TPTP Problem File: SLH0871^1.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 : FOL_Seq_Calc2/0017_Countermodel/prob_00041_001557__12885122_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1557 ( 741 unt; 274 typ; 0 def)
% Number of atoms : 3327 (1555 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9871 ( 443 ~; 62 |; 329 &;7881 @)
% ( 0 <=>;1156 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 25 ( 24 usr)
% Number of type conns : 642 ( 642 >; 0 *; 0 +; 0 <<)
% Number of symbols : 253 ( 250 usr; 20 con; 0-3 aty)
% Number of variables : 3447 ( 216 ^;2990 !; 241 ?;3447 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:44:41.186
%------------------------------------------------------------------------------
% Could-be-implicit typings (24)
thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
list_list_set_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__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
list_list_tm: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
list_list_fm: $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__SeCaV__Otm_J_J,type,
set_list_tm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
set_list_fm: $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__SeCaV__Otm_J_J,type,
set_set_tm: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__SeCaV__Ofm_J_J,type,
set_set_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__SeCaV__Otm_J,type,
list_tm: $tType ).
thf(ty_n_t__List__Olist_It__SeCaV__Ofm_J,type,
list_fm: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__SeCaV__Otm_J,type,
set_tm: $tType ).
thf(ty_n_t__Set__Oset_It__SeCaV__Ofm_J,type,
set_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Extended____Nat__Oenat,type,
extended_enat: $tType ).
thf(ty_n_t__SeCaV__Otm,type,
tm: $tType ).
thf(ty_n_t__SeCaV__Ofm,type,
fm: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (250)
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_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
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__SeCaV__Ofm_J,type,
minus_minus_set_fm: set_fm > set_fm > set_fm ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__SeCaV__Otm_J,type,
minus_minus_set_tm: set_tm > set_tm > set_tm ).
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_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
zero_z5237406670263579293d_enat: extended_enat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Hintikka_Oterms,type,
terms: set_fm > set_tm ).
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__SeCaV__Ofm_J,type,
if_list_fm: $o > list_fm > list_fm > list_fm ).
thf(sy_c_If_001t__List__Olist_It__SeCaV__Otm_J,type,
if_list_tm: $o > list_tm > list_tm > list_tm ).
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_Lattices_Osup__class_Osup_001t__Extended____Nat__Oenat,type,
sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__SeCaV__Ofm_J,type,
sup_sup_set_fm: set_fm > set_fm > set_fm ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__SeCaV__Otm_J,type,
sup_sup_set_tm: set_tm > set_tm > set_tm ).
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_List_OListMem_001t__Nat__Onat,type,
listMem_nat: nat > list_nat > $o ).
thf(sy_c_List_OListMem_001t__SeCaV__Ofm,type,
listMem_fm: fm > list_fm > $o ).
thf(sy_c_List_OListMem_001t__SeCaV__Otm,type,
listMem_tm: tm > list_tm > $o ).
thf(sy_c_List_OListMem_001t__Set__Oset_It__Nat__Onat_J,type,
listMem_set_nat: set_nat > list_set_nat > $o ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
append_list_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oappend_001t__List__Olist_It__SeCaV__Otm_J,type,
append_list_tm: list_list_tm > list_list_tm > list_list_tm ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__SeCaV__Ofm,type,
append_fm: list_fm > list_fm > list_fm ).
thf(sy_c_List_Oappend_001t__SeCaV__Otm,type,
append_tm: list_tm > list_tm > list_tm ).
thf(sy_c_List_Oappend_001t__Set__Oset_It__Nat__Onat_J,type,
append_set_nat: list_set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__SeCaV__Otm,type,
bind_nat_tm: list_nat > ( nat > list_tm ) > list_tm ).
thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__Nat__Onat,type,
bind_tm_nat: list_tm > ( tm > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
bind_tm_tm: list_tm > ( tm > list_tm ) > list_tm ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Obutlast_001t__SeCaV__Otm,type,
butlast_tm: list_tm > list_tm ).
thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
can_select_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_List_Ocan__select_001t__SeCaV__Ofm,type,
can_select_fm: ( fm > $o ) > set_fm > $o ).
thf(sy_c_List_Ocan__select_001t__SeCaV__Otm,type,
can_select_tm: ( tm > $o ) > set_tm > $o ).
thf(sy_c_List_Ocan__select_001t__Set__Oset_It__Nat__Onat_J,type,
can_select_set_nat: ( set_nat > $o ) > set_set_nat > $o ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Oconcat_001t__SeCaV__Otm,type,
concat_tm: list_list_tm > list_tm ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Ocoset_001t__SeCaV__Ofm,type,
coset_fm: list_fm > set_fm ).
thf(sy_c_List_Ocoset_001t__SeCaV__Otm,type,
coset_tm: list_tm > set_tm ).
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_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001t__SeCaV__Ofm,type,
insert_fm: fm > list_fm > list_fm ).
thf(sy_c_List_Oinsert_001t__SeCaV__Otm,type,
insert_tm: tm > list_tm > list_tm ).
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_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olast_001t__SeCaV__Otm,type,
last_tm: list_tm > tm ).
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_It__SeCaV__Otm_J,type,
cons_list_tm: list_tm > list_list_tm > list_list_tm ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__SeCaV__Ofm,type,
cons_fm: fm > list_fm > list_fm ).
thf(sy_c_List_Olist_OCons_001t__SeCaV__Otm,type,
cons_tm: tm > list_tm > list_tm ).
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_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__SeCaV__Otm_J,type,
nil_list_tm: list_list_tm ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__SeCaV__Ofm,type,
nil_fm: list_fm ).
thf(sy_c_List_Olist_ONil_001t__SeCaV__Otm,type,
nil_tm: list_tm ).
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_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__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Otm_J,type,
map_list_tm_list_tm: ( list_tm > list_tm ) > list_list_tm > list_list_tm ).
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__SeCaV__Otm_001t__List__Olist_It__SeCaV__Otm_J,type,
map_tm_list_tm: ( tm > list_tm ) > list_tm > list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__Set__Oset_It__Nat__Onat_J,type,
map_tm_set_nat: ( tm > set_nat ) > list_tm > list_set_nat ).
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_It__SeCaV__Ofm_J,type,
set_list_fm2: list_list_fm > set_list_fm ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__SeCaV__Otm_J,type,
set_list_tm2: list_list_tm > set_list_tm ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
set_list_set_nat2: list_list_set_nat > set_list_set_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__SeCaV__Ofm,type,
set_fm2: list_fm > set_fm ).
thf(sy_c_List_Olist_Oset_001t__SeCaV__Otm,type,
set_tm2: list_tm > set_tm ).
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__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__SeCaV__Ofm,type,
list_ex1_fm: ( fm > $o ) > list_fm > $o ).
thf(sy_c_List_Olist__ex1_001t__SeCaV__Otm,type,
list_ex1_tm: ( tm > $o ) > list_tm > $o ).
thf(sy_c_List_Olist__ex1_001t__Set__Oset_It__Nat__Onat_J,type,
list_ex1_set_nat: ( set_nat > $o ) > list_set_nat > $o ).
thf(sy_c_List_Omap__tailrec__rev_001t__Nat__Onat_001t__Nat__Onat,type,
map_ta7164188454487880599at_nat: ( nat > nat ) > list_nat > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec__rev_001t__Nat__Onat_001t__SeCaV__Otm,type,
map_ta389968950240100318nat_tm: ( nat > tm ) > list_nat > list_tm > list_tm ).
thf(sy_c_List_Omap__tailrec__rev_001t__SeCaV__Otm_001t__Nat__Onat,type,
map_ta7807370561492357248tm_nat: ( tm > nat ) > list_tm > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec__rev_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
map_ta4789309763159252277_tm_tm: ( tm > tm ) > list_tm > list_tm > list_tm ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__SeCaV__Otm,type,
maps_nat_tm: ( nat > list_tm ) > list_nat > list_tm ).
thf(sy_c_List_Omaps_001t__SeCaV__Otm_001t__Nat__Onat,type,
maps_tm_nat: ( tm > list_nat ) > list_tm > list_nat ).
thf(sy_c_List_Omaps_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
maps_tm_tm: ( tm > list_tm ) > list_tm > list_tm ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_Omember_001t__SeCaV__Ofm,type,
member_fm: list_fm > fm > $o ).
thf(sy_c_List_Omember_001t__SeCaV__Otm,type,
member_tm: list_tm > tm > $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_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_On__lists_001t__SeCaV__Otm,type,
n_lists_tm: nat > list_tm > list_list_tm ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onths_001t__Nat__Onat,type,
nths_nat: list_nat > set_nat > list_nat ).
thf(sy_c_List_Onths_001t__SeCaV__Ofm,type,
nths_fm: list_fm > set_nat > list_fm ).
thf(sy_c_List_Onths_001t__SeCaV__Otm,type,
nths_tm: list_tm > set_nat > list_tm ).
thf(sy_c_List_Onths_001t__Set__Oset_It__Nat__Onat_J,type,
nths_set_nat: list_set_nat > set_nat > list_set_nat ).
thf(sy_c_List_Onull_001t__Nat__Onat,type,
null_nat: list_nat > $o ).
thf(sy_c_List_Onull_001t__SeCaV__Ofm,type,
null_fm: list_fm > $o ).
thf(sy_c_List_Onull_001t__SeCaV__Otm,type,
null_tm: list_tm > $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_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oproduct__lists_001t__SeCaV__Otm,type,
product_lists_tm: list_list_tm > list_list_tm ).
thf(sy_c_List_Oremdups_001t__Nat__Onat,type,
remdups_nat: list_nat > list_nat ).
thf(sy_c_List_Oremdups_001t__SeCaV__Otm,type,
remdups_tm: list_tm > list_tm ).
thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001t__SeCaV__Ofm,type,
removeAll_fm: fm > list_fm > list_fm ).
thf(sy_c_List_OremoveAll_001t__SeCaV__Otm,type,
removeAll_tm: tm > list_tm > list_tm ).
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_Oreplicate_001t__List__Olist_It__Nat__Onat_J,type,
replicate_list_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Oreplicate_001t__List__Olist_It__SeCaV__Otm_J,type,
replicate_list_tm: nat > list_tm > list_list_tm ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Oreplicate_001t__SeCaV__Ofm,type,
replicate_fm: nat > fm > list_fm ).
thf(sy_c_List_Oreplicate_001t__SeCaV__Otm,type,
replicate_tm: nat > tm > list_tm ).
thf(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
replicate_set_nat: nat > set_nat > list_set_nat ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001t__SeCaV__Ofm,type,
rotate1_fm: list_fm > list_fm ).
thf(sy_c_List_Orotate1_001t__SeCaV__Otm,type,
rotate1_tm: list_tm > list_tm ).
thf(sy_c_List_Orotate1_001t__Set__Oset_It__Nat__Onat_J,type,
rotate1_set_nat: list_set_nat > list_set_nat ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Osubseqs_001t__SeCaV__Ofm,type,
subseqs_fm: list_fm > list_list_fm ).
thf(sy_c_List_Osubseqs_001t__SeCaV__Otm,type,
subseqs_tm: list_tm > list_list_tm ).
thf(sy_c_List_Osubseqs_001t__Set__Oset_It__Nat__Onat_J,type,
subseqs_set_nat: list_set_nat > list_list_set_nat ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Ounion_001t__SeCaV__Ofm,type,
union_fm: list_fm > list_fm > list_fm ).
thf(sy_c_List_Ounion_001t__SeCaV__Otm,type,
union_tm: list_tm > list_tm > list_tm ).
thf(sy_c_List_Ounion_001t__Set__Oset_It__Nat__Onat_J,type,
union_set_nat: list_set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__SeCaV__Otm,type,
size_size_tm: tm > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
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__SeCaV__Ofm_M_Eo_J,type,
bot_bot_fm_o: fm > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__SeCaV__Otm_M_Eo_J,type,
bot_bot_tm_o: tm > $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_001t__Extended____Nat__Oenat,type,
bot_bo4199563552545308370d_enat: extended_enat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__SeCaV__Ofm_J,type,
bot_bot_set_fm: set_fm ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__SeCaV__Otm_J,type,
bot_bot_set_tm: set_tm ).
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_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Extended____Nat__Oenat_J,type,
ord_le2787558655864224659d_enat: ( $o > extended_enat ) > ( $o > extended_enat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
ord_le7022414076629706543et_nat: ( $o > set_nat ) > ( $o > set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__SeCaV__Otm_J_J,type,
ord_less_eq_o_set_tm: ( $o > set_tm ) > ( $o > set_tm ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__SeCaV__Ofm_J,type,
ord_less_eq_set_fm: set_fm > set_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__SeCaV__Otm_J,type,
ord_less_eq_set_tm: set_tm > set_tm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Extended____Nat__Oenat,type,
order_2428742583041560895d_enat: ( extended_enat > $o ) > extended_enat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Nat__Onat_J,type,
order_5724808138429204845et_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__SeCaV__Otm_J,type,
order_4248741476211579294set_tm: ( set_tm > $o ) > set_tm ).
thf(sy_c_ProverLemmas_Opreds,type,
preds: fm > set_fm ).
thf(sy_c_Prover_OlistFunTm,type,
listFunTm: tm > list_nat ).
thf(sy_c_Prover_OlistFunTms,type,
listFunTms: list_tm > list_nat ).
thf(sy_c_Prover_OsubtermFm,type,
subtermFm: fm > list_tm ).
thf(sy_c_Prover_OsubtermTm,type,
subtermTm: tm > list_tm ).
thf(sy_c_SeCaV_Oext_001t__Nat__Onat,type,
ext_nat: list_nat > list_nat > $o ).
thf(sy_c_SeCaV_Oext_001t__SeCaV__Ofm,type,
ext_fm: list_fm > list_fm > $o ).
thf(sy_c_SeCaV_Oext_001t__SeCaV__Otm,type,
ext_tm: list_tm > list_tm > $o ).
thf(sy_c_SeCaV_Oext_001t__Set__Oset_It__Nat__Onat_J,type,
ext_set_nat: list_set_nat > list_set_nat > $o ).
thf(sy_c_SeCaV_Ofm_OPre,type,
pre: nat > list_tm > fm ).
thf(sy_c_SeCaV_Oinc__list,type,
inc_list: list_tm > list_tm ).
thf(sy_c_SeCaV_Oinc__term,type,
inc_term: tm > tm ).
thf(sy_c_SeCaV_Oliftt,type,
liftt: tm > tm ).
thf(sy_c_SeCaV_Oliftts,type,
liftts: list_tm > list_tm ).
thf(sy_c_SeCaV_Omember_001t__Nat__Onat,type,
member_nat2: nat > list_nat > $o ).
thf(sy_c_SeCaV_Omember_001t__SeCaV__Ofm,type,
member_fm2: fm > list_fm > $o ).
thf(sy_c_SeCaV_Omember_001t__SeCaV__Otm,type,
member_tm2: tm > list_tm > $o ).
thf(sy_c_SeCaV_Omember_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat2: set_nat > list_set_nat > $o ).
thf(sy_c_SeCaV_Onew,type,
new: nat > fm > $o ).
thf(sy_c_SeCaV_Onew__list,type,
new_list: nat > list_tm > $o ).
thf(sy_c_SeCaV_Onew__term,type,
new_term: nat > tm > $o ).
thf(sy_c_SeCaV_Oparams,type,
params: fm > set_nat ).
thf(sy_c_SeCaV_Oparams_H,type,
params2: fm > set_nat ).
thf(sy_c_SeCaV_Oparams_H_H,type,
params3: fm > set_nat ).
thf(sy_c_SeCaV_Oparamst,type,
paramst: tm > set_nat ).
thf(sy_c_SeCaV_Oparamst_H,type,
paramst2: tm > set_nat ).
thf(sy_c_SeCaV_Oparamst_H_H,type,
paramst3: tm > set_nat ).
thf(sy_c_SeCaV_Oparamst_H_H__rel,type,
paramst_rel: tm > tm > $o ).
thf(sy_c_SeCaV_Oparamsts,type,
paramsts: list_tm > set_nat ).
thf(sy_c_SeCaV_Osub,type,
sub: nat > tm > fm > fm ).
thf(sy_c_SeCaV_Osub__list,type,
sub_list: nat > tm > list_tm > list_tm ).
thf(sy_c_SeCaV_Osub__term,type,
sub_term: nat > tm > tm > tm ).
thf(sy_c_SeCaV_Osubstt,type,
substt: tm > tm > nat > tm ).
thf(sy_c_SeCaV_Osubstts,type,
substts: list_tm > tm > nat > list_tm ).
thf(sy_c_SeCaV_Otm_OFun,type,
fun: nat > list_tm > tm ).
thf(sy_c_SeCaV_Otm_OVar,type,
var: nat > tm ).
thf(sy_c_SeCaV_Otm_Osize__tm,type,
size_tm: tm > nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__SeCaV__Ofm,type,
collect_fm: ( fm > $o ) > set_fm ).
thf(sy_c_Set_OCollect_001t__SeCaV__Otm,type,
collect_tm: ( tm > $o ) > set_tm ).
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_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_It__SeCaV__Ofm_J_001t__Set__Oset_It__SeCaV__Ofm_J,type,
image_list_fm_set_fm: ( list_fm > set_fm ) > set_list_fm > set_set_fm ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Otm_J_001t__Set__Oset_It__SeCaV__Otm_J,type,
image_list_tm_set_tm: ( list_tm > set_tm ) > set_list_tm > set_set_tm ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_8726355809080528601et_nat: ( list_set_nat > set_set_nat ) > set_list_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__Set__Oset_It__Nat__Onat_J,type,
image_tm_set_nat: ( tm > set_nat ) > set_tm > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__SeCaV__Ofm,type,
image_set_nat_fm: ( set_nat > fm ) > set_set_nat > set_fm ).
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_Oinsert_001t__Nat__Onat,type,
insert_nat2: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__SeCaV__Ofm,type,
insert_fm2: fm > set_fm > set_fm ).
thf(sy_c_Set_Oinsert_001t__SeCaV__Otm,type,
insert_tm2: tm > set_tm > set_tm ).
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_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__empty_001t__SeCaV__Ofm,type,
is_empty_fm: set_fm > $o ).
thf(sy_c_Set_Ois__empty_001t__SeCaV__Otm,type,
is_empty_tm: set_tm > $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__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__SeCaV__Ofm,type,
is_singleton_fm: set_fm > $o ).
thf(sy_c_Set_Ois__singleton_001t__SeCaV__Otm,type,
is_singleton_tm: set_tm > $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_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001t__SeCaV__Ofm,type,
remove_fm: fm > set_fm > set_fm ).
thf(sy_c_Set_Oremove_001t__SeCaV__Otm,type,
remove_tm: tm > set_tm > set_tm ).
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_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__SeCaV__Ofm,type,
the_elem_fm: set_fm > fm ).
thf(sy_c_Set_Othe__elem_001t__SeCaV__Otm,type,
the_elem_tm: set_tm > tm ).
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_Sublist_Oprefixes_001t__Nat__Onat,type,
prefixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Oprefixes_001t__SeCaV__Otm,type,
prefixes_tm: list_tm > list_list_tm ).
thf(sy_c_Sublist_Osublists_001t__Nat__Onat,type,
sublists_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osublists_001t__SeCaV__Otm,type,
sublists_tm: list_tm > list_list_tm ).
thf(sy_c_Sublist_Osuffixes_001t__Nat__Onat,type,
suffixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osuffixes_001t__SeCaV__Otm,type,
suffixes_tm: list_tm > list_list_tm ).
thf(sy_c_Wellfounded_Oaccp_001t__SeCaV__Otm,type,
accp_tm: ( tm > tm > $o ) > tm > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__SeCaV__Otm_J,type,
member_list_tm: list_tm > set_list_tm > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat3: nat > set_nat > $o ).
thf(sy_c_member_001t__SeCaV__Ofm,type,
member_fm3: fm > set_fm > $o ).
thf(sy_c_member_001t__SeCaV__Otm,type,
member_tm3: tm > set_tm > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat3: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__SeCaV__Ofm_J,type,
member_set_fm: set_fm > set_set_fm > $o ).
thf(sy_c_member_001t__Set__Oset_It__SeCaV__Otm_J,type,
member_set_tm: set_tm > set_set_tm > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_v_S,type,
s: set_fm ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_ts____,type,
ts: list_tm ).
% Relevant facts (1273)
thf(fact_0_Fun_Oprems,axiom,
member_tm3 @ ( fun @ n @ ts ) @ ( terms @ s ) ).
% Fun.prems
thf(fact_1_Fun_Ohyps,axiom,
! [X2aa: tm] :
( ( member_tm3 @ X2aa @ ( set_tm2 @ ts ) )
=> ( ( member_tm3 @ X2aa @ ( terms @ s ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermTm @ X2aa ) ) @ ( terms @ s ) ) ) ) ).
% Fun.hyps
thf(fact_2_member,axiom,
( member_fm2
= ( ^ [P: fm,Z: list_fm] : ( member_fm3 @ P @ ( set_fm2 @ Z ) ) ) ) ).
% member
thf(fact_3_member,axiom,
( member_tm2
= ( ^ [P: tm,Z: list_tm] : ( member_tm3 @ P @ ( set_tm2 @ Z ) ) ) ) ).
% member
thf(fact_4_member,axiom,
( member_nat2
= ( ^ [P: nat,Z: list_nat] : ( member_nat3 @ P @ ( set_nat2 @ Z ) ) ) ) ).
% member
thf(fact_5_member,axiom,
( member_set_nat2
= ( ^ [P: set_nat,Z: list_set_nat] : ( member_set_nat3 @ P @ ( set_set_nat2 @ Z ) ) ) ) ).
% member
thf(fact_6_terms__ne,axiom,
! [S: set_fm] :
( ( terms @ S )
!= bot_bot_set_tm ) ).
% terms_ne
thf(fact_7_in__set__member,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
= ( member_fm @ Xs @ X ) ) ).
% in_set_member
thf(fact_8_in__set__member,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
= ( member_tm @ Xs @ X ) ) ).
% in_set_member
thf(fact_9_in__set__member,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_10_in__set__member,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat3 @ X @ ( set_set_nat2 @ Xs ) )
= ( member_set_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_11_in__set__insert,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_12_in__set__insert,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ( ( insert_tm @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_13_in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_14_in__set__insert,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat3 @ X @ ( set_set_nat2 @ Xs ) )
=> ( ( insert_set_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_15_list__ex1__iff,axiom,
( list_ex1_fm
= ( ^ [P2: fm > $o,Xs2: list_fm] :
? [X2: fm] :
( ( member_fm3 @ X2 @ ( set_fm2 @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y: fm] :
( ( ( member_fm3 @ Y @ ( set_fm2 @ Xs2 ) )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_16_list__ex1__iff,axiom,
( list_ex1_tm
= ( ^ [P2: tm > $o,Xs2: list_tm] :
? [X2: tm] :
( ( member_tm3 @ X2 @ ( set_tm2 @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y: tm] :
( ( ( member_tm3 @ Y @ ( set_tm2 @ Xs2 ) )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_17_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P2: nat > $o,Xs2: list_nat] :
? [X2: nat] :
( ( member_nat3 @ X2 @ ( set_nat2 @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y: nat] :
( ( ( member_nat3 @ Y @ ( set_nat2 @ Xs2 ) )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_18_list__ex1__iff,axiom,
( list_ex1_set_nat
= ( ^ [P2: set_nat > $o,Xs2: list_set_nat] :
? [X2: set_nat] :
( ( member_set_nat3 @ X2 @ ( set_set_nat2 @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y: set_nat] :
( ( ( member_set_nat3 @ Y @ ( set_set_nat2 @ Xs2 ) )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_19_subtermTm__refl,axiom,
! [T: tm] : ( member_tm3 @ T @ ( set_tm2 @ ( subtermTm @ T ) ) ) ).
% subtermTm_refl
thf(fact_20_ListMem__iff,axiom,
( listMem_tm
= ( ^ [X2: tm,Xs2: list_tm] : ( member_tm3 @ X2 @ ( set_tm2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_21_ListMem__iff,axiom,
( listMem_nat
= ( ^ [X2: nat,Xs2: list_nat] : ( member_nat3 @ X2 @ ( set_nat2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_22_ListMem__iff,axiom,
( listMem_set_nat
= ( ^ [X2: set_nat,Xs2: list_set_nat] : ( member_set_nat3 @ X2 @ ( set_set_nat2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_23_ListMem__iff,axiom,
( listMem_fm
= ( ^ [X2: fm,Xs2: list_fm] : ( member_fm3 @ X2 @ ( set_fm2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_24_set__rotate1,axiom,
! [Xs: list_tm] :
( ( set_tm2 @ ( rotate1_tm @ Xs ) )
= ( set_tm2 @ Xs ) ) ).
