,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ X6 @ Xa )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1115_finite__has__minimal,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ Xa @ X6 )
=> ( X6 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1116_finite__surj,axiom,
! [A3: set_nat,B3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A3 ) )
=> ( finite_finite_nat @ B3 ) ) ) ).
% finite_surj
thf(fact_1117_finite__subset__image,axiom,
! [B3: set_nat,F: nat > nat,A3: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A3 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A3 )
& ( finite_finite_nat @ C4 )
& ( B3
= ( image_nat_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1118_ex__finite__subset__image,axiom,
! [F: nat > nat,A3: set_nat,P: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A3 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A3 )
& ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1119_all__finite__subset__image,axiom,
! [F: nat > nat,A3: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A3 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A3 ) )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1120_Finite__Set_Ofinite__set,axiom,
( ( finite_finite_set_a @ top_top_set_set_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% Finite_Set.finite_set
thf(fact_1121_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_1122_finite__option__UNIV,axiom,
( ( finite1674126218327898605tion_a @ top_top_set_option_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_option_UNIV
thf(fact_1123_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_1124_infinite__enumerate,axiom,
! [S4: set_nat] :
( ~ ( finite_finite_nat @ S4 )
=> ? [R: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ R )
& ! [N7: nat] : ( member_nat @ ( R @ N7 ) @ S4 ) ) ) ).
% infinite_enumerate
thf(fact_1125_infinite__nat__iff__unbounded__le,axiom,
! [S4: set_nat] :
( ( ~ ( finite_finite_nat @ S4 ) )
= ( ! [M3: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
& ( member_nat @ N @ S4 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1126_arg__min__least,axiom,
! [S4: set_a,Y: a,F: a > nat] :
( ( finite_finite_a @ S4 )
=> ( ( S4 != bot_bot_set_a )
=> ( ( member_a @ Y @ S4 )
=> ( ord_less_eq_nat @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S4 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1127_arg__min__least,axiom,
! [S4: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S4 )
=> ( ( S4 != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S4 )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S4 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1128_mono__Max__commute,axiom,
! [F: nat > nat,A3: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( F @ ( lattic8265883725875713057ax_nat @ A3 ) )
= ( lattic8265883725875713057ax_nat @ ( image_nat_nat @ F @ A3 ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_1129_Max_Obounded__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ X )
= ( ! [X7: nat] :
( ( member_nat @ X7 @ A3 )
=> ( ord_less_eq_nat @ X7 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_1130_Max__ge,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A3 ) ) ) ) ).
% Max_ge
thf(fact_1131_Max__eqI,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ A3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( member_nat @ X @ A3 )
=> ( ( lattic8265883725875713057ax_nat @ A3 )
= X ) ) ) ) ).
% Max_eqI
thf(fact_1132_Max__eq__if,axiom,
! [A3: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( ord_less_eq_nat @ X6 @ Xa ) ) )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ B3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ A3 )
& ( ord_less_eq_nat @ X6 @ Xa ) ) )
=> ( ( lattic8265883725875713057ax_nat @ A3 )
= ( lattic8265883725875713057ax_nat @ B3 ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_1133_Max_OcoboundedI,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ( ord_less_eq_nat @ A @ ( lattic8265883725875713057ax_nat @ A3 ) ) ) ) ).
