TPTP Problem File: SLH0974^1.p
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%------------------------------------------------------------------------------
% 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 : Safe_Range_RC/0017_Preliminaries/prob_00038_001195__16349070_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1654 ( 757 unt; 360 typ; 0 def)
% Number of atoms : 3513 (2031 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 10783 ( 644 ~; 54 |; 317 &;8475 @)
% ( 0 <=>;1293 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 5 avg)
% Number of types : 30 ( 29 usr)
% Number of type conns : 885 ( 885 >; 0 *; 0 +; 0 <<)
% Number of symbols : 334 ( 331 usr; 37 con; 0-4 aty)
% Number of variables : 3217 ( 253 ^;2869 !; 95 ?;3217 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:21:02.288
%------------------------------------------------------------------------------
% Could-be-implicit typings (29)
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
set_list_set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Sum_sum_nat_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_list_set_nat: $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__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
set_list_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
list_set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
set_set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
set_list_set_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__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_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_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__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
list_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
filter_nat: $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 (331)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__List__Olist_It__Nat__Onat_J,type,
bNF_Gr9051742241863529473st_nat: set_list_list_nat > list_nat > set_list_list_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__List__Olist_Itf__a_J,type,
bNF_Gr7042794125918077091list_a: set_list_list_a > list_a > set_list_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Nat__Onat,type,
bNF_Gr1872714664788909425ft_nat: set_list_nat > nat > set_list_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bNF_Gr5580213925211368579list_a: set_list_set_list_a > set_list_a > set_list_set_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Set__Oset_It__Nat__Onat_J,type,
bNF_Gr2891354507007493415et_nat: set_list_set_nat > set_nat > set_list_set_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Set__Oset_Itf__a_J,type,
bNF_Gr641101480264723709_set_a: set_list_set_a > set_a > set_list_set_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__a,type,
bNF_Greatest_Shift_a: set_list_a > a > set_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__List__Olist_It__Nat__Onat_J,type,
bNF_Gr3053708287304744325st_nat: set_list_list_nat > list_list_nat > set_list_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__List__Olist_Itf__a_J,type,
bNF_Gr4634511371912843295list_a: set_list_list_a > list_list_a > set_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bNF_Gr8877554613853555711list_a: set_list_set_list_a > list_set_list_a > set_set_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Set__Oset_It__Nat__Onat_J,type,
bNF_Gr3282828795834814635et_nat: set_list_set_nat > list_set_nat > set_set_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Set__Oset_Itf__a_J,type,
bNF_Gr5263945959978596985_set_a: set_list_set_a > list_set_a > set_set_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).
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_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
comple8404747032580312297st_nat: set_set_list_nat > set_list_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
comple6928918032620976721list_a: set_set_list_a > set_list_a ).
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_Filter_Ocofinite_001t__Nat__Onat,type,
cofinite_nat: filter_nat ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $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__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6187706683773761046at_nat: set_Sum_sum_nat_nat > $o ).
thf(sy_c_Fun_Ofun__upd_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
fun_upd_list_a_nat: ( list_a > nat ) > list_a > nat > list_a > nat ).
thf(sy_c_Fun_Ofun__upd_001t__List__Olist_Itf__a_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
fun_up8218348934346411485list_a: ( list_a > set_list_a ) > list_a > set_list_a > list_a > set_list_a ).
thf(sy_c_Fun_Ofun__upd_001t__List__Olist_Itf__a_J_001t__Set__Oset_It__Nat__Onat_J,type,
fun_up3731164839266413645et_nat: ( list_a > set_nat ) > list_a > set_nat > list_a > set_nat ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Nat__Onat,type,
fun_upd_nat_nat: ( nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
fun_up1966476811263071829list_a: ( nat > set_list_a ) > nat > set_list_a > nat > set_list_a ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
fun_upd_nat_set_nat: ( nat > set_nat ) > nat > set_nat > nat > set_nat ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on3049792774292151987st_nat: ( list_nat > list_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
inj_on_list_a_list_a: ( list_a > list_a ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
inj_on_list_a_nat: ( list_a > nat ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
inj_on1264545500884751569list_a: ( list_a > set_list_a ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
inj_on_nat_list_a: ( nat > list_a ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on_nat_set_nat: ( nat > set_nat ) > set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_Eo,type,
minus_minus_o: $o > $o > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
minus_5335179877275218001list_a: set_list_list_a > set_list_list_a > set_list_list_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
minus_646659088055828811list_a: set_list_a > set_list_a > set_list_a ).
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_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
minus_4782336368215558443list_a: set_set_list_a > set_set_list_a > set_set_list_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
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_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
uminus7925729386456332763list_a: set_list_a > set_list_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_HOL_Oundefined_001t__Nat__Onat,type,
undefined_nat: nat ).
thf(sy_c_HOL_Oundefined_001tf__a,type,
undefined_a: a ).
thf(sy_c_If_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
if_list_list_nat: $o > list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_If_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
if_list_list_a: $o > list_list_a > list_list_a > list_list_a ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_If_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
if_list_set_list_a: $o > list_set_list_a > list_set_list_a > list_set_list_a ).
thf(sy_c_If_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
if_list_set_nat: $o > list_set_nat > list_set_nat > list_set_nat ).
thf(sy_c_If_001t__List__Olist_Itf__a_J,type,
if_list_a: $o > list_a > list_a > list_a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
if_set_list_a: $o > set_list_a > set_list_a > set_list_a ).
thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
if_set_nat: $o > set_nat > set_nat > set_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_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
inf_inf_set_list_a: set_list_a > set_list_a > set_list_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_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $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__List__Olist_It__Nat__Onat_J_J,type,
sup_sup_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
sup_sup_set_list_a: set_list_a > set_list_a > set_list_a ).
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_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_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_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
lattic5238388535129920115in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
lattic1093996805478795353in_nat: set_nat > nat ).
thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
can_select_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__Nat__Onat_J,type,
concat_list_nat: list_list_list_nat > list_list_nat ).
thf(sy_c_List_Oconcat_001t__List__Olist_Itf__a_J,type,
concat_list_a: list_list_list_a > list_list_a ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Ocoset_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
coset_list_list_a: list_list_list_a > set_list_list_a ).
thf(sy_c_List_Ocoset_001t__List__Olist_It__Nat__Onat_J,type,
coset_list_nat: list_list_nat > set_list_nat ).
thf(sy_c_List_Ocoset_001t__List__Olist_Itf__a_J,type,
coset_list_a: list_list_a > set_list_a ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Ocoset_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
coset_set_list_a: list_set_list_a > set_set_list_a ).
thf(sy_c_List_Ocoset_001t__Set__Oset_It__Nat__Onat_J,type,
coset_set_nat: list_set_nat > set_set_nat ).
thf(sy_c_List_Ocoset_001tf__a,type,
coset_a: list_a > set_a ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
distinct_list_list_a: list_list_list_a > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
distinct_list_nat: list_list_nat > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_Itf__a_J,type,
distinct_list_a: list_list_a > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Set__Oset_It__Nat__Onat_J,type,
distinct_set_nat: list_set_nat > $o ).
thf(sy_c_List_Odistinct_001tf__a,type,
distinct_a: list_a > $o ).
thf(sy_c_List_Odistinct__adj_001t__Nat__Onat,type,
distinct_adj_nat: list_nat > $o ).
thf(sy_c_List_Odistinct__adj_001tf__a,type,
distinct_adj_a: list_a > $o ).
thf(sy_c_List_Ofold_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
fold_l59423398878476163st_nat: ( list_nat > set_list_nat > set_list_nat ) > list_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_List_Ofold_001t__List__Olist_Itf__a_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
fold_l7894317391402898027list_a: ( list_a > set_list_a > set_list_a ) > list_list_a > set_list_a > set_list_a ).
thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
fold_nat_set_nat: ( nat > set_nat > set_nat ) > list_nat > set_nat > set_nat ).
thf(sy_c_List_Ofold_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
fold_s5931075695703335563list_a: ( set_list_a > set_list_a > set_list_a ) > list_set_list_a > set_list_a > set_list_a ).
thf(sy_c_List_Ofold_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
fold_s6172198821268269675list_a: ( set_list_a > set_set_list_a > set_set_list_a ) > list_set_list_a > set_set_list_a > set_set_list_a ).
thf(sy_c_List_Ofold_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
fold_set_nat_set_nat: ( set_nat > set_nat > set_nat ) > list_set_nat > set_nat > set_nat ).
thf(sy_c_List_Ofold_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
fold_s4794219702148550607et_nat: ( set_nat > set_set_nat > set_set_nat ) > list_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_List_Ofold_001tf__a_001t__Set__Oset_Itf__a_J,type,
fold_a_set_a: ( a > set_a > set_a ) > list_a > set_a > set_a ).
thf(sy_c_List_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
insert_set_list_a: set_list_a > list_set_list_a > list_set_list_a ).
thf(sy_c_List_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > list_set_a > list_set_a ).
thf(sy_c_List_Oinsert_001tf__a,type,
insert_a: a > list_a > list_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_It__Nat__Onat_J_J,type,
cons_list_list_nat: list_list_nat > list_list_list_nat > list_list_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_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
cons_set_list_a: set_list_a > list_set_list_a > list_set_list_a ).
thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__Nat__Onat_J,type,
cons_set_nat: set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Olist_OCons_001t__Set__Oset_Itf__a_J,type,
cons_set_a: set_a > list_set_a > list_set_a ).
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_It__Nat__Onat_J_J,type,
nil_list_list_nat: list_list_list_nat ).
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_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
nil_set_list_a: list_set_list_a ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__Nat__Onat_J,type,
nil_set_nat: list_set_nat ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_Itf__a_J,type,
nil_set_a: list_set_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_Itf__a_J_001tf__a,type,
case_list_list_a_a: list_a > ( a > list_a > list_a ) > list_a > list_a ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_li7225945977422193158st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
map_list_nat_set_nat: ( list_nat > set_nat ) > list_list_nat > list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
map_list_a_list_a: ( list_a > list_a ) > list_list_a > list_list_a ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_na6205611841492582150st_nat: ( nat > list_list_nat ) > list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001tf__a,type,
map_nat_a: ( nat > a ) > list_nat > list_a ).
thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
map_a_list_list_a: ( a > list_list_a ) > list_a > list_list_list_a ).
thf(sy_c_List_Olist_Omap_001tf__a_001t__Nat__Onat,type,
map_a_nat: ( a > nat ) > list_a > list_nat ).
thf(sy_c_List_Olist_Omap_001tf__a_001tf__a,type,
map_a_a: ( a > a ) > list_a > list_a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
set_list_list_a2: list_list_list_a > set_list_list_a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_set_list_a2: list_set_list_a > set_set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
set_set_nat2: list_set_nat > set_set_nat ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_Itf__a_J,type,
set_set_a2: list_set_a > set_set_a ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist_Otl_001t__List__Olist_It__Nat__Onat_J,type,
tl_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Otl_001t__List__Olist_Itf__a_J,type,
tl_list_a: list_list_a > list_list_a ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olists_001t__List__Olist_Itf__a_J,type,
lists_list_a: set_list_a > set_list_list_a ).
thf(sy_c_List_Olists_001t__Nat__Onat,type,
lists_nat: set_nat > set_list_nat ).
thf(sy_c_List_Olists_001tf__a,type,
lists_a: set_a > set_list_a ).
thf(sy_c_List_Olistset_001tf__a,type,
listset_a: list_set_a > set_list_a ).
thf(sy_c_List_Omember_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_list_nat > list_nat > $o ).
thf(sy_c_List_Omember_001t__List__Olist_Itf__a_J,type,
member_list_a: list_list_a > list_a > $o ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_Omember_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: list_set_list_a > set_list_a > $o ).
thf(sy_c_List_Omember_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: list_set_nat > set_nat > $o ).
thf(sy_c_List_Omember_001tf__a,type,
member_a: list_a > a > $o ).
thf(sy_c_List_Omin__list_001t__Nat__Onat,type,
min_list_nat: list_nat > nat ).
thf(sy_c_List_Onull_001t__List__Olist_It__Nat__Onat_J,type,
null_list_nat: list_list_nat > $o ).
thf(sy_c_List_Onull_001t__List__Olist_Itf__a_J,type,
null_list_a: list_list_a > $o ).
thf(sy_c_List_Onull_001t__Nat__Onat,type,
null_nat: list_nat > $o ).
thf(sy_c_List_Onull_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
null_set_list_a: list_set_list_a > $o ).
thf(sy_c_List_Onull_001t__Set__Oset_It__Nat__Onat_J,type,
null_set_nat: list_set_nat > $o ).
thf(sy_c_List_Onull_001t__Set__Oset_Itf__a_J,type,
null_set_a: list_set_a > $o ).
thf(sy_c_List_Onull_001tf__a,type,
null_a: list_a > $o ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Oremove1_001t__List__Olist_Itf__a_J,type,
remove1_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oremove1_001t__Nat__Onat,type,
remove1_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oremove1_001tf__a,type,
remove1_a: a > list_a > list_a ).
thf(sy_c_List_OremoveAll_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
remove8017980289111491990list_a: list_list_a > list_list_list_a > list_list_list_a ).
thf(sy_c_List_OremoveAll_001t__List__Olist_It__Nat__Onat_J,type,
removeAll_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_OremoveAll_001t__List__Olist_Itf__a_J,type,
removeAll_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
removeAll_set_list_a: set_list_a > list_set_list_a > list_set_list_a ).
thf(sy_c_List_OremoveAll_001t__Set__Oset_It__Nat__Onat_J,type,
removeAll_set_nat: set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_OremoveAll_001tf__a,type,
removeAll_a: a > list_a > list_a ).
thf(sy_c_List_Oshuffles_001t__List__Olist_Itf__a_J,type,
shuffles_list_a: list_list_a > list_list_a > set_list_list_a ).
thf(sy_c_List_Oshuffles_001t__Nat__Onat,type,
shuffles_nat: list_nat > list_nat > set_list_nat ).
thf(sy_c_List_Oshuffles_001tf__a,type,
shuffles_a: list_a > list_a > set_list_a ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_list_nat_o: list_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
bot_bot_list_a_o: list_a > $o ).
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_It__Set__Oset_It__List__Olist_Itf__a_J_J_M_Eo_J,type,
bot_bot_set_list_a_o: set_list_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_set_nat_o: set_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_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
bot_bot_filter_nat: filter_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
bot_bo1875519244922727510list_a: set_list_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bot_bot_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bot_bot_set_list_a: set_list_a ).
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_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
bot_bo3186585308812441520list_a: set_set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
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__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $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_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_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_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
top_top_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
top_top_set_list_a: set_list_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__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__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__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_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Preliminaries_Oextend_001t__Nat__Onat,type,
extend_nat: set_nat > list_nat > list_nat > set_list_nat ).
thf(sy_c_Preliminaries_Oextend_001tf__a,type,
extend_a: set_nat > list_nat > list_a > set_list_a ).
thf(sy_c_Preliminaries_Olookup_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
lookup_list_a_nat: list_list_a > list_nat > list_a > nat ).
thf(sy_c_Preliminaries_Olookup_001t__List__Olist_Itf__a_J_001tf__a,type,
lookup_list_a_a: list_list_a > list_a > list_a > a ).
thf(sy_c_Preliminaries_Olookup_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
lookup_nat_list_nat: list_nat > list_list_nat > nat > list_nat ).
thf(sy_c_Preliminaries_Olookup_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
lookup_nat_list_a: list_nat > list_list_a > nat > list_a ).
thf(sy_c_Preliminaries_Olookup_001t__Nat__Onat_001t__Nat__Onat,type,
lookup_nat_nat: list_nat > list_nat > nat > nat ).
thf(sy_c_Preliminaries_Olookup_001t__Nat__Onat_001tf__a,type,
lookup_nat_a: list_nat > list_a > nat > a ).
thf(sy_c_Preliminaries_Olookup_001tf__a_001t__List__Olist_It__Nat__Onat_J,type,
lookup_a_list_nat: list_a > list_list_nat > a > list_nat ).
thf(sy_c_Preliminaries_Olookup_001tf__a_001t__List__Olist_Itf__a_J,type,
lookup_a_list_a: list_a > list_list_a > a > list_a ).
thf(sy_c_Preliminaries_Olookup_001tf__a_001t__Nat__Onat,type,
lookup_a_nat: list_a > list_nat > a > nat ).
thf(sy_c_Preliminaries_Olookup_001tf__a_001tf__a,type,
lookup_a_a: list_a > list_a > a > a ).
thf(sy_c_Preliminaries_Orestrict_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
restri6853610114029608726at_nat: set_list_nat > list_list_nat > list_nat > list_nat ).
thf(sy_c_Preliminaries_Orestrict_001t__List__Olist_It__Nat__Onat_J_001tf__a,type,
restrict_list_nat_a: set_list_nat > list_list_nat > list_a > list_a ).
thf(sy_c_Preliminaries_Orestrict_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
restrict_list_a_nat: set_list_a > list_list_a > list_nat > list_nat ).
thf(sy_c_Preliminaries_Orestrict_001t__List__Olist_Itf__a_J_001tf__a,type,
restrict_list_a_a: set_list_a > list_list_a > list_a > list_a ).
thf(sy_c_Preliminaries_Orestrict_001t__Nat__Onat_001t__Nat__Onat,type,
restrict_nat_nat: set_nat > list_nat > list_nat > list_nat ).
thf(sy_c_Preliminaries_Orestrict_001t__Nat__Onat_001tf__a,type,
restrict_nat_a: set_nat > list_nat > list_a > list_a ).
thf(sy_c_Preliminaries_Orestrict_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
restrict_set_nat_nat: set_set_nat > list_set_nat > list_nat > list_nat ).
thf(sy_c_Preliminaries_Orestrict_001t__Set__Oset_It__Nat__Onat_J_001tf__a,type,
restrict_set_nat_a: set_set_nat > list_set_nat > list_a > list_a ).
thf(sy_c_Preliminaries_Orestrict_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
restrict_set_a_nat: set_set_a > list_set_a > list_nat > list_nat ).
thf(sy_c_Preliminaries_Orestrict_001t__Set__Oset_Itf__a_J_001tf__a,type,
restrict_set_a_a: set_set_a > list_set_a > list_a > list_a ).
thf(sy_c_Preliminaries_Orestrict_001tf__a_001t__Nat__Onat,type,
restrict_a_nat: set_a > list_a > list_nat > list_nat ).
thf(sy_c_Preliminaries_Orestrict_001tf__a_001tf__a,type,
restrict_a_a: set_a > list_a > list_a > list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_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__List__Olist_Itf__a_J_J,type,
collect_set_list_a: ( set_list_a > $o ) > set_set_list_a ).
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_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OPow_001t__List__Olist_Itf__a_J,type,
pow_list_a: set_list_a > set_set_list_a ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001tf__a,type,
pow_a: set_a > set_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_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
image_7976474329151083847st_nat: ( list_nat > list_nat ) > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
image_8532145185254316925st_nat: ( list_nat > set_list_nat ) > set_list_nat > set_set_list_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_1775855109352712557et_nat: ( list_nat > set_nat ) > set_list_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
image_list_a_list_a: ( list_a > list_a ) > set_list_a > set_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
image_list_a_nat: ( list_a > nat ) > set_list_a > set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
image_5464838071766335845list_a: ( list_a > set_list_a ) > set_list_a > set_set_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_list_a_set_nat: ( list_a > set_nat ) > set_list_a > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
image_nat_list_a: ( nat > list_a ) > set_nat > set_list_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_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
image_2883343038133793645st_nat: ( nat > set_list_nat ) > set_nat > set_set_list_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
image_nat_set_list_a: ( nat > set_list_a ) > set_nat > set_set_list_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
image_5749939591322298757list_a: ( set_list_a > set_list_a ) > set_set_list_a > set_set_list_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J,type,
image_set_nat_nat_o: ( set_nat > nat > $o ) > set_set_nat > set_nat_o ).
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__Set__Oset_It__List__Olist_Itf__a_J_J,type,
image_a_set_list_a: ( a > set_list_a ) > set_a > set_set_list_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
insert_list_list_a: list_list_a > set_list_list_a > set_list_list_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat2: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a2: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat2: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
insert_set_list_a2: set_list_a > set_set_list_a > set_set_list_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat2: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a2: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a2: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001t__List__Olist_It__Nat__Onat_J,type,
is_empty_list_nat: set_list_nat > $o ).
thf(sy_c_Set_Ois__empty_001t__List__Olist_Itf__a_J,type,
is_empty_list_a: set_list_a > $o ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__empty_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
is_empty_set_list_a: set_set_list_a > $o ).
thf(sy_c_Set_Ois__empty_001t__Set__Oset_It__Nat__Onat_J,type,
is_empty_set_nat: set_set_nat > $o ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__List__Olist_It__Nat__Onat_J,type,
is_sin2641923865335537900st_nat: set_list_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__List__Olist_Itf__a_J,type,
is_singleton_list_a: set_list_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
is_sin8525870043004244056list_a: set_set_list_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
is_singleton_set_nat: set_set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Opairwise_001t__List__Olist_It__Nat__Onat_J,type,
pairwise_list_nat: ( list_nat > list_nat > $o ) > set_list_nat > $o ).
thf(sy_c_Set_Opairwise_001t__List__Olist_Itf__a_J,type,
pairwise_list_a: ( list_a > list_a > $o ) > set_list_a > $o ).
thf(sy_c_Set_Opairwise_001t__Nat__Onat,type,
pairwise_nat: ( nat > nat > $o ) > set_nat > $o ).
thf(sy_c_Set_Opairwise_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
pairwise_set_list_a: ( set_list_a > set_list_a > $o ) > set_set_list_a > $o ).
thf(sy_c_Set_Opairwise_001t__Set__Oset_It__Nat__Onat_J,type,
pairwise_set_nat: ( set_nat > set_nat > $o ) > set_set_nat > $o ).
thf(sy_c_Set_Opairwise_001tf__a,type,
pairwise_a: ( a > a > $o ) > set_a > $o ).
thf(sy_c_Set_Oremove_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
remove_list_list_a: list_list_a > set_list_list_a > set_list_list_a ).
thf(sy_c_Set_Oremove_001t__List__Olist_It__Nat__Onat_J,type,
remove_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oremove_001t__List__Olist_Itf__a_J,type,
remove_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
remove_set_list_a: set_list_a > set_set_list_a > set_set_list_a ).
thf(sy_c_Set_Oremove_001t__Set__Oset_It__Nat__Onat_J,type,
remove_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_It__Nat__Onat_J,type,
the_elem_list_nat: set_list_nat > list_nat ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_Itf__a_J,type,
the_elem_list_a: set_list_a > list_a ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
the_elem_set_list_a: set_set_list_a > set_list_a ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Nat__Onat_J,type,
the_elem_set_nat: set_set_nat > set_nat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_Itf__a_J,type,
the_elem_set_a: set_set_a > set_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
member_list_list_nat: list_list_nat > set_list_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat2: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member5524387281408368019list_a: list_set_list_a > set_list_set_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
member_list_set_nat: list_set_nat > set_list_set_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
member_list_set_a: list_set_a > set_list_set_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a2: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a2: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat2: 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_a2: a > set_a > $o ).
thf(sy_v_Aa____,type,
aa: set_nat ).
thf(sy_v_a____,type,
a2: nat ).
thf(sy_v_d,type,
d: a ).
thf(sy_v_x,type,
x: nat ).
thf(sy_v_xsa____,type,
xsa: list_nat ).
thf(sy_v_ysa____,type,
ysa: list_a ).
% Relevant facts (1274)
thf(fact_0_Cons_Oprems_I1_J,axiom,
~ ( member_nat2 @ x @ aa ) ).
% Cons.prems(1)
thf(fact_1_Cons_Oprems_I2_J,axiom,
member_nat2 @ x @ ( set_nat2 @ ( cons_nat @ a2 @ xsa ) ) ).
% Cons.prems(2)
thf(fact_2__092_060open_062_092_060lbrakk_062x_A_092_060notin_062_AA_059_Ax_A_092_060in_062_Aset_Axs_092_060rbrakk_062_A_092_060Longrightarrow_062_A_092_060exists_062zs_092_060in_062extend_AA_Axs_Ays_O_Alookup_Axs_Azs_Ax_A_061_Ad_092_060close_062,axiom,
( ~ ( member_nat2 @ x @ aa )
=> ( ( member_nat2 @ x @ ( set_nat2 @ xsa ) )
=> ? [X: list_a] :
( ( member_list_a2 @ X @ ( extend_a @ aa @ xsa @ ysa ) )
& ( ( lookup_nat_a @ xsa @ X @ x )
= d ) ) ) ) ).
% \<open>\<lbrakk>x \<notin> A; x \<in> set xs\<rbrakk> \<Longrightarrow> \<exists>zs\<in>extend A xs ys. lookup xs zs x = d\<close>
thf(fact_3_Cons_Ohyps,axiom,
! [A: set_nat,Ys: list_a] :
( ~ ( member_nat2 @ x @ A )
=> ( ( member_nat2 @ x @ ( set_nat2 @ xsa ) )
=> ? [X: list_a] :
( ( member_list_a2 @ X @ ( extend_a @ A @ xsa @ Ys ) )
& ( ( lookup_nat_a @ xsa @ X @ x )
= d ) ) ) ) ).
% Cons.hyps
thf(fact_4__092_060open_062_092_060lbrakk_062x_A_092_060notin_062_AA_A_N_A_123a_125_059_Ax_A_092_060in_062_Aset_Axs_092_060rbrakk_062_A_092_060Longrightarrow_062_A_092_060exists_062zs_092_060in_062extend_A_IA_A_N_A_123a_125_J_Axs_A_Itl_Ays_J_O_Alookup_Axs_Azs_Ax_A_061_Ad_092_060close_062,axiom,
( ~ ( member_nat2 @ x @ ( minus_minus_set_nat @ aa @ ( insert_nat2 @ a2 @ bot_bot_set_nat ) ) )
=> ( ( member_nat2 @ x @ ( set_nat2 @ xsa ) )
=> ? [X: list_a] :
( ( member_list_a2 @ X @ ( extend_a @ ( minus_minus_set_nat @ aa @ ( insert_nat2 @ a2 @ bot_bot_set_nat ) ) @ xsa @ ( tl_a @ ysa ) ) )
& ( ( lookup_nat_a @ xsa @ X @ x )
= d ) ) ) ) ).
% \<open>\<lbrakk>x \<notin> A - {a}; x \<in> set xs\<rbrakk> \<Longrightarrow> \<exists>zs\<in>extend (A - {a}) xs (tl ys). lookup xs zs x = d\<close>
thf(fact_5_lookup_Osimps_I1_J,axiom,
! [X2: nat,Z: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( X2 = Z )
=> ( ( lookup_nat_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) @ Z )
= Y ) )
& ( ( X2 != Z )
=> ( ( lookup_nat_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) @ Z )
= ( lookup_nat_nat @ Xs @ Ys @ Z ) ) ) ) ).
% lookup.simps(1)
thf(fact_6_lookup_Osimps_I1_J,axiom,
! [X2: nat,Z: nat,Xs: list_nat,Y: a,Ys: list_a] :
( ( ( X2 = Z )
=> ( ( lookup_nat_a @ ( cons_nat @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) @ Z )
= Y ) )
& ( ( X2 != Z )
=> ( ( lookup_nat_a @ ( cons_nat @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) @ Z )
= ( lookup_nat_a @ Xs @ Ys @ Z ) ) ) ) ).
% lookup.simps(1)
thf(fact_7_lookup_Osimps_I1_J,axiom,
! [X2: a,Z: a,Xs: list_a,Y: nat,Ys: list_nat] :
( ( ( X2 = Z )
=> ( ( lookup_a_nat @ ( cons_a @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) @ Z )
= Y ) )
& ( ( X2 != Z )
=> ( ( lookup_a_nat @ ( cons_a @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) @ Z )
= ( lookup_a_nat @ Xs @ Ys @ Z ) ) ) ) ).
% lookup.simps(1)
thf(fact_8_lookup_Osimps_I1_J,axiom,
! [X2: a,Z: a,Xs: list_a,Y: a,Ys: list_a] :
( ( ( X2 = Z )
=> ( ( lookup_a_a @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) @ Z )
= Y ) )
& ( ( X2 != Z )
=> ( ( lookup_a_a @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) @ Z )
= ( lookup_a_a @ Xs @ Ys @ Z ) ) ) ) ).
% lookup.simps(1)
thf(fact_9_lookup_Osimps_I1_J,axiom,
! [X2: nat,Z: nat,Xs: list_nat,Y: list_a,Ys: list_list_a] :
( ( ( X2 = Z )
=> ( ( lookup_nat_list_a @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_a @ Y @ Ys ) @ Z )
= Y ) )
& ( ( X2 != Z )
=> ( ( lookup_nat_list_a @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_a @ Y @ Ys ) @ Z )
= ( lookup_nat_list_a @ Xs @ Ys @ Z ) ) ) ) ).
% lookup.simps(1)
thf(fact_10_lookup_Osimps_I1_J,axiom,
! [X2: nat,Z: nat,Xs: list_nat,Y: list_nat,Ys: list_list_nat] :
( ( ( X2 = Z )
=> ( ( lookup_nat_list_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y @ Ys ) @ Z )
= Y ) )
& ( ( X2 != Z )
=> ( ( lookup_nat_list_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y @ Ys ) @ Z )
= ( lookup_nat_list_nat @ Xs @ Ys @ Z ) ) ) ) ).
% lookup.simps(1)
thf(fact_11_lookup_Osimps_I1_J,axiom,
! [X2: a,Z: a,Xs: list_a,Y: list_a,Ys: list_list_a] :
( ( ( X2 = Z )
=> ( ( lookup_a_list_a @ ( cons_a @ X2 @ Xs ) @ ( cons_list_a @ Y @ Ys ) @ Z )
= Y ) )
& ( ( X2 != Z )
=> ( ( lookup_a_list_a @ ( cons_a @ X2 @ Xs ) @ ( cons_list_a @ Y @ Ys ) @ Z )
= ( lookup_a_list_a @ Xs @ Ys @ Z ) ) ) ) ).
% lookup.simps(1)
thf(fact_12_lookup_Osimps_I1_J,axiom,
! [X2: a,Z: a,Xs: list_a,Y: list_nat,Ys: list_list_nat] :
( ( ( X2 = Z )
=> ( ( lookup_a_list_nat @ ( cons_a @ X2 @ Xs ) @ ( cons_list_nat @ Y @ Ys ) @ Z )
= Y ) )
& ( ( X2 != Z )
=> ( ( lookup_a_list_nat @ ( cons_a @ X2 @ Xs ) @ ( cons_list_nat @ Y @ Ys ) @ Z )
= ( lookup_a_list_nat @ Xs @ Ys @ Z ) ) ) ) ).
% lookup.simps(1)
thf(fact_13_lookup_Osimps_I1_J,axiom,
! [X2: list_a,Z: list_a,Xs: list_list_a,Y: nat,Ys: list_nat] :
( ( ( X2 = Z )
=> ( ( lookup_list_a_nat @ ( cons_list_a @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) @ Z )
= Y ) )
& ( ( X2 != Z )
=> ( ( lookup_list_a_nat @ ( cons_list_a @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) @ Z )
= ( lookup_list_a_nat @ Xs @ Ys @ Z ) ) ) ) ).
% lookup.simps(1)
thf(fact_14_lookup_Osimps_I1_J,axiom,
! [X2: list_a,Z: list_a,Xs: list_list_a,Y: a,Ys: list_a] :
( ( ( X2 = Z )
=> ( ( lookup_list_a_a @ ( cons_list_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) @ Z )
= Y ) )
& ( ( X2 != Z )
=> ( ( lookup_list_a_a @ ( cons_list_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) @ Z )
= ( lookup_list_a_a @ Xs @ Ys @ Z ) ) ) ) ).
% lookup.simps(1)
thf(fact_15_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_16_list_Oinject,axiom,
! [X21: list_nat,X22: list_list_nat,Y21: list_nat,Y22: list_list_nat] :
( ( ( cons_list_nat @ X21 @ X22 )
= ( cons_list_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_17_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_18_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_19_not__Cons__self2,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( cons_list_a @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_20_not__Cons__self2,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( cons_list_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_21_not__Cons__self2,axiom,
! [X2: a,Xs: list_a] :
( ( cons_a @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_22_not__Cons__self2,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_23_extend__nonempty,axiom,
! [A: set_nat,Xs: list_nat,Ys: list_nat] :
( ( extend_nat @ A @ Xs @ Ys )
!= bot_bot_set_list_nat ) ).
% extend_nonempty
thf(fact_24_extend__nonempty,axiom,
! [A: set_nat,Xs: list_nat,Ys: list_a] :
( ( extend_a @ A @ Xs @ Ys )
!= bot_bot_set_list_a ) ).