% set_rotate1
thf(fact_25_set__rotate1,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( rotate1_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_26_set__rotate1,axiom,
! [Xs: list_set_nat] :
( ( set_set_nat2 @ ( rotate1_set_nat @ Xs ) )
= ( set_set_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_27_set__rotate1,axiom,
! [Xs: list_fm] :
( ( set_fm2 @ ( rotate1_fm @ Xs ) )
= ( set_fm2 @ Xs ) ) ).
% set_rotate1
thf(fact_28_removeAll__id,axiom,
! [X: tm,Xs: list_tm] :
( ~ ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ( ( removeAll_tm @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_29_removeAll__id,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_30_removeAll__id,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ~ ( member_set_nat3 @ X @ ( set_set_nat2 @ Xs ) )
=> ( ( removeAll_set_nat @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_31_removeAll__id,axiom,
! [X: fm,Xs: list_fm] :
( ~ ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ( ( removeAll_fm @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_32_tm_Oinject_I1_J,axiom,
! [X11: nat,X12: list_tm,Y11: nat,Y12: list_tm] :
( ( ( fun @ X11 @ X12 )
= ( fun @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% tm.inject(1)
thf(fact_33_subterm__Fun__refl,axiom,
! [Ts: list_tm,N: nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermTm @ ( fun @ N @ Ts ) ) ) ) ).
% subterm_Fun_refl
thf(fact_34_subtermTm__le,axiom,
! [T: tm,S2: tm] :
( ( member_tm3 @ T @ ( set_tm2 @ ( subtermTm @ S2 ) ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermTm @ T ) ) @ ( set_tm2 @ ( subtermTm @ S2 ) ) ) ) ).
% subtermTm_le
thf(fact_35_subset__code_I1_J,axiom,
! [Xs: list_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B )
= ( ! [X2: set_nat] :
( ( member_set_nat3 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( member_set_nat3 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_36_subset__code_I1_J,axiom,
! [Xs: list_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B )
= ( ! [X2: fm] :
( ( member_fm3 @ X2 @ ( set_fm2 @ Xs ) )
=> ( member_fm3 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_37_subset__code_I1_J,axiom,
! [Xs: list_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ B )
= ( ! [X2: tm] :
( ( member_tm3 @ X2 @ ( set_tm2 @ Xs ) )
=> ( member_tm3 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_38_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X2: nat] :
( ( member_nat3 @ X2 @ ( set_nat2 @ Xs ) )
=> ( member_nat3 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_39_subset__empty,axiom,
! [A: set_fm] :
( ( ord_less_eq_set_fm @ A @ bot_bot_set_fm )
= ( A = bot_bot_set_fm ) ) ).
% subset_empty
thf(fact_40_subset__empty,axiom,
! [A: set_tm] :
( ( ord_less_eq_set_tm @ A @ bot_bot_set_tm )
= ( A = bot_bot_set_tm ) ) ).
% subset_empty
thf(fact_41_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_42_empty__subsetI,axiom,
! [A: set_fm] : ( ord_less_eq_set_fm @ bot_bot_set_fm @ A ) ).
% empty_subsetI
thf(fact_43_empty__subsetI,axiom,
! [A: set_tm] : ( ord_less_eq_set_tm @ bot_bot_set_tm @ A ) ).
% empty_subsetI
thf(fact_44_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_45_can__select__set__list__ex1,axiom,
! [P3: set_nat > $o,A: list_set_nat] :
( ( can_select_set_nat @ P3 @ ( set_set_nat2 @ A ) )
= ( list_ex1_set_nat @ P3 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_46_can__select__set__list__ex1,axiom,
! [P3: nat > $o,A: list_nat] :
( ( can_select_nat @ P3 @ ( set_nat2 @ A ) )
= ( list_ex1_nat @ P3 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_47_can__select__set__list__ex1,axiom,
! [P3: tm > $o,A: list_tm] :
( ( can_select_tm @ P3 @ ( set_tm2 @ A ) )
= ( list_ex1_tm @ P3 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_48_can__select__set__list__ex1,axiom,
! [P3: fm > $o,A: list_fm] :
( ( can_select_fm @ P3 @ ( set_fm2 @ A ) )
= ( list_ex1_fm @ P3 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_49_SeCaV_Oext,axiom,
( ext_set_nat
= ( ^ [Y: list_set_nat,Z: list_set_nat] : ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Z ) @ ( set_set_nat2 @ Y ) ) ) ) ).
% SeCaV.ext
thf(fact_50_SeCaV_Oext,axiom,
( ext_fm
= ( ^ [Y: list_fm,Z: list_fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Z ) @ ( set_fm2 @ Y ) ) ) ) ).
% SeCaV.ext
thf(fact_51_SeCaV_Oext,axiom,
( ext_tm
= ( ^ [Y: list_tm,Z: list_tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ Z ) @ ( set_tm2 @ Y ) ) ) ) ).
% SeCaV.ext
thf(fact_52_SeCaV_Oext,axiom,
( ext_nat
= ( ^ [Y: list_nat,Z: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Z ) @ ( set_nat2 @ Y ) ) ) ) ).
% SeCaV.ext
thf(fact_53_subsetI,axiom,
! [A: set_fm,B: set_fm] :
( ! [X3: fm] :
( ( member_fm3 @ X3 @ A )
=> ( member_fm3 @ X3 @ B ) )
=> ( ord_less_eq_set_fm @ A @ B ) ) ).
% subsetI
thf(fact_54_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat3 @ X3 @ A )
=> ( member_set_nat3 @ X3 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_55_subsetI,axiom,
! [A: set_tm,B: set_tm] :
( ! [X3: tm] :
( ( member_tm3 @ X3 @ A )
=> ( member_tm3 @ X3 @ B ) )
=> ( ord_less_eq_set_tm @ A @ B ) ) ).
% subsetI
thf(fact_56_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat3 @ X3 @ A )
=> ( member_nat3 @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_57_subset__antisym,axiom,
! [A: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_eq_set_tm @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_58_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_59_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat3 @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_60_empty__iff,axiom,
! [C: tm] :
~ ( member_tm3 @ C @ bot_bot_set_tm ) ).
% empty_iff
thf(fact_61_empty__iff,axiom,
! [C: nat] :
~ ( member_nat3 @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_62_empty__iff,axiom,
! [C: fm] :
~ ( member_fm3 @ C @ bot_bot_set_fm ) ).
% empty_iff
thf(fact_63_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X2: set_nat] :
~ ( member_set_nat3 @ X2 @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_64_all__not__in__conv,axiom,
! [A: set_tm] :
( ( ! [X2: tm] :
~ ( member_tm3 @ X2 @ A ) )
= ( A = bot_bot_set_tm ) ) ).
% all_not_in_conv
thf(fact_65_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat3 @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_66_all__not__in__conv,axiom,
! [A: set_fm] :
( ( ! [X2: fm] :
~ ( member_fm3 @ X2 @ A ) )
= ( A = bot_bot_set_fm ) ) ).
% all_not_in_conv
thf(fact_67_Collect__empty__eq,axiom,
! [P3: tm > $o] :
( ( ( collect_tm @ P3 )
= bot_bot_set_tm )
= ( ! [X2: tm] :
~ ( P3 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_68_Collect__empty__eq,axiom,
! [P3: nat > $o] :
( ( ( collect_nat @ P3 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P3 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_69_Collect__empty__eq,axiom,
! [P3: fm > $o] :
( ( ( collect_fm @ P3 )
= bot_bot_set_fm )
= ( ! [X2: fm] :
~ ( P3 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_70_empty__Collect__eq,axiom,
! [P3: tm > $o] :
( ( bot_bot_set_tm
= ( collect_tm @ P3 ) )
= ( ! [X2: tm] :
~ ( P3 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_71_empty__Collect__eq,axiom,
! [P3: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P3 ) )
= ( ! [X2: nat] :
~ ( P3 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_72_empty__Collect__eq,axiom,
! [P3: fm > $o] :
( ( bot_bot_set_fm
= ( collect_fm @ P3 ) )
= ( ! [X2: fm] :
~ ( P3 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_73_order__refl,axiom,
! [X: set_tm] : ( ord_less_eq_set_tm @ X @ X ) ).
% order_refl
thf(fact_74_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_75_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_76_order__refl,axiom,
! [X: extended_enat] : ( ord_le2932123472753598470d_enat @ X @ X ) ).
% order_refl
thf(fact_77_dual__order_Orefl,axiom,
! [A2: set_tm] : ( ord_less_eq_set_tm @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_78_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_79_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_80_dual__order_Orefl,axiom,
! [A2: extended_enat] : ( ord_le2932123472753598470d_enat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_81_bot__set__def,axiom,
( bot_bot_set_tm
= ( collect_tm @ bot_bot_tm_o ) ) ).
% bot_set_def
thf(fact_82_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_83_bot__set__def,axiom,
( bot_bot_set_fm
= ( collect_fm @ bot_bot_fm_o ) ) ).
% bot_set_def
thf(fact_84_can__select__def,axiom,
( can_select_tm
= ( ^ [P2: tm > $o,A3: set_tm] :
? [X2: tm] :
( ( member_tm3 @ X2 @ A3 )
& ( P2 @ X2 )
& ! [Y: tm] :
( ( ( member_tm3 @ Y @ A3 )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_85_can__select__def,axiom,
( can_select_nat
= ( ^ [P2: nat > $o,A3: set_nat] :
? [X2: nat] :
( ( member_nat3 @ X2 @ A3 )
& ( P2 @ X2 )
& ! [Y: nat] :
( ( ( member_nat3 @ Y @ A3 )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_86_can__select__def,axiom,
( can_select_fm
= ( ^ [P2: fm > $o,A3: set_fm] :
? [X2: fm] :
( ( member_fm3 @ X2 @ A3 )
& ( P2 @ X2 )
& ! [Y: fm] :
( ( ( member_fm3 @ Y @ A3 )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_87_can__select__def,axiom,
( can_select_set_nat
= ( ^ [P2: set_nat > $o,A3: set_set_nat] :
? [X2: set_nat] :
( ( member_set_nat3 @ X2 @ A3 )
& ( P2 @ X2 )
& ! [Y: set_nat] :
( ( ( member_set_nat3 @ Y @ A3 )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_88_order__antisym__conv,axiom,
! [Y2: set_tm,X: set_tm] :
( ( ord_less_eq_set_tm @ Y2 @ X )
=> ( ( ord_less_eq_set_tm @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_89_order__antisym__conv,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_90_order__antisym__conv,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_91_order__antisym__conv,axiom,
! [Y2: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y2 @ X )
=> ( ( ord_le2932123472753598470d_enat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_92_linorder__le__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_93_linorder__le__cases,axiom,
! [X: extended_enat,Y2: extended_enat] :
( ~ ( ord_le2932123472753598470d_enat @ X @ Y2 )
=> ( ord_le2932123472753598470d_enat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_94_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_95_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_96_ord__le__eq__subst,axiom,
! [A2: extended_enat,B2: extended_enat,F: extended_enat > nat,C: nat] :
( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: extended_enat,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_97_ord__le__eq__subst,axiom,
! [A2: extended_enat,B2: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: extended_enat,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_98_ord__le__eq__subst,axiom,
! [A2: set_tm,B2: set_tm,F: set_tm > nat,C: nat] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_99_ord__le__eq__subst,axiom,
! [A2: set_tm,B2: set_tm,F: set_tm > extended_enat,C: extended_enat] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_100_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_101_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > extended_enat,C: extended_enat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_102_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_tm,C: set_tm] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_103_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_104_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_105_ord__eq__le__subst,axiom,
! [A2: extended_enat,F: nat > extended_enat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_106_ord__eq__le__subst,axiom,
! [A2: nat,F: extended_enat > nat,B2: extended_enat,C: extended_enat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le2932123472753598470d_enat @ B2 @ C )
=> ( ! [X3: extended_enat,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_107_ord__eq__le__subst,axiom,
! [A2: extended_enat,F: extended_enat > extended_enat,B2: extended_enat,C: extended_enat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le2932123472753598470d_enat @ B2 @ C )
=> ( ! [X3: extended_enat,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_108_ord__eq__le__subst,axiom,
! [A2: nat,F: set_tm > nat,B2: set_tm,C: set_tm] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ! [X3: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_109_ord__eq__le__subst,axiom,
! [A2: extended_enat,F: set_tm > extended_enat,B2: set_tm,C: set_tm] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ! [X3: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_110_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_111_ord__eq__le__subst,axiom,
! [A2: extended_enat,F: set_nat > extended_enat,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_112_ord__eq__le__subst,axiom,
! [A2: set_tm,F: nat > set_tm,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_tm @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_113_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_114_linorder__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_115_linorder__linear,axiom,
! [X: extended_enat,Y2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y2 )
| ( ord_le2932123472753598470d_enat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_116_order__eq__refl,axiom,
! [X: set_tm,Y2: set_tm] :
( ( X = Y2 )
=> ( ord_less_eq_set_tm @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_117_order__eq__refl,axiom,
! [X: set_nat,Y2: set_nat] :
( ( X = Y2 )
=> ( ord_less_eq_set_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_118_order__eq__refl,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_119_order__eq__refl,axiom,
! [X: extended_enat,Y2: extended_enat] :
( ( X = Y2 )
=> ( ord_le2932123472753598470d_enat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_120_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_121_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_122_order__subst2,axiom,
! [A2: extended_enat,B2: extended_enat,F: extended_enat > nat,C: nat] :
( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: extended_enat,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_123_order__subst2,axiom,
! [A2: extended_enat,B2: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B2 ) @ C )
=> ( ! [X3: extended_enat,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_124_order__subst2,axiom,
! [A2: set_tm,B2: set_tm,F: set_tm > nat,C: nat] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_125_order__subst2,axiom,
! [A2: set_tm,B2: set_tm,F: set_tm > extended_enat,C: extended_enat] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_126_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_127_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > extended_enat,C: extended_enat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_128_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_tm,C: set_tm] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_tm @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_129_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_130_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_131_order__subst1,axiom,
! [A2: nat,F: extended_enat > nat,B2: extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le2932123472753598470d_enat @ B2 @ C )
=> ( ! [X3: extended_enat,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_132_order__subst1,axiom,
! [A2: extended_enat,F: nat > extended_enat,B2: nat,C: nat] :
( ( ord_le2932123472753598470d_enat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_133_order__subst1,axiom,
! [A2: extended_enat,F: extended_enat > extended_enat,B2: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le2932123472753598470d_enat @ B2 @ C )
=> ( ! [X3: extended_enat,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_134_order__subst1,axiom,
! [A2: set_tm,F: nat > set_tm,B2: nat,C: nat] :
( ( ord_less_eq_set_tm @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_tm @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_135_order__subst1,axiom,
! [A2: set_tm,F: extended_enat > set_tm,B2: extended_enat,C: extended_enat] :
( ( ord_less_eq_set_tm @ A2 @ ( F @ B2 ) )
=> ( ( ord_le2932123472753598470d_enat @ B2 @ C )
=> ( ! [X3: extended_enat,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_tm @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_136_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_137_order__subst1,axiom,
! [A2: set_nat,F: extended_enat > set_nat,B2: extended_enat,C: extended_enat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le2932123472753598470d_enat @ B2 @ C )
=> ( ! [X3: extended_enat,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_138_order__subst1,axiom,
! [A2: nat,F: set_tm > nat,B2: set_tm,C: set_tm] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ! [X3: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_139_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_140_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_tm,Z2: set_tm] : ( Y4 = Z2 ) )
= ( ^ [A4: set_tm,B3: set_tm] :
( ( ord_less_eq_set_tm @ A4 @ B3 )
& ( ord_less_eq_set_tm @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_141_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_142_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_143_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: extended_enat,Z2: extended_enat] : ( Y4 = Z2 ) )
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A4 @ B3 )
& ( ord_le2932123472753598470d_enat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_144_mem__Collect__eq,axiom,
! [A2: tm,P3: tm > $o] :
( ( member_tm3 @ A2 @ ( collect_tm @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_145_mem__Collect__eq,axiom,
! [A2: nat,P3: nat > $o] :
( ( member_nat3 @ A2 @ ( collect_nat @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_146_mem__Collect__eq,axiom,
! [A2: fm,P3: fm > $o] :
( ( member_fm3 @ A2 @ ( collect_fm @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_147_mem__Collect__eq,axiom,
! [A2: set_nat,P3: set_nat > $o] :
( ( member_set_nat3 @ A2 @ ( collect_set_nat @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_148_Collect__mem__eq,axiom,
! [A: set_tm] :
( ( collect_tm
@ ^ [X2: tm] : ( member_tm3 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_149_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat3 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_150_Collect__mem__eq,axiom,
! [A: set_fm] :
( ( collect_fm
@ ^ [X2: fm] : ( member_fm3 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_151_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X2: set_nat] : ( member_set_nat3 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_152_antisym,axiom,
! [A2: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_153_antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_154_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_155_antisym,axiom,
! [A2: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
=> ( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_156_dual__order_Otrans,axiom,
! [B2: set_tm,A2: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A2 )
=> ( ( ord_less_eq_set_tm @ C @ B2 )
=> ( ord_less_eq_set_tm @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_157_dual__order_Otrans,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_158_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_159_dual__order_Otrans,axiom,
! [B2: extended_enat,A2: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
=> ( ( ord_le2932123472753598470d_enat @ C @ B2 )
=> ( ord_le2932123472753598470d_enat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_160_dual__order_Oantisym,axiom,
! [B2: set_tm,A2: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A2 )
=> ( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_161_dual__order_Oantisym,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_162_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_163_dual__order_Oantisym,axiom,
! [B2: extended_enat,A2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
=> ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_164_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_tm,Z2: set_tm] : ( Y4 = Z2 ) )
= ( ^ [A4: set_tm,B3: set_tm] :
( ( ord_less_eq_set_tm @ B3 @ A4 )
& ( ord_less_eq_set_tm @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_165_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_166_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_167_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: extended_enat,Z2: extended_enat] : ( Y4 = Z2 ) )
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B3 @ A4 )
& ( ord_le2932123472753598470d_enat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_168_linorder__wlog,axiom,
! [P3: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
=> ( P3 @ A5 @ B4 ) )
=> ( ! [A5: nat,B4: nat] :
( ( P3 @ B4 @ A5 )
=> ( P3 @ A5 @ B4 ) )
=> ( P3 @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_169_linorder__wlog,axiom,
! [P3: extended_enat > extended_enat > $o,A2: extended_enat,B2: extended_enat] :
( ! [A5: extended_enat,B4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A5 @ B4 )
=> ( P3 @ A5 @ B4 ) )
=> ( ! [A5: extended_enat,B4: extended_enat] :
( ( P3 @ B4 @ A5 )
=> ( P3 @ A5 @ B4 ) )
=> ( P3 @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_170_order__trans,axiom,
! [X: set_tm,Y2: set_tm,Z3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y2 )
=> ( ( ord_less_eq_set_tm @ Y2 @ Z3 )
=> ( ord_less_eq_set_tm @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_171_order__trans,axiom,
! [X: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z3 )
=> ( ord_less_eq_set_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_172_order__trans,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_173_order__trans,axiom,
! [X: extended_enat,Y2: extended_enat,Z3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y2 )
=> ( ( ord_le2932123472753598470d_enat @ Y2 @ Z3 )
=> ( ord_le2932123472753598470d_enat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_174_order_Otrans,axiom,
! [A2: set_tm,B2: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ord_less_eq_set_tm @ A2 @ C ) ) ) ).
% order.trans
thf(fact_175_order_Otrans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_176_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_177_order_Otrans,axiom,
! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
=> ( ( ord_le2932123472753598470d_enat @ B2 @ C )
=> ( ord_le2932123472753598470d_enat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_178_order__antisym,axiom,
! [X: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y2 )
=> ( ( ord_less_eq_set_tm @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_179_order__antisym,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_180_order__antisym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_181_order__antisym,axiom,
! [X: extended_enat,Y2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y2 )
=> ( ( ord_le2932123472753598470d_enat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_182_ord__le__eq__trans,axiom,
! [A2: set_tm,B2: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_tm @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_183_ord__le__eq__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_184_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_185_ord__le__eq__trans,axiom,
! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le2932123472753598470d_enat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_186_ord__eq__le__trans,axiom,
! [A2: set_tm,B2: set_tm,C: set_tm] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ord_less_eq_set_tm @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_187_ord__eq__le__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_188_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_189_ord__eq__le__trans,axiom,
! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
( ( A2 = B2 )
=> ( ( ord_le2932123472753598470d_enat @ B2 @ C )
=> ( ord_le2932123472753598470d_enat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_190_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_tm,Z2: set_tm] : ( Y4 = Z2 ) )
= ( ^ [X2: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y )
& ( ord_less_eq_set_tm @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_191_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
& ( ord_less_eq_set_nat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_192_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_193_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: extended_enat,Z2: extended_enat] : ( Y4 = Z2 ) )
= ( ^ [X2: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X2 @ Y )
& ( ord_le2932123472753598470d_enat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_194_le__cases3,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_195_le__cases3,axiom,
! [X: extended_enat,Y2: extended_enat,Z3: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ X @ Y2 )
=> ~ ( ord_le2932123472753598470d_enat @ Y2 @ Z3 ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Y2 @ X )
=> ~ ( ord_le2932123472753598470d_enat @ X @ Z3 ) )
=> ( ( ( ord_le2932123472753598470d_enat @ X @ Z3 )
=> ~ ( ord_le2932123472753598470d_enat @ Z3 @ Y2 ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Z3 @ Y2 )
=> ~ ( ord_le2932123472753598470d_enat @ Y2 @ X ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Y2 @ Z3 )
=> ~ ( ord_le2932123472753598470d_enat @ Z3 @ X ) )
=> ~ ( ( ord_le2932123472753598470d_enat @ Z3 @ X )
=> ~ ( ord_le2932123472753598470d_enat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_196_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_197_nle__le,axiom,
! [A2: extended_enat,B2: extended_enat] :
( ( ~ ( ord_le2932123472753598470d_enat @ A2 @ B2 ) )
= ( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_198_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X2: set_nat] : ( member_set_nat3 @ X2 @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_199_ex__in__conv,axiom,
! [A: set_tm] :
( ( ? [X2: tm] : ( member_tm3 @ X2 @ A ) )
= ( A != bot_bot_set_tm ) ) ).
% ex_in_conv
thf(fact_200_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat3 @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_201_ex__in__conv,axiom,
! [A: set_fm] :
( ( ? [X2: fm] : ( member_fm3 @ X2 @ A ) )
= ( A != bot_bot_set_fm ) ) ).
% ex_in_conv
thf(fact_202_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y3: set_nat] :
~ ( member_set_nat3 @ Y3 @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_203_equals0I,axiom,
! [A: set_tm] :
( ! [Y3: tm] :
~ ( member_tm3 @ Y3 @ A )
=> ( A = bot_bot_set_tm ) ) ).
% equals0I
thf(fact_204_equals0I,axiom,
! [A: set_nat] :
( ! [Y3: nat] :
~ ( member_nat3 @ Y3 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_205_equals0I,axiom,
! [A: set_fm] :
( ! [Y3: fm] :
~ ( member_fm3 @ Y3 @ A )
=> ( A = bot_bot_set_fm ) ) ).
% equals0I
thf(fact_206_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat3 @ A2 @ A ) ) ).
% equals0D
thf(fact_207_equals0D,axiom,
! [A: set_tm,A2: tm] :
( ( A = bot_bot_set_tm )
=> ~ ( member_tm3 @ A2 @ A ) ) ).
% equals0D
thf(fact_208_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat3 @ A2 @ A ) ) ).
% equals0D
thf(fact_209_equals0D,axiom,
! [A: set_fm,A2: fm] :
( ( A = bot_bot_set_fm )
=> ~ ( member_fm3 @ A2 @ A ) ) ).
% equals0D
thf(fact_210_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat3 @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_211_emptyE,axiom,
! [A2: tm] :
~ ( member_tm3 @ A2 @ bot_bot_set_tm ) ).
% emptyE
thf(fact_212_emptyE,axiom,
! [A2: nat] :
~ ( member_nat3 @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_213_emptyE,axiom,
! [A2: fm] :
~ ( member_fm3 @ A2 @ bot_bot_set_fm ) ).
% emptyE
thf(fact_214_Collect__mono__iff,axiom,
! [P3: tm > $o,Q: tm > $o] :
( ( ord_less_eq_set_tm @ ( collect_tm @ P3 ) @ ( collect_tm @ Q ) )
= ( ! [X2: tm] :
( ( P3 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_215_Collect__mono__iff,axiom,
! [P3: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P3 ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P3 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_216_set__eq__subset,axiom,
( ( ^ [Y4: set_tm,Z2: set_tm] : ( Y4 = Z2 ) )
= ( ^ [A3: set_tm,B5: set_tm] :
( ( ord_less_eq_set_tm @ A3 @ B5 )
& ( ord_less_eq_set_tm @ B5 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_217_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A3: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_218_subset__trans,axiom,
! [A: set_tm,B: set_tm,C2: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_eq_set_tm @ B @ C2 )
=> ( ord_less_eq_set_tm @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_219_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_220_Collect__mono,axiom,
! [P3: tm > $o,Q: tm > $o] :
( ! [X3: tm] :
( ( P3 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_tm @ ( collect_tm @ P3 ) @ ( collect_tm @ Q ) ) ) ).