% Max.coboundedI
thf(fact_1134_Max_OboundedI,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A3 )
=> ( ord_less_eq_nat @ A6 @ X ) )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_1135_Max_OboundedE,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ X )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A3 )
=> ( ord_less_eq_nat @ A8 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_1136_eq__Max__iff,axiom,
! [A3: set_nat,M4: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( M4
= ( lattic8265883725875713057ax_nat @ A3 ) )
= ( ( member_nat @ M4 @ A3 )
& ! [X7: nat] :
( ( member_nat @ X7 @ A3 )
=> ( ord_less_eq_nat @ X7 @ M4 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_1137_Max__ge__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A3 ) )
= ( ? [X7: nat] :
( ( member_nat @ X7 @ A3 )
& ( ord_less_eq_nat @ X @ X7 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_1138_Max__eq__iff,axiom,
! [A3: set_nat,M4: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ( lattic8265883725875713057ax_nat @ A3 )
= M4 )
= ( ( member_nat @ M4 @ A3 )
& ! [X7: nat] :
( ( member_nat @ X7 @ A3 )
=> ( ord_less_eq_nat @ X7 @ M4 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_1139_Max_Osubset__imp,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B3 )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ ( lattic8265883725875713057ax_nat @ B3 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_1140_Max__mono,axiom,
! [M: set_nat,N5: set_nat] :
( ( ord_less_eq_set_nat @ M @ N5 )
=> ( ( M != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N5 )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M ) @ ( lattic8265883725875713057ax_nat @ N5 ) ) ) ) ) ).
% Max_mono
thf(fact_1141_mono__Min__commute,axiom,
! [F: nat > nat,A3: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( F @ ( lattic8721135487736765967in_nat @ A3 ) )
= ( lattic8721135487736765967in_nat @ ( image_nat_nat @ F @ A3 ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_1142_image__Fpow__mono,axiom,
! [F: nat > nat,A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ B3 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A3 ) ) @ ( finite_Fpow_nat @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_1143_Min_Obounded__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A3 ) )
= ( ! [X7: nat] :
( ( member_nat @ X7 @ A3 )
=> ( ord_less_eq_nat @ X @ X7 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_1144_Min_OcoboundedI,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A3 ) @ A ) ) ) ).
% Min.coboundedI
thf(fact_1145_Min__eqI,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ A3 )
=> ( ord_less_eq_nat @ X @ Y3 ) )
=> ( ( member_nat @ X @ A3 )
=> ( ( lattic8721135487736765967in_nat @ A3 )
= X ) ) ) ) ).
% Min_eqI
thf(fact_1146_Min__le,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A3 ) @ X ) ) ) ).
% Min_le
thf(fact_1147_Min__eq__iff,axiom,
! [A3: set_nat,M4: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A3 )
= M4 )
= ( ( member_nat @ M4 @ A3 )
& ! [X7: nat] :
( ( member_nat @ X7 @ A3 )
=> ( ord_less_eq_nat @ M4 @ X7 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_1148_Min__le__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A3 ) @ X )
= ( ? [X7: nat] :
( ( member_nat @ X7 @ A3 )
& ( ord_less_eq_nat @ X7 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_1149_eq__Min__iff,axiom,
! [A3: set_nat,M4: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( M4
= ( lattic8721135487736765967in_nat @ A3 ) )
= ( ( member_nat @ M4 @ A3 )
& ! [X7: nat] :
( ( member_nat @ X7 @ A3 )
=> ( ord_less_eq_nat @ M4 @ X7 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_1150_Min_OboundedE,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A3 ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A3 )
=> ( ord_less_eq_nat @ X @ A8 ) ) ) ) ) ).
% Min.boundedE
thf(fact_1151_Min_OboundedI,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A3 )
=> ( ord_less_eq_nat @ X @ A6 ) )
=> ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A3 ) ) ) ) ) ).
% Min.boundedI
thf(fact_1152_Min__antimono,axiom,
! [M: set_nat,N5: set_nat] :
( ( ord_less_eq_set_nat @ M @ N5 )
=> ( ( M != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N5 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N5 ) @ ( lattic8721135487736765967in_nat @ M ) ) ) ) ) ).