% extend_nonempty
thf(fact_25_list_Osel_I3_J,axiom,
! [X21: list_a,X22: list_list_a] :
( ( tl_list_a @ ( cons_list_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_26_list_Osel_I3_J,axiom,
! [X21: list_nat,X22: list_list_nat] :
( ( tl_list_nat @ ( cons_list_nat @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_27_list_Osel_I3_J,axiom,
! [X21: nat,X22: list_nat] :
( ( tl_nat @ ( cons_nat @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_28_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_29_list_Oset__intros_I2_J,axiom,
! [Y: set_list_a,X22: list_set_list_a,X21: set_list_a] :
( ( member_set_list_a2 @ Y @ ( set_set_list_a2 @ X22 ) )
=> ( member_set_list_a2 @ Y @ ( set_set_list_a2 @ ( cons_set_list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_30_list_Oset__intros_I2_J,axiom,
! [Y: set_nat,X22: list_set_nat,X21: set_nat] :
( ( member_set_nat2 @ Y @ ( set_set_nat2 @ X22 ) )
=> ( member_set_nat2 @ Y @ ( set_set_nat2 @ ( cons_set_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_31_list_Oset__intros_I2_J,axiom,
! [Y: list_a,X22: list_list_a,X21: list_a] :
( ( member_list_a2 @ Y @ ( set_list_a2 @ X22 ) )
=> ( member_list_a2 @ Y @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_32_list_Oset__intros_I2_J,axiom,
! [Y: list_nat,X22: list_list_nat,X21: list_nat] :
( ( member_list_nat2 @ Y @ ( set_list_nat2 @ X22 ) )
=> ( member_list_nat2 @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_33_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_34_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a2 @ Y @ ( set_a2 @ X22 ) )
=> ( member_a2 @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_35_list_Oset__intros_I1_J,axiom,
! [X21: set_list_a,X22: list_set_list_a] : ( member_set_list_a2 @ X21 @ ( set_set_list_a2 @ ( cons_set_list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_36_list_Oset__intros_I1_J,axiom,
! [X21: set_nat,X22: list_set_nat] : ( member_set_nat2 @ X21 @ ( set_set_nat2 @ ( cons_set_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_37_list_Oset__intros_I1_J,axiom,
! [X21: list_a,X22: list_list_a] : ( member_list_a2 @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_38_list_Oset__intros_I1_J,axiom,
! [X21: list_nat,X22: list_list_nat] : ( member_list_nat2 @ X21 @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_39_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat2 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_40_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a2 @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_41_list_Oset__cases,axiom,
! [E: set_list_a,A2: list_set_list_a] :
( ( member_set_list_a2 @ E @ ( set_set_list_a2 @ A2 ) )
=> ( ! [Z2: list_set_list_a] :
( A2
!= ( cons_set_list_a @ E @ Z2 ) )
=> ~ ! [Z1: set_list_a,Z2: list_set_list_a] :
( ( A2
= ( cons_set_list_a @ Z1 @ Z2 ) )
=> ~ ( member_set_list_a2 @ E @ ( set_set_list_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_42_list_Oset__cases,axiom,
! [E: set_nat,A2: list_set_nat] :
( ( member_set_nat2 @ E @ ( set_set_nat2 @ A2 ) )
=> ( ! [Z2: list_set_nat] :
( A2
!= ( cons_set_nat @ E @ Z2 ) )
=> ~ ! [Z1: set_nat,Z2: list_set_nat] :
( ( A2
= ( cons_set_nat @ Z1 @ Z2 ) )
=> ~ ( member_set_nat2 @ E @ ( set_set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_43_list_Oset__cases,axiom,
! [E: list_a,A2: list_list_a] :
( ( member_list_a2 @ E @ ( set_list_a2 @ A2 ) )
=> ( ! [Z2: list_list_a] :
( A2
!= ( cons_list_a @ E @ Z2 ) )
=> ~ ! [Z1: list_a,Z2: list_list_a] :
( ( A2
= ( cons_list_a @ Z1 @ Z2 ) )
=> ~ ( member_list_a2 @ E @ ( set_list_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_44_list_Oset__cases,axiom,
! [E: list_nat,A2: list_list_nat] :
( ( member_list_nat2 @ E @ ( set_list_nat2 @ A2 ) )
=> ( ! [Z2: list_list_nat] :
( A2
!= ( cons_list_nat @ E @ Z2 ) )
=> ~ ! [Z1: list_nat,Z2: list_list_nat] :
( ( A2
= ( cons_list_nat @ Z1 @ Z2 ) )
=> ~ ( member_list_nat2 @ E @ ( set_list_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_45_list_Oset__cases,axiom,
! [E: nat,A2: list_nat] :
( ( member_nat2 @ E @ ( set_nat2 @ A2 ) )
=> ( ! [Z2: list_nat] :
( A2
!= ( cons_nat @ E @ Z2 ) )
=> ~ ! [Z1: nat,Z2: list_nat] :
( ( A2
= ( cons_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat2 @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_46_list_Oset__cases,axiom,
! [E: a,A2: list_a] :
( ( member_a2 @ E @ ( set_a2 @ A2 ) )
=> ( ! [Z2: list_a] :
( A2
!= ( cons_a @ E @ Z2 ) )
=> ~ ! [Z1: a,Z2: list_a] :
( ( A2
= ( cons_a @ Z1 @ Z2 ) )
=> ~ ( member_a2 @ E @ ( set_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_47_set__ConsD,axiom,
! [Y: set_list_a,X2: set_list_a,Xs: list_set_list_a] :
( ( member_set_list_a2 @ Y @ ( set_set_list_a2 @ ( cons_set_list_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_set_list_a2 @ Y @ ( set_set_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_48_set__ConsD,axiom,
! [Y: set_nat,X2: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ Y @ ( set_set_nat2 @ ( cons_set_nat @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_set_nat2 @ Y @ ( set_set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_49_set__ConsD,axiom,
! [Y: list_a,X2: list_a,Xs: list_list_a] :
( ( member_list_a2 @ Y @ ( set_list_a2 @ ( cons_list_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_list_a2 @ Y @ ( set_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_50_set__ConsD,axiom,
! [Y: list_nat,X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_list_nat2 @ Y @ ( set_list_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_51_set__ConsD,axiom,
! [Y: nat,X2: nat,Xs: list_nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_nat2 @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_52_set__ConsD,axiom,
! [Y: a,X2: a,Xs: list_a] :
( ( member_a2 @ Y @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_a2 @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_53_list_Osimps_I15_J,axiom,
! [X21: set_list_a,X22: list_set_list_a] :
( ( set_set_list_a2 @ ( cons_set_list_a @ X21 @ X22 ) )
= ( insert_set_list_a2 @ X21 @ ( set_set_list_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_54_list_Osimps_I15_J,axiom,
! [X21: set_nat,X22: list_set_nat] :
( ( set_set_nat2 @ ( cons_set_nat @ X21 @ X22 ) )
= ( insert_set_nat2 @ X21 @ ( set_set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_55_list_Osimps_I15_J,axiom,
! [X21: list_a,X22: list_list_a] :
( ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) )
= ( insert_list_a2 @ X21 @ ( set_list_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_56_list_Osimps_I15_J,axiom,
! [X21: list_nat,X22: list_list_nat] :
( ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) )
= ( insert_list_nat2 @ X21 @ ( set_list_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_57_list_Osimps_I15_J,axiom,
! [X21: nat,X22: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
= ( insert_nat2 @ X21 @ ( set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_58_list_Osimps_I15_J,axiom,
! [X21: a,X22: list_a] :
( ( set_a2 @ ( cons_a @ X21 @ X22 ) )
= ( insert_a2 @ X21 @ ( set_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_59_member__rec_I1_J,axiom,
! [X2: list_a,Xs: list_list_a,Y: list_a] :
( ( member_list_a @ ( cons_list_a @ X2 @ Xs ) @ Y )
= ( ( X2 = Y )
| ( member_list_a @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_60_member__rec_I1_J,axiom,
! [X2: list_nat,Xs: list_list_nat,Y: list_nat] :
( ( member_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ Y )
= ( ( X2 = Y )
| ( member_list_nat @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_61_member__rec_I1_J,axiom,
! [X2: nat,Xs: list_nat,Y: nat] :
( ( member_nat @ ( cons_nat @ X2 @ Xs ) @ Y )
= ( ( X2 = Y )
| ( member_nat @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_62_member__rec_I1_J,axiom,
! [X2: a,Xs: list_a,Y: a] :
( ( member_a @ ( cons_a @ X2 @ Xs ) @ Y )
= ( ( X2 = Y )
| ( member_a @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_63_ShiftD,axiom,
! [Kl: list_list_a,Kl2: set_list_list_a,K: list_a] :
( ( member_list_list_a @ Kl @ ( bNF_Gr7042794125918077091list_a @ Kl2 @ K ) )
=> ( member_list_list_a @ ( cons_list_a @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_64_ShiftD,axiom,
! [Kl: list_list_nat,Kl2: set_list_list_nat,K: list_nat] :
( ( member_list_list_nat @ Kl @ ( bNF_Gr9051742241863529473st_nat @ Kl2 @ K ) )
=> ( member_list_list_nat @ ( cons_list_nat @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_65_ShiftD,axiom,
! [Kl: list_nat,Kl2: set_list_nat,K: nat] :
( ( member_list_nat2 @ Kl @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) )
=> ( member_list_nat2 @ ( cons_nat @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_66_ShiftD,axiom,
! [Kl: list_a,Kl2: set_list_a,K: a] :
( ( member_list_a2 @ Kl @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) )
=> ( member_list_a2 @ ( cons_a @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_67_in__set__member,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
= ( member_list_nat @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_68_in__set__member,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
= ( member_list_a @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_69_in__set__member,axiom,
! [X2: set_list_a,Xs: list_set_list_a] :
( ( member_set_list_a2 @ X2 @ ( set_set_list_a2 @ Xs ) )
= ( member_set_list_a @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_70_in__set__member,axiom,
! [X2: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
= ( member_set_nat @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_71_in__set__member,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
= ( member_a @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_72_in__set__member,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_73_insert__Diff__single,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( insert_list_nat2 @ A2 @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat2 @ A2 @ bot_bot_set_list_nat ) ) )
= ( insert_list_nat2 @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_74_insert__Diff__single,axiom,
! [A2: set_list_a,A: set_set_list_a] :
( ( insert_set_list_a2 @ A2 @ ( minus_4782336368215558443list_a @ A @ ( insert_set_list_a2 @ A2 @ bot_bo3186585308812441520list_a ) ) )
= ( insert_set_list_a2 @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_75_insert__Diff__single,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( insert_set_nat2 @ A2 @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat2 @ A2 @ bot_bot_set_set_nat ) ) )
= ( insert_set_nat2 @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_76_insert__Diff__single,axiom,
! [A2: a,A: set_a] :
( ( insert_a2 @ A2 @ ( minus_minus_set_a @ A @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) )
= ( insert_a2 @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_77_insert__Diff__single,axiom,
! [A2: list_a,A: set_list_a] :
( ( insert_list_a2 @ A2 @ ( minus_646659088055828811list_a @ A @ ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) ) )
= ( insert_list_a2 @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_78_insert__Diff__single,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat2 @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
= ( insert_nat2 @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_79_Diff__insert0,axiom,
! [X2: list_nat,A: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ X2 @ A )
=> ( ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat2 @ X2 @ B ) )
= ( minus_7954133019191499631st_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_80_Diff__insert0,axiom,
! [X2: set_list_a,A: set_set_list_a,B: set_set_list_a] :
( ~ ( member_set_list_a2 @ X2 @ A )
=> ( ( minus_4782336368215558443list_a @ A @ ( insert_set_list_a2 @ X2 @ B ) )
= ( minus_4782336368215558443list_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_81_Diff__insert0,axiom,
! [X2: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat2 @ X2 @ A )
=> ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat2 @ X2 @ B ) )
= ( minus_2163939370556025621et_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_82_Diff__insert0,axiom,
! [X2: list_a,A: set_list_a,B: set_list_a] :
( ~ ( member_list_a2 @ X2 @ A )
=> ( ( minus_646659088055828811list_a @ A @ ( insert_list_a2 @ X2 @ B ) )
= ( minus_646659088055828811list_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_83_Diff__insert0,axiom,
! [X2: a,A: set_a,B: set_a] :
( ~ ( member_a2 @ X2 @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a2 @ X2 @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_84_Diff__insert0,axiom,
! [X2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat2 @ X2 @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ B ) )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_85_insert__Diff1,axiom,
! [X2: list_nat,B: set_list_nat,A: set_list_nat] :
( ( member_list_nat2 @ X2 @ B )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat2 @ X2 @ A ) @ B )
= ( minus_7954133019191499631st_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_86_insert__Diff1,axiom,
! [X2: set_list_a,B: set_set_list_a,A: set_set_list_a] :
( ( member_set_list_a2 @ X2 @ B )
=> ( ( minus_4782336368215558443list_a @ ( insert_set_list_a2 @ X2 @ A ) @ B )
= ( minus_4782336368215558443list_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_87_insert__Diff1,axiom,
! [X2: set_nat,B: set_set_nat,A: set_set_nat] :
( ( member_set_nat2 @ X2 @ B )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat2 @ X2 @ A ) @ B )
= ( minus_2163939370556025621et_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_88_insert__Diff1,axiom,
! [X2: list_a,B: set_list_a,A: set_list_a] :
( ( member_list_a2 @ X2 @ B )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a2 @ X2 @ A ) @ B )
= ( minus_646659088055828811list_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_89_insert__Diff1,axiom,
! [X2: a,B: set_a,A: set_a] :
( ( member_a2 @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X2 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_90_insert__Diff1,axiom,
! [X2: nat,B: set_nat,A: set_nat] :
( ( member_nat2 @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X2 @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_91_Diff__empty,axiom,
! [A: set_set_list_a] :
( ( minus_4782336368215558443list_a @ A @ bot_bo3186585308812441520list_a )
= A ) ).
% Diff_empty
thf(fact_92_Diff__empty,axiom,
! [A: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ bot_bot_set_set_nat )
= A ) ).
% Diff_empty
thf(fact_93_Diff__empty,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ) ).
% Diff_empty
thf(fact_94_Diff__empty,axiom,
! [A: set_list_a] :
( ( minus_646659088055828811list_a @ A @ bot_bot_set_list_a )
= A ) ).
% Diff_empty
thf(fact_95_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_96_empty__Diff,axiom,
! [A: set_set_list_a] :
( ( minus_4782336368215558443list_a @ bot_bo3186585308812441520list_a @ A )
= bot_bo3186585308812441520list_a ) ).
% empty_Diff
thf(fact_97_empty__Diff,axiom,
! [A: set_set_nat] :
( ( minus_2163939370556025621et_nat @ bot_bot_set_set_nat @ A )
= bot_bot_set_set_nat ) ).
% empty_Diff
thf(fact_98_empty__Diff,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_99_empty__Diff,axiom,
! [A: set_list_a] :
( ( minus_646659088055828811list_a @ bot_bot_set_list_a @ A )
= bot_bot_set_list_a ) ).
% empty_Diff
thf(fact_100_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_101_Diff__cancel,axiom,
! [A: set_set_list_a] :
( ( minus_4782336368215558443list_a @ A @ A )
= bot_bo3186585308812441520list_a ) ).
% Diff_cancel
thf(fact_102_Diff__cancel,axiom,
! [A: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ A )
= bot_bot_set_set_nat ) ).
% Diff_cancel
thf(fact_103_Diff__cancel,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_104_Diff__cancel,axiom,
! [A: set_list_a] :
( ( minus_646659088055828811list_a @ A @ A )
= bot_bot_set_list_a ) ).
% Diff_cancel
thf(fact_105_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_106_singletonI,axiom,
! [A2: list_nat] : ( member_list_nat2 @ A2 @ ( insert_list_nat2 @ A2 @ bot_bot_set_list_nat ) ) ).
% singletonI
thf(fact_107_singletonI,axiom,
! [A2: set_list_a] : ( member_set_list_a2 @ A2 @ ( insert_set_list_a2 @ A2 @ bot_bo3186585308812441520list_a ) ) ).
% singletonI
thf(fact_108_singletonI,axiom,
! [A2: set_nat] : ( member_set_nat2 @ A2 @ ( insert_set_nat2 @ A2 @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_109_singletonI,axiom,
! [A2: a] : ( member_a2 @ A2 @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_110_singletonI,axiom,
! [A2: nat] : ( member_nat2 @ A2 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_111_singletonI,axiom,
! [A2: list_a] : ( member_list_a2 @ A2 @ ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_112_Diff__insert,axiom,
! [A: set_list_nat,A2: list_nat,B: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat2 @ A2 @ B ) )
= ( minus_7954133019191499631st_nat @ ( minus_7954133019191499631st_nat @ A @ B ) @ ( insert_list_nat2 @ A2 @ bot_bot_set_list_nat ) ) ) ).
% Diff_insert
thf(fact_113_Diff__insert,axiom,
! [A: set_set_list_a,A2: set_list_a,B: set_set_list_a] :
( ( minus_4782336368215558443list_a @ A @ ( insert_set_list_a2 @ A2 @ B ) )
= ( minus_4782336368215558443list_a @ ( minus_4782336368215558443list_a @ A @ B ) @ ( insert_set_list_a2 @ A2 @ bot_bo3186585308812441520list_a ) ) ) ).
% Diff_insert
thf(fact_114_Diff__insert,axiom,
! [A: set_set_nat,A2: set_nat,B: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat2 @ A2 @ B ) )
= ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ ( insert_set_nat2 @ A2 @ bot_bot_set_set_nat ) ) ) ).
% Diff_insert
thf(fact_115_Diff__insert,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a2 @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_116_Diff__insert,axiom,
! [A: set_list_a,A2: list_a,B: set_list_a] :
( ( minus_646659088055828811list_a @ A @ ( insert_list_a2 @ A2 @ B ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A @ B ) @ ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) ) ) ).
% Diff_insert
thf(fact_117_Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_118_insert__Diff,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A2 @ A )
=> ( ( insert_list_nat2 @ A2 @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat2 @ A2 @ bot_bot_set_list_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_119_insert__Diff,axiom,
! [A2: set_list_a,A: set_set_list_a] :
( ( member_set_list_a2 @ A2 @ A )
=> ( ( insert_set_list_a2 @ A2 @ ( minus_4782336368215558443list_a @ A @ ( insert_set_list_a2 @ A2 @ bot_bo3186585308812441520list_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_120_insert__Diff,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat2 @ A2 @ A )
=> ( ( insert_set_nat2 @ A2 @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat2 @ A2 @ bot_bot_set_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_121_insert__Diff,axiom,
! [A2: a,A: set_a] :
( ( member_a2 @ A2 @ A )
=> ( ( insert_a2 @ A2 @ ( minus_minus_set_a @ A @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_122_insert__Diff,axiom,
! [A2: list_a,A: set_list_a] :
( ( member_list_a2 @ A2 @ A )
=> ( ( insert_list_a2 @ A2 @ ( minus_646659088055828811list_a @ A @ ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_123_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat2 @ A2 @ A )
=> ( ( insert_nat2 @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_124_Diff__insert2,axiom,
! [A: set_list_nat,A2: list_nat,B: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat2 @ A2 @ B ) )
= ( minus_7954133019191499631st_nat @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat2 @ A2 @ bot_bot_set_list_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_125_Diff__insert2,axiom,
! [A: set_set_list_a,A2: set_list_a,B: set_set_list_a] :
( ( minus_4782336368215558443list_a @ A @ ( insert_set_list_a2 @ A2 @ B ) )
= ( minus_4782336368215558443list_a @ ( minus_4782336368215558443list_a @ A @ ( insert_set_list_a2 @ A2 @ bot_bo3186585308812441520list_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_126_Diff__insert2,axiom,
! [A: set_set_nat,A2: set_nat,B: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat2 @ A2 @ B ) )
= ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat2 @ A2 @ bot_bot_set_set_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_127_Diff__insert2,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a2 @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_128_Diff__insert2,axiom,
! [A: set_list_a,A2: list_a,B: set_list_a] :
( ( minus_646659088055828811list_a @ A @ ( insert_list_a2 @ A2 @ B ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A @ ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_129_Diff__insert2,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_130_Diff__insert__absorb,axiom,
! [X2: list_nat,A: set_list_nat] :
( ~ ( member_list_nat2 @ X2 @ A )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat2 @ X2 @ A ) @ ( insert_list_nat2 @ X2 @ bot_bot_set_list_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_131_Diff__insert__absorb,axiom,
! [X2: set_list_a,A: set_set_list_a] :
( ~ ( member_set_list_a2 @ X2 @ A )
=> ( ( minus_4782336368215558443list_a @ ( insert_set_list_a2 @ X2 @ A ) @ ( insert_set_list_a2 @ X2 @ bot_bo3186585308812441520list_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_132_Diff__insert__absorb,axiom,
! [X2: set_nat,A: set_set_nat] :
( ~ ( member_set_nat2 @ X2 @ A )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat2 @ X2 @ A ) @ ( insert_set_nat2 @ X2 @ bot_bot_set_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_133_Diff__insert__absorb,axiom,
! [X2: a,A: set_a] :
( ~ ( member_a2 @ X2 @ A )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X2 @ A ) @ ( insert_a2 @ X2 @ bot_bot_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_134_Diff__insert__absorb,axiom,
! [X2: list_a,A: set_list_a] :
( ~ ( member_list_a2 @ X2 @ A )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a2 @ X2 @ A ) @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_135_Diff__insert__absorb,axiom,
! [X2: nat,A: set_nat] :
( ~ ( member_nat2 @ X2 @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X2 @ A ) @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_136_DiffI,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ A )
=> ( ~ ( member_list_nat2 @ C @ B )
=> ( member_list_nat2 @ C @ ( minus_7954133019191499631st_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_137_DiffI,axiom,
! [C: set_list_a,A: set_set_list_a,B: set_set_list_a] :
( ( member_set_list_a2 @ C @ A )
=> ( ~ ( member_set_list_a2 @ C @ B )
=> ( member_set_list_a2 @ C @ ( minus_4782336368215558443list_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_138_DiffI,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat2 @ C @ A )
=> ( ~ ( member_set_nat2 @ C @ B )
=> ( member_set_nat2 @ C @ ( minus_2163939370556025621et_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_139_DiffI,axiom,
! [C: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a2 @ C @ A )
=> ( ~ ( member_list_a2 @ C @ B )
=> ( member_list_a2 @ C @ ( minus_646659088055828811list_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_140_DiffI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a2 @ C @ A )
=> ( ~ ( member_a2 @ C @ B )
=> ( member_a2 @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_141_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ A )
=> ( ~ ( member_nat2 @ C @ B )
=> ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_142_empty__Collect__eq,axiom,
! [P: list_nat > $o] :
( ( bot_bot_set_list_nat
= ( collect_list_nat @ P ) )
= ( ! [X3: list_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_143_empty__Collect__eq,axiom,
! [P: set_list_a > $o] :
( ( bot_bo3186585308812441520list_a
= ( collect_set_list_a @ P ) )
= ( ! [X3: set_list_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_144_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X3: set_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_145_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_146_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_147_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_148_Collect__empty__eq,axiom,
! [P: list_nat > $o] :
( ( ( collect_list_nat @ P )
= bot_bot_set_list_nat )
= ( ! [X3: list_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_149_Collect__empty__eq,axiom,
! [P: set_list_a > $o] :
( ( ( collect_set_list_a @ P )
= bot_bo3186585308812441520list_a )
= ( ! [X3: set_list_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_150_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_151_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_152_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_153_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_154_all__not__in__conv,axiom,
! [A: set_list_nat] :
( ( ! [X3: list_nat] :
~ ( member_list_nat2 @ X3 @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% all_not_in_conv
thf(fact_155_all__not__in__conv,axiom,
! [A: set_set_list_a] :
( ( ! [X3: set_list_a] :
~ ( member_set_list_a2 @ X3 @ A ) )
= ( A = bot_bo3186585308812441520list_a ) ) ).
% all_not_in_conv
thf(fact_156_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat2 @ X3 @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_157_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X3: a] :
~ ( member_a2 @ X3 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_158_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat2 @ X3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_159_all__not__in__conv,axiom,
! [A: set_list_a] :
( ( ! [X3: list_a] :
~ ( member_list_a2 @ X3 @ A ) )
= ( A = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_160_empty__iff,axiom,
! [C: list_nat] :
~ ( member_list_nat2 @ C @ bot_bot_set_list_nat ) ).
% empty_iff
thf(fact_161_empty__iff,axiom,
! [C: set_list_a] :
~ ( member_set_list_a2 @ C @ bot_bo3186585308812441520list_a ) ).
% empty_iff
thf(fact_162_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat2 @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_163_empty__iff,axiom,
! [C: a] :
~ ( member_a2 @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_164_empty__iff,axiom,
! [C: nat] :
~ ( member_nat2 @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_165_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a2 @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_166_insert__absorb2,axiom,
! [X2: list_nat,A: set_list_nat] :
( ( insert_list_nat2 @ X2 @ ( insert_list_nat2 @ X2 @ A ) )
= ( insert_list_nat2 @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_167_insert__absorb2,axiom,
! [X2: list_a,A: set_list_a] :
( ( insert_list_a2 @ X2 @ ( insert_list_a2 @ X2 @ A ) )
= ( insert_list_a2 @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_168_insert__absorb2,axiom,
! [X2: set_list_a,A: set_set_list_a] :
( ( insert_set_list_a2 @ X2 @ ( insert_set_list_a2 @ X2 @ A ) )
= ( insert_set_list_a2 @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_169_insert__absorb2,axiom,
! [X2: set_nat,A: set_set_nat] :
( ( insert_set_nat2 @ X2 @ ( insert_set_nat2 @ X2 @ A ) )
= ( insert_set_nat2 @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_170_insert__absorb2,axiom,
! [X2: a,A: set_a] :
( ( insert_a2 @ X2 @ ( insert_a2 @ X2 @ A ) )
= ( insert_a2 @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_171_insert__absorb2,axiom,
! [X2: nat,A: set_nat] :
( ( insert_nat2 @ X2 @ ( insert_nat2 @ X2 @ A ) )
= ( insert_nat2 @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_172_insert__iff,axiom,
! [A2: list_nat,B2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A2 @ ( insert_list_nat2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_list_nat2 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_173_insert__iff,axiom,
! [A2: list_a,B2: list_a,A: set_list_a] :
( ( member_list_a2 @ A2 @ ( insert_list_a2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_list_a2 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_174_insert__iff,axiom,
! [A2: set_list_a,B2: set_list_a,A: set_set_list_a] :
( ( member_set_list_a2 @ A2 @ ( insert_set_list_a2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_set_list_a2 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_175_insert__iff,axiom,
! [A2: set_nat,B2: set_nat,A: set_set_nat] :
( ( member_set_nat2 @ A2 @ ( insert_set_nat2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_set_nat2 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_176_insert__iff,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a2 @ A2 @ ( insert_a2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_a2 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_177_insert__iff,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat2 @ A2 @ ( insert_nat2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_nat2 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_178_insertCI,axiom,
! [A2: list_nat,B: set_list_nat,B2: list_nat] :
( ( ~ ( member_list_nat2 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_list_nat2 @ A2 @ ( insert_list_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_179_insertCI,axiom,
! [A2: list_a,B: set_list_a,B2: list_a] :
( ( ~ ( member_list_a2 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_list_a2 @ A2 @ ( insert_list_a2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_180_insertCI,axiom,
! [A2: set_list_a,B: set_set_list_a,B2: set_list_a] :
( ( ~ ( member_set_list_a2 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_set_list_a2 @ A2 @ ( insert_set_list_a2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_181_insertCI,axiom,
! [A2: set_nat,B: set_set_nat,B2: set_nat] :
( ( ~ ( member_set_nat2 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_set_nat2 @ A2 @ ( insert_set_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_182_insertCI,axiom,
! [A2: a,B: set_a,B2: a] :
( ( ~ ( member_a2 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_a2 @ A2 @ ( insert_a2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_183_insertCI,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat2 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_nat2 @ A2 @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_184_Diff__idemp,axiom,
! [A: set_list_a,B: set_list_a] :
( ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A @ B ) @ B )
= ( minus_646659088055828811list_a @ A @ B ) ) ).
% Diff_idemp
thf(fact_185_Diff__idemp,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ).
% Diff_idemp
thf(fact_186_Diff__idemp,axiom,
! [A: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ).
% Diff_idemp
thf(fact_187_Diff__iff,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ ( minus_7954133019191499631st_nat @ A @ B ) )
= ( ( member_list_nat2 @ C @ A )
& ~ ( member_list_nat2 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_188_Diff__iff,axiom,
! [C: set_list_a,A: set_set_list_a,B: set_set_list_a] :
( ( member_set_list_a2 @ C @ ( minus_4782336368215558443list_a @ A @ B ) )
= ( ( member_set_list_a2 @ C @ A )
& ~ ( member_set_list_a2 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_189_Diff__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat2 @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
= ( ( member_set_nat2 @ C @ A )
& ~ ( member_set_nat2 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_190_Diff__iff,axiom,
! [C: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a2 @ C @ ( minus_646659088055828811list_a @ A @ B ) )
= ( ( member_list_a2 @ C @ A )
& ~ ( member_list_a2 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_191_Diff__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a2 @ C @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a2 @ C @ A )
& ~ ( member_a2 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_192_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat2 @ C @ A )
& ~ ( member_nat2 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_193_ex__in__conv,axiom,
! [A: set_list_nat] :
( ( ? [X3: list_nat] : ( member_list_nat2 @ X3 @ A ) )
= ( A != bot_bot_set_list_nat ) ) ).
% ex_in_conv
thf(fact_194_ex__in__conv,axiom,
! [A: set_set_list_a] :
( ( ? [X3: set_list_a] : ( member_set_list_a2 @ X3 @ A ) )
= ( A != bot_bo3186585308812441520list_a ) ) ).
% ex_in_conv
thf(fact_195_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat2 @ X3 @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_196_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X3: a] : ( member_a2 @ X3 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_197_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X3: nat] : ( member_nat2 @ X3 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_198_ex__in__conv,axiom,
! [A: set_list_a] :
( ( ? [X3: list_a] : ( member_list_a2 @ X3 @ A ) )
= ( A != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_199_equals0I,axiom,
! [A: set_list_nat] :
( ! [Y2: list_nat] :
~ ( member_list_nat2 @ Y2 @ A )
=> ( A = bot_bot_set_list_nat ) ) ).
% equals0I
thf(fact_200_equals0I,axiom,
! [A: set_set_list_a] :
( ! [Y2: set_list_a] :
~ ( member_set_list_a2 @ Y2 @ A )
=> ( A = bot_bo3186585308812441520list_a ) ) ).
% equals0I
thf(fact_201_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y2: set_nat] :
~ ( member_set_nat2 @ Y2 @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_202_equals0I,axiom,
! [A: set_a] :
( ! [Y2: a] :
~ ( member_a2 @ Y2 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_203_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat2 @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_204_equals0I,axiom,
! [A: set_list_a] :
( ! [Y2: list_a] :
~ ( member_list_a2 @ Y2 @ A )
=> ( A = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_205_equals0D,axiom,
! [A: set_list_nat,A2: list_nat] :
( ( A = bot_bot_set_list_nat )
=> ~ ( member_list_nat2 @ A2 @ A ) ) ).
% equals0D
thf(fact_206_equals0D,axiom,
! [A: set_set_list_a,A2: set_list_a] :
( ( A = bot_bo3186585308812441520list_a )
=> ~ ( member_set_list_a2 @ A2 @ A ) ) ).
% equals0D
thf(fact_207_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat2 @ A2 @ A ) ) ).
% equals0D
thf(fact_208_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a2 @ A2 @ A ) ) ).
% equals0D
thf(fact_209_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat2 @ A2 @ A ) ) ).
% equals0D
thf(fact_210_equals0D,axiom,
! [A: set_list_a,A2: list_a] :
( ( A = bot_bot_set_list_a )
=> ~ ( member_list_a2 @ A2 @ A ) ) ).
% equals0D
thf(fact_211_emptyE,axiom,
! [A2: list_nat] :
~ ( member_list_nat2 @ A2 @ bot_bot_set_list_nat ) ).
% emptyE
thf(fact_212_emptyE,axiom,
! [A2: set_list_a] :
~ ( member_set_list_a2 @ A2 @ bot_bo3186585308812441520list_a ) ).
% emptyE
thf(fact_213_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat2 @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_214_emptyE,axiom,
! [A2: a] :
~ ( member_a2 @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_215_emptyE,axiom,
! [A2: nat] :
~ ( member_nat2 @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_216_emptyE,axiom,
! [A2: list_a] :
~ ( member_list_a2 @ A2 @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_217_mk__disjoint__insert,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A2 @ A )
=> ? [B3: set_list_nat] :
( ( A
= ( insert_list_nat2 @ A2 @ B3 ) )
& ~ ( member_list_nat2 @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_218_mk__disjoint__insert,axiom,
! [A2: list_a,A: set_list_a] :
( ( member_list_a2 @ A2 @ A )
=> ? [B3: set_list_a] :
( ( A
= ( insert_list_a2 @ A2 @ B3 ) )
& ~ ( member_list_a2 @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_219_mk__disjoint__insert,axiom,
! [A2: set_list_a,A: set_set_list_a] :
( ( member_set_list_a2 @ A2 @ A )
=> ? [B3: set_set_list_a] :
( ( A
= ( insert_set_list_a2 @ A2 @ B3 ) )
& ~ ( member_set_list_a2 @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_220_mk__disjoint__insert,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat2 @ A2 @ A )
=> ? [B3: set_set_nat] :
( ( A
= ( insert_set_nat2 @ A2 @ B3 ) )
& ~ ( member_set_nat2 @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_221_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a2 @ A2 @ A )
=> ? [B3: set_a] :
( ( A
= ( insert_a2 @ A2 @ B3 ) )
& ~ ( member_a2 @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_222_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat2 @ A2 @ A )
=> ? [B3: set_nat] :
( ( A
= ( insert_nat2 @ A2 @ B3 ) )
& ~ ( member_nat2 @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_223_mem__Collect__eq,axiom,
! [A2: set_list_a,P: set_list_a > $o] :
( ( member_set_list_a2 @ A2 @ ( collect_set_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_224_mem__Collect__eq,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( member_set_nat2 @ A2 @ ( collect_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_225_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a2 @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_226_mem__Collect__eq,axiom,
! [A2: list_a,P: list_a > $o] :
( ( member_list_a2 @ A2 @ ( collect_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_227_mem__Collect__eq,axiom,
! [A2: list_nat,P: list_nat > $o] :
( ( member_list_nat2 @ A2 @ ( collect_list_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_228_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat2 @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_229_Collect__mem__eq,axiom,
! [A: set_set_list_a] :
( ( collect_set_list_a
@ ^ [X3: set_list_a] : ( member_set_list_a2 @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_230_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat2 @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_231_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a2 @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_232_Collect__mem__eq,axiom,
! [A: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a2 @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_233_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X3: list_nat] : ( member_list_nat2 @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_234_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_235_Collect__cong,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X: list_a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_list_a @ P )
= ( collect_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_236_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_237_Collect__cong,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X: list_nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_list_nat @ P )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_238_insert__commute,axiom,
! [X2: list_nat,Y: list_nat,A: set_list_nat] :
( ( insert_list_nat2 @ X2 @ ( insert_list_nat2 @ Y @ A ) )
= ( insert_list_nat2 @ Y @ ( insert_list_nat2 @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_239_insert__commute,axiom,
! [X2: list_a,Y: list_a,A: set_list_a] :
( ( insert_list_a2 @ X2 @ ( insert_list_a2 @ Y @ A ) )
= ( insert_list_a2 @ Y @ ( insert_list_a2 @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_240_insert__commute,axiom,
! [X2: set_list_a,Y: set_list_a,A: set_set_list_a] :
( ( insert_set_list_a2 @ X2 @ ( insert_set_list_a2 @ Y @ A ) )
= ( insert_set_list_a2 @ Y @ ( insert_set_list_a2 @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_241_insert__commute,axiom,
! [X2: set_nat,Y: set_nat,A: set_set_nat] :
( ( insert_set_nat2 @ X2 @ ( insert_set_nat2 @ Y @ A ) )
= ( insert_set_nat2 @ Y @ ( insert_set_nat2 @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_242_insert__commute,axiom,
! [X2: a,Y: a,A: set_a] :
( ( insert_a2 @ X2 @ ( insert_a2 @ Y @ A ) )
= ( insert_a2 @ Y @ ( insert_a2 @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_243_insert__commute,axiom,
! [X2: nat,Y: nat,A: set_nat] :
( ( insert_nat2 @ X2 @ ( insert_nat2 @ Y @ A ) )
= ( insert_nat2 @ Y @ ( insert_nat2 @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_244_insert__eq__iff,axiom,
! [A2: list_nat,A: set_list_nat,B2: list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ A2 @ A )
=> ( ~ ( member_list_nat2 @ B2 @ B )
=> ( ( ( insert_list_nat2 @ A2 @ A )
= ( insert_list_nat2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C2: set_list_nat] :
( ( A
= ( insert_list_nat2 @ B2 @ C2 ) )
& ~ ( member_list_nat2 @ B2 @ C2 )
& ( B
= ( insert_list_nat2 @ A2 @ C2 ) )
& ~ ( member_list_nat2 @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_245_insert__eq__iff,axiom,
! [A2: list_a,A: set_list_a,B2: list_a,B: set_list_a] :
( ~ ( member_list_a2 @ A2 @ A )
=> ( ~ ( member_list_a2 @ B2 @ B )
=> ( ( ( insert_list_a2 @ A2 @ A )
= ( insert_list_a2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C2: set_list_a] :
( ( A
= ( insert_list_a2 @ B2 @ C2 ) )
& ~ ( member_list_a2 @ B2 @ C2 )
& ( B
= ( insert_list_a2 @ A2 @ C2 ) )
& ~ ( member_list_a2 @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_246_insert__eq__iff,axiom,
! [A2: set_list_a,A: set_set_list_a,B2: set_list_a,B: set_set_list_a] :
( ~ ( member_set_list_a2 @ A2 @ A )
=> ( ~ ( member_set_list_a2 @ B2 @ B )
=> ( ( ( insert_set_list_a2 @ A2 @ A )
= ( insert_set_list_a2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C2: set_set_list_a] :
( ( A
= ( insert_set_list_a2 @ B2 @ C2 ) )
& ~ ( member_set_list_a2 @ B2 @ C2 )
& ( B
= ( insert_set_list_a2 @ A2 @ C2 ) )
& ~ ( member_set_list_a2 @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_247_insert__eq__iff,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_nat,B: set_set_nat] :
( ~ ( member_set_nat2 @ A2 @ A )
=> ( ~ ( member_set_nat2 @ B2 @ B )
=> ( ( ( insert_set_nat2 @ A2 @ A )
= ( insert_set_nat2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C2: set_set_nat] :
( ( A
= ( insert_set_nat2 @ B2 @ C2 ) )
& ~ ( member_set_nat2 @ B2 @ C2 )
& ( B
= ( insert_set_nat2 @ A2 @ C2 ) )
& ~ ( member_set_nat2 @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_248_insert__eq__iff,axiom,
! [A2: a,A: set_a,B2: a,B: set_a] :
( ~ ( member_a2 @ A2 @ A )
=> ( ~ ( member_a2 @ B2 @ B )
=> ( ( ( insert_a2 @ A2 @ A )
= ( insert_a2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C2: set_a] :
( ( A
= ( insert_a2 @ B2 @ C2 ) )
& ~ ( member_a2 @ B2 @ C2 )
& ( B
= ( insert_a2 @ A2 @ C2 ) )
& ~ ( member_a2 @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_249_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat2 @ A2 @ A )
=> ( ~ ( member_nat2 @ B2 @ B )
=> ( ( ( insert_nat2 @ A2 @ A )
= ( insert_nat2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C2: set_nat] :
( ( A
= ( insert_nat2 @ B2 @ C2 ) )
& ~ ( member_nat2 @ B2 @ C2 )
& ( B
= ( insert_nat2 @ A2 @ C2 ) )
& ~ ( member_nat2 @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_250_insert__absorb,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A2 @ A )
=> ( ( insert_list_nat2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_251_insert__absorb,axiom,
! [A2: list_a,A: set_list_a] :
( ( member_list_a2 @ A2 @ A )
=> ( ( insert_list_a2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_252_insert__absorb,axiom,
! [A2: set_list_a,A: set_set_list_a] :
( ( member_set_list_a2 @ A2 @ A )
=> ( ( insert_set_list_a2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_253_insert__absorb,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat2 @ A2 @ A )
=> ( ( insert_set_nat2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_254_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a2 @ A2 @ A )
=> ( ( insert_a2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_255_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat2 @ A2 @ A )
=> ( ( insert_nat2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_256_insert__ident,axiom,
! [X2: list_nat,A: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ X2 @ A )
=> ( ~ ( member_list_nat2 @ X2 @ B )
=> ( ( ( insert_list_nat2 @ X2 @ A )
= ( insert_list_nat2 @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_257_insert__ident,axiom,
! [X2: list_a,A: set_list_a,B: set_list_a] :
( ~ ( member_list_a2 @ X2 @ A )
=> ( ~ ( member_list_a2 @ X2 @ B )
=> ( ( ( insert_list_a2 @ X2 @ A )
= ( insert_list_a2 @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_258_insert__ident,axiom,
! [X2: set_list_a,A: set_set_list_a,B: set_set_list_a] :
( ~ ( member_set_list_a2 @ X2 @ A )
=> ( ~ ( member_set_list_a2 @ X2 @ B )
=> ( ( ( insert_set_list_a2 @ X2 @ A )
= ( insert_set_list_a2 @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_259_insert__ident,axiom,
! [X2: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat2 @ X2 @ A )
=> ( ~ ( member_set_nat2 @ X2 @ B )
=> ( ( ( insert_set_nat2 @ X2 @ A )
= ( insert_set_nat2 @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_260_insert__ident,axiom,
! [X2: a,A: set_a,B: set_a] :
( ~ ( member_a2 @ X2 @ A )
=> ( ~ ( member_a2 @ X2 @ B )
=> ( ( ( insert_a2 @ X2 @ A )
= ( insert_a2 @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_261_insert__ident,axiom,
! [X2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat2 @ X2 @ A )
=> ( ~ ( member_nat2 @ X2 @ B )
=> ( ( ( insert_nat2 @ X2 @ A )
= ( insert_nat2 @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_262_Set_Oset__insert,axiom,
! [X2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ X2 @ A )
=> ~ ! [B3: set_list_nat] :
( ( A
= ( insert_list_nat2 @ X2 @ B3 ) )
=> ( member_list_nat2 @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_263_Set_Oset__insert,axiom,
! [X2: list_a,A: set_list_a] :
( ( member_list_a2 @ X2 @ A )
=> ~ ! [B3: set_list_a] :
( ( A
= ( insert_list_a2 @ X2 @ B3 ) )
=> ( member_list_a2 @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_264_Set_Oset__insert,axiom,
! [X2: set_list_a,A: set_set_list_a] :
( ( member_set_list_a2 @ X2 @ A )
=> ~ ! [B3: set_set_list_a] :
( ( A
= ( insert_set_list_a2 @ X2 @ B3 ) )
=> ( member_set_list_a2 @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_265_Set_Oset__insert,axiom,
! [X2: set_nat,A: set_set_nat] :
( ( member_set_nat2 @ X2 @ A )
=> ~ ! [B3: set_set_nat] :
( ( A
= ( insert_set_nat2 @ X2 @ B3 ) )
=> ( member_set_nat2 @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_266_Set_Oset__insert,axiom,
! [X2: a,A: set_a] :
( ( member_a2 @ X2 @ A )
=> ~ ! [B3: set_a] :
( ( A
= ( insert_a2 @ X2 @ B3 ) )
=> ( member_a2 @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_267_Set_Oset__insert,axiom,
! [X2: nat,A: set_nat] :
( ( member_nat2 @ X2 @ A )
=> ~ ! [B3: set_nat] :
( ( A
= ( insert_nat2 @ X2 @ B3 ) )
=> ( member_nat2 @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_268_insertI2,axiom,
! [A2: list_nat,B: set_list_nat,B2: list_nat] :
( ( member_list_nat2 @ A2 @ B )
=> ( member_list_nat2 @ A2 @ ( insert_list_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_269_insertI2,axiom,
! [A2: list_a,B: set_list_a,B2: list_a] :
( ( member_list_a2 @ A2 @ B )
=> ( member_list_a2 @ A2 @ ( insert_list_a2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_270_insertI2,axiom,
! [A2: set_list_a,B: set_set_list_a,B2: set_list_a] :
( ( member_set_list_a2 @ A2 @ B )
=> ( member_set_list_a2 @ A2 @ ( insert_set_list_a2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_271_insertI2,axiom,
! [A2: set_nat,B: set_set_nat,B2: set_nat] :
( ( member_set_nat2 @ A2 @ B )
=> ( member_set_nat2 @ A2 @ ( insert_set_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_272_insertI2,axiom,
! [A2: a,B: set_a,B2: a] :
( ( member_a2 @ A2 @ B )
=> ( member_a2 @ A2 @ ( insert_a2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_273_insertI2,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( member_nat2 @ A2 @ B )
=> ( member_nat2 @ A2 @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_274_insertI1,axiom,
! [A2: list_nat,B: set_list_nat] : ( member_list_nat2 @ A2 @ ( insert_list_nat2 @ A2 @ B ) ) ).