% Collect_mono
thf(fact_221_Collect__mono,axiom,
! [P3: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P3 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P3 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_222_subset__refl,axiom,
! [A: set_tm] : ( ord_less_eq_set_tm @ A @ A ) ).
% subset_refl
thf(fact_223_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_224_subset__iff,axiom,
( ord_less_eq_set_fm
= ( ^ [A3: set_fm,B5: set_fm] :
! [T2: fm] :
( ( member_fm3 @ T2 @ A3 )
=> ( member_fm3 @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_225_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
! [T2: set_nat] :
( ( member_set_nat3 @ T2 @ A3 )
=> ( member_set_nat3 @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_226_subset__iff,axiom,
( ord_less_eq_set_tm
= ( ^ [A3: set_tm,B5: set_tm] :
! [T2: tm] :
( ( member_tm3 @ T2 @ A3 )
=> ( member_tm3 @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_227_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat3 @ T2 @ A3 )
=> ( member_nat3 @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_228_equalityD2,axiom,
! [A: set_tm,B: set_tm] :
( ( A = B )
=> ( ord_less_eq_set_tm @ B @ A ) ) ).
% equalityD2
thf(fact_229_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_230_equalityD1,axiom,
! [A: set_tm,B: set_tm] :
( ( A = B )
=> ( ord_less_eq_set_tm @ A @ B ) ) ).
% equalityD1
thf(fact_231_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_232_subset__eq,axiom,
( ord_less_eq_set_fm
= ( ^ [A3: set_fm,B5: set_fm] :
! [X2: fm] :
( ( member_fm3 @ X2 @ A3 )
=> ( member_fm3 @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_233_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
! [X2: set_nat] :
( ( member_set_nat3 @ X2 @ A3 )
=> ( member_set_nat3 @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_234_subset__eq,axiom,
( ord_less_eq_set_tm
= ( ^ [A3: set_tm,B5: set_tm] :
! [X2: tm] :
( ( member_tm3 @ X2 @ A3 )
=> ( member_tm3 @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_235_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
! [X2: nat] :
( ( member_nat3 @ X2 @ A3 )
=> ( member_nat3 @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_236_equalityE,axiom,
! [A: set_tm,B: set_tm] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_tm @ A @ B )
=> ~ ( ord_less_eq_set_tm @ B @ A ) ) ) ).
% equalityE
thf(fact_237_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_238_subsetD,axiom,
! [A: set_fm,B: set_fm,C: fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( member_fm3 @ C @ A )
=> ( member_fm3 @ C @ B ) ) ) ).
% subsetD
thf(fact_239_subsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat3 @ C @ A )
=> ( member_set_nat3 @ C @ B ) ) ) ).
% subsetD
thf(fact_240_subsetD,axiom,
! [A: set_tm,B: set_tm,C: tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( member_tm3 @ C @ A )
=> ( member_tm3 @ C @ B ) ) ) ).
% subsetD
thf(fact_241_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat3 @ C @ A )
=> ( member_nat3 @ C @ B ) ) ) ).
% subsetD
thf(fact_242_in__mono,axiom,
! [A: set_fm,B: set_fm,X: fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( member_fm3 @ X @ A )
=> ( member_fm3 @ X @ B ) ) ) ).
% in_mono
thf(fact_243_in__mono,axiom,
! [A: set_set_nat,B: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat3 @ X @ A )
=> ( member_set_nat3 @ X @ B ) ) ) ).
% in_mono
thf(fact_244_in__mono,axiom,
! [A: set_tm,B: set_tm,X: tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( member_tm3 @ X @ A )
=> ( member_tm3 @ X @ B ) ) ) ).
% in_mono
thf(fact_245_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat3 @ X @ A )
=> ( member_nat3 @ X @ B ) ) ) ).
% in_mono
thf(fact_246_bot_Oextremum__uniqueI,axiom,
! [A2: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ bot_bot_set_fm )
=> ( A2 = bot_bot_set_fm ) ) ).
% bot.extremum_uniqueI
thf(fact_247_bot_Oextremum__uniqueI,axiom,
! [A2: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ bot_bot_set_tm )
=> ( A2 = bot_bot_set_tm ) ) ).
% bot.extremum_uniqueI
thf(fact_248_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_249_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_250_bot_Oextremum__uniqueI,axiom,
! [A2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ bot_bo4199563552545308370d_enat )
=> ( A2 = bot_bo4199563552545308370d_enat ) ) ).
% bot.extremum_uniqueI
thf(fact_251_bot_Oextremum__unique,axiom,
! [A2: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ bot_bot_set_fm )
= ( A2 = bot_bot_set_fm ) ) ).
% bot.extremum_unique
thf(fact_252_bot_Oextremum__unique,axiom,
! [A2: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ bot_bot_set_tm )
= ( A2 = bot_bot_set_tm ) ) ).
% bot.extremum_unique
thf(fact_253_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_254_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_255_bot_Oextremum__unique,axiom,
! [A2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ bot_bo4199563552545308370d_enat )
= ( A2 = bot_bo4199563552545308370d_enat ) ) ).
% bot.extremum_unique
thf(fact_256_bot_Oextremum,axiom,
! [A2: set_fm] : ( ord_less_eq_set_fm @ bot_bot_set_fm @ A2 ) ).
% bot.extremum
thf(fact_257_bot_Oextremum,axiom,
! [A2: set_tm] : ( ord_less_eq_set_tm @ bot_bot_set_tm @ A2 ) ).
% bot.extremum
thf(fact_258_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_259_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_260_bot_Oextremum,axiom,
! [A2: extended_enat] : ( ord_le2932123472753598470d_enat @ bot_bo4199563552545308370d_enat @ A2 ) ).
% bot.extremum
thf(fact_261_subset__emptyI,axiom,
! [A: set_set_nat] :
( ! [X3: set_nat] :
~ ( member_set_nat3 @ X3 @ A )
=> ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat ) ) ).
% subset_emptyI
thf(fact_262_subset__emptyI,axiom,
! [A: set_fm] :
( ! [X3: fm] :
~ ( member_fm3 @ X3 @ A )
=> ( ord_less_eq_set_fm @ A @ bot_bot_set_fm ) ) ).
% subset_emptyI
thf(fact_263_subset__emptyI,axiom,
! [A: set_tm] :
( ! [X3: tm] :
~ ( member_tm3 @ X3 @ A )
=> ( ord_less_eq_set_tm @ A @ bot_bot_set_tm ) ) ).
% subset_emptyI
thf(fact_264_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X3: nat] :
~ ( member_nat3 @ X3 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_265_paramst__subtermTm_I1_J,axiom,
! [T: tm,X4: nat] :
( ( member_nat3 @ X4 @ ( paramst @ T ) )
=> ? [L: list_tm] : ( member_tm3 @ ( fun @ X4 @ L ) @ ( set_tm2 @ ( subtermTm @ T ) ) ) ) ).
% paramst_subtermTm(1)
thf(fact_266_Set_Ois__empty__def,axiom,
( is_empty_tm
= ( ^ [A3: set_tm] : ( A3 = bot_bot_set_tm ) ) ) ).
% Set.is_empty_def
thf(fact_267_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_268_Set_Ois__empty__def,axiom,
( is_empty_fm
= ( ^ [A3: set_fm] : ( A3 = bot_bot_set_fm ) ) ) ).
% Set.is_empty_def
thf(fact_269_remove__code_I1_J,axiom,
! [X: tm,Xs: list_tm] :
( ( remove_tm @ X @ ( set_tm2 @ Xs ) )
= ( set_tm2 @ ( removeAll_tm @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_270_remove__code_I1_J,axiom,
! [X: nat,Xs: list_nat] :
( ( remove_nat @ X @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( removeAll_nat @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_271_remove__code_I1_J,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( remove_set_nat @ X @ ( set_set_nat2 @ Xs ) )
= ( set_set_nat2 @ ( removeAll_set_nat @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_272_remove__code_I1_J,axiom,
! [X: fm,Xs: list_fm] :
( ( remove_fm @ X @ ( set_fm2 @ Xs ) )
= ( set_fm2 @ ( removeAll_fm @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_273_fun__arguments__subterm,axiom,
! [N: nat,Ts: list_tm,P4: fm] :
( ( member_tm3 @ ( fun @ N @ Ts ) @ ( set_tm2 @ ( subtermFm @ P4 ) ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermFm @ P4 ) ) ) ) ).
% fun_arguments_subterm
thf(fact_274_paramsts__subset,axiom,
! [A: list_tm,B: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ A ) @ ( set_tm2 @ B ) )
=> ( ord_less_eq_set_nat @ ( paramsts @ A ) @ ( paramsts @ B ) ) ) ).
% paramsts_subset
thf(fact_275_Greatest__equality,axiom,
! [P3: set_tm > $o,X: set_tm] :
( ( P3 @ X )
=> ( ! [Y3: set_tm] :
( ( P3 @ Y3 )
=> ( ord_less_eq_set_tm @ Y3 @ X ) )
=> ( ( order_4248741476211579294set_tm @ P3 )
= X ) ) ) ).
% Greatest_equality
thf(fact_276_Greatest__equality,axiom,
! [P3: set_nat > $o,X: set_nat] :
( ( P3 @ X )
=> ( ! [Y3: set_nat] :
( ( P3 @ Y3 )
=> ( ord_less_eq_set_nat @ Y3 @ X ) )
=> ( ( order_5724808138429204845et_nat @ P3 )
= X ) ) ) ).
% Greatest_equality
thf(fact_277_Greatest__equality,axiom,
! [P3: extended_enat > $o,X: extended_enat] :
( ( P3 @ X )
=> ( ! [Y3: extended_enat] :
( ( P3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ Y3 @ X ) )
=> ( ( order_2428742583041560895d_enat @ P3 )
= X ) ) ) ).
% Greatest_equality
thf(fact_278_Greatest__equality,axiom,
! [P3: nat > $o,X: nat] :
( ( P3 @ X )
=> ( ! [Y3: nat] :
( ( P3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( order_Greatest_nat @ P3 )
= X ) ) ) ).
% Greatest_equality
thf(fact_279_GreatestI2__order,axiom,
! [P3: set_tm > $o,X: set_tm,Q: set_tm > $o] :
( ( P3 @ X )
=> ( ! [Y3: set_tm] :
( ( P3 @ Y3 )
=> ( ord_less_eq_set_tm @ Y3 @ X ) )
=> ( ! [X3: set_tm] :
( ( P3 @ X3 )
=> ( ! [Y5: set_tm] :
( ( P3 @ Y5 )
=> ( ord_less_eq_set_tm @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_4248741476211579294set_tm @ P3 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_280_GreatestI2__order,axiom,
! [P3: set_nat > $o,X: set_nat,Q: set_nat > $o] :
( ( P3 @ X )
=> ( ! [Y3: set_nat] :
( ( P3 @ Y3 )
=> ( ord_less_eq_set_nat @ Y3 @ X ) )
=> ( ! [X3: set_nat] :
( ( P3 @ X3 )
=> ( ! [Y5: set_nat] :
( ( P3 @ Y5 )
=> ( ord_less_eq_set_nat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P3 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_281_GreatestI2__order,axiom,
! [P3: extended_enat > $o,X: extended_enat,Q: extended_enat > $o] :
( ( P3 @ X )
=> ( ! [Y3: extended_enat] :
( ( P3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ Y3 @ X ) )
=> ( ! [X3: extended_enat] :
( ( P3 @ X3 )
=> ( ! [Y5: extended_enat] :
( ( P3 @ Y5 )
=> ( ord_le2932123472753598470d_enat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_2428742583041560895d_enat @ P3 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_282_GreatestI2__order,axiom,
! [P3: nat > $o,X: nat,Q: nat > $o] :
( ( P3 @ X )
=> ( ! [Y3: nat] :
( ( P3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ! [X3: nat] :
( ( P3 @ X3 )
=> ( ! [Y5: nat] :
( ( P3 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_nat @ P3 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_283_subset__code_I2_J,axiom,
! [A: set_set_nat,Ys: list_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( coset_set_nat @ Ys ) )
= ( ! [X2: set_nat] :
( ( member_set_nat3 @ X2 @ ( set_set_nat2 @ Ys ) )
=> ~ ( member_set_nat3 @ X2 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_284_subset__code_I2_J,axiom,
! [A: set_fm,Ys: list_fm] :
( ( ord_less_eq_set_fm @ A @ ( coset_fm @ Ys ) )
= ( ! [X2: fm] :
( ( member_fm3 @ X2 @ ( set_fm2 @ Ys ) )
=> ~ ( member_fm3 @ X2 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_285_subset__code_I2_J,axiom,
! [A: set_tm,Ys: list_tm] :
( ( ord_less_eq_set_tm @ A @ ( coset_tm @ Ys ) )
= ( ! [X2: tm] :
( ( member_tm3 @ X2 @ ( set_tm2 @ Ys ) )
=> ~ ( member_tm3 @ X2 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_286_subset__code_I2_J,axiom,
! [A: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A @ ( coset_nat @ Ys ) )
= ( ! [X2: nat] :
( ( member_nat3 @ X2 @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat3 @ X2 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_287_Collect__empty__eq__bot,axiom,
! [P3: tm > $o] :
( ( ( collect_tm @ P3 )
= bot_bot_set_tm )
= ( P3 = bot_bot_tm_o ) ) ).
% Collect_empty_eq_bot
thf(fact_288_Collect__empty__eq__bot,axiom,
! [P3: nat > $o] :
( ( ( collect_nat @ P3 )
= bot_bot_set_nat )
= ( P3 = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_289_Collect__empty__eq__bot,axiom,
! [P3: fm > $o] :
( ( ( collect_fm @ P3 )
= bot_bot_set_fm )
= ( P3 = bot_bot_fm_o ) ) ).
% Collect_empty_eq_bot
thf(fact_290_bot__empty__eq,axiom,
( bot_bot_set_nat_o
= ( ^ [X2: set_nat] : ( member_set_nat3 @ X2 @ bot_bot_set_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_291_bot__empty__eq,axiom,
( bot_bot_tm_o
= ( ^ [X2: tm] : ( member_tm3 @ X2 @ bot_bot_set_tm ) ) ) ).
% bot_empty_eq
thf(fact_292_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat3 @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_293_bot__empty__eq,axiom,
( bot_bot_fm_o
= ( ^ [X2: fm] : ( member_fm3 @ X2 @ bot_bot_set_fm ) ) ) ).
% bot_empty_eq
thf(fact_294_member__remove,axiom,
! [X: tm,Y2: tm,A: set_tm] :
( ( member_tm3 @ X @ ( remove_tm @ Y2 @ A ) )
= ( ( member_tm3 @ X @ A )
& ( X != Y2 ) ) ) ).
% member_remove
thf(fact_295_member__remove,axiom,
! [X: nat,Y2: nat,A: set_nat] :
( ( member_nat3 @ X @ ( remove_nat @ Y2 @ A ) )
= ( ( member_nat3 @ X @ A )
& ( X != Y2 ) ) ) ).
% member_remove
thf(fact_296_member__remove,axiom,
! [X: fm,Y2: fm,A: set_fm] :
( ( member_fm3 @ X @ ( remove_fm @ Y2 @ A ) )
= ( ( member_fm3 @ X @ A )
& ( X != Y2 ) ) ) ).
% member_remove
thf(fact_297_member__remove,axiom,
! [X: set_nat,Y2: set_nat,A: set_set_nat] :
( ( member_set_nat3 @ X @ ( remove_set_nat @ Y2 @ A ) )
= ( ( member_set_nat3 @ X @ A )
& ( X != Y2 ) ) ) ).
% member_remove
thf(fact_298_remove__code_I2_J,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( remove_set_nat @ X @ ( coset_set_nat @ Xs ) )
= ( coset_set_nat @ ( insert_set_nat @ X @ Xs ) ) ) ).
% remove_code(2)
thf(fact_299_remove__code_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( remove_nat @ X @ ( coset_nat @ Xs ) )
= ( coset_nat @ ( insert_nat @ X @ Xs ) ) ) ).
% remove_code(2)
thf(fact_300_remove__code_I2_J,axiom,
! [X: tm,Xs: list_tm] :
( ( remove_tm @ X @ ( coset_tm @ Xs ) )
= ( coset_tm @ ( insert_tm @ X @ Xs ) ) ) ).
% remove_code(2)
thf(fact_301_remove__code_I2_J,axiom,
! [X: fm,Xs: list_fm] :
( ( remove_fm @ X @ ( coset_fm @ Xs ) )
= ( coset_fm @ ( insert_fm @ X @ Xs ) ) ) ).
% remove_code(2)
thf(fact_302_subtermFm__subset__params,axiom,
! [P4: fm,A: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermFm @ P4 ) ) @ ( set_tm2 @ A ) )
=> ( ord_less_eq_set_nat @ ( params @ P4 ) @ ( paramsts @ A ) ) ) ).
% subtermFm_subset_params
thf(fact_303_s1_I1_J,axiom,
( new_term
= ( ^ [C3: nat,T2: tm] :
~ ( member_nat3 @ C3 @ ( paramst @ T2 ) ) ) ) ).
% s1(1)
thf(fact_304_is__empty__set,axiom,
! [Xs: list_tm] :
( ( is_empty_tm @ ( set_tm2 @ Xs ) )
= ( null_tm @ Xs ) ) ).
% is_empty_set
thf(fact_305_is__empty__set,axiom,
! [Xs: list_nat] :
( ( is_empty_nat @ ( set_nat2 @ Xs ) )
= ( null_nat @ Xs ) ) ).
% is_empty_set
thf(fact_306_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_307_is__empty__set,axiom,
! [Xs: list_fm] :
( ( is_empty_fm @ ( set_fm2 @ Xs ) )
= ( null_fm @ Xs ) ) ).
% is_empty_set
thf(fact_308_p1,axiom,
paramst2 = paramst ).
% p1
thf(fact_309_s1_I2_J,axiom,
( new_list
= ( ^ [C3: nat,L2: list_tm] :
~ ( member_nat3 @ C3 @ ( paramsts @ L2 ) ) ) ) ).
% s1(2)
thf(fact_310_paramst__liftt_I2_J,axiom,
! [Ts: list_tm] :
( ( paramsts @ ( liftts @ Ts ) )
= ( paramsts @ Ts ) ) ).
% paramst_liftt(2)
thf(fact_311_params__subtermFm,axiom,
! [P4: fm,X4: nat] :
( ( member_nat3 @ X4 @ ( params @ P4 ) )
=> ? [L: list_tm] : ( member_tm3 @ ( fun @ X4 @ L ) @ ( set_tm2 @ ( subtermFm @ P4 ) ) ) ) ).
% params_subtermFm
thf(fact_312_paramst__liftt_I1_J,axiom,
! [T: tm] :
( ( paramst @ ( liftt @ T ) )
= ( paramst @ T ) ) ).
% paramst_liftt(1)
thf(fact_313_listFunTm__paramst_I1_J,axiom,
! [T: tm] :
( ( set_nat2 @ ( listFunTm @ T ) )
= ( paramst @ T ) ) ).
% listFunTm_paramst(1)
thf(fact_314_subterm__Pre__refl,axiom,
! [Ts: list_tm,N: nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermFm @ ( pre @ N @ Ts ) ) ) ) ).
% subterm_Pre_refl
thf(fact_315_fm_Oinject_I1_J,axiom,
! [X11: nat,X12: list_tm,Y11: nat,Y12: list_tm] :
( ( ( pre @ X11 @ X12 )
= ( pre @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% fm.inject(1)
thf(fact_316_params_Osimps_I1_J,axiom,
! [B2: nat,Ts: list_tm] :
( ( params @ ( pre @ B2 @ Ts ) )
= ( paramsts @ Ts ) ) ).
% params.simps(1)
thf(fact_317_liftt_Osimps_I2_J,axiom,
! [A2: nat,Ts: list_tm] :
( ( liftt @ ( fun @ A2 @ Ts ) )
= ( fun @ A2 @ ( liftts @ Ts ) ) ) ).
% liftt.simps(2)
thf(fact_318_new__term_Osimps_I2_J,axiom,
! [C: nat,I: nat,L3: list_tm] :
( ( new_term @ C @ ( fun @ I @ L3 ) )
= ( ( I != C )
& ( ( I != C )
=> ( new_list @ C @ L3 ) ) ) ) ).
% new_term.simps(2)
thf(fact_319_s4_I2_J,axiom,
inc_list = liftts ).
% s4(2)
thf(fact_320_listFunTm__paramst_I2_J,axiom,
! [Ts: list_tm] :
( ( set_nat2 @ ( listFunTms @ Ts ) )
= ( paramsts @ Ts ) ) ).
% listFunTm_paramst(2)
thf(fact_321_s4_I1_J,axiom,
inc_term = liftt ).
% s4(1)
thf(fact_322_new_Osimps_I1_J,axiom,
! [C: nat,I: nat,L3: list_tm] :
( ( new @ C @ ( pre @ I @ L3 ) )
= ( new_list @ C @ L3 ) ) ).
% new.simps(1)
thf(fact_323_p1_H,axiom,
paramst3 = paramst ).
% p1'
thf(fact_324_subset__code_I3_J,axiom,
~ ( ord_le6893508408891458716et_nat @ ( coset_set_nat @ nil_set_nat ) @ ( set_set_nat2 @ nil_set_nat ) ) ).
% subset_code(3)
thf(fact_325_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_fm @ ( coset_fm @ nil_fm ) @ ( set_fm2 @ nil_fm ) ) ).
% subset_code(3)
thf(fact_326_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_tm @ ( coset_tm @ nil_tm ) @ ( set_tm2 @ nil_tm ) ) ).
% subset_code(3)
thf(fact_327_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_328_subtermFm__preds,axiom,
! [T: tm,P4: fm] :
( ( member_tm3 @ T @ ( set_tm2 @ ( subtermFm @ P4 ) ) )
= ( ? [X2: fm] :
( ( member_fm3 @ X2 @ ( preds @ P4 ) )
& ( member_tm3 @ T @ ( set_tm2 @ ( subtermFm @ X2 ) ) ) ) ) ) ).
% subtermFm_preds
thf(fact_329_new__list_Osimps_I2_J,axiom,
! [C: nat,T: tm,L3: list_tm] :
( ( new_list @ C @ ( cons_tm @ T @ L3 ) )
= ( ( ( new_term @ C @ T )
=> ( new_list @ C @ L3 ) )
& ( new_term @ C @ T ) ) ) ).
% new_list.simps(2)
thf(fact_330_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_set_tm
= ( ^ [X5: $o > set_tm,Y6: $o > set_tm] :
( ( ord_less_eq_set_tm @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_tm @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_331_le__rel__bool__arg__iff,axiom,
( ord_le7022414076629706543et_nat
= ( ^ [X5: $o > set_nat,Y6: $o > set_nat] :
( ( ord_less_eq_set_nat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_nat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_332_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X5: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_333_le__rel__bool__arg__iff,axiom,
( ord_le2787558655864224659d_enat
= ( ^ [X5: $o > extended_enat,Y6: $o > extended_enat] :
( ( ord_le2932123472753598470d_enat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le2932123472753598470d_enat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_334_list_Oinject,axiom,
! [X21: tm,X22: list_tm,Y21: tm,Y22: list_tm] :
( ( ( cons_tm @ X21 @ X22 )
= ( cons_tm @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_335_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_336_rotate1__is__Nil__conv,axiom,
! [Xs: list_tm] :
( ( ( rotate1_tm @ Xs )
= nil_tm )
= ( Xs = nil_tm ) ) ).
% rotate1_is_Nil_conv
thf(fact_337_rotate1__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rotate1_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_338_bind__simps_I1_J,axiom,
! [F: tm > list_tm] :
( ( bind_tm_tm @ nil_tm @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_339_bind__simps_I1_J,axiom,
! [F: tm > list_nat] :
( ( bind_tm_nat @ nil_tm @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_340_bind__simps_I1_J,axiom,
! [F: nat > list_tm] :
( ( bind_nat_tm @ nil_nat @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_341_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_342_s2,axiom,
( new
= ( ^ [C3: nat,P: fm] :
~ ( member_nat3 @ C3 @ ( params @ P ) ) ) ) ).
% s2
thf(fact_343_list__ex1__simps_I1_J,axiom,
! [P3: set_nat > $o] :
~ ( list_ex1_set_nat @ P3 @ nil_set_nat ) ).
% list_ex1_simps(1)
thf(fact_344_list__ex1__simps_I1_J,axiom,
! [P3: nat > $o] :
~ ( list_ex1_nat @ P3 @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_345_list__ex1__simps_I1_J,axiom,
! [P3: tm > $o] :
~ ( list_ex1_tm @ P3 @ nil_tm ) ).