% Min_antimono
thf(fact_1153_Min_Osubset__imp,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B3 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B3 ) @ ( lattic8721135487736765967in_nat @ A3 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_1154_the__elem__image__unique,axiom,
! [A3: set_nat,F: nat > nat,X: nat] :
( ( A3 != bot_bot_set_nat )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ A3 )
=> ( ( F @ Y3 )
= ( F @ X ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat @ F @ A3 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1155_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ( linord2614967742042102400et_nat @ A3 )
= nil_nat )
= ( A3 = bot_bot_set_nat ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_1156_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A3: set_nat] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( linord2614967742042102400et_nat @ A3 )
= nil_nat ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_1157_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ( linord2614967742042102400et_nat @ A3 )
= ( linord2614967742042102400et_nat @ B3 ) )
=> ( ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_1158_strict__mono__enumerate,axiom,
! [S4: set_nat] :
( ~ ( finite_finite_nat @ S4 )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ ( infini8530281810654367211te_nat @ S4 ) ) ) ).
% strict_mono_enumerate
thf(fact_1159_semilattice__order__set_Osubset__imp,axiom,
! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A3: set_nat,B3: set_nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
=> ( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B3 )
=> ( Less_eq @ ( lattic7742739596368939638_F_nat @ F @ B3 ) @ ( lattic7742739596368939638_F_nat @ F @ A3 ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_1160_enumerate__mono__le__iff,axiom,
! [S4: set_nat,M4: nat,N2: nat] :
( ~ ( finite_finite_nat @ S4 )
=> ( ( ord_less_eq_nat @ ( infini8530281810654367211te_nat @ S4 @ M4 ) @ ( infini8530281810654367211te_nat @ S4 @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% enumerate_mono_le_iff
thf(fact_1161_le__enumerate,axiom,
! [S4: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ S4 )
=> ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S4 @ N2 ) ) ) ).
% le_enumerate
thf(fact_1162_semilattice__set_Osubset,axiom,
! [F: nat > nat > nat,A3: set_nat,B3: set_nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A3 )
=> ( ( B3 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( ( F @ ( lattic7742739596368939638_F_nat @ F @ B3 ) @ ( lattic7742739596368939638_F_nat @ F @ A3 ) )
= ( lattic7742739596368939638_F_nat @ F @ A3 ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_1163_finite__enum__subset,axiom,
! [X9: set_nat,Y7: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( finite_card_nat @ X9 ) )
=> ( ( infini8530281810654367211te_nat @ X9 @ I2 )
= ( infini8530281810654367211te_nat @ Y7 @ I2 ) ) )
=> ( ( finite_finite_nat @ X9 )
=> ( ( finite_finite_nat @ Y7 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ X9 ) @ ( finite_card_nat @ Y7 ) )
=> ( ord_less_eq_set_nat @ X9 @ Y7 ) ) ) ) ) ).
% finite_enum_subset
thf(fact_1164_card__image__le,axiom,
! [A3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A3 ) ) @ ( finite_card_nat @ A3 ) ) ) ).
% card_image_le
thf(fact_1165_card__mono,axiom,
! [B3: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B3 ) ) ) ) ).
% card_mono
thf(fact_1166_card__seteq,axiom,
! [B3: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B3 ) @ ( finite_card_nat @ A3 ) )
=> ( A3 = B3 ) ) ) ) ).