% insertI1
thf(fact_275_insertI1,axiom,
! [A2: list_a,B: set_list_a] : ( member_list_a2 @ A2 @ ( insert_list_a2 @ A2 @ B ) ) ).
% insertI1
thf(fact_276_insertI1,axiom,
! [A2: set_list_a,B: set_set_list_a] : ( member_set_list_a2 @ A2 @ ( insert_set_list_a2 @ A2 @ B ) ) ).
% insertI1
thf(fact_277_insertI1,axiom,
! [A2: set_nat,B: set_set_nat] : ( member_set_nat2 @ A2 @ ( insert_set_nat2 @ A2 @ B ) ) ).
% insertI1
thf(fact_278_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a2 @ A2 @ ( insert_a2 @ A2 @ B ) ) ).
% insertI1
thf(fact_279_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat2 @ A2 @ ( insert_nat2 @ A2 @ B ) ) ).
% insertI1
thf(fact_280_insertE,axiom,
! [A2: list_nat,B2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A2 @ ( insert_list_nat2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_list_nat2 @ A2 @ A ) ) ) ).
% insertE
thf(fact_281_insertE,axiom,
! [A2: list_a,B2: list_a,A: set_list_a] :
( ( member_list_a2 @ A2 @ ( insert_list_a2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_list_a2 @ A2 @ A ) ) ) ).
% insertE
thf(fact_282_insertE,axiom,
! [A2: set_list_a,B2: set_list_a,A: set_set_list_a] :
( ( member_set_list_a2 @ A2 @ ( insert_set_list_a2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_set_list_a2 @ A2 @ A ) ) ) ).
% insertE
thf(fact_283_insertE,axiom,
! [A2: set_nat,B2: set_nat,A: set_set_nat] :
( ( member_set_nat2 @ A2 @ ( insert_set_nat2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_set_nat2 @ A2 @ A ) ) ) ).
% insertE
thf(fact_284_insertE,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a2 @ A2 @ ( insert_a2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_a2 @ A2 @ A ) ) ) ).
% insertE
thf(fact_285_insertE,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat2 @ A2 @ ( insert_nat2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_nat2 @ A2 @ A ) ) ) ).
% insertE
thf(fact_286_DiffD2,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ ( minus_7954133019191499631st_nat @ A @ B ) )
=> ~ ( member_list_nat2 @ C @ B ) ) ).
% DiffD2
thf(fact_287_DiffD2,axiom,
! [C: set_list_a,A: set_set_list_a,B: set_set_list_a] :
( ( member_set_list_a2 @ C @ ( minus_4782336368215558443list_a @ A @ B ) )
=> ~ ( member_set_list_a2 @ C @ B ) ) ).
% DiffD2
thf(fact_288_DiffD2,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat2 @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ~ ( member_set_nat2 @ C @ B ) ) ).
% DiffD2
thf(fact_289_DiffD2,axiom,
! [C: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a2 @ C @ ( minus_646659088055828811list_a @ A @ B ) )
=> ~ ( member_list_a2 @ C @ B ) ) ).
% DiffD2
thf(fact_290_DiffD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a2 @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a2 @ C @ B ) ) ).
% DiffD2
thf(fact_291_DiffD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat2 @ C @ B ) ) ).
% DiffD2
thf(fact_292_DiffD1,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ ( minus_7954133019191499631st_nat @ A @ B ) )
=> ( member_list_nat2 @ C @ A ) ) ).
% DiffD1
thf(fact_293_DiffD1,axiom,
! [C: set_list_a,A: set_set_list_a,B: set_set_list_a] :
( ( member_set_list_a2 @ C @ ( minus_4782336368215558443list_a @ A @ B ) )
=> ( member_set_list_a2 @ C @ A ) ) ).
% DiffD1
thf(fact_294_DiffD1,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat2 @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ( member_set_nat2 @ C @ A ) ) ).
% DiffD1
thf(fact_295_DiffD1,axiom,
! [C: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a2 @ C @ ( minus_646659088055828811list_a @ A @ B ) )
=> ( member_list_a2 @ C @ A ) ) ).
% DiffD1
thf(fact_296_DiffD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a2 @ C @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a2 @ C @ A ) ) ).
% DiffD1
thf(fact_297_DiffD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat2 @ C @ A ) ) ).
% DiffD1
thf(fact_298_DiffE,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ ( minus_7954133019191499631st_nat @ A @ B ) )
=> ~ ( ( member_list_nat2 @ C @ A )
=> ( member_list_nat2 @ C @ B ) ) ) ).
% DiffE
thf(fact_299_DiffE,axiom,
! [C: set_list_a,A: set_set_list_a,B: set_set_list_a] :
( ( member_set_list_a2 @ C @ ( minus_4782336368215558443list_a @ A @ B ) )
=> ~ ( ( member_set_list_a2 @ C @ A )
=> ( member_set_list_a2 @ C @ B ) ) ) ).
% DiffE
thf(fact_300_DiffE,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat2 @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ~ ( ( member_set_nat2 @ C @ A )
=> ( member_set_nat2 @ C @ B ) ) ) ).
% DiffE
thf(fact_301_DiffE,axiom,
! [C: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a2 @ C @ ( minus_646659088055828811list_a @ A @ B ) )
=> ~ ( ( member_list_a2 @ C @ A )
=> ( member_list_a2 @ C @ B ) ) ) ).
% DiffE
thf(fact_302_DiffE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a2 @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a2 @ C @ A )
=> ( member_a2 @ C @ B ) ) ) ).
% DiffE
thf(fact_303_DiffE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ B ) ) ) ).
% DiffE
thf(fact_304_singleton__inject,axiom,
! [A2: list_nat,B2: list_nat] :
( ( ( insert_list_nat2 @ A2 @ bot_bot_set_list_nat )
= ( insert_list_nat2 @ B2 @ bot_bot_set_list_nat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_305_singleton__inject,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ( insert_set_list_a2 @ A2 @ bot_bo3186585308812441520list_a )
= ( insert_set_list_a2 @ B2 @ bot_bo3186585308812441520list_a ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_306_singleton__inject,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( insert_set_nat2 @ A2 @ bot_bot_set_set_nat )
= ( insert_set_nat2 @ B2 @ bot_bot_set_set_nat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_307_singleton__inject,axiom,
! [A2: a,B2: a] :
( ( ( insert_a2 @ A2 @ bot_bot_set_a )
= ( insert_a2 @ B2 @ bot_bot_set_a ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_308_singleton__inject,axiom,
! [A2: nat,B2: nat] :
( ( ( insert_nat2 @ A2 @ bot_bot_set_nat )
= ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_309_singleton__inject,axiom,
! [A2: list_a,B2: list_a] :
( ( ( insert_list_a2 @ A2 @ bot_bot_set_list_a )
= ( insert_list_a2 @ B2 @ bot_bot_set_list_a ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_310_insert__not__empty,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( insert_list_nat2 @ A2 @ A )
!= bot_bot_set_list_nat ) ).
% insert_not_empty
thf(fact_311_insert__not__empty,axiom,
! [A2: set_list_a,A: set_set_list_a] :
( ( insert_set_list_a2 @ A2 @ A )
!= bot_bo3186585308812441520list_a ) ).
% insert_not_empty
thf(fact_312_insert__not__empty,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( insert_set_nat2 @ A2 @ A )
!= bot_bot_set_set_nat ) ).
% insert_not_empty
thf(fact_313_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a2 @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_314_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat2 @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_315_insert__not__empty,axiom,
! [A2: list_a,A: set_list_a] :
( ( insert_list_a2 @ A2 @ A )
!= bot_bot_set_list_a ) ).
% insert_not_empty
thf(fact_316_doubleton__eq__iff,axiom,
! [A2: list_nat,B2: list_nat,C: list_nat,D: list_nat] :
( ( ( insert_list_nat2 @ A2 @ ( insert_list_nat2 @ B2 @ bot_bot_set_list_nat ) )
= ( insert_list_nat2 @ C @ ( insert_list_nat2 @ D @ bot_bot_set_list_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_317_doubleton__eq__iff,axiom,
! [A2: set_list_a,B2: set_list_a,C: set_list_a,D: set_list_a] :
( ( ( insert_set_list_a2 @ A2 @ ( insert_set_list_a2 @ B2 @ bot_bo3186585308812441520list_a ) )
= ( insert_set_list_a2 @ C @ ( insert_set_list_a2 @ D @ bot_bo3186585308812441520list_a ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_318_doubleton__eq__iff,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
( ( ( insert_set_nat2 @ A2 @ ( insert_set_nat2 @ B2 @ bot_bot_set_set_nat ) )
= ( insert_set_nat2 @ C @ ( insert_set_nat2 @ D @ bot_bot_set_set_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_319_doubleton__eq__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ( insert_a2 @ A2 @ ( insert_a2 @ B2 @ bot_bot_set_a ) )
= ( insert_a2 @ C @ ( insert_a2 @ D @ bot_bot_set_a ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_320_doubleton__eq__iff,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ( insert_nat2 @ A2 @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
= ( insert_nat2 @ C @ ( insert_nat2 @ D @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_321_doubleton__eq__iff,axiom,
! [A2: list_a,B2: list_a,C: list_a,D: list_a] :
( ( ( insert_list_a2 @ A2 @ ( insert_list_a2 @ B2 @ bot_bot_set_list_a ) )
= ( insert_list_a2 @ C @ ( insert_list_a2 @ D @ bot_bot_set_list_a ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_322_singleton__iff,axiom,
! [B2: list_nat,A2: list_nat] :
( ( member_list_nat2 @ B2 @ ( insert_list_nat2 @ A2 @ bot_bot_set_list_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_323_singleton__iff,axiom,
! [B2: set_list_a,A2: set_list_a] :
( ( member_set_list_a2 @ B2 @ ( insert_set_list_a2 @ A2 @ bot_bo3186585308812441520list_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_324_singleton__iff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( member_set_nat2 @ B2 @ ( insert_set_nat2 @ A2 @ bot_bot_set_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_325_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a2 @ B2 @ ( insert_a2 @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_326_singleton__iff,axiom,
! [B2: nat,A2: nat] :
( ( member_nat2 @ B2 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_327_singleton__iff,axiom,
! [B2: list_a,A2: list_a] :
( ( member_list_a2 @ B2 @ ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_328_singletonD,axiom,
! [B2: list_nat,A2: list_nat] :
( ( member_list_nat2 @ B2 @ ( insert_list_nat2 @ A2 @ bot_bot_set_list_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_329_singletonD,axiom,
! [B2: set_list_a,A2: set_list_a] :
( ( member_set_list_a2 @ B2 @ ( insert_set_list_a2 @ A2 @ bot_bo3186585308812441520list_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_330_singletonD,axiom,
! [B2: set_nat,A2: set_nat] :
( ( member_set_nat2 @ B2 @ ( insert_set_nat2 @ A2 @ bot_bot_set_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_331_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a2 @ B2 @ ( insert_a2 @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_332_singletonD,axiom,
! [B2: nat,A2: nat] :
( ( member_nat2 @ B2 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_333_singletonD,axiom,
! [B2: list_a,A2: list_a] :
( ( member_list_a2 @ B2 @ ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_334_insert__Diff__if,axiom,
! [X2: list_nat,B: set_list_nat,A: set_list_nat] :
( ( ( member_list_nat2 @ X2 @ B )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat2 @ X2 @ A ) @ B )
= ( minus_7954133019191499631st_nat @ A @ B ) ) )
& ( ~ ( member_list_nat2 @ X2 @ B )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat2 @ X2 @ A ) @ B )
= ( insert_list_nat2 @ X2 @ ( minus_7954133019191499631st_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_335_insert__Diff__if,axiom,
! [X2: set_list_a,B: set_set_list_a,A: set_set_list_a] :
( ( ( member_set_list_a2 @ X2 @ B )
=> ( ( minus_4782336368215558443list_a @ ( insert_set_list_a2 @ X2 @ A ) @ B )
= ( minus_4782336368215558443list_a @ A @ B ) ) )
& ( ~ ( member_set_list_a2 @ X2 @ B )
=> ( ( minus_4782336368215558443list_a @ ( insert_set_list_a2 @ X2 @ A ) @ B )
= ( insert_set_list_a2 @ X2 @ ( minus_4782336368215558443list_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_336_insert__Diff__if,axiom,
! [X2: set_nat,B: set_set_nat,A: set_set_nat] :
( ( ( member_set_nat2 @ X2 @ B )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat2 @ X2 @ A ) @ B )
= ( minus_2163939370556025621et_nat @ A @ B ) ) )
& ( ~ ( member_set_nat2 @ X2 @ B )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat2 @ X2 @ A ) @ B )
= ( insert_set_nat2 @ X2 @ ( minus_2163939370556025621et_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_337_insert__Diff__if,axiom,
! [X2: list_a,B: set_list_a,A: set_list_a] :
( ( ( member_list_a2 @ X2 @ B )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a2 @ X2 @ A ) @ B )
= ( minus_646659088055828811list_a @ A @ B ) ) )
& ( ~ ( member_list_a2 @ X2 @ B )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a2 @ X2 @ A ) @ B )
= ( insert_list_a2 @ X2 @ ( minus_646659088055828811list_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_338_insert__Diff__if,axiom,
! [X2: a,B: set_a,A: set_a] :
( ( ( member_a2 @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X2 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) )
& ( ~ ( member_a2 @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X2 @ A ) @ B )
= ( insert_a2 @ X2 @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_339_insert__Diff__if,axiom,
! [X2: nat,B: set_nat,A: set_nat] :
( ( ( member_nat2 @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X2 @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) )
& ( ~ ( member_nat2 @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X2 @ A ) @ B )
= ( insert_nat2 @ X2 @ ( minus_minus_set_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_340_the__elem__eq,axiom,
! [X2: list_nat] :
( ( the_elem_list_nat @ ( insert_list_nat2 @ X2 @ bot_bot_set_list_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_341_the__elem__eq,axiom,
! [X2: set_list_a] :
( ( the_elem_set_list_a @ ( insert_set_list_a2 @ X2 @ bot_bo3186585308812441520list_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_342_the__elem__eq,axiom,
! [X2: set_nat] :
( ( the_elem_set_nat @ ( insert_set_nat2 @ X2 @ bot_bot_set_set_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_343_the__elem__eq,axiom,
! [X2: a] :
( ( the_elem_a @ ( insert_a2 @ X2 @ bot_bot_set_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_344_the__elem__eq,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_345_the__elem__eq,axiom,
! [X2: list_a] :
( ( the_elem_list_a @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_346_minus__apply,axiom,
( minus_minus_nat_o
= ( ^ [A3: nat > $o,B4: nat > $o,X3: nat] : ( minus_minus_o @ ( A3 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).
% minus_apply
thf(fact_347_bot__apply,axiom,
( bot_bot_list_a_o
= ( ^ [X3: list_a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_348_bot__apply,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : bot_bot_o ) ) ).
% bot_apply
thf(fact_349_restrict_Osimps_I1_J,axiom,
! [X2: a,A: set_a,Xs: list_a,Y: nat,Ys: list_nat] :
( ( ( member_a2 @ X2 @ A )
=> ( ( restrict_a_nat @ A @ ( cons_a @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_nat @ Y @ ( restrict_a_nat @ ( minus_minus_set_a @ A @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member_a2 @ X2 @ A )
=> ( ( restrict_a_nat @ A @ ( cons_a @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( restrict_a_nat @ A @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_350_restrict_Osimps_I1_J,axiom,
! [X2: a,A: set_a,Xs: list_a,Y: a,Ys: list_a] :
( ( ( member_a2 @ X2 @ A )
=> ( ( restrict_a_a @ A @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( cons_a @ Y @ ( restrict_a_a @ ( minus_minus_set_a @ A @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member_a2 @ X2 @ A )
=> ( ( restrict_a_a @ A @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( restrict_a_a @ A @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_351_restrict_Osimps_I1_J,axiom,
! [X2: nat,A: set_nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( member_nat2 @ X2 @ A )
=> ( ( restrict_nat_nat @ A @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_nat @ Y @ ( restrict_nat_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member_nat2 @ X2 @ A )
=> ( ( restrict_nat_nat @ A @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( restrict_nat_nat @ A @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_352_restrict_Osimps_I1_J,axiom,
! [X2: nat,A: set_nat,Xs: list_nat,Y: a,Ys: list_a] :
( ( ( member_nat2 @ X2 @ A )
=> ( ( restrict_nat_a @ A @ ( cons_nat @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( cons_a @ Y @ ( restrict_nat_a @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member_nat2 @ X2 @ A )
=> ( ( restrict_nat_a @ A @ ( cons_nat @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( restrict_nat_a @ A @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_353_restrict_Osimps_I1_J,axiom,
! [X2: list_a,A: set_list_a,Xs: list_list_a,Y: nat,Ys: list_nat] :
( ( ( member_list_a2 @ X2 @ A )
=> ( ( restrict_list_a_nat @ A @ ( cons_list_a @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_nat @ Y @ ( restrict_list_a_nat @ ( minus_646659088055828811list_a @ A @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member_list_a2 @ X2 @ A )
=> ( ( restrict_list_a_nat @ A @ ( cons_list_a @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( restrict_list_a_nat @ A @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_354_restrict_Osimps_I1_J,axiom,
! [X2: list_a,A: set_list_a,Xs: list_list_a,Y: a,Ys: list_a] :
( ( ( member_list_a2 @ X2 @ A )
=> ( ( restrict_list_a_a @ A @ ( cons_list_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( cons_a @ Y @ ( restrict_list_a_a @ ( minus_646659088055828811list_a @ A @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member_list_a2 @ X2 @ A )
=> ( ( restrict_list_a_a @ A @ ( cons_list_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( restrict_list_a_a @ A @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_355_restrict_Osimps_I1_J,axiom,
! [X2: list_nat,A: set_list_nat,Xs: list_list_nat,Y: nat,Ys: list_nat] :
( ( ( member_list_nat2 @ X2 @ A )
=> ( ( restri6853610114029608726at_nat @ A @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_nat @ Y @ ( restri6853610114029608726at_nat @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat2 @ X2 @ bot_bot_set_list_nat ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member_list_nat2 @ X2 @ A )
=> ( ( restri6853610114029608726at_nat @ A @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( restri6853610114029608726at_nat @ A @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_356_restrict_Osimps_I1_J,axiom,
! [X2: list_nat,A: set_list_nat,Xs: list_list_nat,Y: a,Ys: list_a] :
( ( ( member_list_nat2 @ X2 @ A )
=> ( ( restrict_list_nat_a @ A @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( cons_a @ Y @ ( restrict_list_nat_a @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat2 @ X2 @ bot_bot_set_list_nat ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member_list_nat2 @ X2 @ A )
=> ( ( restrict_list_nat_a @ A @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( restrict_list_nat_a @ A @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_357_restrict_Osimps_I1_J,axiom,
! [X2: set_nat,A: set_set_nat,Xs: list_set_nat,Y: nat,Ys: list_nat] :
( ( ( member_set_nat2 @ X2 @ A )
=> ( ( restrict_set_nat_nat @ A @ ( cons_set_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_nat @ Y @ ( restrict_set_nat_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat2 @ X2 @ bot_bot_set_set_nat ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member_set_nat2 @ X2 @ A )
=> ( ( restrict_set_nat_nat @ A @ ( cons_set_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( restrict_set_nat_nat @ A @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_358_restrict_Osimps_I1_J,axiom,
! [X2: set_nat,A: set_set_nat,Xs: list_set_nat,Y: a,Ys: list_a] :
( ( ( member_set_nat2 @ X2 @ A )
=> ( ( restrict_set_nat_a @ A @ ( cons_set_nat @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( cons_a @ Y @ ( restrict_set_nat_a @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat2 @ X2 @ bot_bot_set_set_nat ) ) @ Xs @ Ys ) ) ) )
& ( ~ ( member_set_nat2 @ X2 @ A )
=> ( ( restrict_set_nat_a @ A @ ( cons_set_nat @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( restrict_set_nat_a @ A @ Xs @ Ys ) ) ) ) ).
% restrict.simps(1)
thf(fact_359_is__singletonI,axiom,
! [X2: list_nat] : ( is_sin2641923865335537900st_nat @ ( insert_list_nat2 @ X2 @ bot_bot_set_list_nat ) ) ).
% is_singletonI
thf(fact_360_is__singletonI,axiom,
! [X2: set_list_a] : ( is_sin8525870043004244056list_a @ ( insert_set_list_a2 @ X2 @ bot_bo3186585308812441520list_a ) ) ).
% is_singletonI
thf(fact_361_is__singletonI,axiom,
! [X2: set_nat] : ( is_singleton_set_nat @ ( insert_set_nat2 @ X2 @ bot_bot_set_set_nat ) ) ).
% is_singletonI
thf(fact_362_is__singletonI,axiom,
! [X2: a] : ( is_singleton_a @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_363_is__singletonI,axiom,
! [X2: nat] : ( is_singleton_nat @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_364_is__singletonI,axiom,
! [X2: list_a] : ( is_singleton_list_a @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) ).
% is_singletonI
thf(fact_365_set__removeAll,axiom,
! [X2: list_list_a,Xs: list_list_list_a] :
( ( set_list_list_a2 @ ( remove8017980289111491990list_a @ X2 @ Xs ) )
= ( minus_5335179877275218001list_a @ ( set_list_list_a2 @ Xs ) @ ( insert_list_list_a @ X2 @ bot_bo1875519244922727510list_a ) ) ) ).
% set_removeAll
thf(fact_366_set__removeAll,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( set_list_nat2 @ ( removeAll_list_nat @ X2 @ Xs ) )
= ( minus_7954133019191499631st_nat @ ( set_list_nat2 @ Xs ) @ ( insert_list_nat2 @ X2 @ bot_bot_set_list_nat ) ) ) ).
% set_removeAll
thf(fact_367_set__removeAll,axiom,
! [X2: set_list_a,Xs: list_set_list_a] :
( ( set_set_list_a2 @ ( removeAll_set_list_a @ X2 @ Xs ) )
= ( minus_4782336368215558443list_a @ ( set_set_list_a2 @ Xs ) @ ( insert_set_list_a2 @ X2 @ bot_bo3186585308812441520list_a ) ) ) ).
% set_removeAll
thf(fact_368_set__removeAll,axiom,
! [X2: set_nat,Xs: list_set_nat] :
( ( set_set_nat2 @ ( removeAll_set_nat @ X2 @ Xs ) )
= ( minus_2163939370556025621et_nat @ ( set_set_nat2 @ Xs ) @ ( insert_set_nat2 @ X2 @ bot_bot_set_set_nat ) ) ) ).
% set_removeAll
thf(fact_369_set__removeAll,axiom,
! [X2: a,Xs: list_a] :
( ( set_a2 @ ( removeAll_a @ X2 @ Xs ) )
= ( minus_minus_set_a @ ( set_a2 @ Xs ) @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) ) ).
% set_removeAll
thf(fact_370_set__removeAll,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( set_list_a2 @ ( removeAll_list_a @ X2 @ Xs ) )
= ( minus_646659088055828811list_a @ ( set_list_a2 @ Xs ) @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) ) ).
% set_removeAll
thf(fact_371_set__removeAll,axiom,
! [X2: nat,Xs: list_nat] :
( ( set_nat2 @ ( removeAll_nat @ X2 @ Xs ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ).
% set_removeAll
thf(fact_372_Succ__Shift,axiom,
! [Kl2: set_list_list_a,K: list_a,Kl: list_list_a] :
( ( bNF_Gr4634511371912843295list_a @ ( bNF_Gr7042794125918077091list_a @ Kl2 @ K ) @ Kl )
= ( bNF_Gr4634511371912843295list_a @ Kl2 @ ( cons_list_a @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_373_Succ__Shift,axiom,
! [Kl2: set_list_list_nat,K: list_nat,Kl: list_list_nat] :
( ( bNF_Gr3053708287304744325st_nat @ ( bNF_Gr9051742241863529473st_nat @ Kl2 @ K ) @ Kl )
= ( bNF_Gr3053708287304744325st_nat @ Kl2 @ ( cons_list_nat @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_374_Succ__Shift,axiom,
! [Kl2: set_list_nat,K: nat,Kl: list_nat] :
( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) @ Kl )
= ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ ( cons_nat @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_375_Succ__Shift,axiom,
! [Kl2: set_list_a,K: a,Kl: list_a] :
( ( bNF_Greatest_Succ_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) @ Kl )
= ( bNF_Greatest_Succ_a @ Kl2 @ ( cons_a @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_376_remove__def,axiom,
( remove_list_nat
= ( ^ [X3: list_nat,A3: set_list_nat] : ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat2 @ X3 @ bot_bot_set_list_nat ) ) ) ) ).
% remove_def
thf(fact_377_remove__def,axiom,
( remove_set_list_a
= ( ^ [X3: set_list_a,A3: set_set_list_a] : ( minus_4782336368215558443list_a @ A3 @ ( insert_set_list_a2 @ X3 @ bot_bo3186585308812441520list_a ) ) ) ) ).
% remove_def
thf(fact_378_remove__def,axiom,
( remove_set_nat
= ( ^ [X3: set_nat,A3: set_set_nat] : ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat2 @ X3 @ bot_bot_set_set_nat ) ) ) ) ).
% remove_def
thf(fact_379_remove__def,axiom,
( remove_a
= ( ^ [X3: a,A3: set_a] : ( minus_minus_set_a @ A3 @ ( insert_a2 @ X3 @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_380_remove__def,axiom,
( remove_list_a
= ( ^ [X3: list_a,A3: set_list_a] : ( minus_646659088055828811list_a @ A3 @ ( insert_list_a2 @ X3 @ bot_bot_set_list_a ) ) ) ) ).
% remove_def
thf(fact_381_remove__def,axiom,
( remove_nat
= ( ^ [X3: nat,A3: set_nat] : ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_382_List_Oset__insert,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( set_list_nat2 @ ( insert_list_nat @ X2 @ Xs ) )
= ( insert_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_383_List_Oset__insert,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( set_list_a2 @ ( insert_list_a @ X2 @ Xs ) )
= ( insert_list_a2 @ X2 @ ( set_list_a2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_384_List_Oset__insert,axiom,
! [X2: set_list_a,Xs: list_set_list_a] :
( ( set_set_list_a2 @ ( insert_set_list_a @ X2 @ Xs ) )
= ( insert_set_list_a2 @ X2 @ ( set_set_list_a2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_385_List_Oset__insert,axiom,
! [X2: set_nat,Xs: list_set_nat] :
( ( set_set_nat2 @ ( insert_set_nat @ X2 @ Xs ) )
= ( insert_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_386_List_Oset__insert,axiom,
! [X2: a,Xs: list_a] :
( ( set_a2 @ ( insert_a @ X2 @ Xs ) )
= ( insert_a2 @ X2 @ ( set_a2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_387_List_Oset__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ( set_nat2 @ ( insert_nat @ X2 @ Xs ) )
= ( insert_nat2 @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_388_not__in__set__insert,axiom,
! [X2: set_list_a,Xs: list_set_list_a] :
( ~ ( member_set_list_a2 @ X2 @ ( set_set_list_a2 @ Xs ) )
=> ( ( insert_set_list_a @ X2 @ Xs )
= ( cons_set_list_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_389_not__in__set__insert,axiom,
! [X2: set_nat,Xs: list_set_nat] :
( ~ ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( ( insert_set_nat @ X2 @ Xs )
= ( cons_set_nat @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_390_not__in__set__insert,axiom,
! [X2: list_a,Xs: list_list_a] :
( ~ ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ( insert_list_a @ X2 @ Xs )
= ( cons_list_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_391_not__in__set__insert,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( insert_list_nat @ X2 @ Xs )
= ( cons_list_nat @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_392_not__in__set__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X2 @ Xs )
= ( cons_nat @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_393_not__in__set__insert,axiom,
! [X2: a,Xs: list_a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X2 @ Xs )
= ( cons_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_394_Set_Ois__empty__def,axiom,
( is_empty_set_list_a
= ( ^ [A3: set_set_list_a] : ( A3 = bot_bo3186585308812441520list_a ) ) ) ).
% Set.is_empty_def
thf(fact_395_Set_Ois__empty__def,axiom,
( is_empty_set_nat
= ( ^ [A3: set_set_nat] : ( A3 = bot_bot_set_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_396_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A3: set_a] : ( A3 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_397_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_398_Set_Ois__empty__def,axiom,
( is_empty_list_a
= ( ^ [A3: set_list_a] : ( A3 = bot_bot_set_list_a ) ) ) ).
% Set.is_empty_def
thf(fact_399_member__remove,axiom,
! [X2: list_nat,Y: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ X2 @ ( remove_list_nat @ Y @ A ) )
= ( ( member_list_nat2 @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_400_member__remove,axiom,
! [X2: set_list_a,Y: set_list_a,A: set_set_list_a] :
( ( member_set_list_a2 @ X2 @ ( remove_set_list_a @ Y @ A ) )
= ( ( member_set_list_a2 @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_401_member__remove,axiom,
! [X2: set_nat,Y: set_nat,A: set_set_nat] :
( ( member_set_nat2 @ X2 @ ( remove_set_nat @ Y @ A ) )
= ( ( member_set_nat2 @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_402_member__remove,axiom,
! [X2: a,Y: a,A: set_a] :
( ( member_a2 @ X2 @ ( remove_a @ Y @ A ) )
= ( ( member_a2 @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_403_member__remove,axiom,
! [X2: list_a,Y: list_a,A: set_list_a] :
( ( member_list_a2 @ X2 @ ( remove_list_a @ Y @ A ) )
= ( ( member_list_a2 @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_404_member__remove,axiom,
! [X2: nat,Y: nat,A: set_nat] :
( ( member_nat2 @ X2 @ ( remove_nat @ Y @ A ) )
= ( ( member_nat2 @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_405_removeAll__id,axiom,
! [X2: set_list_a,Xs: list_set_list_a] :
( ~ ( member_set_list_a2 @ X2 @ ( set_set_list_a2 @ Xs ) )
=> ( ( removeAll_set_list_a @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_406_removeAll__id,axiom,
! [X2: set_nat,Xs: list_set_nat] :
( ~ ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( ( removeAll_set_nat @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_407_removeAll__id,axiom,
! [X2: list_a,Xs: list_list_a] :
( ~ ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ( removeAll_list_a @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_408_removeAll__id,axiom,
! [X2: a,Xs: list_a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( removeAll_a @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_409_removeAll__id,axiom,
! [X2: list_list_a,Xs: list_list_list_a] :
( ~ ( member_list_list_a @ X2 @ ( set_list_list_a2 @ Xs ) )
=> ( ( remove8017980289111491990list_a @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_410_removeAll__id,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( removeAll_list_nat @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_411_removeAll__id,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_412_in__set__insert,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( insert_list_nat @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_413_in__set__insert,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a2 @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ( insert_list_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_414_in__set__insert,axiom,
! [X2: set_list_a,Xs: list_set_list_a] :
( ( member_set_list_a2 @ X2 @ ( set_set_list_a2 @ Xs ) )
=> ( ( insert_set_list_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_415_in__set__insert,axiom,
! [X2: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( ( insert_set_nat @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_416_in__set__insert,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_417_in__set__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_418_remove__code_I1_J,axiom,
! [X2: set_list_a,Xs: list_set_list_a] :
( ( remove_set_list_a @ X2 @ ( set_set_list_a2 @ Xs ) )
= ( set_set_list_a2 @ ( removeAll_set_list_a @ X2 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_419_remove__code_I1_J,axiom,
! [X2: set_nat,Xs: list_set_nat] :
( ( remove_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
= ( set_set_nat2 @ ( removeAll_set_nat @ X2 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_420_remove__code_I1_J,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( remove_list_a @ X2 @ ( set_list_a2 @ Xs ) )
= ( set_list_a2 @ ( removeAll_list_a @ X2 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_421_remove__code_I1_J,axiom,
! [X2: a,Xs: list_a] :
( ( remove_a @ X2 @ ( set_a2 @ Xs ) )
= ( set_a2 @ ( removeAll_a @ X2 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_422_remove__code_I1_J,axiom,
! [X2: list_list_a,Xs: list_list_list_a] :
( ( remove_list_list_a @ X2 @ ( set_list_list_a2 @ Xs ) )
= ( set_list_list_a2 @ ( remove8017980289111491990list_a @ X2 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_423_remove__code_I1_J,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( remove_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
= ( set_list_nat2 @ ( removeAll_list_nat @ X2 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_424_remove__code_I1_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( remove_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( removeAll_nat @ X2 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_425_bot__set__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% bot_set_def
thf(fact_426_bot__set__def,axiom,
( bot_bo3186585308812441520list_a
= ( collect_set_list_a @ bot_bot_set_list_a_o ) ) ).
% bot_set_def
thf(fact_427_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_428_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_429_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_430_bot__set__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a @ bot_bot_list_a_o ) ) ).