% list_ex1_simps(1)
thf(fact_346_list__ex1__simps_I1_J,axiom,
! [P3: fm > $o] :
~ ( list_ex1_fm @ P3 @ nil_fm ) ).
% list_ex1_simps(1)
thf(fact_347_set__empty,axiom,
! [Xs: list_set_nat] :
( ( ( set_set_nat2 @ Xs )
= bot_bot_set_set_nat )
= ( Xs = nil_set_nat ) ) ).
% set_empty
thf(fact_348_set__empty,axiom,
! [Xs: list_tm] :
( ( ( set_tm2 @ Xs )
= bot_bot_set_tm )
= ( Xs = nil_tm ) ) ).
% set_empty
thf(fact_349_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_350_set__empty,axiom,
! [Xs: list_fm] :
( ( ( set_fm2 @ Xs )
= bot_bot_set_fm )
= ( Xs = nil_fm ) ) ).
% set_empty
thf(fact_351_set__empty2,axiom,
! [Xs: list_set_nat] :
( ( bot_bot_set_set_nat
= ( set_set_nat2 @ Xs ) )
= ( Xs = nil_set_nat ) ) ).
% set_empty2
thf(fact_352_set__empty2,axiom,
! [Xs: list_tm] :
( ( bot_bot_set_tm
= ( set_tm2 @ Xs ) )
= ( Xs = nil_tm ) ) ).
% set_empty2
thf(fact_353_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_354_set__empty2,axiom,
! [Xs: list_fm] :
( ( bot_bot_set_fm
= ( set_fm2 @ Xs ) )
= ( Xs = nil_fm ) ) ).
% set_empty2
thf(fact_355_insert__Nil,axiom,
! [X: set_nat] :
( ( insert_set_nat @ X @ nil_set_nat )
= ( cons_set_nat @ X @ nil_set_nat ) ) ).
% insert_Nil
thf(fact_356_insert__Nil,axiom,
! [X: fm] :
( ( insert_fm @ X @ nil_fm )
= ( cons_fm @ X @ nil_fm ) ) ).
% insert_Nil
thf(fact_357_insert__Nil,axiom,
! [X: tm] :
( ( insert_tm @ X @ nil_tm )
= ( cons_tm @ X @ nil_tm ) ) ).
% insert_Nil
thf(fact_358_insert__Nil,axiom,
! [X: nat] :
( ( insert_nat @ X @ nil_nat )
= ( cons_nat @ X @ nil_nat ) ) ).
% insert_Nil
thf(fact_359_not__in__set__insert,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ~ ( member_set_nat3 @ X @ ( set_set_nat2 @ Xs ) )
=> ( ( insert_set_nat @ X @ Xs )
= ( cons_set_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_360_not__in__set__insert,axiom,
! [X: fm,Xs: list_fm] :
( ~ ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X @ Xs )
= ( cons_fm @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_361_not__in__set__insert,axiom,
! [X: tm,Xs: list_tm] :
( ~ ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ( ( insert_tm @ X @ Xs )
= ( cons_tm @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_362_not__in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= ( cons_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_363_list__nonempty__induct,axiom,
! [Xs: list_tm,P3: list_tm > $o] :
( ( Xs != nil_tm )
=> ( ! [X3: tm] : ( P3 @ ( cons_tm @ X3 @ nil_tm ) )
=> ( ! [X3: tm,Xs3: list_tm] :
( ( Xs3 != nil_tm )
=> ( ( P3 @ Xs3 )
=> ( P3 @ ( cons_tm @ X3 @ Xs3 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_364_list__nonempty__induct,axiom,
! [Xs: list_nat,P3: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P3 @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P3 @ Xs3 )
=> ( P3 @ ( cons_nat @ X3 @ Xs3 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_365_not__Cons__self2,axiom,
! [X: tm,Xs: list_tm] :
( ( cons_tm @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_366_not__Cons__self2,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_367_list__induct2_H,axiom,
! [P3: list_tm > list_tm > $o,Xs: list_tm,Ys: list_tm] :
( ( P3 @ nil_tm @ nil_tm )
=> ( ! [X3: tm,Xs3: list_tm] : ( P3 @ ( cons_tm @ X3 @ Xs3 ) @ nil_tm )
=> ( ! [Y3: tm,Ys2: list_tm] : ( P3 @ nil_tm @ ( cons_tm @ Y3 @ Ys2 ) )
=> ( ! [X3: tm,Xs3: list_tm,Y3: tm,Ys2: list_tm] :
( ( P3 @ Xs3 @ Ys2 )
=> ( P3 @ ( cons_tm @ X3 @ Xs3 ) @ ( cons_tm @ Y3 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_368_list__induct2_H,axiom,
! [P3: list_tm > list_nat > $o,Xs: list_tm,Ys: list_nat] :
( ( P3 @ nil_tm @ nil_nat )
=> ( ! [X3: tm,Xs3: list_tm] : ( P3 @ ( cons_tm @ X3 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys2: list_nat] : ( P3 @ nil_tm @ ( cons_nat @ Y3 @ Ys2 ) )
=> ( ! [X3: tm,Xs3: list_tm,Y3: nat,Ys2: list_nat] :
( ( P3 @ Xs3 @ Ys2 )
=> ( P3 @ ( cons_tm @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_369_list__induct2_H,axiom,
! [P3: list_nat > list_tm > $o,Xs: list_nat,Ys: list_tm] :
( ( P3 @ nil_nat @ nil_tm )
=> ( ! [X3: nat,Xs3: list_nat] : ( P3 @ ( cons_nat @ X3 @ Xs3 ) @ nil_tm )
=> ( ! [Y3: tm,Ys2: list_tm] : ( P3 @ nil_nat @ ( cons_tm @ Y3 @ Ys2 ) )
=> ( ! [X3: nat,Xs3: list_nat,Y3: tm,Ys2: list_tm] :
( ( P3 @ Xs3 @ Ys2 )
=> ( P3 @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_tm @ Y3 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_370_list__induct2_H,axiom,
! [P3: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P3 @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat] : ( P3 @ ( cons_nat @ X3 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys2: list_nat] : ( P3 @ nil_nat @ ( cons_nat @ Y3 @ Ys2 ) )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys2: list_nat] :
( ( P3 @ Xs3 @ Ys2 )
=> ( P3 @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_371_neq__Nil__conv,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
= ( ? [Y: tm,Ys3: list_tm] :
( Xs
= ( cons_tm @ Y @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_372_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_373_remdups__adj_Ocases,axiom,
! [X: list_tm] :
( ( X != nil_tm )
=> ( ! [X3: tm] :
( X
!= ( cons_tm @ X3 @ nil_tm ) )
=> ~ ! [X3: tm,Y3: tm,Xs3: list_tm] :
( X
!= ( cons_tm @ X3 @ ( cons_tm @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_374_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X3: nat] :
( X
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_375_transpose_Ocases,axiom,
! [X: list_list_tm] :
( ( X != nil_list_tm )
=> ( ! [Xss: list_list_tm] :
( X
!= ( cons_list_tm @ nil_tm @ Xss ) )
=> ~ ! [X3: tm,Xs3: list_tm,Xss: list_list_tm] :
( X
!= ( cons_list_tm @ ( cons_tm @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_376_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X3: nat,Xs3: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_377_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X3: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs3 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_378_list_Oexhaust,axiom,
! [Y2: list_tm] :
( ( Y2 != nil_tm )
=> ~ ! [X212: tm,X222: list_tm] :
( Y2
!= ( cons_tm @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_379_list_Oexhaust,axiom,
! [Y2: list_nat] :
( ( Y2 != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y2
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_380_list_OdiscI,axiom,
! [List: list_tm,X21: tm,X22: list_tm] :
( ( List
= ( cons_tm @ X21 @ X22 ) )
=> ( List != nil_tm ) ) ).
% list.discI
thf(fact_381_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_382_inc__list_Osimps_I1_J,axiom,
( ( inc_list @ nil_tm )
= nil_tm ) ).
% inc_list.simps(1)
thf(fact_383_inc__list_Osimps_I2_J,axiom,
! [T: tm,L3: list_tm] :
( ( inc_list @ ( cons_tm @ T @ L3 ) )
= ( cons_tm @ ( inc_term @ T ) @ ( inc_list @ L3 ) ) ) ).
% inc_list.simps(2)
thf(fact_384_list_Odistinct_I1_J,axiom,
! [X21: tm,X22: list_tm] :
( nil_tm
!= ( cons_tm @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_385_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_386_set__ConsD,axiom,
! [Y2: set_nat,X: set_nat,Xs: list_set_nat] :
( ( member_set_nat3 @ Y2 @ ( set_set_nat2 @ ( cons_set_nat @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_set_nat3 @ Y2 @ ( set_set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_387_set__ConsD,axiom,
! [Y2: fm,X: fm,Xs: list_fm] :
( ( member_fm3 @ Y2 @ ( set_fm2 @ ( cons_fm @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_fm3 @ Y2 @ ( set_fm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_388_set__ConsD,axiom,
! [Y2: tm,X: tm,Xs: list_tm] :
( ( member_tm3 @ Y2 @ ( set_tm2 @ ( cons_tm @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_tm3 @ Y2 @ ( set_tm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_389_set__ConsD,axiom,
! [Y2: nat,X: nat,Xs: list_nat] :
( ( member_nat3 @ Y2 @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_nat3 @ Y2 @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_390_list_Oset__cases,axiom,
! [E: set_nat,A2: list_set_nat] :
( ( member_set_nat3 @ E @ ( set_set_nat2 @ A2 ) )
=> ( ! [Z22: list_set_nat] :
( A2
!= ( cons_set_nat @ E @ Z22 ) )
=> ~ ! [Z1: set_nat,Z22: list_set_nat] :
( ( A2
= ( cons_set_nat @ Z1 @ Z22 ) )
=> ~ ( member_set_nat3 @ E @ ( set_set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_391_list_Oset__cases,axiom,
! [E: fm,A2: list_fm] :
( ( member_fm3 @ E @ ( set_fm2 @ A2 ) )
=> ( ! [Z22: list_fm] :
( A2
!= ( cons_fm @ E @ Z22 ) )
=> ~ ! [Z1: fm,Z22: list_fm] :
( ( A2
= ( cons_fm @ Z1 @ Z22 ) )
=> ~ ( member_fm3 @ E @ ( set_fm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_392_list_Oset__cases,axiom,
! [E: tm,A2: list_tm] :
( ( member_tm3 @ E @ ( set_tm2 @ A2 ) )
=> ( ! [Z22: list_tm] :
( A2
!= ( cons_tm @ E @ Z22 ) )
=> ~ ! [Z1: tm,Z22: list_tm] :
( ( A2
= ( cons_tm @ Z1 @ Z22 ) )
=> ~ ( member_tm3 @ E @ ( set_tm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_393_list_Oset__cases,axiom,
! [E: nat,A2: list_nat] :
( ( member_nat3 @ E @ ( set_nat2 @ A2 ) )
=> ( ! [Z22: list_nat] :
( A2
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A2
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat3 @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_394_list_Oset__intros_I1_J,axiom,
! [X21: set_nat,X22: list_set_nat] : ( member_set_nat3 @ X21 @ ( set_set_nat2 @ ( cons_set_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_395_list_Oset__intros_I1_J,axiom,
! [X21: fm,X22: list_fm] : ( member_fm3 @ X21 @ ( set_fm2 @ ( cons_fm @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_396_list_Oset__intros_I1_J,axiom,
! [X21: tm,X22: list_tm] : ( member_tm3 @ X21 @ ( set_tm2 @ ( cons_tm @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_397_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat3 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_398_list_Oset__intros_I2_J,axiom,
! [Y2: set_nat,X22: list_set_nat,X21: set_nat] :
( ( member_set_nat3 @ Y2 @ ( set_set_nat2 @ X22 ) )
=> ( member_set_nat3 @ Y2 @ ( set_set_nat2 @ ( cons_set_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_399_list_Oset__intros_I2_J,axiom,
! [Y2: fm,X22: list_fm,X21: fm] :
( ( member_fm3 @ Y2 @ ( set_fm2 @ X22 ) )
=> ( member_fm3 @ Y2 @ ( set_fm2 @ ( cons_fm @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_400_list_Oset__intros_I2_J,axiom,
! [Y2: tm,X22: list_tm,X21: tm] :
( ( member_tm3 @ Y2 @ ( set_tm2 @ X22 ) )
=> ( member_tm3 @ Y2 @ ( set_tm2 @ ( cons_tm @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_401_list_Oset__intros_I2_J,axiom,
! [Y2: nat,X22: list_nat,X21: nat] :
( ( member_nat3 @ Y2 @ ( set_nat2 @ X22 ) )
=> ( member_nat3 @ Y2 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_402_removeAll_Osimps_I2_J,axiom,
! [X: tm,Y2: tm,Xs: list_tm] :
( ( ( X = Y2 )
=> ( ( removeAll_tm @ X @ ( cons_tm @ Y2 @ Xs ) )
= ( removeAll_tm @ X @ Xs ) ) )
& ( ( X != Y2 )
=> ( ( removeAll_tm @ X @ ( cons_tm @ Y2 @ Xs ) )
= ( cons_tm @ Y2 @ ( removeAll_tm @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_403_removeAll_Osimps_I2_J,axiom,
! [X: nat,Y2: nat,Xs: list_nat] :
( ( ( X = Y2 )
=> ( ( removeAll_nat @ X @ ( cons_nat @ Y2 @ Xs ) )
= ( removeAll_nat @ X @ Xs ) ) )
& ( ( X != Y2 )
=> ( ( removeAll_nat @ X @ ( cons_nat @ Y2 @ Xs ) )
= ( cons_nat @ Y2 @ ( removeAll_nat @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_404_inc__term_Osimps_I2_J,axiom,
! [I: nat,L3: list_tm] :
( ( inc_term @ ( fun @ I @ L3 ) )
= ( fun @ I @ ( inc_list @ L3 ) ) ) ).
% inc_term.simps(2)
thf(fact_405_removeAll_Osimps_I1_J,axiom,
! [X: tm] :
( ( removeAll_tm @ X @ nil_tm )
= nil_tm ) ).
% removeAll.simps(1)
thf(fact_406_removeAll_Osimps_I1_J,axiom,
! [X: nat] :
( ( removeAll_nat @ X @ nil_nat )
= nil_nat ) ).
% removeAll.simps(1)
thf(fact_407_rotate1_Osimps_I1_J,axiom,
( ( rotate1_tm @ nil_tm )
= nil_tm ) ).
% rotate1.simps(1)
thf(fact_408_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_409_paramsts_Osimps_I1_J,axiom,
( ( paramsts @ nil_tm )
= bot_bot_set_nat ) ).
% paramsts.simps(1)
thf(fact_410_elem,axiom,
! [X: tm,Xs: list_tm] : ( listMem_tm @ X @ ( cons_tm @ X @ Xs ) ) ).
% elem
thf(fact_411_elem,axiom,
! [X: nat,Xs: list_nat] : ( listMem_nat @ X @ ( cons_nat @ X @ Xs ) ) ).
% elem
thf(fact_412_ListMem_Ocases,axiom,
! [A1: tm,A22: list_tm] :
( ( listMem_tm @ A1 @ A22 )
=> ( ! [Xs3: list_tm] :
( A22
!= ( cons_tm @ A1 @ Xs3 ) )
=> ~ ! [Xs3: list_tm] :
( ? [Y3: tm] :
( A22
= ( cons_tm @ Y3 @ Xs3 ) )
=> ~ ( listMem_tm @ A1 @ Xs3 ) ) ) ) ).
% ListMem.cases
thf(fact_413_ListMem_Ocases,axiom,
! [A1: nat,A22: list_nat] :
( ( listMem_nat @ A1 @ A22 )
=> ( ! [Xs3: list_nat] :
( A22
!= ( cons_nat @ A1 @ Xs3 ) )
=> ~ ! [Xs3: list_nat] :
( ? [Y3: nat] :
( A22
= ( cons_nat @ Y3 @ Xs3 ) )
=> ~ ( listMem_nat @ A1 @ Xs3 ) ) ) ) ).
% ListMem.cases
thf(fact_414_ListMem_Osimps,axiom,
( listMem_tm
= ( ^ [A12: tm,A23: list_tm] :
( ? [X2: tm,Xs2: list_tm] :
( ( A12 = X2 )
& ( A23
= ( cons_tm @ X2 @ Xs2 ) ) )
| ? [X2: tm,Xs2: list_tm,Y: tm] :
( ( A12 = X2 )
& ( A23
= ( cons_tm @ Y @ Xs2 ) )
& ( listMem_tm @ X2 @ Xs2 ) ) ) ) ) ).
% ListMem.simps
thf(fact_415_ListMem_Osimps,axiom,
( listMem_nat
= ( ^ [A12: nat,A23: list_nat] :
( ? [X2: nat,Xs2: list_nat] :
( ( A12 = X2 )
& ( A23
= ( cons_nat @ X2 @ Xs2 ) ) )
| ? [X2: nat,Xs2: list_nat,Y: nat] :
( ( A12 = X2 )
& ( A23
= ( cons_nat @ Y @ Xs2 ) )
& ( listMem_nat @ X2 @ Xs2 ) ) ) ) ) ).
% ListMem.simps
thf(fact_416_insert,axiom,
! [X: tm,Xs: list_tm,Y2: tm] :
( ( listMem_tm @ X @ Xs )
=> ( listMem_tm @ X @ ( cons_tm @ Y2 @ Xs ) ) ) ).
% insert
thf(fact_417_insert,axiom,
! [X: nat,Xs: list_nat,Y2: nat] :
( ( listMem_nat @ X @ Xs )
=> ( listMem_nat @ X @ ( cons_nat @ Y2 @ Xs ) ) ) ).
% insert
thf(fact_418_SeCaV_Omember_Osimps_I2_J,axiom,
! [P4: set_nat,Q2: set_nat,Z3: list_set_nat] :
( ( member_set_nat2 @ P4 @ ( cons_set_nat @ Q2 @ Z3 ) )
= ( ( P4 != Q2 )
=> ( member_set_nat2 @ P4 @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_419_SeCaV_Omember_Osimps_I2_J,axiom,
! [P4: fm,Q2: fm,Z3: list_fm] :
( ( member_fm2 @ P4 @ ( cons_fm @ Q2 @ Z3 ) )
= ( ( P4 != Q2 )
=> ( member_fm2 @ P4 @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_420_SeCaV_Omember_Osimps_I2_J,axiom,
! [P4: tm,Q2: tm,Z3: list_tm] :
( ( member_tm2 @ P4 @ ( cons_tm @ Q2 @ Z3 ) )
= ( ( P4 != Q2 )
=> ( member_tm2 @ P4 @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_421_SeCaV_Omember_Osimps_I2_J,axiom,
! [P4: nat,Q2: nat,Z3: list_nat] :
( ( member_nat2 @ P4 @ ( cons_nat @ Q2 @ Z3 ) )
= ( ( P4 != Q2 )
=> ( member_nat2 @ P4 @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_422_SeCaV_Omember_Osimps_I1_J,axiom,
! [P4: set_nat] :
~ ( member_set_nat2 @ P4 @ nil_set_nat ) ).
% SeCaV.member.simps(1)
thf(fact_423_SeCaV_Omember_Osimps_I1_J,axiom,
! [P4: fm] :
~ ( member_fm2 @ P4 @ nil_fm ) ).
% SeCaV.member.simps(1)
thf(fact_424_SeCaV_Omember_Osimps_I1_J,axiom,
! [P4: tm] :
~ ( member_tm2 @ P4 @ nil_tm ) ).
% SeCaV.member.simps(1)
thf(fact_425_SeCaV_Omember_Osimps_I1_J,axiom,
! [P4: nat] :
~ ( member_nat2 @ P4 @ nil_nat ) ).
% SeCaV.member.simps(1)
thf(fact_426_ext_Osimps_I1_J,axiom,
! [Y2: list_tm] : ( ext_tm @ Y2 @ nil_tm ) ).
% ext.simps(1)
thf(fact_427_ext_Osimps_I1_J,axiom,
! [Y2: list_nat] : ( ext_nat @ Y2 @ nil_nat ) ).
% ext.simps(1)
thf(fact_428_null__rec_I1_J,axiom,
! [X: tm,Xs: list_tm] :
~ ( null_tm @ ( cons_tm @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_429_null__rec_I1_J,axiom,
! [X: nat,Xs: list_nat] :
~ ( null_nat @ ( cons_nat @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_430_liftts_Osimps_I1_J,axiom,
( ( liftts @ nil_tm )
= nil_tm ) ).
% liftts.simps(1)
thf(fact_431_eq__Nil__null,axiom,
! [Xs: list_tm] :
( ( Xs = nil_tm )
= ( null_tm @ Xs ) ) ).
% eq_Nil_null
thf(fact_432_eq__Nil__null,axiom,
! [Xs: list_nat] :
( ( Xs = nil_nat )
= ( null_nat @ Xs ) ) ).
% eq_Nil_null
thf(fact_433_null__rec_I2_J,axiom,
null_tm @ nil_tm ).
% null_rec(2)
thf(fact_434_null__rec_I2_J,axiom,
null_nat @ nil_nat ).
% null_rec(2)
thf(fact_435_new__list_Osimps_I1_J,axiom,
! [C: nat] : ( new_list @ C @ nil_tm ) ).
% new_list.simps(1)
thf(fact_436_member__rec_I1_J,axiom,
! [X: set_nat,Xs: list_set_nat,Y2: set_nat] :
( ( member_set_nat @ ( cons_set_nat @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_set_nat @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_437_member__rec_I1_J,axiom,
! [X: fm,Xs: list_fm,Y2: fm] :
( ( member_fm @ ( cons_fm @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_fm @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_438_member__rec_I1_J,axiom,
! [X: tm,Xs: list_tm,Y2: tm] :
( ( member_tm @ ( cons_tm @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_tm @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_439_member__rec_I1_J,axiom,
! [X: nat,Xs: list_nat,Y2: nat] :
( ( member_nat @ ( cons_nat @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_nat @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_440_preds__shape,axiom,
! [Pre: fm,P4: fm] :
( ( member_fm3 @ Pre @ ( preds @ P4 ) )
=> ? [N2: nat,Ts2: list_tm] :
( Pre
= ( pre @ N2 @ Ts2 ) ) ) ).
% preds_shape
thf(fact_441_member__rec_I2_J,axiom,
! [Y2: set_nat] :
~ ( member_set_nat @ nil_set_nat @ Y2 ) ).
% member_rec(2)
thf(fact_442_member__rec_I2_J,axiom,
! [Y2: fm] :
~ ( member_fm @ nil_fm @ Y2 ) ).
% member_rec(2)
thf(fact_443_member__rec_I2_J,axiom,
! [Y2: tm] :
~ ( member_tm @ nil_tm @ Y2 ) ).
% member_rec(2)
thf(fact_444_member__rec_I2_J,axiom,
! [Y2: nat] :
~ ( member_nat @ nil_nat @ Y2 ) ).
% member_rec(2)
thf(fact_445_set__subset__Cons,axiom,
! [Xs: list_set_nat,X: set_nat] : ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ ( set_set_nat2 @ ( cons_set_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_446_set__subset__Cons,axiom,
! [Xs: list_fm,X: fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ ( cons_fm @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_447_set__subset__Cons,axiom,
! [Xs: list_tm,X: tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ ( cons_tm @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_448_set__subset__Cons,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_449_empty__set,axiom,
( bot_bot_set_set_nat
= ( set_set_nat2 @ nil_set_nat ) ) ).
% empty_set
thf(fact_450_empty__set,axiom,
( bot_bot_set_tm
= ( set_tm2 @ nil_tm ) ) ).
% empty_set
thf(fact_451_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_452_empty__set,axiom,
( bot_bot_set_fm
= ( set_fm2 @ nil_fm ) ) ).
% empty_set
thf(fact_453_List_Oinsert__def,axiom,
( insert_set_nat
= ( ^ [X2: set_nat,Xs2: list_set_nat] : ( if_list_set_nat @ ( member_set_nat3 @ X2 @ ( set_set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_set_nat @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_454_List_Oinsert__def,axiom,
( insert_fm
= ( ^ [X2: fm,Xs2: list_fm] : ( if_list_fm @ ( member_fm3 @ X2 @ ( set_fm2 @ Xs2 ) ) @ Xs2 @ ( cons_fm @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_455_List_Oinsert__def,axiom,
( insert_tm
= ( ^ [X2: tm,Xs2: list_tm] : ( if_list_tm @ ( member_tm3 @ X2 @ ( set_tm2 @ Xs2 ) ) @ Xs2 @ ( cons_tm @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_456_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X2: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat3 @ X2 @ ( set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_nat @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_457_ext_Osimps_I2_J,axiom,
! [Y2: list_set_nat,P4: set_nat,Z3: list_set_nat] :
( ( ext_set_nat @ Y2 @ ( cons_set_nat @ P4 @ Z3 ) )
= ( ( ( member_set_nat2 @ P4 @ Y2 )
=> ( ext_set_nat @ Y2 @ Z3 ) )
& ( member_set_nat2 @ P4 @ Y2 ) ) ) ).