% card_seteq
thf(fact_1167_exists__subset__between,axiom,
! [A3: set_nat,N2: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A3 @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B8: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B8 )
& ( ord_less_eq_set_nat @ B8 @ C2 )
& ( ( finite_card_nat @ B8 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1168_obtain__subset__with__card__n,axiom,
! [N2: nat,S4: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S4 ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S4 )
=> ( ( ( finite_card_nat @ T4 )
= N2 )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1169_finite__if__finite__subsets__card__bdd,axiom,
! [F4: set_nat,C2: nat] :
( ! [G3: set_nat] :
( ( ord_less_eq_set_nat @ G3 @ F4 )
=> ( ( finite_finite_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G3 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F4 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F4 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1170_infinite__arbitrarily__large,axiom,
! [A3: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ A3 )
=> ? [B8: set_nat] :
( ( finite_finite_nat @ B8 )
& ( ( finite_card_nat @ B8 )
= N2 )
& ( ord_less_eq_set_nat @ B8 @ A3 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1171_card__subset__eq,axiom,
! [B3: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ( finite_card_nat @ A3 )
= ( finite_card_nat @ B3 ) )
=> ( A3 = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_1172_card__le__if__inj__on__rel,axiom,
! [B3: set_a,A3: set_a,R2: a > a > $o] :
( ( finite_finite_a @ B3 )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A3 )
=> ? [B9: a] :
( ( member_a @ B9 @ B3 )
& ( R2 @ A6 @ B9 ) ) )
=> ( ! [A13: a,A23: a,B5: a] :
( ( member_a @ A13 @ A3 )
=> ( ( member_a @ A23 @ A3 )
=> ( ( member_a @ B5 @ B3 )
=> ( ( R2 @ A13 @ B5 )
=> ( ( R2 @ A23 @ B5 )
=> ( A13 = A23 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A3 ) @ ( finite_card_a @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1173_card__le__if__inj__on__rel,axiom,
! [B3: set_a,A3: set_nat,R2: nat > a > $o] :
( ( finite_finite_a @ B3 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A3 )
=> ? [B9: a] :
( ( member_a @ B9 @ B3 )
& ( R2 @ A6 @ B9 ) ) )
=> ( ! [A13: nat,A23: nat,B5: a] :
( ( member_nat @ A13 @ A3 )
=> ( ( member_nat @ A23 @ A3 )
=> ( ( member_a @ B5 @ B3 )
=> ( ( R2 @ A13 @ B5 )
=> ( ( R2 @ A23 @ B5 )
=> ( A13 = A23 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_a @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1174_card__le__if__inj__on__rel,axiom,
! [B3: set_nat,A3: set_a,R2: a > nat > $o] :
( ( finite_finite_nat @ B3 )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A3 )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B3 )
& ( R2 @ A6 @ B9 ) ) )
=> ( ! [A13: a,A23: a,B5: nat] :
( ( member_a @ A13 @ A3 )
=> ( ( member_a @ A23 @ A3 )
=> ( ( member_nat @ B5 @ B3 )
=> ( ( R2 @ A13 @ B5 )
=> ( ( R2 @ A23 @ B5 )
=> ( A13 = A23 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A3 ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1175_card__le__if__inj__on__rel,axiom,
! [B3: set_nat,A3: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ B3 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A3 )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B3 )
& ( R2 @ A6 @ B9 ) ) )
=> ( ! [A13: nat,A23: nat,B5: nat] :
( ( member_nat @ A13 @ A3 )
=> ( ( member_nat @ A23 @ A3 )
=> ( ( member_nat @ B5 @ B3 )
=> ( ( R2 @ A13 @ B5 )
=> ( ( R2 @ A23 @ B5 )
=> ( A13 = A23 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1176_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A3: set_a] :
( ( finite_finite_a @ top_top_set_a )
=> ( ( ( finite_card_a @ A3 )
= ( finite_card_a @ top_top_set_a ) )
=> ( A3 = top_top_set_a ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_1177_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( finite_card_nat @ A3 )
= ( finite_card_nat @ top_top_set_nat ) )
=> ( A3 = top_top_set_nat ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_1178_Inf__fin_Osemilattice__set__axioms,axiom,
lattic1258622339881844972_set_a @ inf_inf_set_a ).
% Inf_fin.semilattice_set_axioms
thf(fact_1179_card__le__Suc__Max,axiom,
! [S4: set_nat] :
( ( finite_finite_nat @ S4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ S4 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S4 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_1180_surj__card__le,axiom,
! [A3: set_nat,B3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A3 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B3 ) @ ( finite_card_nat @ A3 ) ) ) ) ).
% surj_card_le
thf(fact_1181_card__psubset,axiom,
! [B3: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B3 ) )
=> ( ord_less_set_nat @ A3 @ B3 ) ) ) ) ).
% card_psubset
thf(fact_1182_finite__le__enumerate,axiom,
! [S4: set_nat,N2: nat] :
( ( finite_finite_nat @ S4 )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S4 ) )
=> ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S4 @ N2 ) ) ) ) ).