% bot_set_def
thf(fact_431_removeAll_Osimps_I2_J,axiom,
! [X2: list_a,Y: list_a,Xs: list_list_a] :
( ( ( X2 = Y )
=> ( ( removeAll_list_a @ X2 @ ( cons_list_a @ Y @ Xs ) )
= ( removeAll_list_a @ X2 @ Xs ) ) )
& ( ( X2 != Y )
=> ( ( removeAll_list_a @ X2 @ ( cons_list_a @ Y @ Xs ) )
= ( cons_list_a @ Y @ ( removeAll_list_a @ X2 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_432_removeAll_Osimps_I2_J,axiom,
! [X2: list_list_a,Y: list_list_a,Xs: list_list_list_a] :
( ( ( X2 = Y )
=> ( ( remove8017980289111491990list_a @ X2 @ ( cons_list_list_a @ Y @ Xs ) )
= ( remove8017980289111491990list_a @ X2 @ Xs ) ) )
& ( ( X2 != Y )
=> ( ( remove8017980289111491990list_a @ X2 @ ( cons_list_list_a @ Y @ Xs ) )
= ( cons_list_list_a @ Y @ ( remove8017980289111491990list_a @ X2 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_433_removeAll_Osimps_I2_J,axiom,
! [X2: list_nat,Y: list_nat,Xs: list_list_nat] :
( ( ( X2 = Y )
=> ( ( removeAll_list_nat @ X2 @ ( cons_list_nat @ Y @ Xs ) )
= ( removeAll_list_nat @ X2 @ Xs ) ) )
& ( ( X2 != Y )
=> ( ( removeAll_list_nat @ X2 @ ( cons_list_nat @ Y @ Xs ) )
= ( cons_list_nat @ Y @ ( removeAll_list_nat @ X2 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_434_removeAll_Osimps_I2_J,axiom,
! [X2: nat,Y: nat,Xs: list_nat] :
( ( ( X2 = Y )
=> ( ( removeAll_nat @ X2 @ ( cons_nat @ Y @ Xs ) )
= ( removeAll_nat @ X2 @ Xs ) ) )
& ( ( X2 != Y )
=> ( ( removeAll_nat @ X2 @ ( cons_nat @ Y @ Xs ) )
= ( cons_nat @ Y @ ( removeAll_nat @ X2 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_435_removeAll_Osimps_I2_J,axiom,
! [X2: a,Y: a,Xs: list_a] :
( ( ( X2 = Y )
=> ( ( removeAll_a @ X2 @ ( cons_a @ Y @ Xs ) )
= ( removeAll_a @ X2 @ Xs ) ) )
& ( ( X2 != Y )
=> ( ( removeAll_a @ X2 @ ( cons_a @ Y @ Xs ) )
= ( cons_a @ Y @ ( removeAll_a @ X2 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_436_is__singleton__the__elem,axiom,
( is_sin2641923865335537900st_nat
= ( ^ [A3: set_list_nat] :
( A3
= ( insert_list_nat2 @ ( the_elem_list_nat @ A3 ) @ bot_bot_set_list_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_437_is__singleton__the__elem,axiom,
( is_sin8525870043004244056list_a
= ( ^ [A3: set_set_list_a] :
( A3
= ( insert_set_list_a2 @ ( the_elem_set_list_a @ A3 ) @ bot_bo3186585308812441520list_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_438_is__singleton__the__elem,axiom,
( is_singleton_set_nat
= ( ^ [A3: set_set_nat] :
( A3
= ( insert_set_nat2 @ ( the_elem_set_nat @ A3 ) @ bot_bot_set_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_439_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A3: set_a] :
( A3
= ( insert_a2 @ ( the_elem_a @ A3 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_440_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat2 @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_441_is__singleton__the__elem,axiom,
( is_singleton_list_a
= ( ^ [A3: set_list_a] :
( A3
= ( insert_list_a2 @ ( the_elem_list_a @ A3 ) @ bot_bot_set_list_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_442_is__singletonI_H,axiom,
! [A: set_list_nat] :
( ( A != bot_bot_set_list_nat )
=> ( ! [X: list_nat,Y2: list_nat] :
( ( member_list_nat2 @ X @ A )
=> ( ( member_list_nat2 @ Y2 @ A )
=> ( X = Y2 ) ) )
=> ( is_sin2641923865335537900st_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_443_is__singletonI_H,axiom,
! [A: set_set_list_a] :
( ( A != bot_bo3186585308812441520list_a )
=> ( ! [X: set_list_a,Y2: set_list_a] :
( ( member_set_list_a2 @ X @ A )
=> ( ( member_set_list_a2 @ Y2 @ A )
=> ( X = Y2 ) ) )
=> ( is_sin8525870043004244056list_a @ A ) ) ) ).
% is_singletonI'
thf(fact_444_is__singletonI_H,axiom,
! [A: set_set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( member_set_nat2 @ X @ A )
=> ( ( member_set_nat2 @ Y2 @ A )
=> ( X = Y2 ) ) )
=> ( is_singleton_set_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_445_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X: a,Y2: a] :
( ( member_a2 @ X @ A )
=> ( ( member_a2 @ Y2 @ A )
=> ( X = Y2 ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_446_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X: nat,Y2: nat] :
( ( member_nat2 @ X @ A )
=> ( ( member_nat2 @ Y2 @ A )
=> ( X = Y2 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_447_is__singletonI_H,axiom,
! [A: set_list_a] :
( ( A != bot_bot_set_list_a )
=> ( ! [X: list_a,Y2: list_a] :
( ( member_list_a2 @ X @ A )
=> ( ( member_list_a2 @ Y2 @ A )
=> ( X = Y2 ) ) )
=> ( is_singleton_list_a @ A ) ) ) ).
% is_singletonI'
thf(fact_448_bot__fun__def,axiom,
( bot_bot_list_a_o
= ( ^ [X3: list_a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_449_bot__fun__def,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_450_fun__diff__def,axiom,
( minus_minus_nat_o
= ( ^ [A3: nat > $o,B4: nat > $o,X3: nat] : ( minus_minus_o @ ( A3 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).
% fun_diff_def
thf(fact_451_List_Oinsert__def,axiom,
( insert_set_list_a
= ( ^ [X3: set_list_a,Xs2: list_set_list_a] : ( if_list_set_list_a @ ( member_set_list_a2 @ X3 @ ( set_set_list_a2 @ Xs2 ) ) @ Xs2 @ ( cons_set_list_a @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_452_List_Oinsert__def,axiom,
( insert_set_nat
= ( ^ [X3: set_nat,Xs2: list_set_nat] : ( if_list_set_nat @ ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_set_nat @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_453_List_Oinsert__def,axiom,
( insert_list_a
= ( ^ [X3: list_a,Xs2: list_list_a] : ( if_list_list_a @ ( member_list_a2 @ X3 @ ( set_list_a2 @ Xs2 ) ) @ Xs2 @ ( cons_list_a @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_454_List_Oinsert__def,axiom,
( insert_list_nat
= ( ^ [X3: list_nat,Xs2: list_list_nat] : ( if_list_list_nat @ ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_list_nat @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_455_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X3: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat2 @ X3 @ ( set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_456_List_Oinsert__def,axiom,
( insert_a
= ( ^ [X3: a,Xs2: list_a] : ( if_list_a @ ( member_a2 @ X3 @ ( set_a2 @ Xs2 ) ) @ Xs2 @ ( cons_a @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_457_is__singletonE,axiom,
! [A: set_list_nat] :
( ( is_sin2641923865335537900st_nat @ A )
=> ~ ! [X: list_nat] :
( A
!= ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) ) ) ).
% is_singletonE
thf(fact_458_is__singletonE,axiom,
! [A: set_set_list_a] :
( ( is_sin8525870043004244056list_a @ A )
=> ~ ! [X: set_list_a] :
( A
!= ( insert_set_list_a2 @ X @ bot_bo3186585308812441520list_a ) ) ) ).
% is_singletonE
thf(fact_459_is__singletonE,axiom,
! [A: set_set_nat] :
( ( is_singleton_set_nat @ A )
=> ~ ! [X: set_nat] :
( A
!= ( insert_set_nat2 @ X @ bot_bot_set_set_nat ) ) ) ).
% is_singletonE
thf(fact_460_is__singletonE,axiom,
! [A: set_a] :
( ( is_singleton_a @ A )
=> ~ ! [X: a] :
( A
!= ( insert_a2 @ X @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_461_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X: nat] :
( A
!= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_462_is__singletonE,axiom,
! [A: set_list_a] :
( ( is_singleton_list_a @ A )
=> ~ ! [X: list_a] :
( A
!= ( insert_list_a2 @ X @ bot_bot_set_list_a ) ) ) ).
% is_singletonE
thf(fact_463_is__singleton__def,axiom,
( is_sin2641923865335537900st_nat
= ( ^ [A3: set_list_nat] :
? [X3: list_nat] :
( A3
= ( insert_list_nat2 @ X3 @ bot_bot_set_list_nat ) ) ) ) ).
% is_singleton_def
thf(fact_464_is__singleton__def,axiom,
( is_sin8525870043004244056list_a
= ( ^ [A3: set_set_list_a] :
? [X3: set_list_a] :
( A3
= ( insert_set_list_a2 @ X3 @ bot_bo3186585308812441520list_a ) ) ) ) ).
% is_singleton_def
thf(fact_465_is__singleton__def,axiom,
( is_singleton_set_nat
= ( ^ [A3: set_set_nat] :
? [X3: set_nat] :
( A3
= ( insert_set_nat2 @ X3 @ bot_bot_set_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_466_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A3: set_a] :
? [X3: a] :
( A3
= ( insert_a2 @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_467_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X3: nat] :
( A3
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_468_is__singleton__def,axiom,
( is_singleton_list_a
= ( ^ [A3: set_list_a] :
? [X3: list_a] :
( A3
= ( insert_list_a2 @ X3 @ bot_bot_set_list_a ) ) ) ) ).
% is_singleton_def
thf(fact_469_is__empty__set,axiom,
! [Xs: list_list_nat] :
( ( is_empty_list_nat @ ( set_list_nat2 @ Xs ) )
= ( null_list_nat @ Xs ) ) ).
% is_empty_set
thf(fact_470_is__empty__set,axiom,
! [Xs: list_list_a] :
( ( is_empty_list_a @ ( set_list_a2 @ Xs ) )
= ( null_list_a @ Xs ) ) ).
% is_empty_set
thf(fact_471_is__empty__set,axiom,
! [Xs: list_set_list_a] :
( ( is_empty_set_list_a @ ( set_set_list_a2 @ Xs ) )
= ( null_set_list_a @ Xs ) ) ).
% is_empty_set
thf(fact_472_is__empty__set,axiom,
! [Xs: list_set_nat] :
( ( is_empty_set_nat @ ( set_set_nat2 @ Xs ) )
= ( null_set_nat @ Xs ) ) ).
% is_empty_set
thf(fact_473_is__empty__set,axiom,
! [Xs: list_a] :
( ( is_empty_a @ ( set_a2 @ Xs ) )
= ( null_a @ Xs ) ) ).
% is_empty_set
thf(fact_474_is__empty__set,axiom,
! [Xs: list_nat] :
( ( is_empty_nat @ ( set_nat2 @ Xs ) )
= ( null_nat @ Xs ) ) ).
% is_empty_set
thf(fact_475_restrict_Oelims,axiom,
! [X2: set_a,Xa: list_a,Xb: list_nat,Y: list_nat] :
( ( ( restrict_a_nat @ X2 @ Xa @ Xb )
= Y )
=> ( ! [X: a,Xs3: list_a] :
( ( Xa
= ( cons_a @ X @ Xs3 ) )
=> ! [Y2: nat,Ys2: list_nat] :
( ( Xb
= ( cons_nat @ Y2 @ Ys2 ) )
=> ~ ( ( ( member_a2 @ X @ X2 )
=> ( Y
= ( cons_nat @ Y2 @ ( restrict_a_nat @ ( minus_minus_set_a @ X2 @ ( insert_a2 @ X @ bot_bot_set_a ) ) @ Xs3 @ Ys2 ) ) ) )
& ( ~ ( member_a2 @ X @ X2 )
=> ( Y
= ( restrict_a_nat @ X2 @ Xs3 @ Ys2 ) ) ) ) ) )
=> ( ( ( Xa = nil_a )
=> ( Y != nil_nat ) )
=> ~ ( ( Xb = nil_nat )
=> ( Y != nil_nat ) ) ) ) ) ).
% restrict.elims
thf(fact_476_restrict_Oelims,axiom,
! [X2: set_a,Xa: list_a,Xb: list_a,Y: list_a] :
( ( ( restrict_a_a @ X2 @ Xa @ Xb )
= Y )
=> ( ! [X: a,Xs3: list_a] :
( ( Xa
= ( cons_a @ X @ Xs3 ) )
=> ! [Y2: a,Ys2: list_a] :
( ( Xb
= ( cons_a @ Y2 @ Ys2 ) )
=> ~ ( ( ( member_a2 @ X @ X2 )
=> ( Y
= ( cons_a @ Y2 @ ( restrict_a_a @ ( minus_minus_set_a @ X2 @ ( insert_a2 @ X @ bot_bot_set_a ) ) @ Xs3 @ Ys2 ) ) ) )
& ( ~ ( member_a2 @ X @ X2 )
=> ( Y
= ( restrict_a_a @ X2 @ Xs3 @ Ys2 ) ) ) ) ) )
=> ( ( ( Xa = nil_a )
=> ( Y != nil_a ) )
=> ~ ( ( Xb = nil_a )
=> ( Y != nil_a ) ) ) ) ) ).
% restrict.elims
thf(fact_477_restrict_Oelims,axiom,
! [X2: set_nat,Xa: list_nat,Xb: list_nat,Y: list_nat] :
( ( ( restrict_nat_nat @ X2 @ Xa @ Xb )
= Y )
=> ( ! [X: nat,Xs3: list_nat] :
( ( Xa
= ( cons_nat @ X @ Xs3 ) )
=> ! [Y2: nat,Ys2: list_nat] :
( ( Xb
= ( cons_nat @ Y2 @ Ys2 ) )
=> ~ ( ( ( member_nat2 @ X @ X2 )
=> ( Y
= ( cons_nat @ Y2 @ ( restrict_nat_nat @ ( minus_minus_set_nat @ X2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ Xs3 @ Ys2 ) ) ) )
& ( ~ ( member_nat2 @ X @ X2 )
=> ( Y
= ( restrict_nat_nat @ X2 @ Xs3 @ Ys2 ) ) ) ) ) )
=> ( ( ( Xa = nil_nat )
=> ( Y != nil_nat ) )
=> ~ ( ( Xb = nil_nat )
=> ( Y != nil_nat ) ) ) ) ) ).
% restrict.elims
thf(fact_478_restrict_Oelims,axiom,
! [X2: set_nat,Xa: list_nat,Xb: list_a,Y: list_a] :
( ( ( restrict_nat_a @ X2 @ Xa @ Xb )
= Y )
=> ( ! [X: nat,Xs3: list_nat] :
( ( Xa
= ( cons_nat @ X @ Xs3 ) )
=> ! [Y2: a,Ys2: list_a] :
( ( Xb
= ( cons_a @ Y2 @ Ys2 ) )
=> ~ ( ( ( member_nat2 @ X @ X2 )
=> ( Y
= ( cons_a @ Y2 @ ( restrict_nat_a @ ( minus_minus_set_nat @ X2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ Xs3 @ Ys2 ) ) ) )
& ( ~ ( member_nat2 @ X @ X2 )
=> ( Y
= ( restrict_nat_a @ X2 @ Xs3 @ Ys2 ) ) ) ) ) )
=> ( ( ( Xa = nil_nat )
=> ( Y != nil_a ) )
=> ~ ( ( Xb = nil_a )
=> ( Y != nil_a ) ) ) ) ) ).
% restrict.elims
thf(fact_479_restrict_Oelims,axiom,
! [X2: set_list_a,Xa: list_list_a,Xb: list_nat,Y: list_nat] :
( ( ( restrict_list_a_nat @ X2 @ Xa @ Xb )
= Y )
=> ( ! [X: list_a,Xs3: list_list_a] :
( ( Xa
= ( cons_list_a @ X @ Xs3 ) )
=> ! [Y2: nat,Ys2: list_nat] :
( ( Xb
= ( cons_nat @ Y2 @ Ys2 ) )
=> ~ ( ( ( member_list_a2 @ X @ X2 )
=> ( Y
= ( cons_nat @ Y2 @ ( restrict_list_a_nat @ ( minus_646659088055828811list_a @ X2 @ ( insert_list_a2 @ X @ bot_bot_set_list_a ) ) @ Xs3 @ Ys2 ) ) ) )
& ( ~ ( member_list_a2 @ X @ X2 )
=> ( Y
= ( restrict_list_a_nat @ X2 @ Xs3 @ Ys2 ) ) ) ) ) )
=> ( ( ( Xa = nil_list_a )
=> ( Y != nil_nat ) )
=> ~ ( ( Xb = nil_nat )
=> ( Y != nil_nat ) ) ) ) ) ).
% restrict.elims
thf(fact_480_restrict_Oelims,axiom,
! [X2: set_list_a,Xa: list_list_a,Xb: list_a,Y: list_a] :
( ( ( restrict_list_a_a @ X2 @ Xa @ Xb )
= Y )
=> ( ! [X: list_a,Xs3: list_list_a] :
( ( Xa
= ( cons_list_a @ X @ Xs3 ) )
=> ! [Y2: a,Ys2: list_a] :
( ( Xb
= ( cons_a @ Y2 @ Ys2 ) )
=> ~ ( ( ( member_list_a2 @ X @ X2 )
=> ( Y
= ( cons_a @ Y2 @ ( restrict_list_a_a @ ( minus_646659088055828811list_a @ X2 @ ( insert_list_a2 @ X @ bot_bot_set_list_a ) ) @ Xs3 @ Ys2 ) ) ) )
& ( ~ ( member_list_a2 @ X @ X2 )
=> ( Y
= ( restrict_list_a_a @ X2 @ Xs3 @ Ys2 ) ) ) ) ) )
=> ( ( ( Xa = nil_list_a )
=> ( Y != nil_a ) )
=> ~ ( ( Xb = nil_a )
=> ( Y != nil_a ) ) ) ) ) ).
% restrict.elims
thf(fact_481_restrict_Oelims,axiom,
! [X2: set_set_a,Xa: list_set_a,Xb: list_nat,Y: list_nat] :
( ( ( restrict_set_a_nat @ X2 @ Xa @ Xb )
= Y )
=> ( ! [X: set_a,Xs3: list_set_a] :
( ( Xa
= ( cons_set_a @ X @ Xs3 ) )
=> ! [Y2: nat,Ys2: list_nat] :
( ( Xb
= ( cons_nat @ Y2 @ Ys2 ) )
=> ~ ( ( ( member_set_a @ X @ X2 )
=> ( Y
= ( cons_nat @ Y2 @ ( restrict_set_a_nat @ ( minus_5736297505244876581_set_a @ X2 @ ( insert_set_a2 @ X @ bot_bot_set_set_a ) ) @ Xs3 @ Ys2 ) ) ) )
& ( ~ ( member_set_a @ X @ X2 )
=> ( Y
= ( restrict_set_a_nat @ X2 @ Xs3 @ Ys2 ) ) ) ) ) )
=> ( ( ( Xa = nil_set_a )
=> ( Y != nil_nat ) )
=> ~ ( ( Xb = nil_nat )
=> ( Y != nil_nat ) ) ) ) ) ).
% restrict.elims
thf(fact_482_restrict_Oelims,axiom,
! [X2: set_set_a,Xa: list_set_a,Xb: list_a,Y: list_a] :
( ( ( restrict_set_a_a @ X2 @ Xa @ Xb )
= Y )
=> ( ! [X: set_a,Xs3: list_set_a] :
( ( Xa
= ( cons_set_a @ X @ Xs3 ) )
=> ! [Y2: a,Ys2: list_a] :
( ( Xb
= ( cons_a @ Y2 @ Ys2 ) )
=> ~ ( ( ( member_set_a @ X @ X2 )
=> ( Y
= ( cons_a @ Y2 @ ( restrict_set_a_a @ ( minus_5736297505244876581_set_a @ X2 @ ( insert_set_a2 @ X @ bot_bot_set_set_a ) ) @ Xs3 @ Ys2 ) ) ) )
& ( ~ ( member_set_a @ X @ X2 )
=> ( Y
= ( restrict_set_a_a @ X2 @ Xs3 @ Ys2 ) ) ) ) ) )
=> ( ( ( Xa = nil_set_a )
=> ( Y != nil_a ) )
=> ~ ( ( Xb = nil_a )
=> ( Y != nil_a ) ) ) ) ) ).
% restrict.elims
thf(fact_483_restrict_Oelims,axiom,
! [X2: set_list_nat,Xa: list_list_nat,Xb: list_nat,Y: list_nat] :
( ( ( restri6853610114029608726at_nat @ X2 @ Xa @ Xb )
= Y )
=> ( ! [X: list_nat,Xs3: list_list_nat] :
( ( Xa
= ( cons_list_nat @ X @ Xs3 ) )
=> ! [Y2: nat,Ys2: list_nat] :
( ( Xb
= ( cons_nat @ Y2 @ Ys2 ) )
=> ~ ( ( ( member_list_nat2 @ X @ X2 )
=> ( Y
= ( cons_nat @ Y2 @ ( restri6853610114029608726at_nat @ ( minus_7954133019191499631st_nat @ X2 @ ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) ) @ Xs3 @ Ys2 ) ) ) )
& ( ~ ( member_list_nat2 @ X @ X2 )
=> ( Y
= ( restri6853610114029608726at_nat @ X2 @ Xs3 @ Ys2 ) ) ) ) ) )
=> ( ( ( Xa = nil_list_nat )
=> ( Y != nil_nat ) )
=> ~ ( ( Xb = nil_nat )
=> ( Y != nil_nat ) ) ) ) ) ).
% restrict.elims
thf(fact_484_restrict_Oelims,axiom,
! [X2: set_list_nat,Xa: list_list_nat,Xb: list_a,Y: list_a] :
( ( ( restrict_list_nat_a @ X2 @ Xa @ Xb )
= Y )
=> ( ! [X: list_nat,Xs3: list_list_nat] :
( ( Xa
= ( cons_list_nat @ X @ Xs3 ) )
=> ! [Y2: a,Ys2: list_a] :
( ( Xb
= ( cons_a @ Y2 @ Ys2 ) )
=> ~ ( ( ( member_list_nat2 @ X @ X2 )
=> ( Y
= ( cons_a @ Y2 @ ( restrict_list_nat_a @ ( minus_7954133019191499631st_nat @ X2 @ ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) ) @ Xs3 @ Ys2 ) ) ) )
& ( ~ ( member_list_nat2 @ X @ X2 )
=> ( Y
= ( restrict_list_nat_a @ X2 @ Xs3 @ Ys2 ) ) ) ) ) )
=> ( ( ( Xa = nil_list_nat )
=> ( Y != nil_a ) )
=> ~ ( ( Xb = nil_a )
=> ( Y != nil_a ) ) ) ) ) ).
% restrict.elims
thf(fact_485_remove__code_I2_J,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( remove_list_a @ X2 @ ( coset_list_a @ Xs ) )
= ( coset_list_a @ ( insert_list_a @ X2 @ Xs ) ) ) ).
% remove_code(2)
thf(fact_486_remove__code_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( remove_nat @ X2 @ ( coset_nat @ Xs ) )
= ( coset_nat @ ( insert_nat @ X2 @ Xs ) ) ) ).
% remove_code(2)
thf(fact_487_remove__code_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( remove_a @ X2 @ ( coset_a @ Xs ) )
= ( coset_a @ ( insert_a @ X2 @ Xs ) ) ) ).
% remove_code(2)
thf(fact_488_pairwise__alt,axiom,
( pairwise_list_nat
= ( ^ [R: list_nat > list_nat > $o,S: set_list_nat] :
! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ S )
=> ! [Y3: list_nat] :
( ( member_list_nat2 @ Y3 @ ( minus_7954133019191499631st_nat @ S @ ( insert_list_nat2 @ X3 @ bot_bot_set_list_nat ) ) )
=> ( R @ X3 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_489_pairwise__alt,axiom,
( pairwise_set_list_a
= ( ^ [R: set_list_a > set_list_a > $o,S: set_set_list_a] :
! [X3: set_list_a] :
( ( member_set_list_a2 @ X3 @ S )
=> ! [Y3: set_list_a] :
( ( member_set_list_a2 @ Y3 @ ( minus_4782336368215558443list_a @ S @ ( insert_set_list_a2 @ X3 @ bot_bo3186585308812441520list_a ) ) )
=> ( R @ X3 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_490_pairwise__alt,axiom,
( pairwise_set_nat
= ( ^ [R: set_nat > set_nat > $o,S: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ S )
=> ! [Y3: set_nat] :
( ( member_set_nat2 @ Y3 @ ( minus_2163939370556025621et_nat @ S @ ( insert_set_nat2 @ X3 @ bot_bot_set_set_nat ) ) )
=> ( R @ X3 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_491_pairwise__alt,axiom,
( pairwise_a
= ( ^ [R: a > a > $o,S: set_a] :
! [X3: a] :
( ( member_a2 @ X3 @ S )
=> ! [Y3: a] :
( ( member_a2 @ Y3 @ ( minus_minus_set_a @ S @ ( insert_a2 @ X3 @ bot_bot_set_a ) ) )
=> ( R @ X3 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_492_pairwise__alt,axiom,
( pairwise_list_a
= ( ^ [R: list_a > list_a > $o,S: set_list_a] :
! [X3: list_a] :
( ( member_list_a2 @ X3 @ S )
=> ! [Y3: list_a] :
( ( member_list_a2 @ Y3 @ ( minus_646659088055828811list_a @ S @ ( insert_list_a2 @ X3 @ bot_bot_set_list_a ) ) )
=> ( R @ X3 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_493_pairwise__alt,axiom,
( pairwise_nat
= ( ^ [R: nat > nat > $o,S: set_nat] :
! [X3: nat] :
( ( member_nat2 @ X3 @ S )
=> ! [Y3: nat] :
( ( member_nat2 @ Y3 @ ( minus_minus_set_nat @ S @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) )
=> ( R @ X3 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_494_the__elem__set,axiom,
! [X2: set_a] :
( ( the_elem_set_a @ ( set_set_a2 @ ( cons_set_a @ X2 @ nil_set_a ) ) )
= X2 ) ).
% the_elem_set
thf(fact_495_the__elem__set,axiom,
! [X2: set_list_a] :
( ( the_elem_set_list_a @ ( set_set_list_a2 @ ( cons_set_list_a @ X2 @ nil_set_list_a ) ) )
= X2 ) ).
% the_elem_set
thf(fact_496_the__elem__set,axiom,
! [X2: set_nat] :
( ( the_elem_set_nat @ ( set_set_nat2 @ ( cons_set_nat @ X2 @ nil_set_nat ) ) )
= X2 ) ).
% the_elem_set
thf(fact_497_the__elem__set,axiom,
! [X2: list_a] :
( ( the_elem_list_a @ ( set_list_a2 @ ( cons_list_a @ X2 @ nil_list_a ) ) )
= X2 ) ).
% the_elem_set
thf(fact_498_the__elem__set,axiom,
! [X2: list_nat] :
( ( the_elem_list_nat @ ( set_list_nat2 @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= X2 ) ).
% the_elem_set
thf(fact_499_the__elem__set,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X2 @ nil_nat ) ) )
= X2 ) ).
% the_elem_set
thf(fact_500_the__elem__set,axiom,
! [X2: a] :
( ( the_elem_a @ ( set_a2 @ ( cons_a @ X2 @ nil_a ) ) )
= X2 ) ).
% the_elem_set
thf(fact_501_insert__code_I2_J,axiom,
! [X2: set_list_a,Xs: list_set_list_a] :
( ( insert_set_list_a2 @ X2 @ ( coset_set_list_a @ Xs ) )
= ( coset_set_list_a @ ( removeAll_set_list_a @ X2 @ Xs ) ) ) ).
% insert_code(2)
thf(fact_502_insert__code_I2_J,axiom,
! [X2: set_nat,Xs: list_set_nat] :
( ( insert_set_nat2 @ X2 @ ( coset_set_nat @ Xs ) )
= ( coset_set_nat @ ( removeAll_set_nat @ X2 @ Xs ) ) ) ).
% insert_code(2)
thf(fact_503_insert__code_I2_J,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( insert_list_a2 @ X2 @ ( coset_list_a @ Xs ) )
= ( coset_list_a @ ( removeAll_list_a @ X2 @ Xs ) ) ) ).
% insert_code(2)
thf(fact_504_insert__code_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( insert_a2 @ X2 @ ( coset_a @ Xs ) )
= ( coset_a @ ( removeAll_a @ X2 @ Xs ) ) ) ).
% insert_code(2)
thf(fact_505_insert__code_I2_J,axiom,
! [X2: list_list_a,Xs: list_list_list_a] :
( ( insert_list_list_a @ X2 @ ( coset_list_list_a @ Xs ) )
= ( coset_list_list_a @ ( remove8017980289111491990list_a @ X2 @ Xs ) ) ) ).
% insert_code(2)
thf(fact_506_insert__code_I2_J,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( insert_list_nat2 @ X2 @ ( coset_list_nat @ Xs ) )
= ( coset_list_nat @ ( removeAll_list_nat @ X2 @ Xs ) ) ) ).
% insert_code(2)
thf(fact_507_insert__code_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( insert_nat2 @ X2 @ ( coset_nat @ Xs ) )
= ( coset_nat @ ( removeAll_nat @ X2 @ Xs ) ) ) ).
% insert_code(2)
thf(fact_508_minus__set__fold,axiom,
! [A: set_list_nat,Xs: list_list_nat] :
( ( minus_7954133019191499631st_nat @ A @ ( set_list_nat2 @ Xs ) )
= ( fold_l59423398878476163st_nat @ remove_list_nat @ Xs @ A ) ) ).
% minus_set_fold
thf(fact_509_minus__set__fold,axiom,
! [A: set_set_list_a,Xs: list_set_list_a] :
( ( minus_4782336368215558443list_a @ A @ ( set_set_list_a2 @ Xs ) )
= ( fold_s6172198821268269675list_a @ remove_set_list_a @ Xs @ A ) ) ).
% minus_set_fold
thf(fact_510_minus__set__fold,axiom,
! [A: set_set_nat,Xs: list_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ ( set_set_nat2 @ Xs ) )
= ( fold_s4794219702148550607et_nat @ remove_set_nat @ Xs @ A ) ) ).
% minus_set_fold
thf(fact_511_minus__set__fold,axiom,
! [A: set_list_a,Xs: list_list_a] :
( ( minus_646659088055828811list_a @ A @ ( set_list_a2 @ Xs ) )
= ( fold_l7894317391402898027list_a @ remove_list_a @ Xs @ A ) ) ).
% minus_set_fold
thf(fact_512_minus__set__fold,axiom,
! [A: set_a,Xs: list_a] :
( ( minus_minus_set_a @ A @ ( set_a2 @ Xs ) )
= ( fold_a_set_a @ remove_a @ Xs @ A ) ) ).
% minus_set_fold
thf(fact_513_minus__set__fold,axiom,
! [A: set_nat,Xs: list_nat] :
( ( minus_minus_set_nat @ A @ ( set_nat2 @ Xs ) )
= ( fold_nat_set_nat @ remove_nat @ Xs @ A ) ) ).
% minus_set_fold
thf(fact_514_empty__Shift,axiom,
! [Kl2: set_list_set_list_a,K: set_list_a] :
( ( member5524387281408368019list_a @ nil_set_list_a @ Kl2 )
=> ( ( member_set_list_a2 @ K @ ( bNF_Gr8877554613853555711list_a @ Kl2 @ nil_set_list_a ) )
=> ( member5524387281408368019list_a @ nil_set_list_a @ ( bNF_Gr5580213925211368579list_a @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_515_empty__Shift,axiom,
! [Kl2: set_list_set_nat,K: set_nat] :
( ( member_list_set_nat @ nil_set_nat @ Kl2 )
=> ( ( member_set_nat2 @ K @ ( bNF_Gr3282828795834814635et_nat @ Kl2 @ nil_set_nat ) )
=> ( member_list_set_nat @ nil_set_nat @ ( bNF_Gr2891354507007493415et_nat @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_516_empty__Shift,axiom,
! [Kl2: set_list_list_a,K: list_a] :
( ( member_list_list_a @ nil_list_a @ Kl2 )
=> ( ( member_list_a2 @ K @ ( bNF_Gr4634511371912843295list_a @ Kl2 @ nil_list_a ) )
=> ( member_list_list_a @ nil_list_a @ ( bNF_Gr7042794125918077091list_a @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_517_empty__Shift,axiom,
! [Kl2: set_list_list_nat,K: list_nat] :
( ( member_list_list_nat @ nil_list_nat @ Kl2 )
=> ( ( member_list_nat2 @ K @ ( bNF_Gr3053708287304744325st_nat @ Kl2 @ nil_list_nat ) )
=> ( member_list_list_nat @ nil_list_nat @ ( bNF_Gr9051742241863529473st_nat @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_518_empty__Shift,axiom,
! [Kl2: set_list_set_a,K: set_a] :
( ( member_list_set_a @ nil_set_a @ Kl2 )
=> ( ( member_set_a @ K @ ( bNF_Gr5263945959978596985_set_a @ Kl2 @ nil_set_a ) )
=> ( member_list_set_a @ nil_set_a @ ( bNF_Gr641101480264723709_set_a @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_519_empty__Shift,axiom,
! [Kl2: set_list_a,K: a] :
( ( member_list_a2 @ nil_a @ Kl2 )
=> ( ( member_a2 @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ nil_a ) )
=> ( member_list_a2 @ nil_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_520_empty__Shift,axiom,
! [Kl2: set_list_nat,K: nat] :
( ( member_list_nat2 @ nil_nat @ Kl2 )
=> ( ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ nil_nat ) )
=> ( member_list_nat2 @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_521_bot__empty__eq,axiom,
( bot_bot_list_nat_o
= ( ^ [X3: list_nat] : ( member_list_nat2 @ X3 @ bot_bot_set_list_nat ) ) ) ).
% bot_empty_eq
thf(fact_522_bot__empty__eq,axiom,
( bot_bot_set_list_a_o
= ( ^ [X3: set_list_a] : ( member_set_list_a2 @ X3 @ bot_bo3186585308812441520list_a ) ) ) ).
% bot_empty_eq
thf(fact_523_bot__empty__eq,axiom,
( bot_bot_set_nat_o
= ( ^ [X3: set_nat] : ( member_set_nat2 @ X3 @ bot_bot_set_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_524_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a2 @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_525_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat2 @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_526_bot__empty__eq,axiom,
( bot_bot_list_a_o
= ( ^ [X3: list_a] : ( member_list_a2 @ X3 @ bot_bot_set_list_a ) ) ) ).
% bot_empty_eq
thf(fact_527_set__empty2,axiom,
! [Xs: list_set_a] :
( ( bot_bot_set_set_a
= ( set_set_a2 @ Xs ) )
= ( Xs = nil_set_a ) ) ).
% set_empty2
thf(fact_528_set__empty2,axiom,
! [Xs: list_list_nat] :
( ( bot_bot_set_list_nat
= ( set_list_nat2 @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% set_empty2
thf(fact_529_set__empty2,axiom,
! [Xs: list_set_list_a] :
( ( bot_bo3186585308812441520list_a
= ( set_set_list_a2 @ Xs ) )
= ( Xs = nil_set_list_a ) ) ).
% set_empty2
thf(fact_530_set__empty2,axiom,
! [Xs: list_set_nat] :
( ( bot_bot_set_set_nat
= ( set_set_nat2 @ Xs ) )
= ( Xs = nil_set_nat ) ) ).
% set_empty2
thf(fact_531_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_532_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_533_set__empty2,axiom,
! [Xs: list_list_a] :
( ( bot_bot_set_list_a
= ( set_list_a2 @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% set_empty2
thf(fact_534_set__empty,axiom,
! [Xs: list_set_a] :
( ( ( set_set_a2 @ Xs )
= bot_bot_set_set_a )
= ( Xs = nil_set_a ) ) ).
% set_empty
thf(fact_535_set__empty,axiom,
! [Xs: list_list_nat] :
( ( ( set_list_nat2 @ Xs )
= bot_bot_set_list_nat )
= ( Xs = nil_list_nat ) ) ).
% set_empty
thf(fact_536_set__empty,axiom,
! [Xs: list_set_list_a] :
( ( ( set_set_list_a2 @ Xs )
= bot_bo3186585308812441520list_a )
= ( Xs = nil_set_list_a ) ) ).
% set_empty
thf(fact_537_set__empty,axiom,
! [Xs: list_set_nat] :
( ( ( set_set_nat2 @ Xs )
= bot_bot_set_set_nat )
= ( Xs = nil_set_nat ) ) ).
% set_empty
thf(fact_538_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_539_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_540_set__empty,axiom,
! [Xs: list_list_a] :
( ( ( set_list_a2 @ Xs )
= bot_bot_set_list_a )
= ( Xs = nil_list_a ) ) ).
% set_empty
thf(fact_541_insert__Nil,axiom,
! [X2: set_a] :
( ( insert_set_a @ X2 @ nil_set_a )
= ( cons_set_a @ X2 @ nil_set_a ) ) ).
% insert_Nil
thf(fact_542_insert__Nil,axiom,
! [X2: list_a] :
( ( insert_list_a @ X2 @ nil_list_a )
= ( cons_list_a @ X2 @ nil_list_a ) ) ).
% insert_Nil
thf(fact_543_insert__Nil,axiom,
! [X2: list_nat] :
( ( insert_list_nat @ X2 @ nil_list_nat )
= ( cons_list_nat @ X2 @ nil_list_nat ) ) ).
% insert_Nil
thf(fact_544_insert__Nil,axiom,
! [X2: nat] :
( ( insert_nat @ X2 @ nil_nat )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% insert_Nil
thf(fact_545_insert__Nil,axiom,
! [X2: a] :
( ( insert_a @ X2 @ nil_a )
= ( cons_a @ X2 @ nil_a ) ) ).
% insert_Nil
thf(fact_546_null__rec_I2_J,axiom,
null_nat @ nil_nat ).
% null_rec(2)
thf(fact_547_null__rec_I2_J,axiom,
null_a @ nil_a ).
% null_rec(2)
thf(fact_548_null__rec_I2_J,axiom,
null_list_a @ nil_list_a ).
% null_rec(2)
thf(fact_549_null__rec_I2_J,axiom,
null_list_nat @ nil_list_nat ).
% null_rec(2)
thf(fact_550_null__rec_I2_J,axiom,
null_set_a @ nil_set_a ).