% ext.simps(2)
thf(fact_458_ext_Osimps_I2_J,axiom,
! [Y2: list_fm,P4: fm,Z3: list_fm] :
( ( ext_fm @ Y2 @ ( cons_fm @ P4 @ Z3 ) )
= ( ( ( member_fm2 @ P4 @ Y2 )
=> ( ext_fm @ Y2 @ Z3 ) )
& ( member_fm2 @ P4 @ Y2 ) ) ) ).
% ext.simps(2)
thf(fact_459_ext_Osimps_I2_J,axiom,
! [Y2: list_tm,P4: tm,Z3: list_tm] :
( ( ext_tm @ Y2 @ ( cons_tm @ P4 @ Z3 ) )
= ( ( ( member_tm2 @ P4 @ Y2 )
=> ( ext_tm @ Y2 @ Z3 ) )
& ( member_tm2 @ P4 @ Y2 ) ) ) ).
% ext.simps(2)
thf(fact_460_ext_Osimps_I2_J,axiom,
! [Y2: list_nat,P4: nat,Z3: list_nat] :
( ( ext_nat @ Y2 @ ( cons_nat @ P4 @ Z3 ) )
= ( ( ( member_nat2 @ P4 @ Y2 )
=> ( ext_nat @ Y2 @ Z3 ) )
& ( member_nat2 @ P4 @ Y2 ) ) ) ).
% ext.simps(2)
thf(fact_461_liftts_Osimps_I2_J,axiom,
! [T: tm,Ts: list_tm] :
( ( liftts @ ( cons_tm @ T @ Ts ) )
= ( cons_tm @ ( liftt @ T ) @ ( liftts @ Ts ) ) ) ).
% liftts.simps(2)
thf(fact_462_sub__term__const__transfer_I1_J,axiom,
! [M: nat,A2: nat,T: tm,S2: tm] :
( ( ( sub_term @ M @ ( fun @ A2 @ nil_tm ) @ T )
!= ( sub_term @ M @ S2 @ T ) )
=> ( member_tm3 @ ( fun @ A2 @ nil_tm ) @ ( set_tm2 @ ( subtermTm @ ( sub_term @ M @ ( fun @ A2 @ nil_tm ) @ T ) ) ) ) ) ).
% sub_term_const_transfer(1)
thf(fact_463_terms__cases,axiom,
! [T: tm,S: set_fm] :
( ( member_tm3 @ T @ ( terms @ S ) )
=> ( ( T
= ( fun @ zero_zero_nat @ nil_tm ) )
| ? [X3: fm] :
( ( member_fm3 @ X3 @ S )
& ( member_tm3 @ T @ ( set_tm2 @ ( subtermFm @ X3 ) ) ) ) ) ) ).
% terms_cases
thf(fact_464_listFunTms_Osimps_I1_J,axiom,
( ( listFunTms @ nil_tm )
= nil_nat ) ).
% listFunTms.simps(1)
thf(fact_465_the__elem__set,axiom,
! [X: set_nat] :
( ( the_elem_set_nat @ ( set_set_nat2 @ ( cons_set_nat @ X @ nil_set_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_466_the__elem__set,axiom,
! [X: fm] :
( ( the_elem_fm @ ( set_fm2 @ ( cons_fm @ X @ nil_fm ) ) )
= X ) ).
% the_elem_set
thf(fact_467_the__elem__set,axiom,
! [X: tm] :
( ( the_elem_tm @ ( set_tm2 @ ( cons_tm @ X @ nil_tm ) ) )
= X ) ).
% the_elem_set
thf(fact_468_the__elem__set,axiom,
! [X: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_469_paramsts_Osimps_I2_J,axiom,
! [T: tm,Ts: list_tm] :
( ( paramsts @ ( cons_tm @ T @ Ts ) )
= ( sup_sup_set_nat @ ( paramst @ T ) @ ( paramsts @ Ts ) ) ) ).
% paramsts.simps(2)
thf(fact_470_map__tailrec__rev_Oelims,axiom,
! [X: tm > tm,Xa: list_tm,Xb: list_tm,Y2: list_tm] :
( ( ( map_ta4789309763159252277_tm_tm @ X @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_tm )
=> ( Y2 != Xb ) )
=> ~ ! [A5: tm,As: list_tm] :
( ( Xa
= ( cons_tm @ A5 @ As ) )
=> ( Y2
!= ( map_ta4789309763159252277_tm_tm @ X @ As @ ( cons_tm @ ( X @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_471_map__tailrec__rev_Oelims,axiom,
! [X: tm > nat,Xa: list_tm,Xb: list_nat,Y2: list_nat] :
( ( ( map_ta7807370561492357248tm_nat @ X @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_tm )
=> ( Y2 != Xb ) )
=> ~ ! [A5: tm,As: list_tm] :
( ( Xa
= ( cons_tm @ A5 @ As ) )
=> ( Y2
!= ( map_ta7807370561492357248tm_nat @ X @ As @ ( cons_nat @ ( X @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_472_map__tailrec__rev_Oelims,axiom,
! [X: nat > tm,Xa: list_nat,Xb: list_tm,Y2: list_tm] :
( ( ( map_ta389968950240100318nat_tm @ X @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_nat )
=> ( Y2 != Xb ) )
=> ~ ! [A5: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A5 @ As ) )
=> ( Y2
!= ( map_ta389968950240100318nat_tm @ X @ As @ ( cons_tm @ ( X @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_473_map__tailrec__rev_Oelims,axiom,
! [X: nat > nat,Xa: list_nat,Xb: list_nat,Y2: list_nat] :
( ( ( map_ta7164188454487880599at_nat @ X @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa = nil_nat )
=> ( Y2 != Xb ) )
=> ~ ! [A5: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A5 @ As ) )
=> ( Y2
!= ( map_ta7164188454487880599at_nat @ X @ As @ ( cons_nat @ ( X @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_474_UnCI,axiom,
! [C: tm,B: set_tm,A: set_tm] :
( ( ~ ( member_tm3 @ C @ B )
=> ( member_tm3 @ C @ A ) )
=> ( member_tm3 @ C @ ( sup_sup_set_tm @ A @ B ) ) ) ).
% UnCI
thf(fact_475_UnCI,axiom,
! [C: fm,B: set_fm,A: set_fm] :
( ( ~ ( member_fm3 @ C @ B )
=> ( member_fm3 @ C @ A ) )
=> ( member_fm3 @ C @ ( sup_sup_set_fm @ A @ B ) ) ) ).
% UnCI
thf(fact_476_UnCI,axiom,
! [C: set_nat,B: set_set_nat,A: set_set_nat] :
( ( ~ ( member_set_nat3 @ C @ B )
=> ( member_set_nat3 @ C @ A ) )
=> ( member_set_nat3 @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_477_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat3 @ C @ B )
=> ( member_nat3 @ C @ A ) )
=> ( member_nat3 @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_478_Un__iff,axiom,
! [C: tm,A: set_tm,B: set_tm] :
( ( member_tm3 @ C @ ( sup_sup_set_tm @ A @ B ) )
= ( ( member_tm3 @ C @ A )
| ( member_tm3 @ C @ B ) ) ) ).
% Un_iff
thf(fact_479_Un__iff,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm3 @ C @ ( sup_sup_set_fm @ A @ B ) )
= ( ( member_fm3 @ C @ A )
| ( member_fm3 @ C @ B ) ) ) ).
% Un_iff
thf(fact_480_Un__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat3 @ C @ ( sup_sup_set_set_nat @ A @ B ) )
= ( ( member_set_nat3 @ C @ A )
| ( member_set_nat3 @ C @ B ) ) ) ).
% Un_iff
thf(fact_481_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat3 @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat3 @ C @ A )
| ( member_nat3 @ C @ B ) ) ) ).
% Un_iff
thf(fact_482_Un__empty,axiom,
! [A: set_tm,B: set_tm] :
( ( ( sup_sup_set_tm @ A @ B )
= bot_bot_set_tm )
= ( ( A = bot_bot_set_tm )
& ( B = bot_bot_set_tm ) ) ) ).
% Un_empty
thf(fact_483_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_484_Un__empty,axiom,
! [A: set_fm,B: set_fm] :
( ( ( sup_sup_set_fm @ A @ B )
= bot_bot_set_fm )
= ( ( A = bot_bot_set_fm )
& ( B = bot_bot_set_fm ) ) ) ).
% Un_empty
thf(fact_485_Un__subset__iff,axiom,
! [A: set_tm,B: set_tm,C2: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_tm @ A @ C2 )
& ( ord_less_eq_set_tm @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_486_Un__subset__iff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_nat @ A @ C2 )
& ( ord_less_eq_set_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_487_UnE,axiom,
! [C: tm,A: set_tm,B: set_tm] :
( ( member_tm3 @ C @ ( sup_sup_set_tm @ A @ B ) )
=> ( ~ ( member_tm3 @ C @ A )
=> ( member_tm3 @ C @ B ) ) ) ).
% UnE
thf(fact_488_UnE,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm3 @ C @ ( sup_sup_set_fm @ A @ B ) )
=> ( ~ ( member_fm3 @ C @ A )
=> ( member_fm3 @ C @ B ) ) ) ).
% UnE
thf(fact_489_UnE,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat3 @ C @ ( sup_sup_set_set_nat @ A @ B ) )
=> ( ~ ( member_set_nat3 @ C @ A )
=> ( member_set_nat3 @ C @ B ) ) ) ).
% UnE
thf(fact_490_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat3 @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat3 @ C @ A )
=> ( member_nat3 @ C @ B ) ) ) ).
% UnE
thf(fact_491_UnI1,axiom,
! [C: tm,A: set_tm,B: set_tm] :
( ( member_tm3 @ C @ A )
=> ( member_tm3 @ C @ ( sup_sup_set_tm @ A @ B ) ) ) ).
% UnI1
thf(fact_492_UnI1,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm3 @ C @ A )
=> ( member_fm3 @ C @ ( sup_sup_set_fm @ A @ B ) ) ) ).
% UnI1
thf(fact_493_UnI1,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat3 @ C @ A )
=> ( member_set_nat3 @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_494_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat3 @ C @ A )
=> ( member_nat3 @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_495_UnI2,axiom,
! [C: tm,B: set_tm,A: set_tm] :
( ( member_tm3 @ C @ B )
=> ( member_tm3 @ C @ ( sup_sup_set_tm @ A @ B ) ) ) ).
% UnI2
thf(fact_496_UnI2,axiom,
! [C: fm,B: set_fm,A: set_fm] :
( ( member_fm3 @ C @ B )
=> ( member_fm3 @ C @ ( sup_sup_set_fm @ A @ B ) ) ) ).
% UnI2
thf(fact_497_UnI2,axiom,
! [C: set_nat,B: set_set_nat,A: set_set_nat] :
( ( member_set_nat3 @ C @ B )
=> ( member_set_nat3 @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_498_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat3 @ C @ B )
=> ( member_nat3 @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_499_bex__Un,axiom,
! [A: set_nat,B: set_nat,P3: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat3 @ X2 @ ( sup_sup_set_nat @ A @ B ) )
& ( P3 @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat3 @ X2 @ A )
& ( P3 @ X2 ) )
| ? [X2: nat] :
( ( member_nat3 @ X2 @ B )
& ( P3 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_500_ball__Un,axiom,
! [A: set_nat,B: set_nat,P3: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat3 @ X2 @ ( sup_sup_set_nat @ A @ B ) )
=> ( P3 @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat3 @ X2 @ A )
=> ( P3 @ X2 ) )
& ! [X2: nat] :
( ( member_nat3 @ X2 @ B )
=> ( P3 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_501_Un__assoc,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_502_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_503_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A3 ) ) ) ).
% Un_commute
thf(fact_504_Un__left__absorb,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_505_Un__left__commute,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_506_Un__empty__right,axiom,
! [A: set_tm] :
( ( sup_sup_set_tm @ A @ bot_bot_set_tm )
= A ) ).
% Un_empty_right
thf(fact_507_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_508_Un__empty__right,axiom,
! [A: set_fm] :
( ( sup_sup_set_fm @ A @ bot_bot_set_fm )
= A ) ).
% Un_empty_right
thf(fact_509_Un__empty__left,axiom,
! [B: set_tm] :
( ( sup_sup_set_tm @ bot_bot_set_tm @ B )
= B ) ).
% Un_empty_left
thf(fact_510_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_511_Un__empty__left,axiom,
! [B: set_fm] :
( ( sup_sup_set_fm @ bot_bot_set_fm @ B )
= B ) ).
% Un_empty_left
thf(fact_512_subset__Un__eq,axiom,
( ord_less_eq_set_tm
= ( ^ [A3: set_tm,B5: set_tm] :
( ( sup_sup_set_tm @ A3 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_513_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( ( sup_sup_set_nat @ A3 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_514_subset__UnE,axiom,
! [C2: set_tm,A: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ C2 @ ( sup_sup_set_tm @ A @ B ) )
=> ~ ! [A6: set_tm] :
( ( ord_less_eq_set_tm @ A6 @ A )
=> ! [B6: set_tm] :
( ( ord_less_eq_set_tm @ B6 @ B )
=> ( C2
!= ( sup_sup_set_tm @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_515_subset__UnE,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ~ ! [A6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ A )
=> ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ B )
=> ( C2
!= ( sup_sup_set_nat @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_516_Un__absorb2,axiom,
! [B: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ B @ A )
=> ( ( sup_sup_set_tm @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_517_Un__absorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_518_Un__absorb1,axiom,
! [A: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( sup_sup_set_tm @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_519_Un__absorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_520_Un__upper2,axiom,
! [B: set_tm,A: set_tm] : ( ord_less_eq_set_tm @ B @ ( sup_sup_set_tm @ A @ B ) ) ).
% Un_upper2
thf(fact_521_Un__upper2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_522_Un__upper1,axiom,
! [A: set_tm,B: set_tm] : ( ord_less_eq_set_tm @ A @ ( sup_sup_set_tm @ A @ B ) ) ).
% Un_upper1
thf(fact_523_Un__upper1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_524_Un__least,axiom,
! [A: set_tm,C2: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A @ C2 )
=> ( ( ord_less_eq_set_tm @ B @ C2 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_525_Un__least,axiom,
! [A: set_nat,C2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_526_Un__mono,axiom,
! [A: set_tm,C2: set_tm,B: set_tm,D: set_tm] :
( ( ord_less_eq_set_tm @ A @ C2 )
=> ( ( ord_less_eq_set_tm @ B @ D )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A @ B ) @ ( sup_sup_set_tm @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_527_Un__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_528_paramst__sub__term_I1_J,axiom,
! [M: nat,S2: tm,T: tm] : ( ord_less_eq_set_nat @ ( paramst @ ( sub_term @ M @ S2 @ T ) ) @ ( sup_sup_set_nat @ ( paramst @ S2 ) @ ( paramst @ T ) ) ) ).
% paramst_sub_term(1)
thf(fact_529_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: tm > tm,A2: tm,As2: list_tm,Bs: list_tm] :
( ( map_ta4789309763159252277_tm_tm @ F @ ( cons_tm @ A2 @ As2 ) @ Bs )
= ( map_ta4789309763159252277_tm_tm @ F @ As2 @ ( cons_tm @ ( F @ A2 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_530_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: tm > nat,A2: tm,As2: list_tm,Bs: list_nat] :
( ( map_ta7807370561492357248tm_nat @ F @ ( cons_tm @ A2 @ As2 ) @ Bs )
= ( map_ta7807370561492357248tm_nat @ F @ As2 @ ( cons_nat @ ( F @ A2 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_531_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: nat > tm,A2: nat,As2: list_nat,Bs: list_tm] :
( ( map_ta389968950240100318nat_tm @ F @ ( cons_nat @ A2 @ As2 ) @ Bs )
= ( map_ta389968950240100318nat_tm @ F @ As2 @ ( cons_tm @ ( F @ A2 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_532_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: nat > nat,A2: nat,As2: list_nat,Bs: list_nat] :
( ( map_ta7164188454487880599at_nat @ F @ ( cons_nat @ A2 @ As2 ) @ Bs )
= ( map_ta7164188454487880599at_nat @ F @ As2 @ ( cons_nat @ ( F @ A2 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_533_listFunTm_Osimps_I1_J,axiom,
! [N: nat,Ts: list_tm] :
( ( listFunTm @ ( fun @ N @ Ts ) )
= ( cons_nat @ N @ ( listFunTms @ Ts ) ) ) ).
% listFunTm.simps(1)
thf(fact_534_sup__bot_Oright__neutral,axiom,
! [A2: set_tm] :
( ( sup_sup_set_tm @ A2 @ bot_bot_set_tm )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_535_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_536_sup__bot_Oright__neutral,axiom,
! [A2: set_fm] :
( ( sup_sup_set_fm @ A2 @ bot_bot_set_fm )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_537_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_tm,B2: set_tm] :
( ( bot_bot_set_tm
= ( sup_sup_set_tm @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_tm )
& ( B2 = bot_bot_set_tm ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_538_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_539_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_fm,B2: set_fm] :
( ( bot_bot_set_fm
= ( sup_sup_set_fm @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_fm )
& ( B2 = bot_bot_set_fm ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_540_sup__bot_Oleft__neutral,axiom,
! [A2: set_tm] :
( ( sup_sup_set_tm @ bot_bot_set_tm @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_541_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_542_sup__bot_Oleft__neutral,axiom,
! [A2: set_fm] :
( ( sup_sup_set_fm @ bot_bot_set_fm @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_543_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_tm,B2: set_tm] :
( ( ( sup_sup_set_tm @ A2 @ B2 )
= bot_bot_set_tm )
= ( ( A2 = bot_bot_set_tm )
& ( B2 = bot_bot_set_tm ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_544_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_545_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_fm,B2: set_fm] :
( ( ( sup_sup_set_fm @ A2 @ B2 )
= bot_bot_set_fm )
= ( ( A2 = bot_bot_set_fm )
& ( B2 = bot_bot_set_fm ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_546_sup__eq__bot__iff,axiom,
! [X: set_tm,Y2: set_tm] :
( ( ( sup_sup_set_tm @ X @ Y2 )
= bot_bot_set_tm )
= ( ( X = bot_bot_set_tm )
& ( Y2 = bot_bot_set_tm ) ) ) ).
% sup_eq_bot_iff
thf(fact_547_sup__eq__bot__iff,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y2 )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_548_sup__eq__bot__iff,axiom,
! [X: set_fm,Y2: set_fm] :
( ( ( sup_sup_set_fm @ X @ Y2 )
= bot_bot_set_fm )
= ( ( X = bot_bot_set_fm )
& ( Y2 = bot_bot_set_fm ) ) ) ).
% sup_eq_bot_iff
thf(fact_549_bot__eq__sup__iff,axiom,
! [X: set_tm,Y2: set_tm] :
( ( bot_bot_set_tm
= ( sup_sup_set_tm @ X @ Y2 ) )
= ( ( X = bot_bot_set_tm )
& ( Y2 = bot_bot_set_tm ) ) ) ).
% bot_eq_sup_iff
thf(fact_550_bot__eq__sup__iff,axiom,
! [X: set_nat,Y2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y2 ) )
= ( ( X = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_551_bot__eq__sup__iff,axiom,
! [X: set_fm,Y2: set_fm] :
( ( bot_bot_set_fm
= ( sup_sup_set_fm @ X @ Y2 ) )
= ( ( X = bot_bot_set_fm )
& ( Y2 = bot_bot_set_fm ) ) ) ).
% bot_eq_sup_iff
thf(fact_552_sup__bot__right,axiom,
! [X: set_tm] :
( ( sup_sup_set_tm @ X @ bot_bot_set_tm )
= X ) ).
% sup_bot_right
thf(fact_553_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_554_sup__bot__right,axiom,
! [X: set_fm] :
( ( sup_sup_set_fm @ X @ bot_bot_set_fm )
= X ) ).
% sup_bot_right
thf(fact_555_sup__bot__left,axiom,
! [X: set_tm] :
( ( sup_sup_set_tm @ bot_bot_set_tm @ X )
= X ) ).
% sup_bot_left
thf(fact_556_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_557_sup__bot__left,axiom,
! [X: set_fm] :
( ( sup_sup_set_fm @ bot_bot_set_fm @ X )
= X ) ).
% sup_bot_left
thf(fact_558_le__sup__iff,axiom,
! [X: set_tm,Y2: set_tm,Z3: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ X @ Y2 ) @ Z3 )
= ( ( ord_less_eq_set_tm @ X @ Z3 )
& ( ord_less_eq_set_tm @ Y2 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_559_le__sup__iff,axiom,
! [X: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ Z3 )
= ( ( ord_less_eq_set_nat @ X @ Z3 )
& ( ord_less_eq_set_nat @ Y2 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_560_le__sup__iff,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y2 ) @ Z3 )
= ( ( ord_less_eq_nat @ X @ Z3 )
& ( ord_less_eq_nat @ Y2 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_561_le__sup__iff,axiom,
! [X: extended_enat,Y2: extended_enat,Z3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ X @ Y2 ) @ Z3 )
= ( ( ord_le2932123472753598470d_enat @ X @ Z3 )
& ( ord_le2932123472753598470d_enat @ Y2 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_562_sup_Obounded__iff,axiom,
! [B2: set_tm,C: set_tm,A2: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_tm @ B2 @ A2 )
& ( ord_less_eq_set_tm @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_563_sup_Obounded__iff,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_564_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_565_sup_Obounded__iff,axiom,
! [B2: extended_enat,C: extended_enat,A2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ B2 @ C ) @ A2 )
= ( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
& ( ord_le2932123472753598470d_enat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_566_inf__sup__ord_I4_J,axiom,
! [Y2: set_tm,X: set_tm] : ( ord_less_eq_set_tm @ Y2 @ ( sup_sup_set_tm @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_567_inf__sup__ord_I4_J,axiom,
! [Y2: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_568_inf__sup__ord_I4_J,axiom,
! [Y2: nat,X: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_569_inf__sup__ord_I4_J,axiom,
! [Y2: extended_enat,X: extended_enat] : ( ord_le2932123472753598470d_enat @ Y2 @ ( sup_su3973961784419623482d_enat @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_570_inf__sup__ord_I3_J,axiom,
! [X: set_tm,Y2: set_tm] : ( ord_less_eq_set_tm @ X @ ( sup_sup_set_tm @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_571_inf__sup__ord_I3_J,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_572_inf__sup__ord_I3_J,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_573_inf__sup__ord_I3_J,axiom,
! [X: extended_enat,Y2: extended_enat] : ( ord_le2932123472753598470d_enat @ X @ ( sup_su3973961784419623482d_enat @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_574_le__supE,axiom,
! [A2: set_tm,B2: set_tm,X: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_tm @ A2 @ X )
=> ~ ( ord_less_eq_set_tm @ B2 @ X ) ) ) ).
% le_supE
thf(fact_575_le__supE,axiom,
! [A2: set_nat,B2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ X )
=> ~ ( ord_less_eq_set_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_576_le__supE,axiom,
! [A2: nat,B2: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_577_le__supE,axiom,
! [A2: extended_enat,B2: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_le2932123472753598470d_enat @ A2 @ X )
=> ~ ( ord_le2932123472753598470d_enat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_578_le__supI,axiom,
! [A2: set_tm,X: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ X )
=> ( ( ord_less_eq_set_tm @ B2 @ X )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_579_le__supI,axiom,
! [A2: set_nat,X: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X )
=> ( ( ord_less_eq_set_nat @ B2 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_580_le__supI,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_581_le__supI,axiom,
! [A2: extended_enat,X: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ X )
=> ( ( ord_le2932123472753598470d_enat @ B2 @ X )
=> ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_582_sup__ge1,axiom,
! [X: set_tm,Y2: set_tm] : ( ord_less_eq_set_tm @ X @ ( sup_sup_set_tm @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_583_sup__ge1,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_584_sup__ge1,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_585_sup__ge1,axiom,
! [X: extended_enat,Y2: extended_enat] : ( ord_le2932123472753598470d_enat @ X @ ( sup_su3973961784419623482d_enat @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_586_sup__ge2,axiom,
! [Y2: set_tm,X: set_tm] : ( ord_less_eq_set_tm @ Y2 @ ( sup_sup_set_tm @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_587_sup__ge2,axiom,
! [Y2: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_588_sup__ge2,axiom,
! [Y2: nat,X: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_589_sup__ge2,axiom,
! [Y2: extended_enat,X: extended_enat] : ( ord_le2932123472753598470d_enat @ Y2 @ ( sup_su3973961784419623482d_enat @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_590_le__supI1,axiom,
! [X: set_tm,A2: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ X @ A2 )
=> ( ord_less_eq_set_tm @ X @ ( sup_sup_set_tm @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_591_le__supI1,axiom,
! [X: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_592_le__supI1,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_593_le__supI1,axiom,
! [X: extended_enat,A2: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ A2 )
=> ( ord_le2932123472753598470d_enat @ X @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_594_le__supI2,axiom,
! [X: set_tm,B2: set_tm,A2: set_tm] :
( ( ord_less_eq_set_tm @ X @ B2 )
=> ( ord_less_eq_set_tm @ X @ ( sup_sup_set_tm @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_595_le__supI2,axiom,
! [X: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X @ B2 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_596_le__supI2,axiom,
! [X: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_597_le__supI2,axiom,
! [X: extended_enat,B2: extended_enat,A2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ B2 )
=> ( ord_le2932123472753598470d_enat @ X @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_598_sup_Omono,axiom,
! [C: set_tm,A2: set_tm,D2: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ C @ A2 )
=> ( ( ord_less_eq_set_tm @ D2 @ B2 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ C @ D2 ) @ ( sup_sup_set_tm @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_599_sup_Omono,axiom,
! [C: set_nat,A2: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D2 ) @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_600_sup_Omono,axiom,
! [C: nat,A2: nat,D2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_601_sup_Omono,axiom,
! [C: extended_enat,A2: extended_enat,D2: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C @ A2 )
=> ( ( ord_le2932123472753598470d_enat @ D2 @ B2 )
=> ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ C @ D2 ) @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_602_sup__mono,axiom,
! [A2: set_tm,C: set_tm,B2: set_tm,D2: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ C )
=> ( ( ord_less_eq_set_tm @ B2 @ D2 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A2 @ B2 ) @ ( sup_sup_set_tm @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_603_sup__mono,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_604_sup__mono,axiom,
! [A2: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_605_sup__mono,axiom,
! [A2: extended_enat,C: extended_enat,B2: extended_enat,D2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ C )
=> ( ( ord_le2932123472753598470d_enat @ B2 @ D2 )
=> ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) @ ( sup_su3973961784419623482d_enat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_606_sup__least,axiom,
! [Y2: set_tm,X: set_tm,Z3: set_tm] :
( ( ord_less_eq_set_tm @ Y2 @ X )
=> ( ( ord_less_eq_set_tm @ Z3 @ X )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ Y2 @ Z3 ) @ X ) ) ) ).