% finite_le_enumerate
thf(fact_1183_finite__fun__UNIVD1,axiom,
( ( finite_finite_a_a @ top_top_set_a_a )
=> ( ( ( finite_card_a @ top_top_set_a )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_fun_UNIVD1
thf(fact_1184_finite__fun__UNIVD1,axiom,
( ( finite_finite_nat_a @ top_top_set_nat_a )
=> ( ( ( finite_card_a @ top_top_set_a )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_fun_UNIVD1
thf(fact_1185_finite__fun__UNIVD1,axiom,
( ( finite_finite_a_nat @ top_top_set_a_nat )
=> ( ( ( finite_card_nat @ top_top_set_nat )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_fun_UNIVD1
thf(fact_1186_finite__fun__UNIVD1,axiom,
( ( finite2115694454571419734at_nat @ top_top_set_nat_nat )
=> ( ( ( finite_card_nat @ top_top_set_nat )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_fun_UNIVD1
thf(fact_1187_card__partition,axiom,
! [C2: set_set_nat,K: nat] :
( ( finite1152437895449049373et_nat @ C2 )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ C2 ) )
=> ( ! [C3: set_nat] :
( ( member_set_nat @ C3 @ C2 )
=> ( ( finite_card_nat @ C3 )
= K ) )
=> ( ! [C1: set_nat,C22: set_nat] :
( ( member_set_nat @ C1 @ C2 )
=> ( ( member_set_nat @ C22 @ C2 )
=> ( ( C1 != C22 )
=> ( ( inf_inf_set_nat @ C1 @ C22 )
= bot_bot_set_nat ) ) ) )
=> ( ( times_times_nat @ K @ ( finite_card_set_nat @ C2 ) )
= ( finite_card_nat @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_1188_card__partition,axiom,
! [C2: set_set_a,K: nat] :
( ( finite_finite_set_a @ C2 )
=> ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ C2 ) )
=> ( ! [C3: set_a] :
( ( member_set_a @ C3 @ C2 )
=> ( ( finite_card_a @ C3 )
= K ) )
=> ( ! [C1: set_a,C22: set_a] :
( ( member_set_a @ C1 @ C2 )
=> ( ( member_set_a @ C22 @ C2 )
=> ( ( C1 != C22 )
=> ( ( inf_inf_set_a @ C1 @ C22 )
= bot_bot_set_a ) ) ) )
=> ( ( times_times_nat @ K @ ( finite_card_set_a @ C2 ) )
= ( finite_card_a @ ( comple2307003609928055243_set_a @ C2 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_1189_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1190_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_1191_one__le__mult__iff,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M4 @ N2 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M4 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_1192_mult__le__cancel2,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M4 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1193_mult__le__mono2,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1194_mult__le__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1195_mult__le__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1196_le__square,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ).
% le_square
thf(fact_1197_le__cube,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ) ).
% le_cube
thf(fact_1198_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1199_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1200_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1201_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_1202_Suc__mult__le__cancel1,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M4 ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_1203_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1204_mono__times__nat,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( times_times_nat @ N2 ) ) ) ).
% mono_times_nat
thf(fact_1205_zero__notin__Suc__image,axiom,
! [A3: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A3 ) ) ).
% zero_notin_Suc_image
thf(fact_1206_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N2 )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1207_finite__UNIV__card__ge__0,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ top_top_set_a ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_1208_finite__UNIV__card__ge__0,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ top_top_set_nat ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_1209_card__le__Suc0__iff__eq,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X7: nat] :
( ( member_nat @ X7 @ A3 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A3 )
=> ( X7 = Y2 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1210_card__range__greater__zero,axiom,
! [F: a > nat] :
( ( finite_finite_nat @ ( image_a_nat @ F @ top_top_set_a ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( image_a_nat @ F @ top_top_set_a ) ) ) ) ).
% card_range_greater_zero
thf(fact_1211_card__range__greater__zero,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ) ).