% null_rec(2)
thf(fact_551_fold__simps_I1_J,axiom,
! [F: set_list_a > set_list_a > set_list_a,S2: set_list_a] :
( ( fold_s5931075695703335563list_a @ F @ nil_set_list_a @ S2 )
= S2 ) ).
% fold_simps(1)
thf(fact_552_fold__simps_I1_J,axiom,
! [F: set_nat > set_nat > set_nat,S2: set_nat] :
( ( fold_set_nat_set_nat @ F @ nil_set_nat @ S2 )
= S2 ) ).
% fold_simps(1)
thf(fact_553_fold__simps_I1_J,axiom,
! [F: nat > set_nat > set_nat,S2: set_nat] :
( ( fold_nat_set_nat @ F @ nil_nat @ S2 )
= S2 ) ).
% fold_simps(1)
thf(fact_554_fold__simps_I1_J,axiom,
! [F: nat > nat > nat,S2: nat] :
( ( fold_nat_nat @ F @ nil_nat @ S2 )
= S2 ) ).
% fold_simps(1)
thf(fact_555_eq__Nil__null,axiom,
! [Xs: list_nat] :
( ( Xs = nil_nat )
= ( null_nat @ Xs ) ) ).
% eq_Nil_null
thf(fact_556_eq__Nil__null,axiom,
! [Xs: list_a] :
( ( Xs = nil_a )
= ( null_a @ Xs ) ) ).
% eq_Nil_null
thf(fact_557_eq__Nil__null,axiom,
! [Xs: list_list_a] :
( ( Xs = nil_list_a )
= ( null_list_a @ Xs ) ) ).
% eq_Nil_null
thf(fact_558_eq__Nil__null,axiom,
! [Xs: list_list_nat] :
( ( Xs = nil_list_nat )
= ( null_list_nat @ Xs ) ) ).
% eq_Nil_null
thf(fact_559_eq__Nil__null,axiom,
! [Xs: list_set_a] :
( ( Xs = nil_set_a )
= ( null_set_a @ Xs ) ) ).
% eq_Nil_null
thf(fact_560_pairwiseD,axiom,
! [R2: list_nat > list_nat > $o,S3: set_list_nat,X2: list_nat,Y: list_nat] :
( ( pairwise_list_nat @ R2 @ S3 )
=> ( ( member_list_nat2 @ X2 @ S3 )
=> ( ( member_list_nat2 @ Y @ S3 )
=> ( ( X2 != Y )
=> ( R2 @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_561_pairwiseD,axiom,
! [R2: set_list_a > set_list_a > $o,S3: set_set_list_a,X2: set_list_a,Y: set_list_a] :
( ( pairwise_set_list_a @ R2 @ S3 )
=> ( ( member_set_list_a2 @ X2 @ S3 )
=> ( ( member_set_list_a2 @ Y @ S3 )
=> ( ( X2 != Y )
=> ( R2 @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_562_pairwiseD,axiom,
! [R2: set_nat > set_nat > $o,S3: set_set_nat,X2: set_nat,Y: set_nat] :
( ( pairwise_set_nat @ R2 @ S3 )
=> ( ( member_set_nat2 @ X2 @ S3 )
=> ( ( member_set_nat2 @ Y @ S3 )
=> ( ( X2 != Y )
=> ( R2 @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_563_pairwiseD,axiom,
! [R2: a > a > $o,S3: set_a,X2: a,Y: a] :
( ( pairwise_a @ R2 @ S3 )
=> ( ( member_a2 @ X2 @ S3 )
=> ( ( member_a2 @ Y @ S3 )
=> ( ( X2 != Y )
=> ( R2 @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_564_pairwiseD,axiom,
! [R2: list_a > list_a > $o,S3: set_list_a,X2: list_a,Y: list_a] :
( ( pairwise_list_a @ R2 @ S3 )
=> ( ( member_list_a2 @ X2 @ S3 )
=> ( ( member_list_a2 @ Y @ S3 )
=> ( ( X2 != Y )
=> ( R2 @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_565_pairwiseD,axiom,
! [R2: nat > nat > $o,S3: set_nat,X2: nat,Y: nat] :
( ( pairwise_nat @ R2 @ S3 )
=> ( ( member_nat2 @ X2 @ S3 )
=> ( ( member_nat2 @ Y @ S3 )
=> ( ( X2 != Y )
=> ( R2 @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_566_pairwiseI,axiom,
! [S3: set_list_nat,R2: list_nat > list_nat > $o] :
( ! [X: list_nat,Y2: list_nat] :
( ( member_list_nat2 @ X @ S3 )
=> ( ( member_list_nat2 @ Y2 @ S3 )
=> ( ( X != Y2 )
=> ( R2 @ X @ Y2 ) ) ) )
=> ( pairwise_list_nat @ R2 @ S3 ) ) ).
% pairwiseI
thf(fact_567_pairwiseI,axiom,
! [S3: set_set_list_a,R2: set_list_a > set_list_a > $o] :
( ! [X: set_list_a,Y2: set_list_a] :
( ( member_set_list_a2 @ X @ S3 )
=> ( ( member_set_list_a2 @ Y2 @ S3 )
=> ( ( X != Y2 )
=> ( R2 @ X @ Y2 ) ) ) )
=> ( pairwise_set_list_a @ R2 @ S3 ) ) ).
% pairwiseI
thf(fact_568_pairwiseI,axiom,
! [S3: set_set_nat,R2: set_nat > set_nat > $o] :
( ! [X: set_nat,Y2: set_nat] :
( ( member_set_nat2 @ X @ S3 )
=> ( ( member_set_nat2 @ Y2 @ S3 )
=> ( ( X != Y2 )
=> ( R2 @ X @ Y2 ) ) ) )
=> ( pairwise_set_nat @ R2 @ S3 ) ) ).
% pairwiseI
thf(fact_569_pairwiseI,axiom,
! [S3: set_a,R2: a > a > $o] :
( ! [X: a,Y2: a] :
( ( member_a2 @ X @ S3 )
=> ( ( member_a2 @ Y2 @ S3 )
=> ( ( X != Y2 )
=> ( R2 @ X @ Y2 ) ) ) )
=> ( pairwise_a @ R2 @ S3 ) ) ).
% pairwiseI
thf(fact_570_pairwiseI,axiom,
! [S3: set_list_a,R2: list_a > list_a > $o] :
( ! [X: list_a,Y2: list_a] :
( ( member_list_a2 @ X @ S3 )
=> ( ( member_list_a2 @ Y2 @ S3 )
=> ( ( X != Y2 )
=> ( R2 @ X @ Y2 ) ) ) )
=> ( pairwise_list_a @ R2 @ S3 ) ) ).
% pairwiseI
thf(fact_571_pairwiseI,axiom,
! [S3: set_nat,R2: nat > nat > $o] :
( ! [X: nat,Y2: nat] :
( ( member_nat2 @ X @ S3 )
=> ( ( member_nat2 @ Y2 @ S3 )
=> ( ( X != Y2 )
=> ( R2 @ X @ Y2 ) ) ) )
=> ( pairwise_nat @ R2 @ S3 ) ) ).
% pairwiseI
thf(fact_572_pairwise__def,axiom,
( pairwise_nat
= ( ^ [R: nat > nat > $o,S: set_nat] :
! [X3: nat] :
( ( member_nat2 @ X3 @ S )
=> ! [Y3: nat] :
( ( member_nat2 @ Y3 @ S )
=> ( ( X3 != Y3 )
=> ( R @ X3 @ Y3 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_573_pairwise__def,axiom,
( pairwise_list_a
= ( ^ [R: list_a > list_a > $o,S: set_list_a] :
! [X3: list_a] :
( ( member_list_a2 @ X3 @ S )
=> ! [Y3: list_a] :
( ( member_list_a2 @ Y3 @ S )
=> ( ( X3 != Y3 )
=> ( R @ X3 @ Y3 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_574_transpose_Ocases,axiom,
! [X2: list_list_list_nat] :
( ( X2 != nil_list_list_nat )
=> ( ! [Xss: list_list_list_nat] :
( X2
!= ( cons_list_list_nat @ nil_list_nat @ Xss ) )
=> ~ ! [X: list_nat,Xs3: list_list_nat,Xss: list_list_list_nat] :
( X2
!= ( cons_list_list_nat @ ( cons_list_nat @ X @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_575_transpose_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X: nat,Xs3: list_nat,Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ ( cons_nat @ X @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_576_transpose_Ocases,axiom,
! [X2: list_list_a] :
( ( X2 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X2
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X: a,Xs3: list_a,Xss: list_list_a] :
( X2
!= ( cons_list_a @ ( cons_a @ X @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_577_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_578_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_579_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_580_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_581_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_582_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_583_min__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X @ Xs3 ) )
=> ( X2 = nil_nat ) ) ).
% min_list.cases
thf(fact_584_remdups__adj_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X: nat] :
( X2
!= ( cons_nat @ X @ nil_nat ) )
=> ~ ! [X: nat,Y2: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X @ ( cons_nat @ Y2 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_585_remdups__adj_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ( ! [X: a] :
( X2
!= ( cons_a @ X @ nil_a ) )
=> ~ ! [X: a,Y2: a,Xs3: list_a] :
( X2
!= ( cons_a @ X @ ( cons_a @ Y2 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_586_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y3: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_587_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y3: a,Ys3: list_a] :
( Xs
= ( cons_a @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_588_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X @ Xs3 ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X: nat,Xs3: list_nat,Y2: nat,Ys2: list_nat] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_589_list__induct2_H,axiom,
! [P: list_nat > list_a > $o,Xs: list_nat,Ys: list_a] :
( ( P @ nil_nat @ nil_a )
=> ( ! [X: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X @ Xs3 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P @ nil_nat @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X: nat,Xs3: list_nat,Y2: a,Ys2: list_a] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_nat @ X @ Xs3 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_590_list__induct2_H,axiom,
! [P: list_a > list_nat > $o,Xs: list_a,Ys: list_nat] :
( ( P @ nil_a @ nil_nat )
=> ( ! [X: a,Xs3: list_a] : ( P @ ( cons_a @ X @ Xs3 ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_a @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X: a,Xs3: list_a,Y2: nat,Ys2: list_nat] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_a @ X @ Xs3 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_591_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X: a,Xs3: list_a] : ( P @ ( cons_a @ X @ Xs3 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X: a,Xs3: list_a,Y2: a,Ys2: list_a] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_a @ X @ Xs3 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_592_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X: nat] : ( P @ ( cons_nat @ X @ nil_nat ) )
=> ( ! [X: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_593_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X: a] : ( P @ ( cons_a @ X @ nil_a ) )
=> ( ! [X: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_a @ X @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_594_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_595_extend_Osimps_I2_J,axiom,
! [A: set_nat,Uv: list_a] :
( ( extend_a @ A @ nil_nat @ Uv )
= ( insert_list_a2 @ nil_a @ bot_bot_set_list_a ) ) ).
% extend.simps(2)
thf(fact_596_pairwise__empty,axiom,
! [P: nat > nat > $o] : ( pairwise_nat @ P @ bot_bot_set_nat ) ).
% pairwise_empty
thf(fact_597_pairwise__empty,axiom,
! [P: list_a > list_a > $o] : ( pairwise_list_a @ P @ bot_bot_set_list_a ) ).
% pairwise_empty
thf(fact_598_pairwise__insert,axiom,
! [R3: nat > nat > $o,X2: nat,S2: set_nat] :
( ( pairwise_nat @ R3 @ ( insert_nat2 @ X2 @ S2 ) )
= ( ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ S2 )
& ( Y3 != X2 ) )
=> ( ( R3 @ X2 @ Y3 )
& ( R3 @ Y3 @ X2 ) ) )
& ( pairwise_nat @ R3 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_599_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_600_empty__set,axiom,
( bot_bot_set_list_a
= ( set_list_a2 @ nil_list_a ) ) ).
% empty_set
thf(fact_601_Nil__tl,axiom,
! [Xs: list_nat] :
( ( nil_nat
= ( tl_nat @ Xs ) )
= ( ( Xs = nil_nat )
| ? [X3: nat] :
( Xs
= ( cons_nat @ X3 @ nil_nat ) ) ) ) ).
% Nil_tl
thf(fact_602_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X3: a] :
( Xs
= ( cons_a @ X3 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_603_tl__Nil,axiom,
! [Xs: list_nat] :
( ( ( tl_nat @ Xs )
= nil_nat )
= ( ( Xs = nil_nat )
| ? [X3: nat] :
( Xs
= ( cons_nat @ X3 @ nil_nat ) ) ) ) ).
% tl_Nil
thf(fact_604_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X3: a] :
( Xs
= ( cons_a @ X3 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_605_list_Oset__sel_I2_J,axiom,
! [A2: list_a,X2: a] :
( ( A2 != nil_a )
=> ( ( member_a2 @ X2 @ ( set_a2 @ ( tl_a @ A2 ) ) )
=> ( member_a2 @ X2 @ ( set_a2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_606_list_Oset__sel_I2_J,axiom,
! [A2: list_nat,X2: nat] :
( ( A2 != nil_nat )
=> ( ( member_nat2 @ X2 @ ( set_nat2 @ ( tl_nat @ A2 ) ) )
=> ( member_nat2 @ X2 @ ( set_nat2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_607_null__rec_I1_J,axiom,
! [X2: nat,Xs: list_nat] :
~ ( null_nat @ ( cons_nat @ X2 @ Xs ) ) ).
% null_rec(1)
thf(fact_608_null__rec_I1_J,axiom,
! [X2: a,Xs: list_a] :
~ ( null_a @ ( cons_a @ X2 @ Xs ) ) ).
% null_rec(1)
thf(fact_609_pairwise__singleton,axiom,
! [P: nat > nat > $o,A: nat] : ( pairwise_nat @ P @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).
% pairwise_singleton
thf(fact_610_pairwise__singleton,axiom,
! [P: list_a > list_a > $o,A: list_a] : ( pairwise_list_a @ P @ ( insert_list_a2 @ A @ bot_bot_set_list_a ) ) ).
% pairwise_singleton
thf(fact_611_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_612_Collect__empty__eq__bot,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( P = bot_bot_list_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_613_listset_Osimps_I1_J,axiom,
( ( listset_a @ nil_set_a )
= ( insert_list_a2 @ nil_a @ bot_bot_set_list_a ) ) ).
% listset.simps(1)
thf(fact_614_lists__empty,axiom,
( ( lists_a @ bot_bot_set_a )
= ( insert_list_a2 @ nil_a @ bot_bot_set_list_a ) ) ).
% lists_empty
thf(fact_615_lists__empty,axiom,
( ( lists_nat @ bot_bot_set_nat )
= ( insert_list_nat2 @ nil_nat @ bot_bot_set_list_nat ) ) ).
% lists_empty
thf(fact_616_lists__empty,axiom,
( ( lists_list_a @ bot_bot_set_list_a )
= ( insert_list_list_a @ nil_list_a @ bot_bo1875519244922727510list_a ) ) ).
% lists_empty
thf(fact_617_hd__Cons__tl,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ Xs ) @ ( tl_nat @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_618_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_619_list_Ocollapse,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) )
= List ) ) ).
% list.collapse
thf(fact_620_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_621_union__set__fold,axiom,
! [Xs: list_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ A )
= ( fold_nat_set_nat @ insert_nat2 @ Xs @ A ) ) ).
% union_set_fold
thf(fact_622_lookup_Oelims,axiom,
! [X2: list_a,Xa: list_nat,Xb: a,Y: nat] :
( ( ( lookup_a_nat @ X2 @ Xa @ Xb )
= Y )
=> ( ! [X: a,Xs3: list_a] :
( ( X2
= ( cons_a @ X @ Xs3 ) )
=> ! [Y2: nat,Ys2: list_nat] :
( ( Xa
= ( cons_nat @ Y2 @ Ys2 ) )
=> ~ ( ( ( X = Xb )
=> ( Y = Y2 ) )
& ( ( X != Xb )
=> ( Y
= ( lookup_a_nat @ Xs3 @ Ys2 @ Xb ) ) ) ) ) )
=> ( ( ( X2 = nil_a )
=> ( Y != undefined_nat ) )
=> ~ ( ( Xa = nil_nat )
=> ( Y != undefined_nat ) ) ) ) ) ).
% lookup.elims
thf(fact_623_lookup_Oelims,axiom,
! [X2: list_a,Xa: list_a,Xb: a,Y: a] :
( ( ( lookup_a_a @ X2 @ Xa @ Xb )
= Y )
=> ( ! [X: a,Xs3: list_a] :
( ( X2
= ( cons_a @ X @ Xs3 ) )
=> ! [Y2: a,Ys2: list_a] :
( ( Xa
= ( cons_a @ Y2 @ Ys2 ) )
=> ~ ( ( ( X = Xb )
=> ( Y = Y2 ) )
& ( ( X != Xb )
=> ( Y
= ( lookup_a_a @ Xs3 @ Ys2 @ Xb ) ) ) ) ) )
=> ( ( ( X2 = nil_a )
=> ( Y != undefined_a ) )
=> ~ ( ( Xa = nil_a )
=> ( Y != undefined_a ) ) ) ) ) ).
% lookup.elims
thf(fact_624_lookup_Oelims,axiom,
! [X2: list_nat,Xa: list_a,Xb: nat,Y: a] :
( ( ( lookup_nat_a @ X2 @ Xa @ Xb )
= Y )
=> ( ! [X: nat,Xs3: list_nat] :
( ( X2
= ( cons_nat @ X @ Xs3 ) )
=> ! [Y2: a,Ys2: list_a] :
( ( Xa
= ( cons_a @ Y2 @ Ys2 ) )
=> ~ ( ( ( X = Xb )
=> ( Y = Y2 ) )
& ( ( X != Xb )
=> ( Y
= ( lookup_nat_a @ Xs3 @ Ys2 @ Xb ) ) ) ) ) )
=> ( ( ( X2 = nil_nat )
=> ( Y != undefined_a ) )
=> ~ ( ( Xa = nil_a )
=> ( Y != undefined_a ) ) ) ) ) ).
% lookup.elims
thf(fact_625_lookup_Oelims,axiom,
! [X2: list_nat,Xa: list_nat,Xb: nat,Y: nat] :
( ( ( lookup_nat_nat @ X2 @ Xa @ Xb )
= Y )
=> ( ! [X: nat,Xs3: list_nat] :
( ( X2
= ( cons_nat @ X @ Xs3 ) )
=> ! [Y2: nat,Ys2: list_nat] :
( ( Xa
= ( cons_nat @ Y2 @ Ys2 ) )
=> ~ ( ( ( X = Xb )
=> ( Y = Y2 ) )
& ( ( X != Xb )
=> ( Y
= ( lookup_nat_nat @ Xs3 @ Ys2 @ Xb ) ) ) ) ) )
=> ( ( ( X2 = nil_nat )
=> ( Y != undefined_nat ) )
=> ~ ( ( Xa = nil_nat )
=> ( Y != undefined_nat ) ) ) ) ) ).
% lookup.elims
thf(fact_626_UNIV__I,axiom,
! [X2: nat] : ( member_nat2 @ X2 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_627_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat2 @ C @ A )
& ( member_nat2 @ C @ B ) ) ) ).
% Int_iff
thf(fact_628_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ A )
=> ( ( member_nat2 @ C @ B )
=> ( member_nat2 @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_629_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat2 @ C @ A )
| ( member_nat2 @ C @ B ) ) ) ).
% Un_iff
thf(fact_630_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat2 @ C @ B )
=> ( member_nat2 @ C @ A ) )
=> ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_631_inf__bot__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_632_inf__bot__left,axiom,
! [X2: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ X2 )
= bot_bot_set_list_a ) ).
% inf_bot_left
thf(fact_633_inf__bot__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_634_inf__bot__right,axiom,
! [X2: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% inf_bot_right
thf(fact_635_inf__top_Oright__neutral,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
= A2 ) ).
% inf_top.right_neutral
thf(fact_636_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( A2 = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_637_inf__top_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_638_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= top_top_set_nat )
= ( ( A2 = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_639_top__eq__inf__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X2 @ Y ) )
= ( ( X2 = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_640_inf__eq__top__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X2 @ Y )
= top_top_set_nat )
= ( ( X2 = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_641_inf__top__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ top_top_set_nat )
= X2 ) ).
% inf_top_right
thf(fact_642_inf__top__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X2 )
= X2 ) ).
% inf_top_left
thf(fact_643_sup__bot__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_644_sup__bot__left,axiom,
! [X2: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_645_sup__bot__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% sup_bot_right
thf(fact_646_sup__bot__right,axiom,
! [X2: set_list_a] :
( ( sup_sup_set_list_a @ X2 @ bot_bot_set_list_a )
= X2 ) ).
% sup_bot_right
thf(fact_647_bot__eq__sup__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X2 @ Y ) )
= ( ( X2 = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_648_bot__eq__sup__iff,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( bot_bot_set_list_a
= ( sup_sup_set_list_a @ X2 @ Y ) )
= ( ( X2 = bot_bot_set_list_a )
& ( Y = bot_bot_set_list_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_649_sup__eq__bot__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X2 @ Y )
= bot_bot_set_nat )
= ( ( X2 = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_650_sup__eq__bot__iff,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( ( sup_sup_set_list_a @ X2 @ Y )
= bot_bot_set_list_a )
= ( ( X2 = bot_bot_set_list_a )
& ( Y = bot_bot_set_list_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_651_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_652_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ( sup_sup_set_list_a @ A2 @ B2 )
= bot_bot_set_list_a )
= ( ( A2 = bot_bot_set_list_a )
& ( B2 = bot_bot_set_list_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_653_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_654_sup__bot_Oleft__neutral,axiom,
! [A2: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_655_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_656_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( bot_bot_set_list_a
= ( sup_sup_set_list_a @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_list_a )
& ( B2 = bot_bot_set_list_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_657_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_658_sup__bot_Oright__neutral,axiom,
! [A2: set_list_a] :
( ( sup_sup_set_list_a @ A2 @ bot_bot_set_list_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_659_sup__top__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ top_top_set_nat )
= top_top_set_nat ) ).
% sup_top_right
thf(fact_660_sup__top__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X2 )
= top_top_set_nat ) ).
% sup_top_left
thf(fact_661_Int__UNIV,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 ) ) ) ).
% Int_UNIV
thf(fact_662_Un__empty,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_663_Un__empty,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ( sup_sup_set_list_a @ A @ B )
= bot_bot_set_list_a )
= ( ( A = bot_bot_set_list_a )
& ( B = bot_bot_set_list_a ) ) ) ).
% Un_empty
thf(fact_664_Int__insert__left__if0,axiom,
! [A2: nat,C3: set_nat,B: set_nat] :
( ~ ( member_nat2 @ A2 @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A2 @ B ) @ C3 )
= ( inf_inf_set_nat @ B @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_665_Int__insert__left__if1,axiom,
! [A2: nat,C3: set_nat,B: set_nat] :
( ( member_nat2 @ A2 @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A2 @ B ) @ C3 )
= ( insert_nat2 @ A2 @ ( inf_inf_set_nat @ B @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_666_insert__inter__insert,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat2 @ A2 @ A ) @ ( insert_nat2 @ A2 @ B ) )
= ( insert_nat2 @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_667_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat2 @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat2 @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_668_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat2 @ A2 @ B ) )
= ( insert_nat2 @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_669_Un__insert__left,axiom,
! [A2: nat,B: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat2 @ A2 @ B ) @ C3 )
= ( insert_nat2 @ A2 @ ( sup_sup_set_nat @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_670_Un__insert__right,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( insert_nat2 @ A2 @ B ) )
= ( insert_nat2 @ A2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_671_Un__Diff__cancel,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_672_Un__Diff__cancel2,axiom,
! [B: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B @ A ) @ A )
= ( sup_sup_set_nat @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_673_lists__UNIV,axiom,
( ( lists_nat @ top_top_set_nat )
= top_top_set_list_nat ) ).
% lists_UNIV
thf(fact_674_Cons__in__lists__iff,axiom,
! [X2: nat,Xs: list_nat,A: set_nat] :
( ( member_list_nat2 @ ( cons_nat @ X2 @ Xs ) @ ( lists_nat @ A ) )
= ( ( member_nat2 @ X2 @ A )
& ( member_list_nat2 @ Xs @ ( lists_nat @ A ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_675_Cons__in__lists__iff,axiom,
! [X2: a,Xs: list_a,A: set_a] :
( ( member_list_a2 @ ( cons_a @ X2 @ Xs ) @ ( lists_a @ A ) )
= ( ( member_a2 @ X2 @ A )
& ( member_list_a2 @ Xs @ ( lists_a @ A ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_676_in__listsI,axiom,
! [Xs: list_nat,A: set_nat] :
( ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X @ A ) )
=> ( member_list_nat2 @ Xs @ ( lists_nat @ A ) ) ) ).
% in_listsI
thf(fact_677_Diff__UNIV,axiom,
! [A: set_list_a] :
( ( minus_646659088055828811list_a @ A @ top_top_set_list_a )
= bot_bot_set_list_a ) ).
% Diff_UNIV
thf(fact_678_Diff__UNIV,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_679_disjoint__insert_I2_J,axiom,
! [A: set_nat,B2: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat2 @ B2 @ B ) ) )
= ( ~ ( member_nat2 @ B2 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_680_disjoint__insert_I2_J,axiom,
! [A: set_list_a,B2: list_a,B: set_list_a] :
( ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A @ ( insert_list_a2 @ B2 @ B ) ) )
= ( ~ ( member_list_a2 @ B2 @ A )
& ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_681_disjoint__insert_I1_J,axiom,
! [B: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat2 @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat2 @ A2 @ B )
& ( ( inf_inf_set_nat @ B @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_682_disjoint__insert_I1_J,axiom,
! [B: set_list_a,A2: list_a,A: set_list_a] :
( ( ( inf_inf_set_list_a @ B @ ( insert_list_a2 @ A2 @ A ) )
= bot_bot_set_list_a )
= ( ~ ( member_list_a2 @ A2 @ B )
& ( ( inf_inf_set_list_a @ B @ A )
= bot_bot_set_list_a ) ) ) ).
% disjoint_insert(1)
thf(fact_683_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat2 @ A2 @ A ) @ B ) )
= ( ~ ( member_nat2 @ A2 @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_684_insert__disjoint_I2_J,axiom,
! [A2: list_a,A: set_list_a,B: set_list_a] :
( ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ ( insert_list_a2 @ A2 @ A ) @ B ) )
= ( ~ ( member_list_a2 @ A2 @ B )
& ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_685_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat2 @ A2 @ A ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat2 @ A2 @ B )
& ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_686_insert__disjoint_I1_J,axiom,
! [A2: list_a,A: set_list_a,B: set_list_a] :
( ( ( inf_inf_set_list_a @ ( insert_list_a2 @ A2 @ A ) @ B )
= bot_bot_set_list_a )
= ( ~ ( member_list_a2 @ A2 @ B )
& ( ( inf_inf_set_list_a @ A @ B )
= bot_bot_set_list_a ) ) ) ).
% insert_disjoint(1)
thf(fact_687_Diff__disjoint,axiom,
! [A: set_list_a,B: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( minus_646659088055828811list_a @ B @ A ) )
= bot_bot_set_list_a ) ).
% Diff_disjoint
thf(fact_688_Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_689_Diff__Un,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( minus_minus_set_nat @ A @ ( sup_sup_set_nat @ B @ C3 ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ C3 ) ) ) ).
% Diff_Un
thf(fact_690_Diff__Int,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( minus_minus_set_nat @ A @ ( inf_inf_set_nat @ B @ C3 ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ C3 ) ) ) ).
% Diff_Int
thf(fact_691_Int__Diff__Un,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_692_Un__Diff__Int,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( inf_inf_set_nat @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_693_Int__UNIV__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% Int_UNIV_right
thf(fact_694_Un__UNIV__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_695_Int__UNIV__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B )
= B ) ).
% Int_UNIV_left
thf(fact_696_Un__UNIV__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_697_UNIV__witness,axiom,
? [X: nat] : ( member_nat2 @ X @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_698_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X: nat] : ( member_nat2 @ X @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_699_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat2 @ C @ B ) ) ).
% IntD2
thf(fact_700_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat2 @ C @ A ) ) ).
% IntD1
thf(fact_701_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat2 @ C @ B )
=> ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_702_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_703_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat2 @ C @ A )
=> ~ ( member_nat2 @ C @ B ) ) ) ).
% IntE
thf(fact_704_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ B ) ) ) ).
% UnE
thf(fact_705_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_706_Un__empty__left,axiom,
! [B: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ B )
= B ) ).
% Un_empty_left
thf(fact_707_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_708_Un__empty__right,axiom,
! [A: set_list_a] :
( ( sup_sup_set_list_a @ A @ bot_bot_set_list_a )
= A ) ).
% Un_empty_right
thf(fact_709_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ~ ( member_nat2 @ X @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_710_Int__emptyI,axiom,
! [A: set_list_a,B: set_list_a] :
( ! [X: list_a] :
( ( member_list_a2 @ X @ A )
=> ~ ( member_list_a2 @ X @ B ) )
=> ( ( inf_inf_set_list_a @ A @ B )
= bot_bot_set_list_a ) ) ).
% Int_emptyI
thf(fact_711_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ~ ( member_nat2 @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_712_disjoint__iff,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
( ( member_list_a2 @ X3 @ A )
=> ~ ( member_list_a2 @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_713_Int__empty__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_714_Int__empty__left,axiom,
! [B: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ B )
= bot_bot_set_list_a ) ).
% Int_empty_left
thf(fact_715_Int__empty__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_716_Int__empty__right,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% Int_empty_right
thf(fact_717_disjoint__iff__not__equal,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ! [Y3: nat] :
( ( member_nat2 @ Y3 @ B )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_718_disjoint__iff__not__equal,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
( ( member_list_a2 @ X3 @ A )
=> ! [Y3: list_a] :
( ( member_list_a2 @ Y3 @ B )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_719_Int__insert__left,axiom,
! [A2: nat,C3: set_nat,B: set_nat] :
( ( ( member_nat2 @ A2 @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A2 @ B ) @ C3 )
= ( insert_nat2 @ A2 @ ( inf_inf_set_nat @ B @ C3 ) ) ) )
& ( ~ ( member_nat2 @ A2 @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A2 @ B ) @ C3 )
= ( inf_inf_set_nat @ B @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_720_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( member_nat2 @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat2 @ A2 @ B ) )
= ( insert_nat2 @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) )
& ( ~ ( member_nat2 @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat2 @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_721_Un__Diff,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C3 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ C3 ) @ ( minus_minus_set_nat @ B @ C3 ) ) ) ).
% Un_Diff
thf(fact_722_list_Osel_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_723_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_724_Int__Diff,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C3 )
= ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ C3 ) ) ) ).
% Int_Diff
thf(fact_725_Diff__Int2,axiom,
! [A: set_nat,C3: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C3 ) @ ( inf_inf_set_nat @ B @ C3 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C3 ) @ B ) ) ).
% Diff_Int2
thf(fact_726_Diff__Diff__Int,axiom,
! [A: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( minus_minus_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_727_Diff__Int__distrib,axiom,
! [C3: set_nat,A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ C3 @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ C3 @ A ) @ ( inf_inf_set_nat @ C3 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_728_Diff__Int__distrib2,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C3 )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C3 ) @ ( inf_inf_set_nat @ B @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_729_empty__not__UNIV,axiom,
bot_bot_set_list_a != top_top_set_list_a ).
% empty_not_UNIV
thf(fact_730_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_731_insert__UNIV,axiom,
! [X2: nat] :
( ( insert_nat2 @ X2 @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_732_listsE,axiom,
! [X2: nat,L: list_nat,A: set_nat] :
( ( member_list_nat2 @ ( cons_nat @ X2 @ L ) @ ( lists_nat @ A ) )
=> ~ ( ( member_nat2 @ X2 @ A )
=> ~ ( member_list_nat2 @ L @ ( lists_nat @ A ) ) ) ) ).
% listsE
thf(fact_733_listsE,axiom,
! [X2: a,L: list_a,A: set_a] :
( ( member_list_a2 @ ( cons_a @ X2 @ L ) @ ( lists_a @ A ) )
=> ~ ( ( member_a2 @ X2 @ A )
=> ~ ( member_list_a2 @ L @ ( lists_a @ A ) ) ) ) ).
% listsE
thf(fact_734_lists_OCons,axiom,
! [A2: nat,A: set_nat,L: list_nat] :
( ( member_nat2 @ A2 @ A )
=> ( ( member_list_nat2 @ L @ ( lists_nat @ A ) )
=> ( member_list_nat2 @ ( cons_nat @ A2 @ L ) @ ( lists_nat @ A ) ) ) ) ).
% lists.Cons
thf(fact_735_lists_OCons,axiom,
! [A2: a,A: set_a,L: list_a] :
( ( member_a2 @ A2 @ A )
=> ( ( member_list_a2 @ L @ ( lists_a @ A ) )
=> ( member_list_a2 @ ( cons_a @ A2 @ L ) @ ( lists_a @ A ) ) ) ) ).
% lists.Cons
thf(fact_736_in__listsD,axiom,
! [Xs: list_nat,A: set_nat] :
( ( member_list_nat2 @ Xs @ ( lists_nat @ A ) )
=> ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X4 @ A ) ) ) ).
% in_listsD
thf(fact_737_in__lists__conv__set,axiom,
! [Xs: list_nat,A: set_nat] :
( ( member_list_nat2 @ Xs @ ( lists_nat @ A ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X3 @ A ) ) ) ) ).
% in_lists_conv_set
thf(fact_738_insert__is__Un,axiom,
( insert_nat2
= ( ^ [A4: nat] : ( sup_sup_set_nat @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_739_insert__is__Un,axiom,
( insert_list_a2
= ( ^ [A4: list_a] : ( sup_sup_set_list_a @ ( insert_list_a2 @ A4 @ bot_bot_set_list_a ) ) ) ) ).
% insert_is_Un
thf(fact_740_Un__singleton__iff,axiom,
! [A: set_nat,B: set_nat,X2: nat] :
( ( ( sup_sup_set_nat @ A @ B )
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_741_Un__singleton__iff,axiom,
! [A: set_list_a,B: set_list_a,X2: list_a] :
( ( ( sup_sup_set_list_a @ A @ B )
= ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) )
= ( ( ( A = bot_bot_set_list_a )
& ( B
= ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) )
| ( ( A
= ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) )
& ( B = bot_bot_set_list_a ) )
| ( ( A
= ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) )
& ( B
= ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_742_singleton__Un__iff,axiom,
! [X2: nat,A: set_nat,B: set_nat] :
( ( ( insert_nat2 @ X2 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A @ B ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_743_singleton__Un__iff,axiom,
! [X2: list_a,A: set_list_a,B: set_list_a] :
( ( ( insert_list_a2 @ X2 @ bot_bot_set_list_a )
= ( sup_sup_set_list_a @ A @ B ) )
= ( ( ( A = bot_bot_set_list_a )
& ( B
= ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) )
| ( ( A
= ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) )
& ( B = bot_bot_set_list_a ) )
| ( ( A
= ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) )
& ( B
= ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_744_Diff__triv,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B )
= bot_bot_set_list_a )
=> ( ( minus_646659088055828811list_a @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_745_Diff__triv,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_746_Int__Diff__disjoint,axiom,
! [A: set_list_a,B: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A @ B ) @ ( minus_646659088055828811list_a @ A @ B ) )
= bot_bot_set_list_a ) ).
% Int_Diff_disjoint
thf(fact_747_Int__Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ B ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_748_list_Oset__sel_I1_J,axiom,
! [A2: list_nat] :
( ( A2 != nil_nat )
=> ( member_nat2 @ ( hd_nat @ A2 ) @ ( set_nat2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_749_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat2 @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_750_list_Oexpand,axiom,
! [List: list_a,List2: list_a] :
( ( ( List = nil_a )
= ( List2 = nil_a ) )
=> ( ( ( List != nil_a )
=> ( ( List2 != nil_a )
=> ( ( ( hd_a @ List )
= ( hd_a @ List2 ) )
& ( ( tl_a @ List )
= ( tl_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_751_lists_Ocases,axiom,
! [A2: list_nat,A: set_nat] :
( ( member_list_nat2 @ A2 @ ( lists_nat @ A ) )
=> ( ( A2 != nil_nat )
=> ~ ! [A5: nat,L2: list_nat] :
( ( A2
= ( cons_nat @ A5 @ L2 ) )
=> ( ( member_nat2 @ A5 @ A )
=> ~ ( member_list_nat2 @ L2 @ ( lists_nat @ A ) ) ) ) ) ) ).
% lists.cases
thf(fact_752_lists_Ocases,axiom,
! [A2: list_a,A: set_a] :
( ( member_list_a2 @ A2 @ ( lists_a @ A ) )
=> ( ( A2 != nil_a )
=> ~ ! [A5: a,L2: list_a] :
( ( A2
= ( cons_a @ A5 @ L2 ) )
=> ( ( member_a2 @ A5 @ A )
=> ~ ( member_list_a2 @ L2 @ ( lists_a @ A ) ) ) ) ) ) ).
% lists.cases
thf(fact_753_lists_Osimps,axiom,
! [A2: list_nat,A: set_nat] :
( ( member_list_nat2 @ A2 @ ( lists_nat @ A ) )
= ( ( A2 = nil_nat )
| ? [A4: nat,L3: list_nat] :
( ( A2
= ( cons_nat @ A4 @ L3 ) )
& ( member_nat2 @ A4 @ A )
& ( member_list_nat2 @ L3 @ ( lists_nat @ A ) ) ) ) ) ).