% sup_least
thf(fact_607_sup__least,axiom,
! [Y2: set_nat,X: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( ord_less_eq_set_nat @ Z3 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y2 @ Z3 ) @ X ) ) ) ).
% sup_least
thf(fact_608_sup__least,axiom,
! [Y2: nat,X: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ Z3 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z3 ) @ X ) ) ) ).
% sup_least
thf(fact_609_sup__least,axiom,
! [Y2: extended_enat,X: extended_enat,Z3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y2 @ X )
=> ( ( ord_le2932123472753598470d_enat @ Z3 @ X )
=> ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ Y2 @ Z3 ) @ X ) ) ) ).
% sup_least
thf(fact_610_le__iff__sup,axiom,
( ord_less_eq_set_tm
= ( ^ [X2: set_tm,Y: set_tm] :
( ( sup_sup_set_tm @ X2 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_611_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X2 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_612_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y: nat] :
( ( sup_sup_nat @ X2 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_613_le__iff__sup,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [X2: extended_enat,Y: extended_enat] :
( ( sup_su3973961784419623482d_enat @ X2 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_614_sup_OorderE,axiom,
! [B2: set_tm,A2: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_tm @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_615_sup_OorderE,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_616_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_617_sup_OorderE,axiom,
! [B2: extended_enat,A2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
=> ( A2
= ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_618_sup_OorderI,axiom,
! [A2: set_tm,B2: set_tm] :
( ( A2
= ( sup_sup_set_tm @ A2 @ B2 ) )
=> ( ord_less_eq_set_tm @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_619_sup_OorderI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_620_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_621_sup_OorderI,axiom,
! [A2: extended_enat,B2: extended_enat] :
( ( A2
= ( sup_su3973961784419623482d_enat @ A2 @ B2 ) )
=> ( ord_le2932123472753598470d_enat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_622_sup__unique,axiom,
! [F: set_tm > set_tm > set_tm,X: set_tm,Y2: set_tm] :
( ! [X3: set_tm,Y3: set_tm] : ( ord_less_eq_set_tm @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_tm,Y3: set_tm] : ( ord_less_eq_set_tm @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_tm,Y3: set_tm,Z4: set_tm] :
( ( ord_less_eq_set_tm @ Y3 @ X3 )
=> ( ( ord_less_eq_set_tm @ Z4 @ X3 )
=> ( ord_less_eq_set_tm @ ( F @ Y3 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_set_tm @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_623_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y2: set_nat] :
( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_nat,Y3: set_nat,Z4: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_set_nat @ Z4 @ X3 )
=> ( ord_less_eq_set_nat @ ( F @ Y3 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_set_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_624_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y2: nat] :
( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ Z4 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y3 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_625_sup__unique,axiom,
! [F: extended_enat > extended_enat > extended_enat,X: extended_enat,Y2: extended_enat] :
( ! [X3: extended_enat,Y3: extended_enat] : ( ord_le2932123472753598470d_enat @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: extended_enat,Y3: extended_enat] : ( ord_le2932123472753598470d_enat @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: extended_enat,Y3: extended_enat,Z4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y3 @ X3 )
=> ( ( ord_le2932123472753598470d_enat @ Z4 @ X3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ Y3 @ Z4 ) @ X3 ) ) )
=> ( ( sup_su3973961784419623482d_enat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_626_sup_Oabsorb1,axiom,
! [B2: set_tm,A2: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A2 )
=> ( ( sup_sup_set_tm @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_627_sup_Oabsorb1,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_628_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_629_sup_Oabsorb1,axiom,
! [B2: extended_enat,A2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
=> ( ( sup_su3973961784419623482d_enat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_630_sup_Oabsorb2,axiom,
! [A2: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ( sup_sup_set_tm @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_631_sup_Oabsorb2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_632_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_633_sup_Oabsorb2,axiom,
! [A2: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
=> ( ( sup_su3973961784419623482d_enat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_634_sup__absorb1,axiom,
! [Y2: set_tm,X: set_tm] :
( ( ord_less_eq_set_tm @ Y2 @ X )
=> ( ( sup_sup_set_tm @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_635_sup__absorb1,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( sup_sup_set_nat @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_636_sup__absorb1,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( sup_sup_nat @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_637_sup__absorb1,axiom,
! [Y2: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y2 @ X )
=> ( ( sup_su3973961784419623482d_enat @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_638_sup__absorb2,axiom,
! [X: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y2 )
=> ( ( sup_sup_set_tm @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_639_sup__absorb2,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( sup_sup_set_nat @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_640_sup__absorb2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( sup_sup_nat @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_641_sup__absorb2,axiom,
! [X: extended_enat,Y2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y2 )
=> ( ( sup_su3973961784419623482d_enat @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_642_sup_OboundedE,axiom,
! [B2: set_tm,C: set_tm,A2: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_tm @ B2 @ A2 )
=> ~ ( ord_less_eq_set_tm @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_643_sup_OboundedE,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_644_sup_OboundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_645_sup_OboundedE,axiom,
! [B2: extended_enat,C: extended_enat,A2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
=> ~ ( ord_le2932123472753598470d_enat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_646_sup_OboundedI,axiom,
! [B2: set_tm,A2: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A2 )
=> ( ( ord_less_eq_set_tm @ C @ A2 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_647_sup_OboundedI,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_648_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_649_sup_OboundedI,axiom,
! [B2: extended_enat,A2: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
=> ( ( ord_le2932123472753598470d_enat @ C @ A2 )
=> ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_650_sup_Oorder__iff,axiom,
( ord_less_eq_set_tm
= ( ^ [B3: set_tm,A4: set_tm] :
( A4
= ( sup_sup_set_tm @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_651_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A4: set_nat] :
( A4
= ( sup_sup_set_nat @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_652_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_653_sup_Oorder__iff,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
( A4
= ( sup_su3973961784419623482d_enat @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_654_sup_Ocobounded1,axiom,
! [A2: set_tm,B2: set_tm] : ( ord_less_eq_set_tm @ A2 @ ( sup_sup_set_tm @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_655_sup_Ocobounded1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_656_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_657_sup_Ocobounded1,axiom,
! [A2: extended_enat,B2: extended_enat] : ( ord_le2932123472753598470d_enat @ A2 @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_658_sup_Ocobounded2,axiom,
! [B2: set_tm,A2: set_tm] : ( ord_less_eq_set_tm @ B2 @ ( sup_sup_set_tm @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_659_sup_Ocobounded2,axiom,
! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_660_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_661_sup_Ocobounded2,axiom,
! [B2: extended_enat,A2: extended_enat] : ( ord_le2932123472753598470d_enat @ B2 @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_662_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_tm
= ( ^ [B3: set_tm,A4: set_tm] :
( ( sup_sup_set_tm @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_663_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_664_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_665_sup_Oabsorb__iff1,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
( ( sup_su3973961784419623482d_enat @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_666_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_tm
= ( ^ [A4: set_tm,B3: set_tm] :
( ( sup_sup_set_tm @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_667_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_668_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_669_sup_Oabsorb__iff2,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( sup_su3973961784419623482d_enat @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_670_sup_OcoboundedI1,axiom,
! [C: set_tm,A2: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ C @ A2 )
=> ( ord_less_eq_set_tm @ C @ ( sup_sup_set_tm @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_671_sup_OcoboundedI1,axiom,
! [C: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_672_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_673_sup_OcoboundedI1,axiom,
! [C: extended_enat,A2: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C @ A2 )
=> ( ord_le2932123472753598470d_enat @ C @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_674_sup_OcoboundedI2,axiom,
! [C: set_tm,B2: set_tm,A2: set_tm] :
( ( ord_less_eq_set_tm @ C @ B2 )
=> ( ord_less_eq_set_tm @ C @ ( sup_sup_set_tm @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_675_sup_OcoboundedI2,axiom,
! [C: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_676_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_677_sup_OcoboundedI2,axiom,
! [C: extended_enat,B2: extended_enat,A2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C @ B2 )
=> ( ord_le2932123472753598470d_enat @ C @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_678_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_679_le__zero__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% le_zero_eq
thf(fact_680_set__union,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( set_tm2 @ ( union_tm @ Xs @ Ys ) )
= ( sup_sup_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ Ys ) ) ) ).
% set_union
thf(fact_681_set__union,axiom,
! [Xs: list_set_nat,Ys: list_set_nat] :
( ( set_set_nat2 @ ( union_set_nat @ Xs @ Ys ) )
= ( sup_sup_set_set_nat @ ( set_set_nat2 @ Xs ) @ ( set_set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_682_set__union,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( set_fm2 @ ( union_fm @ Xs @ Ys ) )
= ( sup_sup_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys ) ) ) ).
% set_union
thf(fact_683_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_684_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_tm @ N @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_tm @ N @ nil_tm )
= nil_list_tm ) ) ) ).
% n_lists_Nil
thf(fact_685_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_686_paramst__sub__term_I2_J,axiom,
! [M: nat,S2: tm,L3: list_tm] : ( ord_less_eq_set_nat @ ( paramsts @ ( sub_list @ M @ S2 @ L3 ) ) @ ( sup_sup_set_nat @ ( paramst @ S2 ) @ ( paramsts @ L3 ) ) ) ).
% paramst_sub_term(2)
thf(fact_687_list__encode_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X3: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs3 ) ) ) ).
% list_encode.cases
thf(fact_688_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_689_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_690_Nat_Oex__has__greatest__nat,axiom,
! [P3: nat > $o,K: nat,B2: nat] :
( ( P3 @ K )
=> ( ! [Y3: nat] :
( ( P3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X3: nat] :
( ( P3 @ X3 )
& ! [Y5: nat] :
( ( P3 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_691_GreatestI__ex__nat,axiom,
! [P3: nat > $o,B2: nat] :
( ? [X_1: nat] : ( P3 @ X_1 )
=> ( ! [Y3: nat] :
( ( P3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ( P3 @ ( order_Greatest_nat @ P3 ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_692_Greatest__le__nat,axiom,
! [P3: nat > $o,K: nat,B2: nat] :
( ( P3 @ K )
=> ( ! [Y3: nat] :
( ( P3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P3 ) ) ) ) ).
% Greatest_le_nat
thf(fact_693_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_694_GreatestI__nat,axiom,
! [P3: nat > $o,K: nat,B2: nat] :
( ( P3 @ K )
=> ( ! [Y3: nat] :
( ( P3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ( P3 @ ( order_Greatest_nat @ P3 ) ) ) ) ).
% GreatestI_nat
thf(fact_695_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_696_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_697_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_698_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_699_sub__list_Osimps_I1_J,axiom,
! [V: nat,S2: tm] :
( ( sub_list @ V @ S2 @ nil_tm )
= nil_tm ) ).
% sub_list.simps(1)
thf(fact_700_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_701_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_702_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_703_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_704_sub__term_Osimps_I2_J,axiom,
! [V: nat,S2: tm,I: nat,L3: list_tm] :
( ( sub_term @ V @ S2 @ ( fun @ I @ L3 ) )
= ( fun @ I @ ( sub_list @ V @ S2 @ L3 ) ) ) ).
% sub_term.simps(2)
thf(fact_705_sub__list_Osimps_I2_J,axiom,
! [V: nat,S2: tm,T: tm,L3: list_tm] :
( ( sub_list @ V @ S2 @ ( cons_tm @ T @ L3 ) )
= ( cons_tm @ ( sub_term @ V @ S2 @ T ) @ ( sub_list @ V @ S2 @ L3 ) ) ) ).
% sub_list.simps(2)
thf(fact_706_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_707_zero__le,axiom,
! [X: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ X ) ).
% zero_le
thf(fact_708_n__lists_Osimps_I1_J,axiom,
! [Xs: list_tm] :
( ( n_lists_tm @ zero_zero_nat @ Xs )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% n_lists.simps(1)
thf(fact_709_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_710_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_tm] :
( ( sup_sup_set_tm @ X @ bot_bot_set_tm )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_711_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_712_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_fm] :
( ( sup_sup_set_fm @ X @ bot_bot_set_fm )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_713_s5_I1_J,axiom,
( sub_term
= ( ^ [V2: nat,S3: tm,T2: tm] : ( substt @ T2 @ S3 @ V2 ) ) ) ).
% s5(1)
thf(fact_714_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_715_le__numeral__extra_I3_J,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ).
% le_numeral_extra(3)
thf(fact_716_sublists_Osimps_I1_J,axiom,
( ( sublists_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% sublists.simps(1)
thf(fact_717_sublists_Osimps_I1_J,axiom,
( ( sublists_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% sublists.simps(1)
thf(fact_718_paramst_Osimps_I2_J,axiom,
! [A2: nat,Ts: list_tm] :
( ( paramst @ ( fun @ A2 @ Ts ) )
= ( sup_sup_set_nat @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) @ ( paramsts @ Ts ) ) ) ).
% paramst.simps(2)
thf(fact_719_product__lists_Osimps_I1_J,axiom,
( ( product_lists_tm @ nil_list_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% product_lists.simps(1)
thf(fact_720_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_721_params__sub,axiom,
! [M: nat,T: tm,P4: fm] : ( ord_less_eq_set_nat @ ( params @ ( sub @ M @ T @ P4 ) ) @ ( sup_sup_set_nat @ ( paramst @ T ) @ ( params @ P4 ) ) ) ).
% params_sub
thf(fact_722_insert__absorb2,axiom,
! [X: nat,A: set_nat] :
( ( insert_nat2 @ X @ ( insert_nat2 @ X @ A ) )
= ( insert_nat2 @ X @ A ) ) ).
% insert_absorb2
thf(fact_723_insert__absorb2,axiom,
! [X: fm,A: set_fm] :
( ( insert_fm2 @ X @ ( insert_fm2 @ X @ A ) )
= ( insert_fm2 @ X @ A ) ) ).
% insert_absorb2
thf(fact_724_insert__iff,axiom,
! [A2: tm,B2: tm,A: set_tm] :
( ( member_tm3 @ A2 @ ( insert_tm2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_tm3 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_725_insert__iff,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat3 @ A2 @ ( insert_nat2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_nat3 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_726_insert__iff,axiom,
! [A2: fm,B2: fm,A: set_fm] :
( ( member_fm3 @ A2 @ ( insert_fm2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_fm3 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_727_insert__iff,axiom,
! [A2: set_nat,B2: set_nat,A: set_set_nat] :
( ( member_set_nat3 @ A2 @ ( insert_set_nat2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_set_nat3 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_728_insertCI,axiom,
! [A2: tm,B: set_tm,B2: tm] :
( ( ~ ( member_tm3 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_tm3 @ A2 @ ( insert_tm2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_729_insertCI,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat3 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_nat3 @ A2 @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_730_insertCI,axiom,
! [A2: fm,B: set_fm,B2: fm] :
( ( ~ ( member_fm3 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_fm3 @ A2 @ ( insert_fm2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_731_insertCI,axiom,
! [A2: set_nat,B: set_set_nat,B2: set_nat] :
( ( ~ ( member_set_nat3 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_set_nat3 @ A2 @ ( insert_set_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_732_singletonI,axiom,
! [A2: set_nat] : ( member_set_nat3 @ A2 @ ( insert_set_nat2 @ A2 @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_733_singletonI,axiom,
! [A2: tm] : ( member_tm3 @ A2 @ ( insert_tm2 @ A2 @ bot_bot_set_tm ) ) ).
% singletonI
thf(fact_734_singletonI,axiom,
! [A2: nat] : ( member_nat3 @ A2 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_735_singletonI,axiom,
! [A2: fm] : ( member_fm3 @ A2 @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) ).
% singletonI
thf(fact_736_insert__subset,axiom,
! [X: fm,A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ ( insert_fm2 @ X @ A ) @ B )
= ( ( member_fm3 @ X @ B )
& ( ord_less_eq_set_fm @ A @ B ) ) ) ).
% insert_subset
thf(fact_737_insert__subset,axiom,
! [X: set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( insert_set_nat2 @ X @ A ) @ B )
= ( ( member_set_nat3 @ X @ B )
& ( ord_le6893508408891458716et_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_738_insert__subset,axiom,
! [X: tm,A: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ ( insert_tm2 @ X @ A ) @ B )
= ( ( member_tm3 @ X @ B )
& ( ord_less_eq_set_tm @ A @ B ) ) ) ).
% insert_subset
thf(fact_739_insert__subset,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ A ) @ B )
= ( ( member_nat3 @ X @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_740_Un__insert__right,axiom,
! [A: set_fm,A2: fm,B: set_fm] :
( ( sup_sup_set_fm @ A @ ( insert_fm2 @ A2 @ B ) )
= ( insert_fm2 @ A2 @ ( sup_sup_set_fm @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_741_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_742_Un__insert__left,axiom,
! [A2: fm,B: set_fm,C2: set_fm] :
( ( sup_sup_set_fm @ ( insert_fm2 @ A2 @ B ) @ C2 )
= ( insert_fm2 @ A2 @ ( sup_sup_set_fm @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_743_Un__insert__left,axiom,
! [A2: nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat2 @ A2 @ B ) @ C2 )
= ( insert_nat2 @ A2 @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_744_singleton__insert__inj__eq,axiom,
! [B2: fm,A2: fm,A: set_fm] :
( ( ( insert_fm2 @ B2 @ bot_bot_set_fm )
= ( insert_fm2 @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ B2 @ bot_bot_set_fm ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_745_singleton__insert__inj__eq,axiom,
! [B2: tm,A2: tm,A: set_tm] :
( ( ( insert_tm2 @ B2 @ bot_bot_set_tm )
= ( insert_tm2 @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_tm @ A @ ( insert_tm2 @ B2 @ bot_bot_set_tm ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_746_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_747_singleton__insert__inj__eq_H,axiom,
! [A2: fm,A: set_fm,B2: fm] :
( ( ( insert_fm2 @ A2 @ A )
= ( insert_fm2 @ B2 @ bot_bot_set_fm ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ B2 @ bot_bot_set_fm ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_748_singleton__insert__inj__eq_H,axiom,
! [A2: tm,A: set_tm,B2: tm] :
( ( ( insert_tm2 @ A2 @ A )
= ( insert_tm2 @ B2 @ bot_bot_set_tm ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_tm @ A @ ( insert_tm2 @ B2 @ bot_bot_set_tm ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_749_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_750_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_751_list_Osimps_I15_J,axiom,
! [X21: fm,X22: list_fm] :
( ( set_fm2 @ ( cons_fm @ X21 @ X22 ) )
= ( insert_fm2 @ X21 @ ( set_fm2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_752_list_Osimps_I15_J,axiom,
! [X21: tm,X22: list_tm] :
( ( set_tm2 @ ( cons_tm @ X21 @ X22 ) )
= ( insert_tm2 @ X21 @ ( set_tm2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_753_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_754_List_Oset__insert,axiom,
! [X: tm,Xs: list_tm] :
( ( set_tm2 @ ( insert_tm @ X @ Xs ) )
= ( insert_tm2 @ X @ ( set_tm2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_755_List_Oset__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( set_nat2 @ ( insert_nat @ X @ Xs ) )
= ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_756_List_Oset__insert,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( set_set_nat2 @ ( insert_set_nat @ X @ Xs ) )
= ( insert_set_nat2 @ X @ ( set_set_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_757_List_Oset__insert,axiom,
! [X: fm,Xs: list_fm] :
( ( set_fm2 @ ( insert_fm @ X @ Xs ) )
= ( insert_fm2 @ X @ ( set_fm2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_758_the__elem__eq,axiom,
! [X: tm] :
( ( the_elem_tm @ ( insert_tm2 @ X @ bot_bot_set_tm ) )
= X ) ).
% the_elem_eq
thf(fact_759_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_760_the__elem__eq,axiom,
! [X: fm] :
( ( the_elem_fm @ ( insert_fm2 @ X @ bot_bot_set_fm ) )
= X ) ).
% the_elem_eq
thf(fact_761_mk__disjoint__insert,axiom,
! [A2: tm,A: set_tm] :
( ( member_tm3 @ A2 @ A )
=> ? [B7: set_tm] :
( ( A
= ( insert_tm2 @ A2 @ B7 ) )
& ~ ( member_tm3 @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_762_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat3 @ A2 @ A )
=> ? [B7: set_nat] :
( ( A
= ( insert_nat2 @ A2 @ B7 ) )
& ~ ( member_nat3 @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_763_mk__disjoint__insert,axiom,
! [A2: fm,A: set_fm] :
( ( member_fm3 @ A2 @ A )
=> ? [B7: set_fm] :
( ( A
= ( insert_fm2 @ A2 @ B7 ) )
& ~ ( member_fm3 @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_764_mk__disjoint__insert,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat3 @ A2 @ A )
=> ? [B7: set_set_nat] :
( ( A
= ( insert_set_nat2 @ A2 @ B7 ) )
& ~ ( member_set_nat3 @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_765_insert__commute,axiom,
! [X: nat,Y2: nat,A: set_nat] :
( ( insert_nat2 @ X @ ( insert_nat2 @ Y2 @ A ) )
= ( insert_nat2 @ Y2 @ ( insert_nat2 @ X @ A ) ) ) ).
% insert_commute
thf(fact_766_insert__commute,axiom,
! [X: fm,Y2: fm,A: set_fm] :
( ( insert_fm2 @ X @ ( insert_fm2 @ Y2 @ A ) )
= ( insert_fm2 @ Y2 @ ( insert_fm2 @ X @ A ) ) ) ).