% card_range_greater_zero
thf(fact_1212_nat__mult__le__cancel__disj,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1213_mono__mult,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( times_times_nat @ A ) ) ) ).
% mono_mult
thf(fact_1214_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1215_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1216_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1217_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1218_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_1219_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1220_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1221_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1222_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1223_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1224_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1225_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1226_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1227_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1228_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1229_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1230_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1231_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1232_nat__mult__le__cancel1,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1233_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1234_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_1235_n__lists__Nil,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( n_lists_a @ N2 @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) )
& ( ( N2 != zero_zero_nat )
=> ( ( n_lists_a @ N2 @ nil_a )
= nil_list_a ) ) ) ).
% n_lists_Nil
thf(fact_1236_n__lists_Osimps_I1_J,axiom,
! [Xs: list_a] :
( ( n_lists_a @ zero_zero_nat @ Xs )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% n_lists.simps(1)
thf(fact_1237_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1238_sum__le__card__Max,axiom,
! [A3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A3 )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( times_times_nat @ ( finite_card_nat @ A3 ) @ ( lattic8265883725875713057ax_nat @ ( image_nat_nat @ F @ A3 ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_1239_card__Min__le__sum,axiom,
! [A3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( finite_card_nat @ A3 ) @ ( lattic8721135487736765967in_nat @ ( image_nat_nat @ F @ A3 ) ) ) @ ( groups3542108847815614940at_nat @ F @ A3 ) ) ) ).
% card_Min_le_sum
thf(fact_1240_sum__pos2,axiom,
! [I4: set_a,I: a,F: a > nat] :
( ( finite_finite_a @ I4 )
=> ( ( member_a @ I @ I4 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups6334556678337121940_a_nat @ F @ I4 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1241_sum__pos2,axiom,
! [I4: set_nat,I: nat,F: nat > nat] :
( ( finite_finite_nat @ I4 )
=> ( ( member_nat @ I @ I4 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ I4 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1242_card__Union__le__sum__card__weak,axiom,
! [U3: set_set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ U3 )
=> ( finite_finite_nat @ X6 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( comple7399068483239264473et_nat @ U3 ) ) @ ( groups8294997508430121362at_nat @ finite_card_nat @ U3 ) ) ) ).
% card_Union_le_sum_card_weak
thf(fact_1243_sum__nonneg,axiom,
! [A3: set_a,F: a > nat] :
( ! [X6: a] :
( ( member_a @ X6 @ A3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X6 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups6334556678337121940_a_nat @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_1244_sum__nonneg,axiom,
! [A3: set_nat,F: nat > nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X6 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_1245_sum__nonpos,axiom,
! [A3: set_a,F: a > nat] :
( ! [X6: a] :
( ( member_a @ X6 @ A3 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F @ A3 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_1246_sum__nonpos,axiom,
! [A3: set_nat,F: nat > nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_1247_sum__mono__inv,axiom,
! [F: a > nat,I4: set_a,G: a > nat,I: a] :
( ( ( groups6334556678337121940_a_nat @ F @ I4 )
= ( groups6334556678337121940_a_nat @ G @ I4 ) )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I4 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_a @ I @ I4 )
=> ( ( finite_finite_a @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1248_sum__mono__inv,axiom,
! [F: nat > nat,I4: set_nat,G: nat > nat,I: nat] :
( ( ( groups3542108847815614940at_nat @ F @ I4 )
= ( groups3542108847815614940at_nat @ G @ I4 ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_nat @ I @ I4 )
=> ( ( finite_finite_nat @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1249_card__Union__le__sum__card,axiom,
! [U3: set_set_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( comple7399068483239264473et_nat @ U3 ) ) @ ( groups8294997508430121362at_nat @ finite_card_nat @ U3 ) ) ).