% lists.simps
thf(fact_754_lists_Osimps,axiom,
! [A2: list_a,A: set_a] :
( ( member_list_a2 @ A2 @ ( lists_a @ A ) )
= ( ( A2 = nil_a )
| ? [A4: a,L3: list_a] :
( ( A2
= ( cons_a @ A4 @ L3 ) )
& ( member_a2 @ A4 @ A )
& ( member_list_a2 @ L3 @ ( lists_a @ A ) ) ) ) ) ).
% lists.simps
thf(fact_755_UNIV__coset,axiom,
( top_top_set_nat
= ( coset_nat @ nil_nat ) ) ).
% UNIV_coset
thf(fact_756_lookup_Osimps_I2_J,axiom,
! [Uv: list_a,Uw: nat] :
( ( lookup_nat_a @ nil_nat @ Uv @ Uw )
= undefined_a ) ).
% lookup.simps(2)
thf(fact_757_lookup_Osimps_I2_J,axiom,
! [Uv: list_nat,Uw: nat] :
( ( lookup_nat_nat @ nil_nat @ Uv @ Uw )
= undefined_nat ) ).
% lookup.simps(2)
thf(fact_758_lookup_Osimps_I3_J,axiom,
! [Uu: list_nat,Uw: nat] :
( ( lookup_nat_a @ Uu @ nil_a @ Uw )
= undefined_a ) ).
% lookup.simps(3)
thf(fact_759_lookup_Osimps_I3_J,axiom,
! [Uu: list_nat,Uw: nat] :
( ( lookup_nat_nat @ Uu @ nil_nat @ Uw )
= undefined_nat ) ).
% lookup.simps(3)
thf(fact_760_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P2: nat > $o,Xs2: list_nat] :
? [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs2 ) )
& ( P2 @ X3 )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ ( set_nat2 @ Xs2 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_761_list_Oexhaust__sel,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( List
= ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_762_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_763_boolean__algebra_Odisj__one__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X2 )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_764_boolean__algebra_Odisj__one__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ top_top_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_765_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_766_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ X2 )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_767_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_768_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_769_set__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_770_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_list_a,X2: set_list_a,Y: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ X2 )
= bot_bot_set_list_a )
=> ( ( ( sup_sup_set_list_a @ A2 @ X2 )
= top_top_set_list_a )
=> ( ( ( inf_inf_set_list_a @ A2 @ Y )
= bot_bot_set_list_a )
=> ( ( ( sup_sup_set_list_a @ A2 @ Y )
= top_top_set_list_a )
=> ( X2 = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_771_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_nat,X2: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ X2 )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A2 @ X2 )
= top_top_set_nat )
=> ( ( ( inf_inf_set_nat @ A2 @ Y )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A2 @ Y )
= top_top_set_nat )
=> ( X2 = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_772_can__select__set__list__ex1,axiom,
! [P: nat > $o,A: list_nat] :
( ( can_select_nat @ P @ ( set_nat2 @ A ) )
= ( list_ex1_nat @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_773_iso__tuple__UNIV__I,axiom,
! [X2: nat] : ( member_nat2 @ X2 @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_774_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_775_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X3: nat] : ( member_nat2 @ X3 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_776_can__select__def,axiom,
( can_select_nat
= ( ^ [P2: nat > $o,A3: set_nat] :
? [X3: nat] :
( ( member_nat2 @ X3 @ A3 )
& ( P2 @ X3 )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ A3 )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_777_boolean__algebra_Oconj__one__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ top_top_set_nat )
= X2 ) ).
% boolean_algebra.conj_one_right
thf(fact_778_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_779_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_list_a] :
( ( sup_sup_set_list_a @ X2 @ bot_bot_set_list_a )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_780_Sup__fin_Oset__eq__fold,axiom,
! [X2: nat,Xs: list_nat] :
( ( lattic1093996805478795353in_nat @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) )
= ( fold_nat_nat @ sup_sup_nat @ Xs @ X2 ) ) ).
% Sup_fin.set_eq_fold
thf(fact_781_Cons__in__subseqsD,axiom,
! [Y: nat,Ys: list_nat,Xs: list_nat] :
( ( member_list_nat2 @ ( cons_nat @ Y @ Ys ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
=> ( member_list_nat2 @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_782_Cons__in__subseqsD,axiom,
! [Y: a,Ys: list_a,Xs: list_a] :
( ( member_list_a2 @ ( cons_a @ Y @ Ys ) @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) )
=> ( member_list_a2 @ Ys @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_783_Sup__fin_Osingleton,axiom,
! [X2: nat] :
( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% Sup_fin.singleton
thf(fact_784_Inf__fin_Oset__eq__fold,axiom,
! [X2: nat,Xs: list_nat] :
( ( lattic5238388535129920115in_nat @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) )
= ( fold_nat_nat @ inf_inf_nat @ Xs @ X2 ) ) ).
% Inf_fin.set_eq_fold
thf(fact_785_Cons__in__shuffles__iff,axiom,
! [Z: nat,Zs: list_nat,Xs: list_nat,Ys: list_nat] :
( ( member_list_nat2 @ ( cons_nat @ Z @ Zs ) @ ( shuffles_nat @ Xs @ Ys ) )
= ( ( ( Xs != nil_nat )
& ( ( hd_nat @ Xs )
= Z )
& ( member_list_nat2 @ Zs @ ( shuffles_nat @ ( tl_nat @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_nat )
& ( ( hd_nat @ Ys )
= Z )
& ( member_list_nat2 @ Zs @ ( shuffles_nat @ Xs @ ( tl_nat @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_786_Cons__in__shuffles__iff,axiom,
! [Z: a,Zs: list_a,Xs: list_a,Ys: list_a] :
( ( member_list_a2 @ ( cons_a @ Z @ Zs ) @ ( shuffles_a @ Xs @ Ys ) )
= ( ( ( Xs != nil_a )
& ( ( hd_a @ Xs )
= Z )
& ( member_list_a2 @ Zs @ ( shuffles_a @ ( tl_a @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_a )
& ( ( hd_a @ Ys )
= Z )
& ( member_list_a2 @ Zs @ ( shuffles_a @ Xs @ ( tl_a @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_787_distinct__adj__Cons,axiom,
! [X2: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X2 @ Xs ) )
= ( ( Xs = nil_nat )
| ( ( X2
!= ( hd_nat @ Xs ) )
& ( distinct_adj_nat @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_788_distinct__adj__Cons,axiom,
! [X2: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X2 @ Xs ) )
= ( ( Xs = nil_a )
| ( ( X2
!= ( hd_a @ Xs ) )
& ( distinct_adj_a @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_789_distinct__adj__Cons__Cons,axiom,
! [X2: nat,Y: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Xs ) ) )
= ( ( X2 != Y )
& ( distinct_adj_nat @ ( cons_nat @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_790_distinct__adj__Cons__Cons,axiom,
! [X2: a,Y: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X2 @ ( cons_a @ Y @ Xs ) ) )
= ( ( X2 != Y )
& ( distinct_adj_a @ ( cons_a @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_791_Inf__fin_Osingleton,axiom,
! [X2: nat] :
( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% Inf_fin.singleton
thf(fact_792_Cons__in__shuffles__leftI,axiom,
! [Zs: list_nat,Xs: list_nat,Ys: list_nat,Z: nat] :
( ( member_list_nat2 @ Zs @ ( shuffles_nat @ Xs @ Ys ) )
=> ( member_list_nat2 @ ( cons_nat @ Z @ Zs ) @ ( shuffles_nat @ ( cons_nat @ Z @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_793_Cons__in__shuffles__leftI,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a,Z: a] :
( ( member_list_a2 @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( member_list_a2 @ ( cons_a @ Z @ Zs ) @ ( shuffles_a @ ( cons_a @ Z @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_794_Cons__in__shuffles__rightI,axiom,
! [Zs: list_nat,Xs: list_nat,Ys: list_nat,Z: nat] :
( ( member_list_nat2 @ Zs @ ( shuffles_nat @ Xs @ Ys ) )
=> ( member_list_nat2 @ ( cons_nat @ Z @ Zs ) @ ( shuffles_nat @ Xs @ ( cons_nat @ Z @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_795_Cons__in__shuffles__rightI,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a,Z: a] :
( ( member_list_a2 @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( member_list_a2 @ ( cons_a @ Z @ Zs ) @ ( shuffles_a @ Xs @ ( cons_a @ Z @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_796_distinct__adj__ConsD,axiom,
! [X2: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X2 @ Xs ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_797_distinct__adj__ConsD,axiom,
! [X2: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X2 @ Xs ) )
=> ( distinct_adj_a @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_798_shufflesE,axiom,
! [Zs: list_nat,Xs: list_nat,Ys: list_nat] :
( ( member_list_nat2 @ Zs @ ( shuffles_nat @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys != nil_nat ) )
=> ( ( ( Zs = Ys )
=> ( Xs != nil_nat ) )
=> ( ! [X: nat,Xs4: list_nat] :
( ( Xs
= ( cons_nat @ X @ Xs4 ) )
=> ! [Z3: nat,Zs2: list_nat] :
( ( Zs
= ( cons_nat @ Z3 @ Zs2 ) )
=> ( ( X = Z3 )
=> ~ ( member_list_nat2 @ Zs2 @ ( shuffles_nat @ Xs4 @ Ys ) ) ) ) )
=> ~ ! [Y2: nat,Ys4: list_nat] :
( ( Ys
= ( cons_nat @ Y2 @ Ys4 ) )
=> ! [Z3: nat,Zs2: list_nat] :
( ( Zs
= ( cons_nat @ Z3 @ Zs2 ) )
=> ( ( Y2 = Z3 )
=> ~ ( member_list_nat2 @ Zs2 @ ( shuffles_nat @ Xs @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_799_shufflesE,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a] :
( ( member_list_a2 @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys != nil_a ) )
=> ( ( ( Zs = Ys )
=> ( Xs != nil_a ) )
=> ( ! [X: a,Xs4: list_a] :
( ( Xs
= ( cons_a @ X @ Xs4 ) )
=> ! [Z3: a,Zs2: list_a] :
( ( Zs
= ( cons_a @ Z3 @ Zs2 ) )
=> ( ( X = Z3 )
=> ~ ( member_list_a2 @ Zs2 @ ( shuffles_a @ Xs4 @ Ys ) ) ) ) )
=> ~ ! [Y2: a,Ys4: list_a] :
( ( Ys
= ( cons_a @ Y2 @ Ys4 ) )
=> ! [Z3: a,Zs2: list_a] :
( ( Zs
= ( cons_a @ Z3 @ Zs2 ) )
=> ( ( Y2 = Z3 )
=> ~ ( member_list_a2 @ Zs2 @ ( shuffles_a @ Xs @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_800_set__shuffles,axiom,
! [Zs: list_nat,Xs: list_nat,Ys: list_nat] :
( ( member_list_nat2 @ Zs @ ( shuffles_nat @ Xs @ Ys ) )
=> ( ( set_nat2 @ Zs )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_801_distinct__adj__singleton,axiom,
! [X2: nat] : ( distinct_adj_nat @ ( cons_nat @ X2 @ nil_nat ) ) ).
% distinct_adj_singleton
thf(fact_802_distinct__adj__singleton,axiom,
! [X2: a] : ( distinct_adj_a @ ( cons_a @ X2 @ nil_a ) ) ).
% distinct_adj_singleton
thf(fact_803_shuffles_Osimps_I1_J,axiom,
! [Ys: list_a] :
( ( shuffles_a @ nil_a @ Ys )
= ( insert_list_a2 @ Ys @ bot_bot_set_list_a ) ) ).
% shuffles.simps(1)
thf(fact_804_shuffles_Osimps_I2_J,axiom,
! [Xs: list_a] :
( ( shuffles_a @ Xs @ nil_a )
= ( insert_list_a2 @ Xs @ bot_bot_set_list_a ) ) ).
% shuffles.simps(2)
thf(fact_805_shuffles_Oelims,axiom,
! [X2: list_nat,Xa: list_nat,Y: set_list_nat] :
( ( ( shuffles_nat @ X2 @ Xa )
= Y )
=> ( ( ( X2 = nil_nat )
=> ( Y
!= ( insert_list_nat2 @ Xa @ bot_bot_set_list_nat ) ) )
=> ( ( ( Xa = nil_nat )
=> ( Y
!= ( insert_list_nat2 @ X2 @ bot_bot_set_list_nat ) ) )
=> ~ ! [X: nat,Xs3: list_nat] :
( ( X2
= ( cons_nat @ X @ Xs3 ) )
=> ! [Y2: nat,Ys2: list_nat] :
( ( Xa
= ( cons_nat @ Y2 @ Ys2 ) )
=> ( Y
!= ( sup_sup_set_list_nat @ ( image_7976474329151083847st_nat @ ( cons_nat @ X ) @ ( shuffles_nat @ Xs3 @ ( cons_nat @ Y2 @ Ys2 ) ) ) @ ( image_7976474329151083847st_nat @ ( cons_nat @ Y2 ) @ ( shuffles_nat @ ( cons_nat @ X @ Xs3 ) @ Ys2 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_806_shuffles_Oelims,axiom,
! [X2: list_a,Xa: list_a,Y: set_list_a] :
( ( ( shuffles_a @ X2 @ Xa )
= Y )
=> ( ( ( X2 = nil_a )
=> ( Y
!= ( insert_list_a2 @ Xa @ bot_bot_set_list_a ) ) )
=> ( ( ( Xa = nil_a )
=> ( Y
!= ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) )
=> ~ ! [X: a,Xs3: list_a] :
( ( X2
= ( cons_a @ X @ Xs3 ) )
=> ! [Y2: a,Ys2: list_a] :
( ( Xa
= ( cons_a @ Y2 @ Ys2 ) )
=> ( Y
!= ( sup_sup_set_list_a @ ( image_list_a_list_a @ ( cons_a @ X ) @ ( shuffles_a @ Xs3 @ ( cons_a @ Y2 @ Ys2 ) ) ) @ ( image_list_a_list_a @ ( cons_a @ Y2 ) @ ( shuffles_a @ ( cons_a @ X @ Xs3 ) @ Ys2 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_807_Sup__fin_Oinsert__remove,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X2 @ A ) )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X2 @ A ) )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_808_Sup__fin_Oremove,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ X2 @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_809_Inf__fin_Oinsert__remove,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X2 @ A ) )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X2 @ A ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_810_Inf__fin_Oremove,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ X2 @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_811_image__eqI,axiom,
! [B2: nat,F: nat > nat,X2: nat,A: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat2 @ X2 @ A )
=> ( member_nat2 @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_812_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_813_image__empty,axiom,
! [F: nat > list_a] :
( ( image_nat_list_a @ F @ bot_bot_set_nat )
= bot_bot_set_list_a ) ).
% image_empty
thf(fact_814_image__empty,axiom,
! [F: list_a > nat] :
( ( image_list_a_nat @ F @ bot_bot_set_list_a )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_815_image__empty,axiom,
! [F: list_a > list_a] :
( ( image_list_a_list_a @ F @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% image_empty
thf(fact_816_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_817_empty__is__image,axiom,
! [F: list_a > nat,A: set_list_a] :
( ( bot_bot_set_nat
= ( image_list_a_nat @ F @ A ) )
= ( A = bot_bot_set_list_a ) ) ).
% empty_is_image
thf(fact_818_empty__is__image,axiom,
! [F: nat > list_a,A: set_nat] :
( ( bot_bot_set_list_a
= ( image_nat_list_a @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_819_empty__is__image,axiom,
! [F: list_a > list_a,A: set_list_a] :
( ( bot_bot_set_list_a
= ( image_list_a_list_a @ F @ A ) )
= ( A = bot_bot_set_list_a ) ) ).
% empty_is_image
thf(fact_820_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_821_image__is__empty,axiom,
! [F: list_a > nat,A: set_list_a] :
( ( ( image_list_a_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_list_a ) ) ).
% image_is_empty
thf(fact_822_image__is__empty,axiom,
! [F: nat > list_a,A: set_nat] :
( ( ( image_nat_list_a @ F @ A )
= bot_bot_set_list_a )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_823_image__is__empty,axiom,
! [F: list_a > list_a,A: set_list_a] :
( ( ( image_list_a_list_a @ F @ A )
= bot_bot_set_list_a )
= ( A = bot_bot_set_list_a ) ) ).
% image_is_empty
thf(fact_824_image__insert,axiom,
! [F: nat > nat,A2: nat,B: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat2 @ A2 @ B ) )
= ( insert_nat2 @ ( F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_825_insert__image,axiom,
! [X2: nat,A: set_nat,F: nat > nat] :
( ( member_nat2 @ X2 @ A )
=> ( ( insert_nat2 @ ( F @ X2 ) @ ( image_nat_nat @ F @ A ) )
= ( image_nat_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_826_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_827_inf__Sup__absorb,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ( ( inf_inf_nat @ A2 @ ( lattic1093996805478795353in_nat @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_828_sup__Inf__absorb,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_829_Inf__fin_Oinsert,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X2 @ A ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_830_Sup__fin_Oinsert,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X2 @ A ) )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_831_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat2 @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat2 @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_832_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > nat] :
( ( member_nat2 @ X2 @ A )
=> ( member_nat2 @ ( F @ X2 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_833_range__eqI,axiom,
! [B2: nat,F: nat > nat,X2: nat] :
( ( B2
= ( F @ X2 ) )
=> ( member_nat2 @ B2 @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_834_rangeI,axiom,
! [F: nat > nat,X2: nat] : ( member_nat2 @ ( F @ X2 ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_835_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_836_Inf__fin_Ohom__commute,axiom,
! [H: nat > nat,N: set_nat] :
( ! [X: nat,Y2: nat] :
( ( H @ ( inf_inf_nat @ X @ Y2 ) )
= ( inf_inf_nat @ ( H @ X ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_nat @ N )
=> ( ( N != bot_bot_set_nat )
=> ( ( H @ ( lattic5238388535129920115in_nat @ N ) )
= ( lattic5238388535129920115in_nat @ ( image_nat_nat @ H @ N ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_837_Sup__fin_Ohom__commute,axiom,
! [H: nat > nat,N: set_nat] :
( ! [X: nat,Y2: nat] :
( ( H @ ( sup_sup_nat @ X @ Y2 ) )
= ( sup_sup_nat @ ( H @ X ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_nat @ N )
=> ( ( N != bot_bot_set_nat )
=> ( ( H @ ( lattic1093996805478795353in_nat @ N ) )
= ( lattic1093996805478795353in_nat @ ( image_nat_nat @ H @ N ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_838_shuffles_Osimps_I3_J,axiom,
! [X2: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( shuffles_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( sup_sup_set_list_nat @ ( image_7976474329151083847st_nat @ ( cons_nat @ X2 ) @ ( shuffles_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) ) @ ( image_7976474329151083847st_nat @ ( cons_nat @ Y ) @ ( shuffles_nat @ ( cons_nat @ X2 @ Xs ) @ Ys ) ) ) ) ).
% shuffles.simps(3)
thf(fact_839_shuffles_Osimps_I3_J,axiom,
! [X2: a,Xs: list_a,Y: a,Ys: list_a] :
( ( shuffles_a @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( sup_sup_set_list_a @ ( image_list_a_list_a @ ( cons_a @ X2 ) @ ( shuffles_a @ Xs @ ( cons_a @ Y @ Ys ) ) ) @ ( image_list_a_list_a @ ( cons_a @ Y ) @ ( shuffles_a @ ( cons_a @ X2 @ Xs ) @ Ys ) ) ) ) ).
% shuffles.simps(3)
thf(fact_840_Inf__fin_Oin__idem,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ X2 @ A )
=> ( ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ A ) )
= ( lattic5238388535129920115in_nat @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_841_Sup__fin_Oin__idem,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ X2 @ A )
=> ( ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_842_range__eq__singletonD,axiom,
! [F: nat > nat,A2: nat,X2: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
=> ( ( F @ X2 )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_843_range__eq__singletonD,axiom,
! [F: nat > list_a,A2: list_a,X2: nat] :
( ( ( image_nat_list_a @ F @ top_top_set_nat )
= ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) )
=> ( ( F @ X2 )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_844_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat2 @ X2 @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X2 @ A ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_845_Inf__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X: nat,Y2: nat] : ( member_nat2 @ ( inf_inf_nat @ X @ Y2 ) @ ( insert_nat2 @ X @ ( insert_nat2 @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat2 @ ( lattic5238388535129920115in_nat @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_846_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat2 @ X2 @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X2 @ A ) )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_847_Sup__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X: nat,Y2: nat] : ( member_nat2 @ ( sup_sup_nat @ X @ Y2 ) @ ( insert_nat2 @ X @ ( insert_nat2 @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat2 @ ( lattic1093996805478795353in_nat @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_848_Inf__fin_Ounion,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic5238388535129920115in_nat @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_849_Sup__fin_Ounion,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_850_finite__Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_851_finite__Un,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_852_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_853_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_854_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_855_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat2 @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_856_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_857_in__image__insert__iff,axiom,
! [B: set_set_list_a,X2: list_a,A: set_list_a] :
( ! [C4: set_list_a] :
( ( member_set_list_a2 @ C4 @ B )
=> ~ ( member_list_a2 @ X2 @ C4 ) )
=> ( ( member_set_list_a2 @ A @ ( image_5749939591322298757list_a @ ( insert_list_a2 @ X2 ) @ B ) )
= ( ( member_list_a2 @ X2 @ A )
& ( member_set_list_a2 @ ( minus_646659088055828811list_a @ A @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_858_in__image__insert__iff,axiom,
! [B: set_set_nat,X2: nat,A: set_nat] :
( ! [C4: set_nat] :
( ( member_set_nat2 @ C4 @ B )
=> ~ ( member_nat2 @ X2 @ C4 ) )
=> ( ( member_set_nat2 @ A @ ( image_7916887816326733075et_nat @ ( insert_nat2 @ X2 ) @ B ) )
= ( ( member_nat2 @ X2 @ A )
& ( member_set_nat2 @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_859_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X: nat] :
( Y
= ( F @ X ) ) ) ).
% surjD
thf(fact_860_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X: nat] :
( Y
!= ( F @ X ) ) ) ).
% surjE
thf(fact_861_surjI,axiom,
! [G2: nat > nat,F: nat > nat] :
( ! [X: nat] :
( ( G2 @ ( F @ X ) )
= X )
=> ( ( image_nat_nat @ G2 @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_862_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X3: nat] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_863_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_864_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_865_ex__new__if__finite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A )
=> ? [A5: nat] :
~ ( member_nat2 @ A5 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_866_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_867_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_868_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_869_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_870_infinite__imp__nonempty,axiom,
! [S3: set_nat] :
( ~ ( finite_finite_nat @ S3 )
=> ( S3 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_871_infinite__imp__nonempty,axiom,
! [S3: set_list_a] :
( ~ ( finite_finite_list_a @ S3 )
=> ( S3 != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_872_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat2 @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_873_Diff__infinite__finite,axiom,
! [T: set_nat,S3: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S3 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_874_infinite__Un,axiom,
! [S3: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S3 @ T ) ) )
= ( ~ ( finite_finite_nat @ S3 )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_875_Un__infinite,axiom,
! [S3: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S3 )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S3 @ T ) ) ) ).
% Un_infinite
thf(fact_876_finite__UnI,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_877_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat2 @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_878_infinite__finite__induct,axiom,
! [P: set_list_a > $o,A: set_list_a] :
( ! [A6: set_list_a] :
( ~ ( finite_finite_list_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ~ ( member_list_a2 @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a2 @ X @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_879_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X: nat] : ( P @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat2 @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_880_finite__ne__induct,axiom,
! [F2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( F2 != bot_bot_set_list_a )
=> ( ! [X: list_a] : ( P @ ( insert_list_a2 @ X @ bot_bot_set_list_a ) )
=> ( ! [X: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( F3 != bot_bot_set_list_a )
=> ( ~ ( member_list_a2 @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a2 @ X @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_881_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat2 @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_882_finite__induct,axiom,
! [F2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ~ ( member_list_a2 @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a2 @ X @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_883_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A3: set_nat,B5: nat] :
( ( A4
= ( insert_nat2 @ B5 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_884_finite_Osimps,axiom,
( finite_finite_list_a
= ( ^ [A4: set_list_a] :
( ( A4 = bot_bot_set_list_a )
| ? [A3: set_list_a,B5: list_a] :
( ( A4
= ( insert_list_a2 @ B5 @ A3 ) )
& ( finite_finite_list_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_885_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A5: nat] :
( A2
= ( insert_nat2 @ A5 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_886_finite_Ocases,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( A2 != bot_bot_set_list_a )
=> ~ ! [A6: set_list_a] :
( ? [A5: list_a] :
( A2
= ( insert_list_a2 @ A5 @ A6 ) )
=> ~ ( finite_finite_list_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_887_finite__empty__induct,axiom,
! [A: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ A )
=> ( ( P @ A )
=> ( ! [A5: list_a,A6: set_list_a] :
( ( finite_finite_list_a @ A6 )
=> ( ( member_list_a2 @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a2 @ A5 @ bot_bot_set_list_a ) ) ) ) ) )
=> ( P @ bot_bot_set_list_a ) ) ) ) ).
% finite_empty_induct
thf(fact_888_finite__empty__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ A )
=> ( ! [A5: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( member_nat2 @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat2 @ A5 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_889_infinite__coinduct,axiom,
! [X5: set_list_a > $o,A: set_list_a] :
( ( X5 @ A )
=> ( ! [A6: set_list_a] :
( ( X5 @ A6 )
=> ? [X4: list_a] :
( ( member_list_a2 @ X4 @ A6 )
& ( ( X5 @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a2 @ X4 @ bot_bot_set_list_a ) ) )
| ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a2 @ X4 @ bot_bot_set_list_a ) ) ) ) ) )
=> ~ ( finite_finite_list_a @ A ) ) ) ).
% infinite_coinduct
thf(fact_890_infinite__coinduct,axiom,
! [X5: set_nat > $o,A: set_nat] :
( ( X5 @ A )
=> ( ! [A6: set_nat] :
( ( X5 @ A6 )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ A6 )
& ( ( X5 @ ( minus_minus_set_nat @ A6 @ ( insert_nat2 @ X4 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat2 @ X4 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_891_infinite__remove,axiom,
! [S3: set_list_a,A2: list_a] :
( ~ ( finite_finite_list_a @ S3 )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S3 @ ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) ) ) ) ).
% infinite_remove
thf(fact_892_infinite__remove,axiom,
! [S3: set_nat,A2: nat] :
( ~ ( finite_finite_nat @ S3 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_893_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_894_cofinite__bot,axiom,
( ( cofinite_nat = bot_bot_filter_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% cofinite_bot
thf(fact_895_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( linord2614967742042102400et_nat @ A )
= nil_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_896_fun__upd__image,axiom,
! [X2: list_a,A: set_list_a,F: list_a > nat,Y: nat] :
( ( ( member_list_a2 @ X2 @ A )
=> ( ( image_list_a_nat @ ( fun_upd_list_a_nat @ F @ X2 @ Y ) @ A )
= ( insert_nat2 @ Y @ ( image_list_a_nat @ F @ ( minus_646659088055828811list_a @ A @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) ) ) ) )
& ( ~ ( member_list_a2 @ X2 @ A )
=> ( ( image_list_a_nat @ ( fun_upd_list_a_nat @ F @ X2 @ Y ) @ A )
= ( image_list_a_nat @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_897_fun__upd__image,axiom,
! [X2: nat,A: set_nat,F: nat > nat,Y: nat] :
( ( ( member_nat2 @ X2 @ A )
=> ( ( image_nat_nat @ ( fun_upd_nat_nat @ F @ X2 @ Y ) @ A )
= ( insert_nat2 @ Y @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat2 @ X2 @ A )
=> ( ( image_nat_nat @ ( fun_upd_nat_nat @ F @ X2 @ Y ) @ A )
= ( image_nat_nat @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_898_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
( ( linord2614967742042102400et_nat @ bot_bot_set_nat )
= nil_nat ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_899_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( linord2614967742042102400et_nat @ A )
= nil_nat ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_900_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A ) )
= A ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_901_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: set_nat,B: set_nat] :
( ( ( linord2614967742042102400et_nat @ A )
= ( linord2614967742042102400et_nat @ B ) )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( A = B ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_902_sorted__list__of__set__nonempty,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( linord2614967742042102400et_nat @ A )
= ( cons_nat @ ( lattic8721135487736765967in_nat @ A ) @ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ ( lattic8721135487736765967in_nat @ A ) @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_903_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) )
= ( remove1_nat @ X2 @ ( linord2614967742042102400et_nat @ A ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_904_inj__on__Un,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( ( inj_on_nat_nat @ F @ A )
& ( inj_on_nat_nat @ F @ B )
& ( ( inf_inf_set_nat @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ B @ A ) ) )
= bot_bot_set_nat ) ) ) ).
% inj_on_Un
thf(fact_905_inj__on__Un,axiom,
! [F: nat > list_a,A: set_nat,B: set_nat] :
( ( inj_on_nat_list_a @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( ( inj_on_nat_list_a @ F @ A )
& ( inj_on_nat_list_a @ F @ B )
& ( ( inf_inf_set_list_a @ ( image_nat_list_a @ F @ ( minus_minus_set_nat @ A @ B ) ) @ ( image_nat_list_a @ F @ ( minus_minus_set_nat @ B @ A ) ) )
= bot_bot_set_list_a ) ) ) ).
% inj_on_Un
thf(fact_906_Inf__fin__le__Sup__fin,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_907_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_908_empty__subsetI,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A ) ).
% empty_subsetI
thf(fact_909_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_910_subset__empty,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
= ( A = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_911_insert__subset,axiom,
! [X2: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat2 @ X2 @ A ) @ B )
= ( ( member_nat2 @ X2 @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_912_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B2: nat] :
( ( ( insert_nat2 @ A2 @ A )
= ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_913_singleton__insert__inj__eq_H,axiom,
! [A2: list_a,A: set_list_a,B2: list_a] :
( ( ( insert_list_a2 @ A2 @ A )
= ( insert_list_a2 @ B2 @ bot_bot_set_list_a ) )
= ( ( A2 = B2 )
& ( ord_le8861187494160871172list_a @ A @ ( insert_list_a2 @ B2 @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_914_singleton__insert__inj__eq,axiom,
! [B2: nat,A2: nat,A: set_nat] :
( ( ( insert_nat2 @ B2 @ bot_bot_set_nat )
= ( insert_nat2 @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_915_singleton__insert__inj__eq,axiom,
! [B2: list_a,A2: list_a,A: set_list_a] :
( ( ( insert_list_a2 @ B2 @ bot_bot_set_list_a )
= ( insert_list_a2 @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le8861187494160871172list_a @ A @ ( insert_list_a2 @ B2 @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_916_Diff__eq__empty__iff,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ( minus_646659088055828811list_a @ A @ B )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_917_Diff__eq__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_918_in__set__remove1,axiom,
! [A2: nat,B2: nat,Xs: list_nat] :
( ( A2 != B2 )
=> ( ( member_nat2 @ A2 @ ( set_nat2 @ ( remove1_nat @ B2 @ Xs ) ) )
= ( member_nat2 @ A2 @ ( set_nat2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_919_Min__singleton,axiom,
! [X2: nat] :
( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% Min_singleton
thf(fact_920_Min_Obounded__iff,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic8721135487736765967in_nat @ A ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_nat @ X2 @ X3 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_921_inj__on__insert,axiom,
! [F: list_a > nat,A2: list_a,A: set_list_a] :
( ( inj_on_list_a_nat @ F @ ( insert_list_a2 @ A2 @ A ) )
= ( ( inj_on_list_a_nat @ F @ A )
& ~ ( member_nat2 @ ( F @ A2 ) @ ( image_list_a_nat @ F @ ( minus_646659088055828811list_a @ A @ ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_922_inj__on__insert,axiom,
! [F: nat > nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ ( insert_nat2 @ A2 @ A ) )
= ( ( inj_on_nat_nat @ F @ A )
& ~ ( member_nat2 @ ( F @ A2 ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_923_Min__insert2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ! [B6: nat] :
( ( member_nat2 @ B6 @ A )
=> ( ord_less_eq_nat @ A2 @ B6 ) )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ A2 @ A ) )
= A2 ) ) ) ).
% Min_insert2
thf(fact_924_Min_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B ) @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% Min.subset_imp
thf(fact_925_Min__antimono,axiom,
! [M: set_nat,N: set_nat] :
( ( ord_less_eq_set_nat @ M @ N )
=> ( ( M != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N ) @ ( lattic8721135487736765967in_nat @ M ) ) ) ) ) ).
% Min_antimono
thf(fact_926_Min_OboundedI,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ A )
=> ( ord_less_eq_nat @ X2 @ A5 ) )
=> ( ord_less_eq_nat @ X2 @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% Min.boundedI
thf(fact_927_Min_OboundedE,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic8721135487736765967in_nat @ A ) )
=> ! [A7: nat] :
( ( member_nat2 @ A7 @ A )
=> ( ord_less_eq_nat @ X2 @ A7 ) ) ) ) ) ).
% Min.boundedE
thf(fact_928_eq__Min__iff,axiom,
! [A: set_nat,M2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( M2
= ( lattic8721135487736765967in_nat @ A ) )
= ( ( member_nat2 @ M2 @ A )
& ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_nat @ M2 @ X3 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_929_Min__le__iff,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ X2 )
= ( ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ( ord_less_eq_nat @ X3 @ X2 ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_930_Min__eq__iff,axiom,
! [A: set_nat,M2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A )
= M2 )
= ( ( member_nat2 @ M2 @ A )
& ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_nat @ M2 @ X3 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_931_inj__on__image__set__diff,axiom,
! [F: nat > nat,C3: set_nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ C3 )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C3 )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_932_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_933_bot_Oextremum__uniqueI,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
=> ( A2 = bot_bot_set_list_a ) ) ).
% bot.extremum_uniqueI
thf(fact_934_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_935_bot_Oextremum__unique,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% bot.extremum_unique
thf(fact_936_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_937_bot_Oextremum,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% bot.extremum
thf(fact_938_top__greatest,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% top_greatest
thf(fact_939_top_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
= ( A2 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_940_top_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
=> ( A2 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_941_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( member_nat2 @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_942_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_943_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_944_Diff__mono,axiom,
! [A: set_nat,C3: set_nat,D2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ( ord_less_eq_set_nat @ D2 @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_945_Diff__subset,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_946_double__diff,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C3 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_947_inj__on__Cons1,axiom,
! [X2: nat,A: set_list_nat] : ( inj_on3049792774292151987st_nat @ ( cons_nat @ X2 ) @ A ) ).
% inj_on_Cons1
thf(fact_948_inj__on__Cons1,axiom,
! [X2: a,A: set_list_a] : ( inj_on_list_a_list_a @ ( cons_a @ X2 ) @ A ) ).
% inj_on_Cons1
thf(fact_949_set__remove1__subset,axiom,
! [X2: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( remove1_nat @ X2 @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_950_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_951_insert__mono,axiom,
! [C3: set_nat,D2: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ C3 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ A2 @ C3 ) @ ( insert_nat2 @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_952_subset__insert,axiom,
! [X2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat2 @ X2 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X2 @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_953_subset__insertI,axiom,
! [B: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat2 @ A2 @ B ) ) ).
% subset_insertI
thf(fact_954_subset__insertI2,axiom,
! [A: set_nat,B: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_955_remove1_Osimps_I2_J,axiom,
! [X2: nat,Y: nat,Xs: list_nat] :
( ( ( X2 = Y )
=> ( ( remove1_nat @ X2 @ ( cons_nat @ Y @ Xs ) )
= Xs ) )
& ( ( X2 != Y )
=> ( ( remove1_nat @ X2 @ ( cons_nat @ Y @ Xs ) )
= ( cons_nat @ Y @ ( remove1_nat @ X2 @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_956_remove1_Osimps_I2_J,axiom,
! [X2: a,Y: a,Xs: list_a] :
( ( ( X2 = Y )
=> ( ( remove1_a @ X2 @ ( cons_a @ Y @ Xs ) )
= Xs ) )
& ( ( X2 != Y )
=> ( ( remove1_a @ X2 @ ( cons_a @ Y @ Xs ) )
= ( cons_a @ Y @ ( remove1_a @ X2 @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_957_notin__set__remove1,axiom,
! [X2: nat,Xs: list_nat,Y: nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ~ ( member_nat2 @ X2 @ ( set_nat2 @ ( remove1_nat @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_958_remove1__idem,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X2 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_959_Inf__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_960_Sup__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_961_range__ex1__eq,axiom,
! [F: nat > nat,B2: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat2 @ B2 @ ( image_nat_nat @ F @ top_top_set_nat ) )
= ( ? [X3: nat] :
( ( B2
= ( F @ X3 ) )
& ! [Y3: nat] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_962_inj__image__mem__iff,axiom,
! [F: nat > nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat2 @ ( F @ A2 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat2 @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_963_inj__img__insertE,axiom,
! [F: nat > nat,A: set_nat,X2: nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ~ ( member_nat2 @ X2 @ B )
=> ( ( ( insert_nat2 @ X2 @ B )
= ( image_nat_nat @ F @ A ) )
=> ~ ! [X6: nat,A8: set_nat] :
( ~ ( member_nat2 @ X6 @ A8 )
=> ( ( A
= ( insert_nat2 @ X6 @ A8 ) )
=> ( ( X2
= ( F @ X6 ) )
=> ( B
!= ( image_nat_nat @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_964_Min__in,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( member_nat2 @ ( lattic8721135487736765967in_nat @ A ) @ A ) ) ) ).