% insert_commute
thf(fact_767_insert__eq__iff,axiom,
! [A2: tm,A: set_tm,B2: tm,B: set_tm] :
( ~ ( member_tm3 @ A2 @ A )
=> ( ~ ( member_tm3 @ B2 @ B )
=> ( ( ( insert_tm2 @ A2 @ A )
= ( insert_tm2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C4: set_tm] :
( ( A
= ( insert_tm2 @ B2 @ C4 ) )
& ~ ( member_tm3 @ B2 @ C4 )
& ( B
= ( insert_tm2 @ A2 @ C4 ) )
& ~ ( member_tm3 @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_768_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat3 @ A2 @ A )
=> ( ~ ( member_nat3 @ B2 @ B )
=> ( ( ( insert_nat2 @ A2 @ A )
= ( insert_nat2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C4: set_nat] :
( ( A
= ( insert_nat2 @ B2 @ C4 ) )
& ~ ( member_nat3 @ B2 @ C4 )
& ( B
= ( insert_nat2 @ A2 @ C4 ) )
& ~ ( member_nat3 @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_769_insert__eq__iff,axiom,
! [A2: fm,A: set_fm,B2: fm,B: set_fm] :
( ~ ( member_fm3 @ A2 @ A )
=> ( ~ ( member_fm3 @ B2 @ B )
=> ( ( ( insert_fm2 @ A2 @ A )
= ( insert_fm2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C4: set_fm] :
( ( A
= ( insert_fm2 @ B2 @ C4 ) )
& ~ ( member_fm3 @ B2 @ C4 )
& ( B
= ( insert_fm2 @ A2 @ C4 ) )
& ~ ( member_fm3 @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_770_insert__eq__iff,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_nat,B: set_set_nat] :
( ~ ( member_set_nat3 @ A2 @ A )
=> ( ~ ( member_set_nat3 @ B2 @ B )
=> ( ( ( insert_set_nat2 @ A2 @ A )
= ( insert_set_nat2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C4: set_set_nat] :
( ( A
= ( insert_set_nat2 @ B2 @ C4 ) )
& ~ ( member_set_nat3 @ B2 @ C4 )
& ( B
= ( insert_set_nat2 @ A2 @ C4 ) )
& ~ ( member_set_nat3 @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_771_insert__absorb,axiom,
! [A2: tm,A: set_tm] :
( ( member_tm3 @ A2 @ A )
=> ( ( insert_tm2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_772_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat3 @ A2 @ A )
=> ( ( insert_nat2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_773_insert__absorb,axiom,
! [A2: fm,A: set_fm] :
( ( member_fm3 @ A2 @ A )
=> ( ( insert_fm2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_774_insert__absorb,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat3 @ A2 @ A )
=> ( ( insert_set_nat2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_775_insert__ident,axiom,
! [X: tm,A: set_tm,B: set_tm] :
( ~ ( member_tm3 @ X @ A )
=> ( ~ ( member_tm3 @ X @ B )
=> ( ( ( insert_tm2 @ X @ A )
= ( insert_tm2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_776_insert__ident,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat3 @ X @ A )
=> ( ~ ( member_nat3 @ X @ B )
=> ( ( ( insert_nat2 @ X @ A )
= ( insert_nat2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_777_insert__ident,axiom,
! [X: fm,A: set_fm,B: set_fm] :
( ~ ( member_fm3 @ X @ A )
=> ( ~ ( member_fm3 @ X @ B )
=> ( ( ( insert_fm2 @ X @ A )
= ( insert_fm2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_778_insert__ident,axiom,
! [X: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat3 @ X @ A )
=> ( ~ ( member_set_nat3 @ X @ B )
=> ( ( ( insert_set_nat2 @ X @ A )
= ( insert_set_nat2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_779_Set_Oset__insert,axiom,
! [X: tm,A: set_tm] :
( ( member_tm3 @ X @ A )
=> ~ ! [B7: set_tm] :
( ( A
= ( insert_tm2 @ X @ B7 ) )
=> ( member_tm3 @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_780_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat3 @ X @ A )
=> ~ ! [B7: set_nat] :
( ( A
= ( insert_nat2 @ X @ B7 ) )
=> ( member_nat3 @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_781_Set_Oset__insert,axiom,
! [X: fm,A: set_fm] :
( ( member_fm3 @ X @ A )
=> ~ ! [B7: set_fm] :
( ( A
= ( insert_fm2 @ X @ B7 ) )
=> ( member_fm3 @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_782_Set_Oset__insert,axiom,
! [X: set_nat,A: set_set_nat] :
( ( member_set_nat3 @ X @ A )
=> ~ ! [B7: set_set_nat] :
( ( A
= ( insert_set_nat2 @ X @ B7 ) )
=> ( member_set_nat3 @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_783_insertI2,axiom,
! [A2: tm,B: set_tm,B2: tm] :
( ( member_tm3 @ A2 @ B )
=> ( member_tm3 @ A2 @ ( insert_tm2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_784_insertI2,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( member_nat3 @ A2 @ B )
=> ( member_nat3 @ A2 @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_785_insertI2,axiom,
! [A2: fm,B: set_fm,B2: fm] :
( ( member_fm3 @ A2 @ B )
=> ( member_fm3 @ A2 @ ( insert_fm2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_786_insertI2,axiom,
! [A2: set_nat,B: set_set_nat,B2: set_nat] :
( ( member_set_nat3 @ A2 @ B )
=> ( member_set_nat3 @ A2 @ ( insert_set_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_787_insertI1,axiom,
! [A2: tm,B: set_tm] : ( member_tm3 @ A2 @ ( insert_tm2 @ A2 @ B ) ) ).
% insertI1
thf(fact_788_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat3 @ A2 @ ( insert_nat2 @ A2 @ B ) ) ).
% insertI1
thf(fact_789_insertI1,axiom,
! [A2: fm,B: set_fm] : ( member_fm3 @ A2 @ ( insert_fm2 @ A2 @ B ) ) ).
% insertI1
thf(fact_790_insertI1,axiom,
! [A2: set_nat,B: set_set_nat] : ( member_set_nat3 @ A2 @ ( insert_set_nat2 @ A2 @ B ) ) ).
% insertI1
thf(fact_791_insertE,axiom,
! [A2: tm,B2: tm,A: set_tm] :
( ( member_tm3 @ A2 @ ( insert_tm2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_tm3 @ A2 @ A ) ) ) ).
% insertE
thf(fact_792_insertE,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat3 @ A2 @ ( insert_nat2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_nat3 @ A2 @ A ) ) ) ).
% insertE
thf(fact_793_insertE,axiom,
! [A2: fm,B2: fm,A: set_fm] :
( ( member_fm3 @ A2 @ ( insert_fm2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_fm3 @ A2 @ A ) ) ) ).
% insertE
thf(fact_794_insertE,axiom,
! [A2: set_nat,B2: set_nat,A: set_set_nat] :
( ( member_set_nat3 @ A2 @ ( insert_set_nat2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_set_nat3 @ A2 @ A ) ) ) ).
% insertE
thf(fact_795_singletonD,axiom,
! [B2: set_nat,A2: set_nat] :
( ( member_set_nat3 @ B2 @ ( insert_set_nat2 @ A2 @ bot_bot_set_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_796_singletonD,axiom,
! [B2: tm,A2: tm] :
( ( member_tm3 @ B2 @ ( insert_tm2 @ A2 @ bot_bot_set_tm ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_797_singletonD,axiom,
! [B2: nat,A2: nat] :
( ( member_nat3 @ B2 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_798_singletonD,axiom,
! [B2: fm,A2: fm] :
( ( member_fm3 @ B2 @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_799_singleton__iff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( member_set_nat3 @ B2 @ ( insert_set_nat2 @ A2 @ bot_bot_set_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_800_singleton__iff,axiom,
! [B2: tm,A2: tm] :
( ( member_tm3 @ B2 @ ( insert_tm2 @ A2 @ bot_bot_set_tm ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_801_singleton__iff,axiom,
! [B2: nat,A2: nat] :
( ( member_nat3 @ B2 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_802_singleton__iff,axiom,
! [B2: fm,A2: fm] :
( ( member_fm3 @ B2 @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_803_doubleton__eq__iff,axiom,
! [A2: tm,B2: tm,C: tm,D2: tm] :
( ( ( insert_tm2 @ A2 @ ( insert_tm2 @ B2 @ bot_bot_set_tm ) )
= ( insert_tm2 @ C @ ( insert_tm2 @ D2 @ bot_bot_set_tm ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_804_doubleton__eq__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ( insert_nat2 @ A2 @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
= ( insert_nat2 @ C @ ( insert_nat2 @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_805_doubleton__eq__iff,axiom,
! [A2: fm,B2: fm,C: fm,D2: fm] :
( ( ( insert_fm2 @ A2 @ ( insert_fm2 @ B2 @ bot_bot_set_fm ) )
= ( insert_fm2 @ C @ ( insert_fm2 @ D2 @ bot_bot_set_fm ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_806_insert__not__empty,axiom,
! [A2: tm,A: set_tm] :
( ( insert_tm2 @ A2 @ A )
!= bot_bot_set_tm ) ).
% insert_not_empty
thf(fact_807_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat2 @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_808_insert__not__empty,axiom,
! [A2: fm,A: set_fm] :
( ( insert_fm2 @ A2 @ A )
!= bot_bot_set_fm ) ).
% insert_not_empty
thf(fact_809_singleton__inject,axiom,
! [A2: tm,B2: tm] :
( ( ( insert_tm2 @ A2 @ bot_bot_set_tm )
= ( insert_tm2 @ B2 @ bot_bot_set_tm ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_810_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_811_singleton__inject,axiom,
! [A2: fm,B2: fm] :
( ( ( insert_fm2 @ A2 @ bot_bot_set_fm )
= ( insert_fm2 @ B2 @ bot_bot_set_fm ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_812_insert__subsetI,axiom,
! [X: fm,A: set_fm,X6: set_fm] :
( ( member_fm3 @ X @ A )
=> ( ( ord_less_eq_set_fm @ X6 @ A )
=> ( ord_less_eq_set_fm @ ( insert_fm2 @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_813_insert__subsetI,axiom,
! [X: set_nat,A: set_set_nat,X6: set_set_nat] :
( ( member_set_nat3 @ X @ A )
=> ( ( ord_le6893508408891458716et_nat @ X6 @ A )
=> ( ord_le6893508408891458716et_nat @ ( insert_set_nat2 @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_814_insert__subsetI,axiom,
! [X: tm,A: set_tm,X6: set_tm] :
( ( member_tm3 @ X @ A )
=> ( ( ord_less_eq_set_tm @ X6 @ A )
=> ( ord_less_eq_set_tm @ ( insert_tm2 @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_815_insert__subsetI,axiom,
! [X: nat,A: set_nat,X6: set_nat] :
( ( member_nat3 @ X @ A )
=> ( ( ord_less_eq_set_nat @ X6 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_816_insert__mono,axiom,
! [C2: set_fm,D: set_fm,A2: fm] :
( ( ord_less_eq_set_fm @ C2 @ D )
=> ( ord_less_eq_set_fm @ ( insert_fm2 @ A2 @ C2 ) @ ( insert_fm2 @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_817_insert__mono,axiom,
! [C2: set_tm,D: set_tm,A2: tm] :
( ( ord_less_eq_set_tm @ C2 @ D )
=> ( ord_less_eq_set_tm @ ( insert_tm2 @ A2 @ C2 ) @ ( insert_tm2 @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_818_insert__mono,axiom,
! [C2: set_nat,D: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ C2 @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ A2 @ C2 ) @ ( insert_nat2 @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_819_subset__insert,axiom,
! [X: fm,A: set_fm,B: set_fm] :
( ~ ( member_fm3 @ X @ A )
=> ( ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X @ B ) )
= ( ord_less_eq_set_fm @ A @ B ) ) ) ).
% subset_insert
thf(fact_820_subset__insert,axiom,
! [X: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat3 @ X @ A )
=> ( ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat2 @ X @ B ) )
= ( ord_le6893508408891458716et_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_821_subset__insert,axiom,
! [X: tm,A: set_tm,B: set_tm] :
( ~ ( member_tm3 @ X @ A )
=> ( ( ord_less_eq_set_tm @ A @ ( insert_tm2 @ X @ B ) )
= ( ord_less_eq_set_tm @ A @ B ) ) ) ).
% subset_insert
thf(fact_822_subset__insert,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat3 @ X @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_823_subset__insertI,axiom,
! [B: set_fm,A2: fm] : ( ord_less_eq_set_fm @ B @ ( insert_fm2 @ A2 @ B ) ) ).
% subset_insertI
thf(fact_824_subset__insertI,axiom,
! [B: set_tm,A2: tm] : ( ord_less_eq_set_tm @ B @ ( insert_tm2 @ A2 @ B ) ) ).
% subset_insertI
thf(fact_825_subset__insertI,axiom,
! [B: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat2 @ A2 @ B ) ) ).
% subset_insertI
thf(fact_826_subset__insertI2,axiom,
! [A: set_fm,B: set_fm,B2: fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_827_subset__insertI2,axiom,
! [A: set_tm,B: set_tm,B2: tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ord_less_eq_set_tm @ A @ ( insert_tm2 @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_828_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_829_subset__singletonD,axiom,
! [A: set_fm,X: fm] :
( ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X @ bot_bot_set_fm ) )
=> ( ( A = bot_bot_set_fm )
| ( A
= ( insert_fm2 @ X @ bot_bot_set_fm ) ) ) ) ).
% subset_singletonD
thf(fact_830_subset__singletonD,axiom,
! [A: set_tm,X: tm] :
( ( ord_less_eq_set_tm @ A @ ( insert_tm2 @ X @ bot_bot_set_tm ) )
=> ( ( A = bot_bot_set_tm )
| ( A
= ( insert_tm2 @ X @ bot_bot_set_tm ) ) ) ) ).
% subset_singletonD
thf(fact_831_subset__singletonD,axiom,
! [A: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_832_subset__singleton__iff,axiom,
! [X6: set_fm,A2: fm] :
( ( ord_less_eq_set_fm @ X6 @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) )
= ( ( X6 = bot_bot_set_fm )
| ( X6
= ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) ) ) ).
% subset_singleton_iff
thf(fact_833_subset__singleton__iff,axiom,
! [X6: set_tm,A2: tm] :
( ( ord_less_eq_set_tm @ X6 @ ( insert_tm2 @ A2 @ bot_bot_set_tm ) )
= ( ( X6 = bot_bot_set_tm )
| ( X6
= ( insert_tm2 @ A2 @ bot_bot_set_tm ) ) ) ) ).
% subset_singleton_iff
thf(fact_834_subset__singleton__iff,axiom,
! [X6: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X6 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
= ( ( X6 = bot_bot_set_nat )
| ( X6
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_835_insert__is__Un,axiom,
( insert_tm2
= ( ^ [A4: tm] : ( sup_sup_set_tm @ ( insert_tm2 @ A4 @ bot_bot_set_tm ) ) ) ) ).
% insert_is_Un
thf(fact_836_insert__is__Un,axiom,
( insert_nat2
= ( ^ [A4: nat] : ( sup_sup_set_nat @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_837_insert__is__Un,axiom,
( insert_fm2
= ( ^ [A4: fm] : ( sup_sup_set_fm @ ( insert_fm2 @ A4 @ bot_bot_set_fm ) ) ) ) ).
% insert_is_Un
thf(fact_838_Un__singleton__iff,axiom,
! [A: set_tm,B: set_tm,X: tm] :
( ( ( sup_sup_set_tm @ A @ B )
= ( insert_tm2 @ X @ bot_bot_set_tm ) )
= ( ( ( A = bot_bot_set_tm )
& ( B
= ( insert_tm2 @ X @ bot_bot_set_tm ) ) )
| ( ( A
= ( insert_tm2 @ X @ bot_bot_set_tm ) )
& ( B = bot_bot_set_tm ) )
| ( ( A
= ( insert_tm2 @ X @ bot_bot_set_tm ) )
& ( B
= ( insert_tm2 @ X @ bot_bot_set_tm ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_839_Un__singleton__iff,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A @ B )
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_840_Un__singleton__iff,axiom,
! [A: set_fm,B: set_fm,X: fm] :
( ( ( sup_sup_set_fm @ A @ B )
= ( insert_fm2 @ X @ bot_bot_set_fm ) )
= ( ( ( A = bot_bot_set_fm )
& ( B
= ( insert_fm2 @ X @ bot_bot_set_fm ) ) )
| ( ( A
= ( insert_fm2 @ X @ bot_bot_set_fm ) )
& ( B = bot_bot_set_fm ) )
| ( ( A
= ( insert_fm2 @ X @ bot_bot_set_fm ) )
& ( B
= ( insert_fm2 @ X @ bot_bot_set_fm ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_841_singleton__Un__iff,axiom,
! [X: tm,A: set_tm,B: set_tm] :
( ( ( insert_tm2 @ X @ bot_bot_set_tm )
= ( sup_sup_set_tm @ A @ B ) )
= ( ( ( A = bot_bot_set_tm )
& ( B
= ( insert_tm2 @ X @ bot_bot_set_tm ) ) )
| ( ( A
= ( insert_tm2 @ X @ bot_bot_set_tm ) )
& ( B = bot_bot_set_tm ) )
| ( ( A
= ( insert_tm2 @ X @ bot_bot_set_tm ) )
& ( B
= ( insert_tm2 @ X @ bot_bot_set_tm ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_842_singleton__Un__iff,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ( ( insert_nat2 @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A @ B ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_843_singleton__Un__iff,axiom,
! [X: fm,A: set_fm,B: set_fm] :
( ( ( insert_fm2 @ X @ bot_bot_set_fm )
= ( sup_sup_set_fm @ A @ B ) )
= ( ( ( A = bot_bot_set_fm )
& ( B
= ( insert_fm2 @ X @ bot_bot_set_fm ) ) )
| ( ( A
= ( insert_fm2 @ X @ bot_bot_set_fm ) )
& ( B = bot_bot_set_fm ) )
| ( ( A
= ( insert_fm2 @ X @ bot_bot_set_fm ) )
& ( B
= ( insert_fm2 @ X @ bot_bot_set_fm ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_844_insert__code_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( insert_nat2 @ X @ ( coset_nat @ Xs ) )
= ( coset_nat @ ( removeAll_nat @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_845_insert__code_I2_J,axiom,
! [X: fm,Xs: list_fm] :
( ( insert_fm2 @ X @ ( coset_fm @ Xs ) )
= ( coset_fm @ ( removeAll_fm @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_846_sub_Osimps_I1_J,axiom,
! [V: nat,S2: tm,I: nat,L3: list_tm] :
( ( sub @ V @ S2 @ ( pre @ I @ L3 ) )
= ( pre @ I @ ( sub_list @ V @ S2 @ L3 ) ) ) ).
% sub.simps(1)
thf(fact_847_sub__const__transfer,axiom,
! [M: nat,A2: nat,P4: fm,T: tm] :
( ( ( sub @ M @ ( fun @ A2 @ nil_tm ) @ P4 )
!= ( sub @ M @ T @ P4 ) )
=> ( member_tm3 @ ( fun @ A2 @ nil_tm ) @ ( set_tm2 @ ( subtermFm @ ( sub @ M @ ( fun @ A2 @ nil_tm ) @ P4 ) ) ) ) ) ).
% sub_const_transfer
thf(fact_848_is__singleton__the__elem,axiom,
( is_singleton_tm
= ( ^ [A3: set_tm] :
( A3
= ( insert_tm2 @ ( the_elem_tm @ A3 ) @ bot_bot_set_tm ) ) ) ) ).
% is_singleton_the_elem
thf(fact_849_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_850_is__singleton__the__elem,axiom,
( is_singleton_fm
= ( ^ [A3: set_fm] :
( A3
= ( insert_fm2 @ ( the_elem_fm @ A3 ) @ bot_bot_set_fm ) ) ) ) ).
% is_singleton_the_elem
thf(fact_851_is__singletonI,axiom,
! [X: tm] : ( is_singleton_tm @ ( insert_tm2 @ X @ bot_bot_set_tm ) ) ).
% is_singletonI
thf(fact_852_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_853_is__singletonI,axiom,
! [X: fm] : ( is_singleton_fm @ ( insert_fm2 @ X @ bot_bot_set_fm ) ) ).
% is_singletonI
thf(fact_854_substts_Osimps_I2_J,axiom,
! [T: tm,Ts: list_tm,S2: tm,K: nat] :
( ( substts @ ( cons_tm @ T @ Ts ) @ S2 @ K )
= ( cons_tm @ ( substt @ T @ S2 @ K ) @ ( substts @ Ts @ S2 @ K ) ) ) ).
% substts.simps(2)
thf(fact_855_substt_Osimps_I2_J,axiom,
! [A2: nat,Ts: list_tm,S2: tm,K: nat] :
( ( substt @ ( fun @ A2 @ Ts ) @ S2 @ K )
= ( fun @ A2 @ ( substts @ Ts @ S2 @ K ) ) ) ).
% substt.simps(2)
thf(fact_856_set__removeAll,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( set_set_nat2 @ ( removeAll_set_nat @ X @ Xs ) )
= ( minus_2163939370556025621et_nat @ ( set_set_nat2 @ Xs ) @ ( insert_set_nat2 @ X @ bot_bot_set_set_nat ) ) ) ).
% set_removeAll
thf(fact_857_set__removeAll,axiom,
! [X: tm,Xs: list_tm] :
( ( set_tm2 @ ( removeAll_tm @ X @ Xs ) )
= ( minus_minus_set_tm @ ( set_tm2 @ Xs ) @ ( insert_tm2 @ X @ bot_bot_set_tm ) ) ) ).
% set_removeAll
thf(fact_858_set__removeAll,axiom,
! [X: nat,Xs: list_nat] :
( ( set_nat2 @ ( removeAll_nat @ X @ Xs ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ).
% set_removeAll
thf(fact_859_set__removeAll,axiom,
! [X: fm,Xs: list_fm] :
( ( set_fm2 @ ( removeAll_fm @ X @ Xs ) )
= ( minus_minus_set_fm @ ( set_fm2 @ Xs ) @ ( insert_fm2 @ X @ bot_bot_set_fm ) ) ) ).
% set_removeAll
thf(fact_860_s5_I2_J,axiom,
( sub_list
= ( ^ [V2: nat,S3: tm,L2: list_tm] : ( substts @ L2 @ S3 @ V2 ) ) ) ).
% s5(2)
thf(fact_861_DiffI,axiom,
! [C: tm,A: set_tm,B: set_tm] :
( ( member_tm3 @ C @ A )
=> ( ~ ( member_tm3 @ C @ B )
=> ( member_tm3 @ C @ ( minus_minus_set_tm @ A @ B ) ) ) ) ).
% DiffI
thf(fact_862_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat3 @ C @ A )
=> ( ~ ( member_nat3 @ C @ B )
=> ( member_nat3 @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_863_DiffI,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm3 @ C @ A )
=> ( ~ ( member_fm3 @ C @ B )
=> ( member_fm3 @ C @ ( minus_minus_set_fm @ A @ B ) ) ) ) ).
% DiffI
thf(fact_864_DiffI,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat3 @ C @ A )
=> ( ~ ( member_set_nat3 @ C @ B )
=> ( member_set_nat3 @ C @ ( minus_2163939370556025621et_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_865_Diff__iff,axiom,
! [C: tm,A: set_tm,B: set_tm] :
( ( member_tm3 @ C @ ( minus_minus_set_tm @ A @ B ) )
= ( ( member_tm3 @ C @ A )
& ~ ( member_tm3 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_866_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat3 @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat3 @ C @ A )
& ~ ( member_nat3 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_867_Diff__iff,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm3 @ C @ ( minus_minus_set_fm @ A @ B ) )
= ( ( member_fm3 @ C @ A )
& ~ ( member_fm3 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_868_Diff__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat3 @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
= ( ( member_set_nat3 @ C @ A )
& ~ ( member_set_nat3 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_869_Diff__cancel,axiom,
! [A: set_tm] :
( ( minus_minus_set_tm @ A @ A )
= bot_bot_set_tm ) ).
% Diff_cancel
thf(fact_870_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_871_Diff__cancel,axiom,
! [A: set_fm] :
( ( minus_minus_set_fm @ A @ A )
= bot_bot_set_fm ) ).
% Diff_cancel
thf(fact_872_empty__Diff,axiom,
! [A: set_tm] :
( ( minus_minus_set_tm @ bot_bot_set_tm @ A )
= bot_bot_set_tm ) ).
% empty_Diff
thf(fact_873_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_874_empty__Diff,axiom,
! [A: set_fm] :
( ( minus_minus_set_fm @ bot_bot_set_fm @ A )
= bot_bot_set_fm ) ).
% empty_Diff
thf(fact_875_Diff__empty,axiom,
! [A: set_tm] :
( ( minus_minus_set_tm @ A @ bot_bot_set_tm )
= A ) ).
% Diff_empty
thf(fact_876_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_877_Diff__empty,axiom,
! [A: set_fm] :
( ( minus_minus_set_fm @ A @ bot_bot_set_fm )
= A ) ).