% card_Union_le_sum_card
thf(fact_1250_sum__nonneg__eq__0__iff,axiom,
! [A3: set_a,F: a > nat] :
( ( finite_finite_a @ A3 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X6 ) ) )
=> ( ( ( groups6334556678337121940_a_nat @ F @ A3 )
= zero_zero_nat )
= ( ! [X7: a] :
( ( member_a @ X7 @ A3 )
=> ( ( F @ X7 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1251_sum__nonneg__eq__0__iff,axiom,
! [A3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A3 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X6 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ A3 )
= zero_zero_nat )
= ( ! [X7: nat] :
( ( member_nat @ X7 @ A3 )
=> ( ( F @ X7 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1252_sum__le__included,axiom,
! [S2: set_nat,T2: set_nat,G: nat > nat,I: nat > nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ T2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X6 ) ) )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T2 )
& ( ( I @ Xa )
= X6 )
& ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S2 ) @ ( groups3542108847815614940at_nat @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1253_sum__strict__mono__ex1,axiom,
! [A3: set_nat,F: nat > nat,G: nat > nat] :
( ( finite_finite_nat @ A3 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
=> ( ? [X8: nat] :
( ( member_nat @ X8 @ A3 )
& ( ord_less_nat @ ( F @ X8 ) @ ( G @ X8 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ G @ A3 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_1254_sum__strict__mono2,axiom,
! [B3: set_a,A3: set_a,B: a,F: a > nat] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_a @ B @ ( minus_minus_set_a @ B3 @ A3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
=> ( ! [X6: a] :
( ( member_a @ X6 @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X6 ) ) )
=> ( ord_less_nat @ ( groups6334556678337121940_a_nat @ F @ A3 ) @ ( groups6334556678337121940_a_nat @ F @ B3 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_1255_sum__strict__mono2,axiom,
! [B3: set_nat,A3: set_nat,B: nat,F: nat > nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( member_nat @ B @ ( minus_minus_set_nat @ B3 @ A3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X6 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ F @ B3 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_1256_DiffI,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ A3 )
=> ( ~ ( member_a @ C @ B3 )
=> ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_1257_DiffI,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ A3 )
=> ( ~ ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_1258_Diff__iff,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) )
= ( ( member_a @ C @ A3 )
& ~ ( member_a @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_1259_Diff__iff,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
= ( ( member_nat @ C @ A3 )
& ~ ( member_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_1260_Diff__UNIV,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ top_top_set_a )
= bot_bot_set_a ) ).
% Diff_UNIV
thf(fact_1261_Diff__UNIV,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ A3 @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_1262_Diff__disjoint,axiom,
! [A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ A3 @ ( minus_minus_set_a @ B3 @ A3 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_1263_card__le__sym__Diff,axiom,
! [A3: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B3 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B3 @ A3 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1264_DiffE,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) )
=> ~ ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B3 ) ) ) ).
% DiffE
thf(fact_1265_DiffE,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
=> ~ ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_1266_DiffD1,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) )
=> ( member_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_1267_DiffD1,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
=> ( member_nat @ C @ A3 ) ) ).
% DiffD1
thf(fact_1268_DiffD2,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) )
=> ~ ( member_a @ C @ B3 ) ) ).
% DiffD2
thf(fact_1269_DiffD2,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
=> ~ ( member_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_1270_psubset__imp__ex__mem,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ? [B5: a] : ( member_a @ B5 @ ( minus_minus_set_a @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1271_psubset__imp__ex__mem,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
=> ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1272_Int__Diff__disjoint,axiom,
! [A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ ( minus_minus_set_a @ A3 @ B3 ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_1273_Diff__triv,axiom,
! [A3: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A3 @ B3 )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A3 @ B3 )
= A3 ) ) ).
% Diff_triv
% Helper facts (3)
thf(help_If_3_1_If_001tf__a_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
% Conjectures (3)
thf(conj_0,hypothesis,
member_a @ a2 @ tfin ).
thf(conj_1,hypothesis,
less @ zero @ c ).
thf(conj_2,conjecture,
less @ a2 @ ( plus @ a2 @ c ) ).
%------------------------------------------------------------------------------