% Min_in
thf(fact_965_diff__shunt__var,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( ( minus_646659088055828811list_a @ X2 @ Y )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_966_diff__shunt__var,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X2 @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_967_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat2 @ X @ A )
& ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ Xa2 @ X )
=> ( X = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_968_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat2 @ X @ A )
& ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ X @ Xa2 )
=> ( X = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_969_range__subsetD,axiom,
! [F: nat > nat,B: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B )
=> ( member_nat2 @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_970_subset__singleton__iff,axiom,
! [X5: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_971_subset__singleton__iff,axiom,
! [X5: set_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ X5 @ ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) )
= ( ( X5 = bot_bot_set_list_a )
| ( X5
= ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_972_subset__singletonD,axiom,
! [A: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_973_subset__singletonD,axiom,
! [A: set_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) )
=> ( ( A = bot_bot_set_list_a )
| ( A
= ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) ) ) ).
% subset_singletonD
thf(fact_974_set__subset__Cons,axiom,
! [Xs: list_nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_975_set__subset__Cons,axiom,
! [Xs: list_a,X2: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_976_image__diff__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_977_subset__Diff__insert,axiom,
! [A: set_nat,B: set_nat,X2: nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat2 @ X2 @ C3 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C3 ) )
& ~ ( member_nat2 @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_978_Diff__subset__conv,axiom,
! [A: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C3 )
= ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_979_Diff__partition,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_980_subset__code_I2_J,axiom,
! [A: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A @ ( coset_nat @ Ys ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat2 @ X3 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_981_finite__UNIV__inj__surj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_982_finite__UNIV__surj__inj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_983_image__set__diff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_984_finite__subset__induct,axiom,
! [F2: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat2 @ A5 @ A )
=> ( ~ ( member_nat2 @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_985_finite__subset__induct,axiom,
! [F2: set_list_a,A: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( ord_le8861187494160871172list_a @ F2 @ A )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A5: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( member_list_a2 @ A5 @ A )
=> ( ~ ( member_list_a2 @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a2 @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_986_finite__subset__induct_H,axiom,
! [F2: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat2 @ A5 @ A )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ~ ( member_nat2 @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_987_finite__subset__induct_H,axiom,
! [F2: set_list_a,A: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( ord_le8861187494160871172list_a @ F2 @ A )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A5: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( member_list_a2 @ A5 @ A )
=> ( ( ord_le8861187494160871172list_a @ F3 @ A )
=> ( ~ ( member_list_a2 @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a2 @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_988_Diff__single__insert,axiom,
! [A: set_list_a,X2: list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) @ B )
=> ( ord_le8861187494160871172list_a @ A @ ( insert_list_a2 @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_989_Diff__single__insert,axiom,
! [A: set_nat,X2: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_990_subset__insert__iff,axiom,
! [A: set_list_a,X2: list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( insert_list_a2 @ X2 @ B ) )
= ( ( ( member_list_a2 @ X2 @ A )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) @ B ) )
& ( ~ ( member_list_a2 @ X2 @ A )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_991_subset__insert__iff,axiom,
! [A: set_nat,X2: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X2 @ B ) )
= ( ( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_992_Sup__fin_OboundedE,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X2 )
=> ! [A7: nat] :
( ( member_nat2 @ A7 @ A )
=> ( ord_less_eq_nat @ A7 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_993_Sup__fin_OboundedI,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ A )
=> ( ord_less_eq_nat @ A5 @ X2 ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_994_Sup__fin_Obounded__iff,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X2 )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_nat @ X3 @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_995_Inf__fin_Obounded__iff,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_nat @ X2 @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_996_Inf__fin_OboundedI,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ A )
=> ( ord_less_eq_nat @ X2 @ A5 ) )
=> ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_997_Inf__fin_OboundedE,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A ) )
=> ! [A7: nat] :
( ( member_nat2 @ A7 @ A )
=> ( ord_less_eq_nat @ X2 @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_998_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_999_Cons__shuffles__subset1,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat] : ( ord_le6045566169113846134st_nat @ ( image_7976474329151083847st_nat @ ( cons_nat @ X2 ) @ ( shuffles_nat @ Xs @ Ys ) ) @ ( shuffles_nat @ ( cons_nat @ X2 @ Xs ) @ Ys ) ) ).
% Cons_shuffles_subset1
thf(fact_1000_Cons__shuffles__subset1,axiom,
! [X2: a,Xs: list_a,Ys: list_a] : ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ ( cons_a @ X2 ) @ ( shuffles_a @ Xs @ Ys ) ) @ ( shuffles_a @ ( cons_a @ X2 @ Xs ) @ Ys ) ) ).
% Cons_shuffles_subset1
thf(fact_1001_Cons__shuffles__subset2,axiom,
! [Y: nat,Xs: list_nat,Ys: list_nat] : ( ord_le6045566169113846134st_nat @ ( image_7976474329151083847st_nat @ ( cons_nat @ Y ) @ ( shuffles_nat @ Xs @ Ys ) ) @ ( shuffles_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) ) ).
% Cons_shuffles_subset2
thf(fact_1002_Cons__shuffles__subset2,axiom,
! [Y: a,Xs: list_a,Ys: list_a] : ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ ( cons_a @ Y ) @ ( shuffles_a @ Xs @ Ys ) ) @ ( shuffles_a @ Xs @ ( cons_a @ Y @ Ys ) ) ) ).
% Cons_shuffles_subset2
thf(fact_1003_subset__subseqs,axiom,
! [X5: set_nat,Xs: list_nat] :
( ( ord_less_eq_set_nat @ X5 @ ( set_nat2 @ Xs ) )
=> ( member_set_nat2 @ X5 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_1004_finite__remove__induct,axiom,
! [B: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ B )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A6: set_list_a] :
( ( finite_finite_list_a @ A6 )
=> ( ( A6 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A6 @ B )
=> ( ! [X4: list_a] :
( ( member_list_a2 @ X4 @ A6 )
=> ( P @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a2 @ X4 @ bot_bot_set_list_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1005_finite__remove__induct,axiom,
! [B: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat2 @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1006_remove__induct,axiom,
! [P: set_list_a > $o,B: set_list_a] :
( ( P @ bot_bot_set_list_a )
=> ( ( ~ ( finite_finite_list_a @ B )
=> ( P @ B ) )
=> ( ! [A6: set_list_a] :
( ( finite_finite_list_a @ A6 )
=> ( ( A6 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A6 @ B )
=> ( ! [X4: list_a] :
( ( member_list_a2 @ X4 @ A6 )
=> ( P @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a2 @ X4 @ bot_bot_set_list_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1007_remove__induct,axiom,
! [P: set_nat > $o,B: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P @ B ) )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat2 @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1008_Inf__fin_Osubset,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) )
= ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1009_Sup__fin_Osubset,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B ) @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1010_min__list__Min,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( min_list_nat @ Xs )
= ( lattic8721135487736765967in_nat @ ( set_nat2 @ Xs ) ) ) ) ).
% min_list_Min
thf(fact_1011_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( member_nat2 @ X @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_1012_in__mono,axiom,
! [A: set_nat,B: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat2 @ X2 @ A )
=> ( member_nat2 @ X2 @ B ) ) ) ).
% in_mono
thf(fact_1013_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ B ) ) ) ).
% subsetD
thf(fact_1014_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
! [X3: nat] :
( ( member_nat2 @ X3 @ A3 )
=> ( member_nat2 @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_1015_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
! [T2: nat] :
( ( member_nat2 @ T2 @ A3 )
=> ( member_nat2 @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1016_infinite__countable__subset,axiom,
! [S3: set_nat] :
( ~ ( finite_finite_nat @ S3 )
=> ? [F4: nat > nat] :
( ( inj_on_nat_nat @ F4 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ top_top_set_nat ) @ S3 ) ) ) ).
% infinite_countable_subset
thf(fact_1017_infinite__iff__countable__subset,axiom,
! [S3: set_nat] :
( ( ~ ( finite_finite_nat @ S3 ) )
= ( ? [F5: nat > nat] :
( ( inj_on_nat_nat @ F5 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F5 @ top_top_set_nat ) @ S3 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_1018_insert__subsetI,axiom,
! [X2: nat,A: set_nat,X5: set_nat] :
( ( member_nat2 @ X2 @ A )
=> ( ( ord_less_eq_set_nat @ X5 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ X2 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1019_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X: nat] :
~ ( member_nat2 @ X @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1020_subset__emptyI,axiom,
! [A: set_list_a] :
( ! [X: list_a] :
~ ( member_list_a2 @ X @ A )
=> ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a ) ) ).
% subset_emptyI
thf(fact_1021_set__remove1__eq,axiom,
! [Xs: list_list_a,X2: list_a] :
( ( distinct_list_a @ Xs )
=> ( ( set_list_a2 @ ( remove1_list_a @ X2 @ Xs ) )
= ( minus_646659088055828811list_a @ ( set_list_a2 @ Xs ) @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) ) ) ).
% set_remove1_eq
thf(fact_1022_set__remove1__eq,axiom,
! [Xs: list_nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( ( set_nat2 @ ( remove1_nat @ X2 @ Xs ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).
% set_remove1_eq
thf(fact_1023_finite__distinct__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs3: list_nat] :
( ( ( set_nat2 @ Xs3 )
= A )
& ( distinct_nat @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_1024_distinct__singleton,axiom,
! [X2: nat] : ( distinct_nat @ ( cons_nat @ X2 @ nil_nat ) ) ).
% distinct_singleton
thf(fact_1025_distinct__singleton,axiom,
! [X2: a] : ( distinct_a @ ( cons_a @ X2 @ nil_a ) ) ).
% distinct_singleton
thf(fact_1026_distinct_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( distinct_nat @ ( cons_nat @ X2 @ Xs ) )
= ( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
& ( distinct_nat @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1027_distinct_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( distinct_a @ ( cons_a @ X2 @ Xs ) )
= ( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
& ( distinct_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1028_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1029_distinct__length__2__or__more,axiom,
! [A2: nat,B2: nat,Xs: list_nat] :
( ( distinct_nat @ ( cons_nat @ A2 @ ( cons_nat @ B2 @ Xs ) ) )
= ( ( A2 != B2 )
& ( distinct_nat @ ( cons_nat @ A2 @ Xs ) )
& ( distinct_nat @ ( cons_nat @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_1030_distinct__length__2__or__more,axiom,
! [A2: a,B2: a,Xs: list_a] :
( ( distinct_a @ ( cons_a @ A2 @ ( cons_a @ B2 @ Xs ) ) )
= ( ( A2 != B2 )
& ( distinct_a @ ( cons_a @ A2 @ Xs ) )
& ( distinct_a @ ( cons_a @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_1031_distinct__tl,axiom,
! [Xs: list_a] :
( ( distinct_a @ Xs )
=> ( distinct_a @ ( tl_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_1032_distinct__disjoint__shuffles,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( distinct_nat @ Ys )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) )
= bot_bot_set_nat )
=> ( ( member_list_nat2 @ Zs @ ( shuffles_nat @ Xs @ Ys ) )
=> ( distinct_nat @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_1033_distinct__disjoint__shuffles,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( distinct_list_a @ Xs )
=> ( ( distinct_list_a @ Ys )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ Ys ) )
= bot_bot_set_list_a )
=> ( ( member_list_list_a @ Zs @ ( shuffles_list_a @ Xs @ Ys ) )
=> ( distinct_list_a @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_1034_distinct__concat__iff,axiom,
! [Xs: list_list_nat] :
( ( distinct_nat @ ( concat_nat @ Xs ) )
= ( ( distinct_list_nat @ ( removeAll_list_nat @ nil_nat @ Xs ) )
& ! [Ys3: list_nat] :
( ( member_list_nat2 @ Ys3 @ ( set_list_nat2 @ Xs ) )
=> ( distinct_nat @ Ys3 ) )
& ! [Ys3: list_nat,Zs3: list_nat] :
( ( ( member_list_nat2 @ Ys3 @ ( set_list_nat2 @ Xs ) )
& ( member_list_nat2 @ Zs3 @ ( set_list_nat2 @ Xs ) )
& ( Ys3 != Zs3 ) )
=> ( ( inf_inf_set_nat @ ( set_nat2 @ Ys3 ) @ ( set_nat2 @ Zs3 ) )
= bot_bot_set_nat ) ) ) ) ).
% distinct_concat_iff
thf(fact_1035_distinct__concat__iff,axiom,
! [Xs: list_list_list_a] :
( ( distinct_list_a @ ( concat_list_a @ Xs ) )
= ( ( distinct_list_list_a @ ( remove8017980289111491990list_a @ nil_list_a @ Xs ) )
& ! [Ys3: list_list_a] :
( ( member_list_list_a @ Ys3 @ ( set_list_list_a2 @ Xs ) )
=> ( distinct_list_a @ Ys3 ) )
& ! [Ys3: list_list_a,Zs3: list_list_a] :
( ( ( member_list_list_a @ Ys3 @ ( set_list_list_a2 @ Xs ) )
& ( member_list_list_a @ Zs3 @ ( set_list_list_a2 @ Xs ) )
& ( Ys3 != Zs3 ) )
=> ( ( inf_inf_set_list_a @ ( set_list_a2 @ Ys3 ) @ ( set_list_a2 @ Zs3 ) )
= bot_bot_set_list_a ) ) ) ) ).
% distinct_concat_iff
thf(fact_1036_distinct__concat,axiom,
! [Xs: list_list_nat] :
( ( distinct_list_nat @ Xs )
=> ( ! [Ys2: list_nat] :
( ( member_list_nat2 @ Ys2 @ ( set_list_nat2 @ Xs ) )
=> ( distinct_nat @ Ys2 ) )
=> ( ! [Ys2: list_nat,Zs4: list_nat] :
( ( member_list_nat2 @ Ys2 @ ( set_list_nat2 @ Xs ) )
=> ( ( member_list_nat2 @ Zs4 @ ( set_list_nat2 @ Xs ) )
=> ( ( Ys2 != Zs4 )
=> ( ( inf_inf_set_nat @ ( set_nat2 @ Ys2 ) @ ( set_nat2 @ Zs4 ) )
= bot_bot_set_nat ) ) ) )
=> ( distinct_nat @ ( concat_nat @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1037_distinct__concat,axiom,
! [Xs: list_list_list_a] :
( ( distinct_list_list_a @ Xs )
=> ( ! [Ys2: list_list_a] :
( ( member_list_list_a @ Ys2 @ ( set_list_list_a2 @ Xs ) )
=> ( distinct_list_a @ Ys2 ) )
=> ( ! [Ys2: list_list_a,Zs4: list_list_a] :
( ( member_list_list_a @ Ys2 @ ( set_list_list_a2 @ Xs ) )
=> ( ( member_list_list_a @ Zs4 @ ( set_list_list_a2 @ Xs ) )
=> ( ( Ys2 != Zs4 )
=> ( ( inf_inf_set_list_a @ ( set_list_a2 @ Ys2 ) @ ( set_list_a2 @ Zs4 ) )
= bot_bot_set_list_a ) ) ) )
=> ( distinct_list_a @ ( concat_list_a @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1038_subseqs__powset,axiom,
! [Xs: list_nat] :
( ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
= ( pow_nat @ ( set_nat2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_1039_Pow__singleton__iff,axiom,
! [X5: set_nat,Y4: set_nat] :
( ( ( pow_nat @ X5 )
= ( insert_set_nat2 @ Y4 @ bot_bot_set_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
& ( Y4 = bot_bot_set_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_1040_Pow__singleton__iff,axiom,
! [X5: set_list_a,Y4: set_list_a] :
( ( ( pow_list_a @ X5 )
= ( insert_set_list_a2 @ Y4 @ bot_bo3186585308812441520list_a ) )
= ( ( X5 = bot_bot_set_list_a )
& ( Y4 = bot_bot_set_list_a ) ) ) ).
% Pow_singleton_iff
thf(fact_1041_Pow__empty,axiom,
( ( pow_nat @ bot_bot_set_nat )
= ( insert_set_nat2 @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_empty
thf(fact_1042_Pow__empty,axiom,
( ( pow_list_a @ bot_bot_set_list_a )
= ( insert_set_list_a2 @ bot_bot_set_list_a @ bot_bo3186585308812441520list_a ) ) ).
% Pow_empty
thf(fact_1043_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_1044_Pow__bottom,axiom,
! [B: set_nat] : ( member_set_nat2 @ bot_bot_set_nat @ ( pow_nat @ B ) ) ).
% Pow_bottom
thf(fact_1045_Pow__bottom,axiom,
! [B: set_list_a] : ( member_set_list_a2 @ bot_bot_set_list_a @ ( pow_list_a @ B ) ) ).
% Pow_bottom
thf(fact_1046_Pow__insert,axiom,
! [A2: nat,A: set_nat] :
( ( pow_nat @ ( insert_nat2 @ A2 @ A ) )
= ( sup_sup_set_set_nat @ ( pow_nat @ A ) @ ( image_7916887816326733075et_nat @ ( insert_nat2 @ A2 ) @ ( pow_nat @ A ) ) ) ) ).
% Pow_insert
thf(fact_1047_Pow__set_I1_J,axiom,
( ( pow_nat @ ( set_nat2 @ nil_nat ) )
= ( insert_set_nat2 @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_set(1)
thf(fact_1048_Pow__set_I1_J,axiom,
( ( pow_list_a @ ( set_list_a2 @ nil_list_a ) )
= ( insert_set_list_a2 @ bot_bot_set_list_a @ bot_bo3186585308812441520list_a ) ) ).
% Pow_set(1)
thf(fact_1049_list_Oset__map,axiom,
! [F: nat > nat,V: list_nat] :
( ( set_nat2 @ ( map_nat_nat @ F @ V ) )
= ( image_nat_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_1050_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z4: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1051_map__eq__Cons__conv,axiom,
! [F: a > nat,Xs: list_a,Y: nat,Ys: list_nat] :
( ( ( map_a_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z4: a,Zs3: list_a] :
( ( Xs
= ( cons_a @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_a_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1052_map__eq__Cons__conv,axiom,
! [F: nat > a,Xs: list_nat,Y: a,Ys: list_a] :
( ( ( map_nat_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( ? [Z4: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_nat_a @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1053_map__eq__Cons__conv,axiom,
! [F: a > a,Xs: list_a,Y: a,Ys: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( ? [Z4: a,Zs3: list_a] :
( ( Xs
= ( cons_a @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_a_a @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1054_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z4: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1055_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: a > nat,Ys: list_a] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_a_nat @ F @ Ys ) )
= ( ? [Z4: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_a_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1056_Cons__eq__map__conv,axiom,
! [X2: a,Xs: list_a,F: nat > a,Ys: list_nat] :
( ( ( cons_a @ X2 @ Xs )
= ( map_nat_a @ F @ Ys ) )
= ( ? [Z4: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_nat_a @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1057_Cons__eq__map__conv,axiom,
! [X2: a,Xs: list_a,F: a > a,Ys: list_a] :
( ( ( cons_a @ X2 @ Xs )
= ( map_a_a @ F @ Ys ) )
= ( ? [Z4: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_a_a @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1058_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z3: nat,Zs4: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs4 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_nat_nat @ F @ Zs4 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1059_map__eq__Cons__D,axiom,
! [F: a > nat,Xs: list_a,Y: nat,Ys: list_nat] :
( ( ( map_a_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z3: a,Zs4: list_a] :
( ( Xs
= ( cons_a @ Z3 @ Zs4 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_a_nat @ F @ Zs4 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1060_map__eq__Cons__D,axiom,
! [F: nat > a,Xs: list_nat,Y: a,Ys: list_a] :
( ( ( map_nat_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
=> ? [Z3: nat,Zs4: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs4 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_nat_a @ F @ Zs4 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1061_map__eq__Cons__D,axiom,
! [F: a > a,Xs: list_a,Y: a,Ys: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
=> ? [Z3: a,Zs4: list_a] :
( ( Xs
= ( cons_a @ Z3 @ Zs4 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_a_a @ F @ Zs4 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1062_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z3: nat,Zs4: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs4 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs4 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1063_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: a > nat,Ys: list_a] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_a_nat @ F @ Ys ) )
=> ? [Z3: a,Zs4: list_a] :
( ( Ys
= ( cons_a @ Z3 @ Zs4 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_a_nat @ F @ Zs4 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1064_Cons__eq__map__D,axiom,
! [X2: a,Xs: list_a,F: nat > a,Ys: list_nat] :
( ( ( cons_a @ X2 @ Xs )
= ( map_nat_a @ F @ Ys ) )
=> ? [Z3: nat,Zs4: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs4 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_a @ F @ Zs4 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1065_Cons__eq__map__D,axiom,
! [X2: a,Xs: list_a,F: a > a,Ys: list_a] :
( ( ( cons_a @ X2 @ Xs )
= ( map_a_a @ F @ Ys ) )
=> ? [Z3: a,Zs4: list_a] :
( ( Ys
= ( cons_a @ Z3 @ Zs4 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_a_a @ F @ Zs4 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1066_list_Osimps_I9_J,axiom,
! [F: nat > nat,X21: nat,X22: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1067_list_Osimps_I9_J,axiom,
! [F: nat > a,X21: nat,X22: list_nat] :
( ( map_nat_a @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_a @ ( F @ X21 ) @ ( map_nat_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1068_list_Osimps_I9_J,axiom,
! [F: a > nat,X21: a,X22: list_a] :
( ( map_a_nat @ F @ ( cons_a @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_a_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1069_list_Osimps_I9_J,axiom,
! [F: a > a,X21: a,X22: list_a] :
( ( map_a_a @ F @ ( cons_a @ X21 @ X22 ) )
= ( cons_a @ ( F @ X21 ) @ ( map_a_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1070_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_1071_list_Omap__ident__strong,axiom,
! [T3: list_nat,F: nat > nat] :
( ! [Z3: nat] :
( ( member_nat2 @ Z3 @ ( set_nat2 @ T3 ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_nat_nat @ F @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_1072_map__tl,axiom,
! [F: a > a,Xs: list_a] :
( ( map_a_a @ F @ ( tl_a @ Xs ) )
= ( tl_a @ ( map_a_a @ F @ Xs ) ) ) ).
% map_tl
thf(fact_1073_list_Omap__sel_I2_J,axiom,
! [A2: list_a,F: a > a] :
( ( A2 != nil_a )
=> ( ( tl_a @ ( map_a_a @ F @ A2 ) )
= ( map_a_a @ F @ ( tl_a @ A2 ) ) ) ) ).
% list.map_sel(2)
thf(fact_1074_image__set,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( image_nat_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( map_nat_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_1075_distinct__set__subseqs,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_set_nat @ ( map_list_nat_set_nat @ set_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_1076_Max__singleton,axiom,
! [X2: nat] :
( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% Max_singleton
thf(fact_1077_Max_Obounded__iff,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X2 )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_nat @ X3 @ X2 ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_1078_Max__in,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( member_nat2 @ ( lattic8265883725875713057ax_nat @ A ) @ A ) ) ) ).
% Max_in
thf(fact_1079_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B ) @ A2 )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ B )
=> ( ( inf_inf_set_nat @ X3 @ A2 )
= bot_bot_set_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1080_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_list_a,A2: set_list_a] :
( ( ( inf_inf_set_list_a @ ( comple6928918032620976721list_a @ B ) @ A2 )
= bot_bot_set_list_a )
= ( ! [X3: set_list_a] :
( ( member_set_list_a2 @ X3 @ B )
=> ( ( inf_inf_set_list_a @ X3 @ A2 )
= bot_bot_set_list_a ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1081_Max__eq__iff,axiom,
! [A: set_nat,M2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ( lattic8265883725875713057ax_nat @ A )
= M2 )
= ( ( member_nat2 @ M2 @ A )
& ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_1082_Max__ge__iff,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic8265883725875713057ax_nat @ A ) )
= ( ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ( ord_less_eq_nat @ X2 @ X3 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_1083_eq__Max__iff,axiom,
! [A: set_nat,M2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( M2
= ( lattic8265883725875713057ax_nat @ A ) )
= ( ( member_nat2 @ M2 @ A )
& ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_1084_Max_OboundedE,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X2 )
=> ! [A7: nat] :
( ( member_nat2 @ A7 @ A )
=> ( ord_less_eq_nat @ A7 @ X2 ) ) ) ) ) ).
% Max.boundedE
thf(fact_1085_Max_OboundedI,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ A )
=> ( ord_less_eq_nat @ A5 @ X2 ) )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X2 ) ) ) ) ).
% Max.boundedI
thf(fact_1086_Max__insert2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ! [B6: nat] :
( ( member_nat2 @ B6 @ A )
=> ( ord_less_eq_nat @ B6 @ A2 ) )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ A2 @ A ) )
= A2 ) ) ) ).
% Max_insert2
thf(fact_1087_insert__partition,axiom,
! [X2: set_nat,F2: set_set_nat] :
( ~ ( member_set_nat2 @ X2 @ F2 )
=> ( ! [X: set_nat] :
( ( member_set_nat2 @ X @ ( insert_set_nat2 @ X2 @ F2 ) )
=> ! [Xa3: set_nat] :
( ( member_set_nat2 @ Xa3 @ ( insert_set_nat2 @ X2 @ F2 ) )
=> ( ( X != Xa3 )
=> ( ( inf_inf_set_nat @ X @ Xa3 )
= bot_bot_set_nat ) ) ) )
=> ( ( inf_inf_set_nat @ X2 @ ( comple7399068483239264473et_nat @ F2 ) )
= bot_bot_set_nat ) ) ) ).
% insert_partition
thf(fact_1088_insert__partition,axiom,
! [X2: set_list_a,F2: set_set_list_a] :
( ~ ( member_set_list_a2 @ X2 @ F2 )
=> ( ! [X: set_list_a] :
( ( member_set_list_a2 @ X @ ( insert_set_list_a2 @ X2 @ F2 ) )
=> ! [Xa3: set_list_a] :
( ( member_set_list_a2 @ Xa3 @ ( insert_set_list_a2 @ X2 @ F2 ) )
=> ( ( X != Xa3 )
=> ( ( inf_inf_set_list_a @ X @ Xa3 )
= bot_bot_set_list_a ) ) ) )
=> ( ( inf_inf_set_list_a @ X2 @ ( comple6928918032620976721list_a @ F2 ) )
= bot_bot_set_list_a ) ) ) ).
% insert_partition
thf(fact_1089_Sup__set__fold,axiom,
! [Xs: list_set_nat] :
( ( comple7399068483239264473et_nat @ ( set_set_nat2 @ Xs ) )
= ( fold_set_nat_set_nat @ sup_sup_set_nat @ Xs @ bot_bot_set_nat ) ) ).
% Sup_set_fold
thf(fact_1090_Sup__set__fold,axiom,
! [Xs: list_set_list_a] :
( ( comple6928918032620976721list_a @ ( set_set_list_a2 @ Xs ) )
= ( fold_s5931075695703335563list_a @ sup_sup_set_list_a @ Xs @ bot_bot_set_list_a ) ) ).
% Sup_set_fold
thf(fact_1091_Max__mono,axiom,
! [M: set_nat,N: set_nat] :
( ( ord_less_eq_set_nat @ M @ N )
=> ( ( M != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M ) @ ( lattic8265883725875713057ax_nat @ N ) ) ) ) ) ).
% Max_mono
thf(fact_1092_Max_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ ( lattic8265883725875713057ax_nat @ B ) ) ) ) ) ).
% Max.subset_imp
thf(fact_1093_Sup__bot__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ A )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1094_Sup__bot__conv_I2_J,axiom,
! [A: set_set_list_a] :
( ( bot_bot_set_list_a
= ( comple6928918032620976721list_a @ A ) )
= ( ! [X3: set_list_a] :
( ( member_set_list_a2 @ X3 @ A )
=> ( X3 = bot_bot_set_list_a ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1095_Sup__bot__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ A )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1096_Sup__bot__conv_I1_J,axiom,
! [A: set_set_list_a] :
( ( ( comple6928918032620976721list_a @ A )
= bot_bot_set_list_a )
= ( ! [X3: set_list_a] :
( ( member_set_list_a2 @ X3 @ A )
=> ( X3 = bot_bot_set_list_a ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1097_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_1098_Sup__empty,axiom,
( ( comple6928918032620976721list_a @ bot_bo3186585308812441520list_a )
= bot_bot_set_list_a ) ).
% Sup_empty
thf(fact_1099_Sup__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_1100_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S: set_nat_o,X3: nat] : ( member_nat2 @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1101_Union__empty__conv,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ A )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_1102_Union__empty__conv,axiom,
! [A: set_set_list_a] :
( ( ( comple6928918032620976721list_a @ A )
= bot_bot_set_list_a )
= ( ! [X3: set_list_a] :
( ( member_set_list_a2 @ X3 @ A )
=> ( X3 = bot_bot_set_list_a ) ) ) ) ).
% Union_empty_conv
thf(fact_1103_empty__Union__conv,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ A )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_1104_empty__Union__conv,axiom,
! [A: set_set_list_a] :
( ( bot_bot_set_list_a
= ( comple6928918032620976721list_a @ A ) )
= ( ! [X3: set_list_a] :
( ( member_set_list_a2 @ X3 @ A )
=> ( X3 = bot_bot_set_list_a ) ) ) ) ).
% empty_Union_conv
thf(fact_1105_Union__disjoint,axiom,
! [C3: set_set_nat,A: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ C3 ) @ A )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ C3 )
=> ( ( inf_inf_set_nat @ X3 @ A )
= bot_bot_set_nat ) ) ) ) ).
% Union_disjoint
thf(fact_1106_Union__disjoint,axiom,
! [C3: set_set_list_a,A: set_list_a] :
( ( ( inf_inf_set_list_a @ ( comple6928918032620976721list_a @ C3 ) @ A )
= bot_bot_set_list_a )
= ( ! [X3: set_list_a] :
( ( member_set_list_a2 @ X3 @ C3 )
=> ( ( inf_inf_set_list_a @ X3 @ A )
= bot_bot_set_list_a ) ) ) ) ).
% Union_disjoint
thf(fact_1107_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_1108_Union__empty,axiom,
( ( comple6928918032620976721list_a @ bot_bo3186585308812441520list_a )
= bot_bot_set_list_a ) ).
% Union_empty
thf(fact_1109_Union__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Union_UNIV
thf(fact_1110_UNION__fun__upd,axiom,
! [A: list_a > set_nat,I: list_a,B: set_nat,J: set_list_a] :
( ( comple7399068483239264473et_nat @ ( image_list_a_set_nat @ ( fun_up3731164839266413645et_nat @ A @ I @ B ) @ J ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_list_a_set_nat @ A @ ( minus_646659088055828811list_a @ J @ ( insert_list_a2 @ I @ bot_bot_set_list_a ) ) ) ) @ ( if_set_nat @ ( member_list_a2 @ I @ J ) @ B @ bot_bot_set_nat ) ) ) ).
% UNION_fun_upd
thf(fact_1111_UNION__fun__upd,axiom,
! [A: list_a > set_list_a,I: list_a,B: set_list_a,J: set_list_a] :
( ( comple6928918032620976721list_a @ ( image_5464838071766335845list_a @ ( fun_up8218348934346411485list_a @ A @ I @ B ) @ J ) )
= ( sup_sup_set_list_a @ ( comple6928918032620976721list_a @ ( image_5464838071766335845list_a @ A @ ( minus_646659088055828811list_a @ J @ ( insert_list_a2 @ I @ bot_bot_set_list_a ) ) ) ) @ ( if_set_list_a @ ( member_list_a2 @ I @ J ) @ B @ bot_bot_set_list_a ) ) ) ).
% UNION_fun_upd
thf(fact_1112_UNION__fun__upd,axiom,
! [A: nat > set_nat,I: nat,B: set_nat,J: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( fun_upd_nat_set_nat @ A @ I @ B ) @ J ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ ( minus_minus_set_nat @ J @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) ) ) @ ( if_set_nat @ ( member_nat2 @ I @ J ) @ B @ bot_bot_set_nat ) ) ) ).
% UNION_fun_upd
thf(fact_1113_UNION__fun__upd,axiom,
! [A: nat > set_list_a,I: nat,B: set_list_a,J: set_nat] :
( ( comple6928918032620976721list_a @ ( image_nat_set_list_a @ ( fun_up1966476811263071829list_a @ A @ I @ B ) @ J ) )
= ( sup_sup_set_list_a @ ( comple6928918032620976721list_a @ ( image_nat_set_list_a @ A @ ( minus_minus_set_nat @ J @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) ) ) @ ( if_set_list_a @ ( member_nat2 @ I @ J ) @ B @ bot_bot_set_list_a ) ) ) ).
% UNION_fun_upd
thf(fact_1114_cSup__singleton,axiom,
! [X2: nat] :
( ( complete_Sup_Sup_nat @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% cSup_singleton
thf(fact_1115_cSup__least,axiom,
! [X5: set_nat,Z: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat2 @ X @ X5 )
=> ( ord_less_eq_nat @ X @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1116_cSup__eq__non__empty,axiom,
! [X5: set_nat,A2: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat2 @ X @ X5 )
=> ( ord_less_eq_nat @ X @ A2 ) )
=> ( ! [Y2: nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ Y2 ) )
=> ( ord_less_eq_nat @ A2 @ Y2 ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1117_cSup__eq__Max,axiom,
! [X5: set_nat] :
( ( finite_finite_nat @ X5 )
=> ( ( X5 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ X5 )
= ( lattic8265883725875713057ax_nat @ X5 ) ) ) ) ).
% cSup_eq_Max
thf(fact_1118_cSup__eq__Sup__fin,axiom,
! [X5: set_nat] :
( ( finite_finite_nat @ X5 )
=> ( ( X5 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ X5 )
= ( lattic1093996805478795353in_nat @ X5 ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_1119_finite__compl,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( uminus5710092332889474511et_nat @ A ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_compl
thf(fact_1120_Compl__disjoint2,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ A )
= bot_bot_set_nat ) ).
% Compl_disjoint2
thf(fact_1121_Compl__disjoint2,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ A ) @ A )
= bot_bot_set_list_a ) ).
% Compl_disjoint2
thf(fact_1122_Compl__disjoint,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
= bot_bot_set_nat ) ).
% Compl_disjoint
thf(fact_1123_Compl__disjoint,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( uminus7925729386456332763list_a @ A ) )
= bot_bot_set_list_a ) ).
% Compl_disjoint
thf(fact_1124_Diff__Compl,axiom,
! [A: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( uminus5710092332889474511et_nat @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% Diff_Compl
thf(fact_1125_Compl__Diff__eq,axiom,
! [A: set_nat,B: set_nat] :
( ( uminus5710092332889474511et_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ B ) ) ).
% Compl_Diff_eq
thf(fact_1126_boolean__algebra_Ocompl__one,axiom,
( ( uminus7925729386456332763list_a @ top_top_set_list_a )
= bot_bot_set_list_a ) ).
% boolean_algebra.compl_one
thf(fact_1127_boolean__algebra_Ocompl__one,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.compl_one
thf(fact_1128_boolean__algebra_Ocompl__zero,axiom,
( ( uminus7925729386456332763list_a @ bot_bot_set_list_a )
= top_top_set_list_a ) ).
% boolean_algebra.compl_zero
thf(fact_1129_boolean__algebra_Ocompl__zero,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.compl_zero
thf(fact_1130_inf__compl__bot__left1,axiom,
! [X2: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ ( inf_inf_set_nat @ X2 @ Y ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left1
thf(fact_1131_inf__compl__bot__left1,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ X2 ) @ ( inf_inf_set_list_a @ X2 @ Y ) )
= bot_bot_set_list_a ) ).
% inf_compl_bot_left1
thf(fact_1132_inf__compl__bot__left2,axiom,
! [X2: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left2
thf(fact_1133_inf__compl__bot__left2,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ X2 ) @ Y ) )
= bot_bot_set_list_a ) ).
% inf_compl_bot_left2
thf(fact_1134_inf__compl__bot__right,axiom,
! [X2: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X2 ) ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_right
thf(fact_1135_inf__compl__bot__right,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ Y @ ( uminus7925729386456332763list_a @ X2 ) ) )
= bot_bot_set_list_a ) ).
% inf_compl_bot_right
thf(fact_1136_boolean__algebra_Oconj__cancel__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ X2 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1137_boolean__algebra_Oconj__cancel__left,axiom,
! [X2: set_list_a] :
( ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ X2 ) @ X2 )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1138_boolean__algebra_Oconj__cancel__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ X2 ) )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1139_boolean__algebra_Oconj__cancel__right,axiom,
! [X2: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ ( uminus7925729386456332763list_a @ X2 ) )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1140_sup__compl__top__left1,axiom,
! [X2: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ ( sup_sup_set_nat @ X2 @ Y ) )
= top_top_set_nat ) ).
% sup_compl_top_left1
thf(fact_1141_sup__compl__top__left2,axiom,
! [X2: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y ) )
= top_top_set_nat ) ).
% sup_compl_top_left2
thf(fact_1142_boolean__algebra_Odisj__cancel__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ X2 )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_left
thf(fact_1143_boolean__algebra_Odisj__cancel__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ X2 ) )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_right
thf(fact_1144_subset__Compl__singleton,axiom,
! [A: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat2 @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_1145_subset__Compl__singleton,axiom,
! [A: set_list_a,B2: list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( uminus7925729386456332763list_a @ ( insert_list_a2 @ B2 @ bot_bot_set_list_a ) ) )
= ( ~ ( member_list_a2 @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_1146_coset__def,axiom,
( coset_nat
= ( ^ [Xs2: list_nat] : ( uminus5710092332889474511et_nat @ ( set_nat2 @ Xs2 ) ) ) ) ).
% coset_def
thf(fact_1147_compl__coset,axiom,
! [Xs: list_nat] :
( ( uminus5710092332889474511et_nat @ ( coset_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% compl_coset
thf(fact_1148_Compl__eq__Diff__UNIV,axiom,
( uminus5710092332889474511et_nat
= ( minus_minus_set_nat @ top_top_set_nat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1149_Compl__empty__eq,axiom,
( ( uminus7925729386456332763list_a @ bot_bot_set_list_a )
= top_top_set_list_a ) ).
% Compl_empty_eq
thf(fact_1150_Compl__empty__eq,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% Compl_empty_eq
thf(fact_1151_Compl__UNIV__eq,axiom,
( ( uminus7925729386456332763list_a @ top_top_set_list_a )
= bot_bot_set_list_a ) ).
% Compl_UNIV_eq
thf(fact_1152_Compl__UNIV__eq,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Compl_UNIV_eq
thf(fact_1153_inf__cancel__left2,axiom,
! [X2: set_nat,A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ A2 ) @ ( inf_inf_set_nat @ X2 @ B2 ) )
= bot_bot_set_nat ) ).
% inf_cancel_left2
thf(fact_1154_inf__cancel__left2,axiom,
! [X2: set_list_a,A2: set_list_a,B2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ X2 ) @ A2 ) @ ( inf_inf_set_list_a @ X2 @ B2 ) )
= bot_bot_set_list_a ) ).
% inf_cancel_left2
thf(fact_1155_inf__cancel__left1,axiom,
! [X2: set_nat,A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X2 @ A2 ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ B2 ) )
= bot_bot_set_nat ) ).
% inf_cancel_left1
thf(fact_1156_inf__cancel__left1,axiom,
! [X2: set_list_a,A2: set_list_a,B2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X2 @ A2 ) @ ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ X2 ) @ B2 ) )
= bot_bot_set_list_a ) ).
% inf_cancel_left1
thf(fact_1157_diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] : ( inf_inf_set_nat @ X3 @ ( uminus5710092332889474511et_nat @ Y3 ) ) ) ) ).
% diff_eq
thf(fact_1158_sup__cancel__left2,axiom,
! [X2: set_nat,A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ A2 ) @ ( sup_sup_set_nat @ X2 @ B2 ) )
= top_top_set_nat ) ).
% sup_cancel_left2
thf(fact_1159_sup__cancel__left1,axiom,
! [X2: set_nat,A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ A2 ) @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ B2 ) )
= top_top_set_nat ) ).
% sup_cancel_left1
thf(fact_1160_subset__Compl__self__eq,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_1161_subset__Compl__self__eq,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( uminus7925729386456332763list_a @ A ) )
= ( A = bot_bot_set_list_a ) ) ).
% subset_Compl_self_eq
thf(fact_1162_Diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B4: set_nat] : ( inf_inf_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ B4 ) ) ) ) ).
% Diff_eq
thf(fact_1163_Compl__partition,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
= top_top_set_nat ) ).
% Compl_partition
thf(fact_1164_Compl__partition2,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ A )
= top_top_set_nat ) ).
% Compl_partition2
thf(fact_1165_inf__shunt,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X2 @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% inf_shunt
thf(fact_1166_inf__shunt,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( ( inf_inf_set_list_a @ X2 @ Y )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ X2 @ ( uminus7925729386456332763list_a @ Y ) ) ) ).
% inf_shunt
thf(fact_1167_sup__shunt,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X2 @ Y )
= top_top_set_nat )
= ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y ) ) ).
% sup_shunt
thf(fact_1168_surj__Compl__image__subset,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A ) ) @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1169_disjoint__eq__subset__Compl,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1170_disjoint__eq__subset__Compl,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A @ ( uminus7925729386456332763list_a @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1171_Compl__insert,axiom,
! [X2: list_a,A: set_list_a] :
( ( uminus7925729386456332763list_a @ ( insert_list_a2 @ X2 @ A ) )
= ( minus_646659088055828811list_a @ ( uminus7925729386456332763list_a @ A ) @ ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) ) ).
% Compl_insert
thf(fact_1172_Compl__insert,axiom,
! [X2: nat,A: set_nat] :
( ( uminus5710092332889474511et_nat @ ( insert_nat2 @ X2 @ A ) )
= ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ).
% Compl_insert
thf(fact_1173_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( ( inf_inf_set_list_a @ X2 @ Y )
= bot_bot_set_list_a )
=> ( ( ( sup_sup_set_list_a @ X2 @ Y )
= top_top_set_list_a )
=> ( ( uminus7925729386456332763list_a @ X2 )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1174_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X2 @ Y )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ X2 @ Y )
= top_top_set_nat )
=> ( ( uminus5710092332889474511et_nat @ X2 )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1175_ComplI,axiom,
! [C: nat,A: set_nat] :
( ~ ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_1176_Compl__iff,axiom,
! [C: nat,A: set_nat] :
( ( member_nat2 @ C @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat2 @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1177_ComplD,axiom,
! [C: nat,A: set_nat] :
( ( member_nat2 @ C @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat2 @ C @ A ) ) ).
% ComplD
thf(fact_1178_extend_Osimps_I1_J,axiom,
! [X2: nat,A: set_nat,Xs: list_nat,Ys: list_a] :
( ( ( member_nat2 @ X2 @ A )
=> ( ( extend_a @ A @ ( cons_nat @ X2 @ Xs ) @ Ys )
= ( comple6928918032620976721list_a
@ ( image_5464838071766335845list_a
@ ^ [Zs3: list_a] : ( insert_list_a2 @ ( cons_a @ ( hd_a @ Ys ) @ Zs3 ) @ bot_bot_set_list_a )
@ ( extend_a @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) @ Xs @ ( tl_a @ Ys ) ) ) ) ) )
& ( ~ ( member_nat2 @ X2 @ A )
=> ( ( extend_a @ A @ ( cons_nat @ X2 @ Xs ) @ Ys )
= ( comple6928918032620976721list_a
@ ( image_a_set_list_a
@ ^ [Z4: a] :
( comple6928918032620976721list_a
@ ( image_5464838071766335845list_a
@ ^ [Zs3: list_a] : ( insert_list_a2 @ ( cons_a @ Z4 @ Zs3 ) @ bot_bot_set_list_a )
@ ( extend_a @ A @ Xs @ Ys ) ) )
@ top_top_set_a ) ) ) ) ) ).
% extend.simps(1)
thf(fact_1179_extend_Osimps_I1_J,axiom,
! [X2: nat,A: set_nat,Xs: list_nat,Ys: list_nat] :
( ( ( member_nat2 @ X2 @ A )
=> ( ( extend_nat @ A @ ( cons_nat @ X2 @ Xs ) @ Ys )
= ( comple8404747032580312297st_nat
@ ( image_8532145185254316925st_nat
@ ^ [Zs3: list_nat] : ( insert_list_nat2 @ ( cons_nat @ ( hd_nat @ Ys ) @ Zs3 ) @ bot_bot_set_list_nat )
@ ( extend_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) @ Xs @ ( tl_nat @ Ys ) ) ) ) ) )
& ( ~ ( member_nat2 @ X2 @ A )
=> ( ( extend_nat @ A @ ( cons_nat @ X2 @ Xs ) @ Ys )
= ( comple8404747032580312297st_nat
@ ( image_2883343038133793645st_nat
@ ^ [Z4: nat] :
( comple8404747032580312297st_nat
@ ( image_8532145185254316925st_nat
@ ^ [Zs3: list_nat] : ( insert_list_nat2 @ ( cons_nat @ Z4 @ Zs3 ) @ bot_bot_set_list_nat )
@ ( extend_nat @ A @ Xs @ Ys ) ) )
@ top_top_set_nat ) ) ) ) ) ).
% extend.simps(1)
thf(fact_1180_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_list_a
@ ^ [S4: list_a] : P )
= top_top_set_list_a ) )
& ( ~ P
=> ( ( collect_list_a
@ ^ [S4: list_a] : P )
= bot_bot_set_list_a ) ) ) ).
% Collect_const
thf(fact_1181_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_nat
@ ^ [S4: nat] : P )
= top_top_set_nat ) )
& ( ~ P
=> ( ( collect_nat
@ ^ [S4: nat] : P )
= bot_bot_set_nat ) ) ) ).
% Collect_const
thf(fact_1182_finite__Collect__not,axiom,
! [P: nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
~ ( P @ X3 ) ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Collect_not
thf(fact_1183_singleton__conv2,axiom,
! [A2: nat] :
( ( collect_nat
@ ( ^ [Y5: nat,Z5: nat] : ( Y5 = Z5 )
@ A2 ) )
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_1184_singleton__conv2,axiom,
! [A2: list_a] :
( ( collect_list_a
@ ( ^ [Y5: list_a,Z5: list_a] : ( Y5 = Z5 )
@ A2 ) )
= ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) ) ).
% singleton_conv2
thf(fact_1185_singleton__conv,axiom,
! [A2: nat] :
( ( collect_nat
@ ^ [X3: nat] : ( X3 = A2 ) )
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_1186_singleton__conv,axiom,
! [A2: list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( X3 = A2 ) )
= ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) ) ).
% singleton_conv
thf(fact_1187_UN__constant,axiom,
! [A: set_nat,C: set_nat] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : C
@ A ) )
= bot_bot_set_nat ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_1188_UN__constant,axiom,
! [A: set_nat,C: set_list_a] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple6928918032620976721list_a
@ ( image_nat_set_list_a
@ ^ [Y3: nat] : C
@ A ) )
= bot_bot_set_list_a ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple6928918032620976721list_a
@ ( image_nat_set_list_a
@ ^ [Y3: nat] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_1189_UN__constant,axiom,
! [A: set_list_a,C: set_nat] :
( ( ( A = bot_bot_set_list_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_list_a_set_nat
@ ^ [Y3: list_a] : C
@ A ) )
= bot_bot_set_nat ) )
& ( ( A != bot_bot_set_list_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_list_a_set_nat
@ ^ [Y3: list_a] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_1190_UN__constant,axiom,
! [A: set_list_a,C: set_list_a] :
( ( ( A = bot_bot_set_list_a )
=> ( ( comple6928918032620976721list_a
@ ( image_5464838071766335845list_a
@ ^ [Y3: list_a] : C
@ A ) )
= bot_bot_set_list_a ) )
& ( ( A != bot_bot_set_list_a )
=> ( ( comple6928918032620976721list_a
@ ( image_5464838071766335845list_a
@ ^ [Y3: list_a] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_1191_range__constant,axiom,
! [X2: nat] :
( ( image_nat_nat
@ ^ [Uu2: nat] : X2
@ top_top_set_nat )
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ).
% range_constant
thf(fact_1192_range__constant,axiom,
! [X2: list_a] :
( ( image_nat_list_a
@ ^ [Uu2: nat] : X2
@ top_top_set_nat )
= ( insert_list_a2 @ X2 @ bot_bot_set_list_a ) ) ).
% range_constant
thf(fact_1193_UN__simps_I1_J,axiom,
! [C3: set_nat,A2: nat,B: nat > set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : ( insert_nat2 @ A2 @ ( B @ X3 ) )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : ( insert_nat2 @ A2 @ ( B @ X3 ) )
@ C3 ) )
= ( insert_nat2 @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_1194_UN__simps_I1_J,axiom,
! [C3: set_nat,A2: list_a,B: nat > set_list_a] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple6928918032620976721list_a
@ ( image_nat_set_list_a
@ ^ [X3: nat] : ( insert_list_a2 @ A2 @ ( B @ X3 ) )
@ C3 ) )
= bot_bot_set_list_a ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple6928918032620976721list_a
@ ( image_nat_set_list_a
@ ^ [X3: nat] : ( insert_list_a2 @ A2 @ ( B @ X3 ) )
@ C3 ) )
= ( insert_list_a2 @ A2 @ ( comple6928918032620976721list_a @ ( image_nat_set_list_a @ B @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_1195_UN__simps_I1_J,axiom,
! [C3: set_list_a,A2: nat,B: list_a > set_nat] :
( ( ( C3 = bot_bot_set_list_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_list_a_set_nat
@ ^ [X3: list_a] : ( insert_nat2 @ A2 @ ( B @ X3 ) )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bot_set_list_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_list_a_set_nat
@ ^ [X3: list_a] : ( insert_nat2 @ A2 @ ( B @ X3 ) )
@ C3 ) )
= ( insert_nat2 @ A2 @ ( comple7399068483239264473et_nat @ ( image_list_a_set_nat @ B @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_1196_UN__simps_I1_J,axiom,
! [C3: set_list_a,A2: list_a,B: list_a > set_list_a] :
( ( ( C3 = bot_bot_set_list_a )
=> ( ( comple6928918032620976721list_a
@ ( image_5464838071766335845list_a
@ ^ [X3: list_a] : ( insert_list_a2 @ A2 @ ( B @ X3 ) )
@ C3 ) )
= bot_bot_set_list_a ) )
& ( ( C3 != bot_bot_set_list_a )
=> ( ( comple6928918032620976721list_a
@ ( image_5464838071766335845list_a
@ ^ [X3: list_a] : ( insert_list_a2 @ A2 @ ( B @ X3 ) )
@ C3 ) )
= ( insert_list_a2 @ A2 @ ( comple6928918032620976721list_a @ ( image_5464838071766335845list_a @ B @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_1197_UN__singleton,axiom,
! [A: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : ( insert_nat2 @ X3 @ bot_bot_set_nat )
@ A ) )
= A ) ).
% UN_singleton
thf(fact_1198_UN__singleton,axiom,
! [A: set_list_a] :
( ( comple6928918032620976721list_a
@ ( image_5464838071766335845list_a
@ ^ [X3: list_a] : ( insert_list_a2 @ X3 @ bot_bot_set_list_a )
@ A ) )
= A ) ).
% UN_singleton
thf(fact_1199_UN__simps_I2_J,axiom,
! [C3: set_nat,A: nat > set_nat,B: set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : ( sup_sup_set_nat @ ( A @ X3 ) @ B )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : ( sup_sup_set_nat @ ( A @ X3 ) @ B )
@ C3 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ C3 ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1200_UN__simps_I2_J,axiom,
! [C3: set_nat,A: nat > set_list_a,B: set_list_a] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple6928918032620976721list_a
@ ( image_nat_set_list_a
@ ^ [X3: nat] : ( sup_sup_set_list_a @ ( A @ X3 ) @ B )
@ C3 ) )
= bot_bot_set_list_a ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple6928918032620976721list_a
@ ( image_nat_set_list_a
@ ^ [X3: nat] : ( sup_sup_set_list_a @ ( A @ X3 ) @ B )
@ C3 ) )
= ( sup_sup_set_list_a @ ( comple6928918032620976721list_a @ ( image_nat_set_list_a @ A @ C3 ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1201_UN__simps_I2_J,axiom,
! [C3: set_list_a,A: list_a > set_nat,B: set_nat] :
( ( ( C3 = bot_bot_set_list_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_list_a_set_nat
@ ^ [X3: list_a] : ( sup_sup_set_nat @ ( A @ X3 ) @ B )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bot_set_list_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_list_a_set_nat
@ ^ [X3: list_a] : ( sup_sup_set_nat @ ( A @ X3 ) @ B )
@ C3 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_list_a_set_nat @ A @ C3 ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1202_UN__simps_I2_J,axiom,
! [C3: set_list_a,A: list_a > set_list_a,B: set_list_a] :
( ( ( C3 = bot_bot_set_list_a )
=> ( ( comple6928918032620976721list_a
@ ( image_5464838071766335845list_a
@ ^ [X3: list_a] : ( sup_sup_set_list_a @ ( A @ X3 ) @ B )
@ C3 ) )
= bot_bot_set_list_a ) )
& ( ( C3 != bot_bot_set_list_a )
=> ( ( comple6928918032620976721list_a
@ ( image_5464838071766335845list_a
@ ^ [X3: list_a] : ( sup_sup_set_list_a @ ( A @ X3 ) @ B )
@ C3 ) )
= ( sup_sup_set_list_a @ ( comple6928918032620976721list_a @ ( image_5464838071766335845list_a @ A @ C3 ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1203_UN__simps_I3_J,axiom,
! [C3: set_nat,A: set_nat,B: nat > set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : ( sup_sup_set_nat @ A @ ( B @ X3 ) )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : ( sup_sup_set_nat @ A @ ( B @ X3 ) )
@ C3 ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1204_UN__simps_I3_J,axiom,
! [C3: set_nat,A: set_list_a,B: nat > set_list_a] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple6928918032620976721list_a
@ ( image_nat_set_list_a
@ ^ [X3: nat] : ( sup_sup_set_list_a @ A @ ( B @ X3 ) )
@ C3 ) )
= bot_bot_set_list_a ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple6928918032620976721list_a
@ ( image_nat_set_list_a
@ ^ [X3: nat] : ( sup_sup_set_list_a @ A @ ( B @ X3 ) )
@ C3 ) )
= ( sup_sup_set_list_a @ A @ ( comple6928918032620976721list_a @ ( image_nat_set_list_a @ B @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1205_UN__simps_I3_J,axiom,
! [C3: set_list_a,A: set_nat,B: list_a > set_nat] :
( ( ( C3 = bot_bot_set_list_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_list_a_set_nat
@ ^ [X3: list_a] : ( sup_sup_set_nat @ A @ ( B @ X3 ) )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bot_set_list_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_list_a_set_nat
@ ^ [X3: list_a] : ( sup_sup_set_nat @ A @ ( B @ X3 ) )
@ C3 ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_list_a_set_nat @ B @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1206_UN__simps_I3_J,axiom,
! [C3: set_list_a,A: set_list_a,B: list_a > set_list_a] :
( ( ( C3 = bot_bot_set_list_a )
=> ( ( comple6928918032620976721list_a
@ ( image_5464838071766335845list_a
@ ^ [X3: list_a] : ( sup_sup_set_list_a @ A @ ( B @ X3 ) )
@ C3 ) )
= bot_bot_set_list_a ) )
& ( ( C3 != bot_bot_set_list_a )
=> ( ( comple6928918032620976721list_a
@ ( image_5464838071766335845list_a
@ ^ [X3: list_a] : ( sup_sup_set_list_a @ A @ ( B @ X3 ) )
@ C3 ) )
= ( sup_sup_set_list_a @ A @ ( comple6928918032620976721list_a @ ( image_5464838071766335845list_a @ B @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1207_set__concat,axiom,
! [Xs: list_list_nat] :
( ( set_nat2 @ ( concat_nat @ Xs ) )
= ( comple7399068483239264473et_nat @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ Xs ) ) ) ) ).
% set_concat
thf(fact_1208_Pow__set_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( pow_nat @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) )
= ( sup_sup_set_set_nat @ ( pow_nat @ ( set_nat2 @ Xs ) ) @ ( image_7916887816326733075et_nat @ ( insert_nat2 @ X2 ) @ ( pow_nat @ ( set_nat2 @ Xs ) ) ) ) ) ).
% Pow_set(2)
thf(fact_1209_Pow__set_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( pow_a @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) )
= ( sup_sup_set_set_a @ ( pow_a @ ( set_a2 @ Xs ) ) @ ( image_set_a_set_a @ ( insert_a2 @ X2 ) @ ( pow_a @ ( set_a2 @ Xs ) ) ) ) ) ).
% Pow_set(2)
thf(fact_1210_subset__CollectI,axiom,
! [B: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ! [X: nat] :
( ( member_nat2 @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1211_subset__Collect__iff,axiom,
! [B: set_nat,A: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1212_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1213_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat2 @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1214_pred__subset__eq,axiom,
! [R2: set_nat,S3: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat2 @ X3 @ R2 )
@ ^ [X3: nat] : ( member_nat2 @ X3 @ S3 ) )
= ( ord_less_eq_set_nat @ R2 @ S3 ) ) ).
% pred_subset_eq
thf(fact_1215_lists__eq__set,axiom,
( lists_nat
= ( ^ [A3: set_nat] :
( collect_list_nat
@ ^ [Xs2: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A3 ) ) ) ) ).
% lists_eq_set
thf(fact_1216_Collect__conv__if2,axiom,
! [P: nat > $o,A2: nat] :
( ( ( P @ A2 )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A2 = X3 )
& ( P @ X3 ) ) )
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A2 = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_1217_Collect__conv__if2,axiom,
! [P: list_a > $o,A2: list_a] :
( ( ( P @ A2 )
=> ( ( collect_list_a
@ ^ [X3: list_a] :
( ( A2 = X3 )
& ( P @ X3 ) ) )
= ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_list_a
@ ^ [X3: list_a] :
( ( A2 = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_list_a ) ) ) ).
% Collect_conv_if2
thf(fact_1218_Collect__conv__if,axiom,
! [P: nat > $o,A2: nat] :
( ( ( P @ A2 )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A2 )
& ( P @ X3 ) ) )
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A2 )
& ( P @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_1219_Collect__conv__if,axiom,
! [P: list_a > $o,A2: list_a] :
( ( ( P @ A2 )
=> ( ( collect_list_a
@ ^ [X3: list_a] :
( ( X3 = A2 )
& ( P @ X3 ) ) )
= ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_list_a
@ ^ [X3: list_a] :
( ( X3 = A2 )
& ( P @ X3 ) ) )
= bot_bot_set_list_a ) ) ) ).
% Collect_conv_if
thf(fact_1220_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X3: nat] : $false ) ) ).
% empty_def
thf(fact_1221_empty__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a
@ ^ [X3: list_a] : $false ) ) ).
% empty_def
thf(fact_1222_sup__Un__eq,axiom,
! [R2: set_nat,S3: set_nat] :
( ( sup_sup_nat_o
@ ^ [X3: nat] : ( member_nat2 @ X3 @ R2 )
@ ^ [X3: nat] : ( member_nat2 @ X3 @ S3 ) )
= ( ^ [X3: nat] : ( member_nat2 @ X3 @ ( sup_sup_set_nat @ R2 @ S3 ) ) ) ) ).
% sup_Un_eq
thf(fact_1223_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat2 @ X3 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_1224_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ A3 )
| ( member_nat2 @ X3 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1225_insert__def,axiom,
( insert_nat2
= ( ^ [A4: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X3: nat] : ( X3 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_1226_insert__compr,axiom,
( insert_nat2
= ( ^ [A4: nat,B4: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( X3 = A4 )
| ( member_nat2 @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_1227_insert__Collect,axiom,
! [A2: nat,P: nat > $o] :
( ( insert_nat2 @ A2 @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U: nat] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_1228_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ A3 )
& ( member_nat2 @ X3 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_1229_Int__Collect,axiom,
! [X2: nat,A: set_nat,P: nat > $o] :
( ( member_nat2 @ X2 @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) )
= ( ( member_nat2 @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_1230_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat2 @ X3 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_1231_inf__Int__eq,axiom,
! [R2: set_nat,S3: set_nat] :
( ( inf_inf_nat_o
@ ^ [X3: nat] : ( member_nat2 @ X3 @ R2 )
@ ^ [X3: nat] : ( member_nat2 @ X3 @ S3 ) )
= ( ^ [X3: nat] : ( member_nat2 @ X3 @ ( inf_inf_set_nat @ R2 @ S3 ) ) ) ) ).
% inf_Int_eq
thf(fact_1232_inj__singleton,axiom,
! [A: set_nat] :
( inj_on_nat_set_nat
@ ^ [X3: nat] : ( insert_nat2 @ X3 @ bot_bot_set_nat )
@ A ) ).
% inj_singleton
thf(fact_1233_inj__singleton,axiom,
! [A: set_list_a] :
( inj_on1264545500884751569list_a
@ ^ [X3: list_a] : ( insert_list_a2 @ X3 @ bot_bot_set_list_a )
@ A ) ).
% inj_singleton
thf(fact_1234_sorted__list__of__set_Oinj__on,axiom,
( inj_on_nat_nat
@ ^ [X3: nat] : X3
@ top_top_set_nat ) ).
% sorted_list_of_set.inj_on
thf(fact_1235_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X3: nat] : $true ) ) ).
% UNIV_def
thf(fact_1236_Shift__def,axiom,
( bNF_Gr1872714664788909425ft_nat
= ( ^ [Kl3: set_list_nat,K2: nat] :
( collect_list_nat
@ ^ [Kl4: list_nat] : ( member_list_nat2 @ ( cons_nat @ K2 @ Kl4 ) @ Kl3 ) ) ) ) ).
% Shift_def
thf(fact_1237_Shift__def,axiom,
( bNF_Greatest_Shift_a
= ( ^ [Kl3: set_list_a,K2: a] :
( collect_list_a
@ ^ [Kl4: list_a] : ( member_list_a2 @ ( cons_a @ K2 @ Kl4 ) @ Kl3 ) ) ) ) ).
% Shift_def
thf(fact_1238_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ A3 )
& ~ ( member_nat2 @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1239_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat2 @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1240_tl__def,axiom,
( tl_a
= ( case_list_list_a_a @ nil_a
@ ^ [X213: a,X223: list_a] : X223 ) ) ).
% tl_def
thf(fact_1241_finite__inverse__image,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [J2: nat] : ( member_nat2 @ ( F @ J2 ) @ A ) ) ) ) ) ).
% finite_inverse_image
thf(fact_1242_product__lists_Osimps_I2_J,axiom,
! [Xs: list_nat,Xss2: list_list_nat] :
( ( product_lists_nat @ ( cons_list_nat @ Xs @ Xss2 ) )
= ( concat_list_nat
@ ( map_na6205611841492582150st_nat
@ ^ [X3: nat] : ( map_li7225945977422193158st_nat @ ( cons_nat @ X3 ) @ ( product_lists_nat @ Xss2 ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_1243_product__lists_Osimps_I2_J,axiom,
! [Xs: list_a,Xss2: list_list_a] :
( ( product_lists_a @ ( cons_list_a @ Xs @ Xss2 ) )
= ( concat_list_a
@ ( map_a_list_list_a
@ ^ [X3: a] : ( map_list_a_list_a @ ( cons_a @ X3 ) @ ( product_lists_a @ Xss2 ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_1244_inj__on__disjoint__Un,axiom,
! [F: nat > nat,A: set_nat,G2: nat > nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat @ G2 @ B )
=> ( ( ( inf_inf_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ G2 @ B ) )
= bot_bot_set_nat )
=> ( inj_on_nat_nat
@ ^ [X3: nat] : ( if_nat @ ( member_nat2 @ X3 @ A ) @ ( F @ X3 ) @ ( G2 @ X3 ) )
@ ( sup_sup_set_nat @ A @ B ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_1245_inj__on__disjoint__Un,axiom,
! [F: nat > list_a,A: set_nat,G2: nat > list_a,B: set_nat] :
( ( inj_on_nat_list_a @ F @ A )
=> ( ( inj_on_nat_list_a @ G2 @ B )
=> ( ( ( inf_inf_set_list_a @ ( image_nat_list_a @ F @ A ) @ ( image_nat_list_a @ G2 @ B ) )
= bot_bot_set_list_a )
=> ( inj_on_nat_list_a
@ ^ [X3: nat] : ( if_list_a @ ( member_nat2 @ X3 @ A ) @ ( F @ X3 ) @ ( G2 @ X3 ) )
@ ( sup_sup_set_nat @ A @ B ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_1246_image__constant,axiom,
! [X2: nat,A: set_nat,C: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ( image_nat_nat
@ ^ [X3: nat] : C
@ A )
= ( insert_nat2 @ C @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_1247_image__constant,axiom,
! [X2: nat,A: set_nat,C: list_a] :
( ( member_nat2 @ X2 @ A )
=> ( ( image_nat_list_a
@ ^ [X3: nat] : C
@ A )
= ( insert_list_a2 @ C @ bot_bot_set_list_a ) ) ) ).
% image_constant
thf(fact_1248_image__constant__conv,axiom,
! [A: set_nat,C: nat] :
( ( ( A = bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X3: nat] : C
@ A )
= bot_bot_set_nat ) )
& ( ( A != bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X3: nat] : C
@ A )
= ( insert_nat2 @ C @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_1249_image__constant__conv,axiom,
! [A: set_nat,C: list_a] :
( ( ( A = bot_bot_set_nat )
=> ( ( image_nat_list_a
@ ^ [X3: nat] : C
@ A )
= bot_bot_set_list_a ) )
& ( ( A != bot_bot_set_nat )
=> ( ( image_nat_list_a
@ ^ [X3: nat] : C
@ A )
= ( insert_list_a2 @ C @ bot_bot_set_list_a ) ) ) ) ).
% image_constant_conv
thf(fact_1250_image__constant__conv,axiom,
! [A: set_list_a,C: nat] :
( ( ( A = bot_bot_set_list_a )
=> ( ( image_list_a_nat
@ ^ [X3: list_a] : C
@ A )
= bot_bot_set_nat ) )
& ( ( A != bot_bot_set_list_a )
=> ( ( image_list_a_nat
@ ^ [X3: list_a] : C
@ A )
= ( insert_nat2 @ C @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_1251_image__constant__conv,axiom,
! [A: set_list_a,C: list_a] :
( ( ( A = bot_bot_set_list_a )
=> ( ( image_list_a_list_a
@ ^ [X3: list_a] : C
@ A )
= bot_bot_set_list_a ) )
& ( ( A != bot_bot_set_list_a )
=> ( ( image_list_a_list_a
@ ^ [X3: list_a] : C
@ A )
= ( insert_list_a2 @ C @ bot_bot_set_list_a ) ) ) ) ).
% image_constant_conv
thf(fact_1252_rangeE,axiom,
! [B2: nat,F: nat > nat] :
( ( member_nat2 @ B2 @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ~ ! [X: nat] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_1253_imageE,axiom,
! [B2: nat,F: nat > nat,A: set_nat] :
( ( member_nat2 @ B2 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [X: nat] :
( ( B2
= ( F @ X ) )
=> ~ ( member_nat2 @ X @ A ) ) ) ).
% imageE
thf(fact_1254_Compr__image__eq,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ ( image_nat_nat @ F @ A ) )
& ( P @ X3 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1255_finite__range__imageI,axiom,
! [G2: nat > nat,F: nat > nat] :
( ( finite_finite_nat @ ( image_nat_nat @ G2 @ top_top_set_nat ) )
=> ( finite_finite_nat
@ ( image_nat_nat
@ ^ [X3: nat] : ( F @ ( G2 @ X3 ) )
@ top_top_set_nat ) ) ) ).
% finite_range_imageI
thf(fact_1256_UN__extend__simps_I1_J,axiom,
! [C3: set_nat,A2: nat,B: nat > set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( insert_nat2 @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C3 ) ) )
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( insert_nat2 @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C3 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : ( insert_nat2 @ A2 @ ( B @ X3 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_1257_UN__extend__simps_I1_J,axiom,
! [C3: set_nat,A2: list_a,B: nat > set_list_a] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( insert_list_a2 @ A2 @ ( comple6928918032620976721list_a @ ( image_nat_set_list_a @ B @ C3 ) ) )
= ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( insert_list_a2 @ A2 @ ( comple6928918032620976721list_a @ ( image_nat_set_list_a @ B @ C3 ) ) )
= ( comple6928918032620976721list_a
@ ( image_nat_set_list_a
@ ^ [X3: nat] : ( insert_list_a2 @ A2 @ ( B @ X3 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_1258_UN__extend__simps_I1_J,axiom,
! [C3: set_list_a,A2: nat,B: list_a > set_nat] :
( ( ( C3 = bot_bot_set_list_a )
=> ( ( insert_nat2 @ A2 @ ( comple7399068483239264473et_nat @ ( image_list_a_set_nat @ B @ C3 ) ) )
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
& ( ( C3 != bot_bot_set_list_a )
=> ( ( insert_nat2 @ A2 @ ( comple7399068483239264473et_nat @ ( image_list_a_set_nat @ B @ C3 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_list_a_set_nat
@ ^ [X3: list_a] : ( insert_nat2 @ A2 @ ( B @ X3 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_1259_UN__extend__simps_I1_J,axiom,
! [C3: set_list_a,A2: list_a,B: list_a > set_list_a] :
( ( ( C3 = bot_bot_set_list_a )
=> ( ( insert_list_a2 @ A2 @ ( comple6928918032620976721list_a @ ( image_5464838071766335845list_a @ B @ C3 ) ) )
= ( insert_list_a2 @ A2 @ bot_bot_set_list_a ) ) )
& ( ( C3 != bot_bot_set_list_a )
=> ( ( insert_list_a2 @ A2 @ ( comple6928918032620976721list_a @ ( image_5464838071766335845list_a @ B @ C3 ) ) )
= ( comple6928918032620976721list_a
@ ( image_5464838071766335845list_a
@ ^ [X3: list_a] : ( insert_list_a2 @ A2 @ ( B @ X3 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_1260_UN__insert__distrib,axiom,
! [U2: nat,A: set_nat,A2: nat,B: nat > set_nat] :
( ( member_nat2 @ U2 @ A )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : ( insert_nat2 @ A2 @ ( B @ X3 ) )
@ A ) )
= ( insert_nat2 @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_1261_SUP__Sup__eq,axiom,
! [S3: set_set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_set_nat_nat_o
@ ^ [I2: set_nat,X3: nat] : ( member_nat2 @ X3 @ I2 )
@ S3 ) )
= ( ^ [X3: nat] : ( member_nat2 @ X3 @ ( comple7399068483239264473et_nat @ S3 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1262_UN__empty,axiom,
! [B: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_1263_UN__empty,axiom,
! [B: nat > set_list_a] :
( ( comple6928918032620976721list_a @ ( image_nat_set_list_a @ B @ bot_bot_set_nat ) )
= bot_bot_set_list_a ) ).
% UN_empty
thf(fact_1264_UN__empty,axiom,
! [B: list_a > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_list_a_set_nat @ B @ bot_bot_set_list_a ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_1265_UN__empty,axiom,
! [B: list_a > set_list_a] :
( ( comple6928918032620976721list_a @ ( image_5464838071766335845list_a @ B @ bot_bot_set_list_a ) )
= bot_bot_set_list_a ) ).
% UN_empty
thf(fact_1266_SUP__constant,axiom,
! [A: set_nat,C: set_nat] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : C
@ A ) )
= bot_bot_set_nat ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1267_SUP__constant,axiom,
! [A: set_nat,C: set_list_a] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple6928918032620976721list_a
@ ( image_nat_set_list_a
@ ^ [Y3: nat] : C
@ A ) )
= bot_bot_set_list_a ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple6928918032620976721list_a
@ ( image_nat_set_list_a
@ ^ [Y3: nat] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1268_SUP__constant,axiom,
! [A: set_list_a,C: set_nat] :
( ( ( A = bot_bot_set_list_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_list_a_set_nat
@ ^ [Y3: list_a] : C
@ A ) )
= bot_bot_set_nat ) )
& ( ( A != bot_bot_set_list_a )
=> ( ( comple7399068483239264473et_nat
@ ( image_list_a_set_nat
@ ^ [Y3: list_a] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1269_SUP__constant,axiom,
! [A: set_list_a,C: set_list_a] :
( ( ( A = bot_bot_set_list_a )
=> ( ( comple6928918032620976721list_a
@ ( image_5464838071766335845list_a
@ ^ [Y3: list_a] : C
@ A ) )
= bot_bot_set_list_a ) )
& ( ( A != bot_bot_set_list_a )
=> ( ( comple6928918032620976721list_a
@ ( image_5464838071766335845list_a
@ ^ [Y3: list_a] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1270_SUP__empty,axiom,
! [F: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_1271_SUP__empty,axiom,
! [F: nat > set_list_a] :
( ( comple6928918032620976721list_a @ ( image_nat_set_list_a @ F @ bot_bot_set_nat ) )
= bot_bot_set_list_a ) ).
% SUP_empty
thf(fact_1272_SUP__empty,axiom,
! [F: list_a > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_list_a_set_nat @ F @ bot_bot_set_list_a ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_1273_SUP__empty,axiom,
! [F: list_a > set_list_a] :
( ( comple6928918032620976721list_a @ ( image_5464838071766335845list_a @ F @ bot_bot_set_list_a ) )
= bot_bot_set_list_a ) ).
% SUP_empty
% Helper facts (19)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X2: list_a,Y: list_a] :
( ( if_list_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X2: list_a,Y: list_a] :
( ( if_list_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X2: set_nat,Y: set_nat] :
( ( if_set_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X2: set_nat,Y: set_nat] :
( ( if_set_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J_T,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( if_set_list_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J_T,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( if_set_list_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
! [X2: list_list_a,Y: list_list_a] :
( ( if_list_list_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
! [X2: list_list_a,Y: list_list_a] :
( ( if_list_list_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [X2: list_set_nat,Y: list_set_nat] :
( ( if_list_set_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [X2: list_set_nat,Y: list_set_nat] :
( ( if_list_set_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
! [X2: list_list_nat,Y: list_list_nat] :
( ( if_list_list_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
! [X2: list_list_nat,Y: list_list_nat] :
( ( if_list_list_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_T,axiom,
! [X2: list_set_list_a,Y: list_set_list_a] :
( ( if_list_set_list_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_T,axiom,
! [X2: list_set_list_a,Y: list_set_list_a] :
( ( if_list_set_list_a @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
? [X4: list_a] :
( ( member_list_a2 @ X4 @ ( extend_a @ aa @ ( cons_nat @ a2 @ xsa ) @ ysa ) )
& ( ( lookup_nat_a @ ( cons_nat @ a2 @ xsa ) @ X4 @ x )
= d ) ) ).
%------------------------------------------------------------------------------