% Diff_empty
thf(fact_878_Diff__insert0,axiom,
! [X: tm,A: set_tm,B: set_tm] :
( ~ ( member_tm3 @ X @ A )
=> ( ( minus_minus_set_tm @ A @ ( insert_tm2 @ X @ B ) )
= ( minus_minus_set_tm @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_879_Diff__insert0,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat3 @ X @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ B ) )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_880_Diff__insert0,axiom,
! [X: fm,A: set_fm,B: set_fm] :
( ~ ( member_fm3 @ X @ A )
=> ( ( minus_minus_set_fm @ A @ ( insert_fm2 @ X @ B ) )
= ( minus_minus_set_fm @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_881_Diff__insert0,axiom,
! [X: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat3 @ X @ A )
=> ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat2 @ X @ B ) )
= ( minus_2163939370556025621et_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_882_insert__Diff1,axiom,
! [X: tm,B: set_tm,A: set_tm] :
( ( member_tm3 @ X @ B )
=> ( ( minus_minus_set_tm @ ( insert_tm2 @ X @ A ) @ B )
= ( minus_minus_set_tm @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_883_insert__Diff1,axiom,
! [X: nat,B: set_nat,A: set_nat] :
( ( member_nat3 @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_884_insert__Diff1,axiom,
! [X: fm,B: set_fm,A: set_fm] :
( ( member_fm3 @ X @ B )
=> ( ( minus_minus_set_fm @ ( insert_fm2 @ X @ A ) @ B )
= ( minus_minus_set_fm @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_885_insert__Diff1,axiom,
! [X: set_nat,B: set_set_nat,A: set_set_nat] :
( ( member_set_nat3 @ X @ B )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat2 @ X @ A ) @ B )
= ( minus_2163939370556025621et_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_886_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_887_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_888_Diff__eq__empty__iff,axiom,
! [A: set_fm,B: set_fm] :
( ( ( minus_minus_set_fm @ A @ B )
= bot_bot_set_fm )
= ( ord_less_eq_set_fm @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_889_Diff__eq__empty__iff,axiom,
! [A: set_tm,B: set_tm] :
( ( ( minus_minus_set_tm @ A @ B )
= bot_bot_set_tm )
= ( ord_less_eq_set_tm @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_890_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_891_insert__Diff__single,axiom,
! [A2: tm,A: set_tm] :
( ( insert_tm2 @ A2 @ ( minus_minus_set_tm @ A @ ( insert_tm2 @ A2 @ bot_bot_set_tm ) ) )
= ( insert_tm2 @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_892_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_893_insert__Diff__single,axiom,
! [A2: fm,A: set_fm] :
( ( insert_fm2 @ A2 @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) )
= ( insert_fm2 @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_894_DiffE,axiom,
! [C: tm,A: set_tm,B: set_tm] :
( ( member_tm3 @ C @ ( minus_minus_set_tm @ A @ B ) )
=> ~ ( ( member_tm3 @ C @ A )
=> ( member_tm3 @ C @ B ) ) ) ).
% DiffE
thf(fact_895_DiffE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat3 @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat3 @ C @ A )
=> ( member_nat3 @ C @ B ) ) ) ).
% DiffE
thf(fact_896_DiffE,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm3 @ C @ ( minus_minus_set_fm @ A @ B ) )
=> ~ ( ( member_fm3 @ C @ A )
=> ( member_fm3 @ C @ B ) ) ) ).
% DiffE
thf(fact_897_DiffE,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat3 @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ~ ( ( member_set_nat3 @ C @ A )
=> ( member_set_nat3 @ C @ B ) ) ) ).
% DiffE
thf(fact_898_DiffD1,axiom,
! [C: tm,A: set_tm,B: set_tm] :
( ( member_tm3 @ C @ ( minus_minus_set_tm @ A @ B ) )
=> ( member_tm3 @ C @ A ) ) ).
% DiffD1
thf(fact_899_DiffD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat3 @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat3 @ C @ A ) ) ).
% DiffD1
thf(fact_900_DiffD1,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm3 @ C @ ( minus_minus_set_fm @ A @ B ) )
=> ( member_fm3 @ C @ A ) ) ).
% DiffD1
thf(fact_901_DiffD1,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat3 @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ( member_set_nat3 @ C @ A ) ) ).
% DiffD1
thf(fact_902_DiffD2,axiom,
! [C: tm,A: set_tm,B: set_tm] :
( ( member_tm3 @ C @ ( minus_minus_set_tm @ A @ B ) )
=> ~ ( member_tm3 @ C @ B ) ) ).
% DiffD2
thf(fact_903_DiffD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat3 @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat3 @ C @ B ) ) ).
% DiffD2
thf(fact_904_DiffD2,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm3 @ C @ ( minus_minus_set_fm @ A @ B ) )
=> ~ ( member_fm3 @ C @ B ) ) ).
% DiffD2
thf(fact_905_DiffD2,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat3 @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ~ ( member_set_nat3 @ C @ B ) ) ).
% DiffD2
thf(fact_906_double__diff,axiom,
! [A: set_tm,B: set_tm,C2: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_eq_set_tm @ B @ C2 )
=> ( ( minus_minus_set_tm @ B @ ( minus_minus_set_tm @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_907_double__diff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_908_Diff__subset,axiom,
! [A: set_tm,B: set_tm] : ( ord_less_eq_set_tm @ ( minus_minus_set_tm @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_909_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_910_Diff__mono,axiom,
! [A: set_tm,C2: set_tm,D: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A @ C2 )
=> ( ( ord_less_eq_set_tm @ D @ B )
=> ( ord_less_eq_set_tm @ ( minus_minus_set_tm @ A @ B ) @ ( minus_minus_set_tm @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_911_Diff__mono,axiom,
! [A: set_nat,C2: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_912_insert__Diff__if,axiom,
! [X: tm,B: set_tm,A: set_tm] :
( ( ( member_tm3 @ X @ B )
=> ( ( minus_minus_set_tm @ ( insert_tm2 @ X @ A ) @ B )
= ( minus_minus_set_tm @ A @ B ) ) )
& ( ~ ( member_tm3 @ X @ B )
=> ( ( minus_minus_set_tm @ ( insert_tm2 @ X @ A ) @ B )
= ( insert_tm2 @ X @ ( minus_minus_set_tm @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_913_insert__Diff__if,axiom,
! [X: nat,B: set_nat,A: set_nat] :
( ( ( member_nat3 @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) )
& ( ~ ( member_nat3 @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ B )
= ( insert_nat2 @ X @ ( minus_minus_set_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_914_insert__Diff__if,axiom,
! [X: fm,B: set_fm,A: set_fm] :
( ( ( member_fm3 @ X @ B )
=> ( ( minus_minus_set_fm @ ( insert_fm2 @ X @ A ) @ B )
= ( minus_minus_set_fm @ A @ B ) ) )
& ( ~ ( member_fm3 @ X @ B )
=> ( ( minus_minus_set_fm @ ( insert_fm2 @ X @ A ) @ B )
= ( insert_fm2 @ X @ ( minus_minus_set_fm @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_915_insert__Diff__if,axiom,
! [X: set_nat,B: set_set_nat,A: set_set_nat] :
( ( ( member_set_nat3 @ X @ B )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat2 @ X @ A ) @ B )
= ( minus_2163939370556025621et_nat @ A @ B ) ) )
& ( ~ ( member_set_nat3 @ X @ B )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat2 @ X @ A ) @ B )
= ( insert_set_nat2 @ X @ ( minus_2163939370556025621et_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_916_Un__Diff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ C2 ) @ ( minus_minus_set_nat @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_917_diff__shunt__var,axiom,
! [X: set_fm,Y2: set_fm] :
( ( ( minus_minus_set_fm @ X @ Y2 )
= bot_bot_set_fm )
= ( ord_less_eq_set_fm @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_918_diff__shunt__var,axiom,
! [X: set_tm,Y2: set_tm] :
( ( ( minus_minus_set_tm @ X @ Y2 )
= bot_bot_set_tm )
= ( ord_less_eq_set_tm @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_919_diff__shunt__var,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_920_Diff__insert,axiom,
! [A: set_tm,A2: tm,B: set_tm] :
( ( minus_minus_set_tm @ A @ ( insert_tm2 @ A2 @ B ) )
= ( minus_minus_set_tm @ ( minus_minus_set_tm @ A @ B ) @ ( insert_tm2 @ A2 @ bot_bot_set_tm ) ) ) ).
% Diff_insert
thf(fact_921_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_922_Diff__insert,axiom,
! [A: set_fm,A2: fm,B: set_fm] :
( ( minus_minus_set_fm @ A @ ( insert_fm2 @ A2 @ B ) )
= ( minus_minus_set_fm @ ( minus_minus_set_fm @ A @ B ) @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) ) ).
% Diff_insert
thf(fact_923_insert__Diff,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat3 @ A2 @ A )
=> ( ( insert_set_nat2 @ A2 @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat2 @ A2 @ bot_bot_set_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_924_insert__Diff,axiom,
! [A2: tm,A: set_tm] :
( ( member_tm3 @ A2 @ A )
=> ( ( insert_tm2 @ A2 @ ( minus_minus_set_tm @ A @ ( insert_tm2 @ A2 @ bot_bot_set_tm ) ) )
= A ) ) ).
% insert_Diff
thf(fact_925_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat3 @ A2 @ A )
=> ( ( insert_nat2 @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_926_insert__Diff,axiom,
! [A2: fm,A: set_fm] :
( ( member_fm3 @ A2 @ A )
=> ( ( insert_fm2 @ A2 @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) )
= A ) ) ).
% insert_Diff
thf(fact_927_Diff__insert2,axiom,
! [A: set_tm,A2: tm,B: set_tm] :
( ( minus_minus_set_tm @ A @ ( insert_tm2 @ A2 @ B ) )
= ( minus_minus_set_tm @ ( minus_minus_set_tm @ A @ ( insert_tm2 @ A2 @ bot_bot_set_tm ) ) @ B ) ) ).
% Diff_insert2
thf(fact_928_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_929_Diff__insert2,axiom,
! [A: set_fm,A2: fm,B: set_fm] :
( ( minus_minus_set_fm @ A @ ( insert_fm2 @ A2 @ B ) )
= ( minus_minus_set_fm @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) @ B ) ) ).
% Diff_insert2
thf(fact_930_Diff__insert__absorb,axiom,
! [X: set_nat,A: set_set_nat] :
( ~ ( member_set_nat3 @ X @ A )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat2 @ X @ A ) @ ( insert_set_nat2 @ X @ bot_bot_set_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_931_Diff__insert__absorb,axiom,
! [X: tm,A: set_tm] :
( ~ ( member_tm3 @ X @ A )
=> ( ( minus_minus_set_tm @ ( insert_tm2 @ X @ A ) @ ( insert_tm2 @ X @ bot_bot_set_tm ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_932_Diff__insert__absorb,axiom,
! [X: nat,A: set_nat] :
( ~ ( member_nat3 @ X @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_933_Diff__insert__absorb,axiom,
! [X: fm,A: set_fm] :
( ~ ( member_fm3 @ X @ A )
=> ( ( minus_minus_set_fm @ ( insert_fm2 @ X @ A ) @ ( insert_fm2 @ X @ bot_bot_set_fm ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_934_subset__Diff__insert,axiom,
! [A: set_fm,B: set_fm,X: fm,C2: set_fm] :
( ( ord_less_eq_set_fm @ A @ ( minus_minus_set_fm @ B @ ( insert_fm2 @ X @ C2 ) ) )
= ( ( ord_less_eq_set_fm @ A @ ( minus_minus_set_fm @ B @ C2 ) )
& ~ ( member_fm3 @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_935_subset__Diff__insert,axiom,
! [A: set_set_nat,B: set_set_nat,X: set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( minus_2163939370556025621et_nat @ B @ ( insert_set_nat2 @ X @ C2 ) ) )
= ( ( ord_le6893508408891458716et_nat @ A @ ( minus_2163939370556025621et_nat @ B @ C2 ) )
& ~ ( member_set_nat3 @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_936_subset__Diff__insert,axiom,
! [A: set_tm,B: set_tm,X: tm,C2: set_tm] :
( ( ord_less_eq_set_tm @ A @ ( minus_minus_set_tm @ B @ ( insert_tm2 @ X @ C2 ) ) )
= ( ( ord_less_eq_set_tm @ A @ ( minus_minus_set_tm @ B @ C2 ) )
& ~ ( member_tm3 @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_937_subset__Diff__insert,axiom,
! [A: set_nat,B: set_nat,X: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat2 @ X @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) )
& ~ ( member_nat3 @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_938_Diff__partition,axiom,
! [A: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( sup_sup_set_tm @ A @ ( minus_minus_set_tm @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_939_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_940_Diff__subset__conv,axiom,
! [A: set_tm,B: set_tm,C2: set_tm] :
( ( ord_less_eq_set_tm @ ( minus_minus_set_tm @ A @ B ) @ C2 )
= ( ord_less_eq_set_tm @ A @ ( sup_sup_set_tm @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_941_Diff__subset__conv,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
= ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_942_preds_Osimps_I1_J,axiom,
! [N: nat,Ts: list_tm] :
( ( preds @ ( pre @ N @ Ts ) )
= ( insert_fm2 @ ( pre @ N @ Ts ) @ bot_bot_set_fm ) ) ).
% preds.simps(1)
thf(fact_943_is__singletonI_H,axiom,
! [A: set_set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( member_set_nat3 @ X3 @ A )
=> ( ( member_set_nat3 @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_singleton_set_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_944_is__singletonI_H,axiom,
! [A: set_tm] :
( ( A != bot_bot_set_tm )
=> ( ! [X3: tm,Y3: tm] :
( ( member_tm3 @ X3 @ A )
=> ( ( member_tm3 @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_singleton_tm @ A ) ) ) ).
% is_singletonI'
thf(fact_945_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y3: nat] :
( ( member_nat3 @ X3 @ A )
=> ( ( member_nat3 @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_946_is__singletonI_H,axiom,
! [A: set_fm] :
( ( A != bot_bot_set_fm )
=> ( ! [X3: fm,Y3: fm] :
( ( member_fm3 @ X3 @ A )
=> ( ( member_fm3 @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_singleton_fm @ A ) ) ) ).
% is_singletonI'
thf(fact_947_substts_Osimps_I1_J,axiom,
! [S2: tm,K: nat] :
( ( substts @ nil_tm @ S2 @ K )
= nil_tm ) ).
% substts.simps(1)
thf(fact_948_Diff__single__insert,axiom,
! [A: set_fm,X: fm,B: set_fm] :
( ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ X @ bot_bot_set_fm ) ) @ B )
=> ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_949_Diff__single__insert,axiom,
! [A: set_tm,X: tm,B: set_tm] :
( ( ord_less_eq_set_tm @ ( minus_minus_set_tm @ A @ ( insert_tm2 @ X @ bot_bot_set_tm ) ) @ B )
=> ( ord_less_eq_set_tm @ A @ ( insert_tm2 @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_950_Diff__single__insert,axiom,
! [A: set_nat,X: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_951_subset__insert__iff,axiom,
! [A: set_set_nat,X: set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat2 @ X @ B ) )
= ( ( ( member_set_nat3 @ X @ A )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat2 @ X @ bot_bot_set_set_nat ) ) @ B ) )
& ( ~ ( member_set_nat3 @ X @ A )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_952_subset__insert__iff,axiom,
! [A: set_fm,X: fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X @ B ) )
= ( ( ( member_fm3 @ X @ A )
=> ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ X @ bot_bot_set_fm ) ) @ B ) )
& ( ~ ( member_fm3 @ X @ A )
=> ( ord_less_eq_set_fm @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_953_subset__insert__iff,axiom,
! [A: set_tm,X: tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A @ ( insert_tm2 @ X @ B ) )
= ( ( ( member_tm3 @ X @ A )
=> ( ord_less_eq_set_tm @ ( minus_minus_set_tm @ A @ ( insert_tm2 @ X @ bot_bot_set_tm ) ) @ B ) )
& ( ~ ( member_tm3 @ X @ A )
=> ( ord_less_eq_set_tm @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_954_subset__insert__iff,axiom,
! [A: set_nat,X: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ B ) )
= ( ( ( member_nat3 @ X @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat3 @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_955_remove__def,axiom,
( remove_tm
= ( ^ [X2: tm,A3: set_tm] : ( minus_minus_set_tm @ A3 @ ( insert_tm2 @ X2 @ bot_bot_set_tm ) ) ) ) ).
% remove_def
thf(fact_956_remove__def,axiom,
( remove_nat
= ( ^ [X2: nat,A3: set_nat] : ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_957_remove__def,axiom,
( remove_fm
= ( ^ [X2: fm,A3: set_fm] : ( minus_minus_set_fm @ A3 @ ( insert_fm2 @ X2 @ bot_bot_set_fm ) ) ) ) ).
% remove_def
thf(fact_958_is__singleton__def,axiom,
( is_singleton_tm
= ( ^ [A3: set_tm] :
? [X2: tm] :
( A3
= ( insert_tm2 @ X2 @ bot_bot_set_tm ) ) ) ) ).
% is_singleton_def
thf(fact_959_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X2: nat] :
( A3
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_960_is__singleton__def,axiom,
( is_singleton_fm
= ( ^ [A3: set_fm] :
? [X2: fm] :
( A3
= ( insert_fm2 @ X2 @ bot_bot_set_fm ) ) ) ) ).
% is_singleton_def
thf(fact_961_is__singletonE,axiom,
! [A: set_tm] :
( ( is_singleton_tm @ A )
=> ~ ! [X3: tm] :
( A
!= ( insert_tm2 @ X3 @ bot_bot_set_tm ) ) ) ).
% is_singletonE
thf(fact_962_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X3: nat] :
( A
!= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_963_is__singletonE,axiom,
! [A: set_fm] :
( ( is_singleton_fm @ A )
=> ~ ! [X3: fm] :
( A
!= ( insert_fm2 @ X3 @ bot_bot_set_fm ) ) ) ).
% is_singletonE
thf(fact_964_subseqs_Osimps_I1_J,axiom,
( ( subseqs_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% subseqs.simps(1)
thf(fact_965_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_966_set__replicate,axiom,
! [N: nat,X: set_nat] :
( ( N != zero_zero_nat )
=> ( ( set_set_nat2 @ ( replicate_set_nat @ N @ X ) )
= ( insert_set_nat2 @ X @ bot_bot_set_set_nat ) ) ) ).
% set_replicate
thf(fact_967_set__replicate,axiom,
! [N: nat,X: tm] :
( ( N != zero_zero_nat )
=> ( ( set_tm2 @ ( replicate_tm @ N @ X ) )
= ( insert_tm2 @ X @ bot_bot_set_tm ) ) ) ).
% set_replicate
thf(fact_968_set__replicate,axiom,
! [N: nat,X: nat] :
( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ).
% set_replicate
thf(fact_969_set__replicate,axiom,
! [N: nat,X: fm] :
( ( N != zero_zero_nat )
=> ( ( set_fm2 @ ( replicate_fm @ N @ X ) )
= ( insert_fm2 @ X @ bot_bot_set_fm ) ) ) ).
% set_replicate
thf(fact_970_nths__singleton,axiom,
! [A: set_nat,X: tm] :
( ( ( member_nat3 @ zero_zero_nat @ A )
=> ( ( nths_tm @ ( cons_tm @ X @ nil_tm ) @ A )
= ( cons_tm @ X @ nil_tm ) ) )
& ( ~ ( member_nat3 @ zero_zero_nat @ A )
=> ( ( nths_tm @ ( cons_tm @ X @ nil_tm ) @ A )
= nil_tm ) ) ) ).
% nths_singleton
thf(fact_971_nths__singleton,axiom,
! [A: set_nat,X: nat] :
( ( ( member_nat3 @ zero_zero_nat @ A )
=> ( ( nths_nat @ ( cons_nat @ X @ nil_nat ) @ A )
= ( cons_nat @ X @ nil_nat ) ) )
& ( ~ ( member_nat3 @ zero_zero_nat @ A )
=> ( ( nths_nat @ ( cons_nat @ X @ nil_nat ) @ A )
= nil_nat ) ) ) ).
% nths_singleton
thf(fact_972_subtermTm_Osimps_I2_J,axiom,
! [N: nat] :
( ( subtermTm @ ( var @ N ) )
= ( cons_tm @ ( var @ N ) @ nil_tm ) ) ).
% subtermTm.simps(2)
thf(fact_973_listFunTms_Osimps_I2_J,axiom,
! [T: tm,Ts: list_tm] :
( ( listFunTms @ ( cons_tm @ T @ Ts ) )
= ( append_nat @ ( listFunTm @ T ) @ ( listFunTms @ Ts ) ) ) ).
% listFunTms.simps(2)
thf(fact_974_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_975_same__append__eq,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_976_append__same__eq,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys @ Xs )
= ( append_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_977_append__assoc,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
= ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_978_append_Oassoc,axiom,
! [A2: list_nat,B2: list_nat,C: list_nat] :
( ( append_nat @ ( append_nat @ A2 @ B2 ) @ C )
= ( append_nat @ A2 @ ( append_nat @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_979_tm_Oinject_I2_J,axiom,
! [X23: nat,Y23: nat] :
( ( ( var @ X23 )
= ( var @ Y23 ) )
= ( X23 = Y23 ) ) ).
% tm.inject(2)
thf(fact_980_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_981_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_982_append__is__Nil__conv,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= nil_tm )
= ( ( Xs = nil_tm )
& ( Ys = nil_tm ) ) ) ).
% append_is_Nil_conv
thf(fact_983_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_984_Nil__is__append__conv,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( nil_tm
= ( append_tm @ Xs @ Ys ) )
= ( ( Xs = nil_tm )
& ( Ys = nil_tm ) ) ) ).
% Nil_is_append_conv
thf(fact_985_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_986_self__append__conv2,axiom,
! [Y2: list_tm,Xs: list_tm] :
( ( Y2
= ( append_tm @ Xs @ Y2 ) )
= ( Xs = nil_tm ) ) ).
% self_append_conv2
thf(fact_987_self__append__conv2,axiom,
! [Y2: list_nat,Xs: list_nat] :
( ( Y2
= ( append_nat @ Xs @ Y2 ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_988_append__self__conv2,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= Ys )
= ( Xs = nil_tm ) ) ).
% append_self_conv2
thf(fact_989_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_990_self__append__conv,axiom,
! [Y2: list_tm,Ys: list_tm] :
( ( Y2
= ( append_tm @ Y2 @ Ys ) )
= ( Ys = nil_tm ) ) ).
% self_append_conv
thf(fact_991_self__append__conv,axiom,
! [Y2: list_nat,Ys: list_nat] :
( ( Y2
= ( append_nat @ Y2 @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_992_append__self__conv,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= Xs )
= ( Ys = nil_tm ) ) ).
% append_self_conv
thf(fact_993_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_994_append__Nil2,axiom,
! [Xs: list_tm] :
( ( append_tm @ Xs @ nil_tm )
= Xs ) ).
% append_Nil2
thf(fact_995_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_996_append_Oright__neutral,axiom,
! [A2: list_tm] :
( ( append_tm @ A2 @ nil_tm )
= A2 ) ).
% append.right_neutral
thf(fact_997_append_Oright__neutral,axiom,
! [A2: list_nat] :
( ( append_nat @ A2 @ nil_nat )
= A2 ) ).
% append.right_neutral
thf(fact_998_nths__nil,axiom,
! [A: set_nat] :
( ( nths_tm @ nil_tm @ A )
= nil_tm ) ).
% nths_nil
thf(fact_999_nths__nil,axiom,
! [A: set_nat] :
( ( nths_nat @ nil_nat @ A )
= nil_nat ) ).
% nths_nil
thf(fact_1000_removeAll__append,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( removeAll_nat @ X @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( removeAll_nat @ X @ Xs ) @ ( removeAll_nat @ X @ Ys ) ) ) ).
% removeAll_append
thf(fact_1001_append1__eq__conv,axiom,
! [Xs: list_tm,X: tm,Ys: list_tm,Y2: tm] :
( ( ( append_tm @ Xs @ ( cons_tm @ X @ nil_tm ) )
= ( append_tm @ Ys @ ( cons_tm @ Y2 @ nil_tm ) ) )
= ( ( Xs = Ys )
& ( X = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_1002_append1__eq__conv,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y2: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_1003_set__append,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( set_tm2 @ ( append_tm @ Xs @ Ys ) )
= ( sup_sup_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ Ys ) ) ) ).
% set_append
thf(fact_1004_set__append,axiom,
! [Xs: list_set_nat,Ys: list_set_nat] :
( ( set_set_nat2 @ ( append_set_nat @ Xs @ Ys ) )
= ( sup_sup_set_set_nat @ ( set_set_nat2 @ Xs ) @ ( set_set_nat2 @ Ys ) ) ) ).
% set_append
thf(fact_1005_set__append,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( set_fm2 @ ( append_fm @ Xs @ Ys ) )
= ( sup_sup_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys ) ) ) ).
% set_append
thf(fact_1006_set__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( append_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_append
thf(fact_1007_replicate__empty,axiom,
! [N: nat,X: tm] :
( ( ( replicate_tm @ N @ X )
= nil_tm )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_1008_replicate__empty,axiom,
! [N: nat,X: nat] :
( ( ( replicate_nat @ N @ X )
= nil_nat )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_1009_empty__replicate,axiom,
! [N: nat,X: tm] :
( ( nil_tm
= ( replicate_tm @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_1010_empty__replicate,axiom,
! [N: nat,X: nat] :
( ( nil_nat
= ( replicate_nat @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_1011_in__set__replicate,axiom,
! [X: tm,N: nat,Y2: tm] :
( ( member_tm3 @ X @ ( set_tm2 @ ( replicate_tm @ N @ Y2 ) ) )
= ( ( X = Y2 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1012_in__set__replicate,axiom,
! [X: nat,N: nat,Y2: nat] :
( ( member_nat3 @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y2 ) ) )
= ( ( X = Y2 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1013_in__set__replicate,axiom,
! [X: set_nat,N: nat,Y2: set_nat] :
( ( member_set_nat3 @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N @ Y2 ) ) )
= ( ( X = Y2 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1014_in__set__replicate,axiom,
! [X: fm,N: nat,Y2: fm] :
( ( member_fm3 @ X @ ( set_fm2 @ ( replicate_fm @ N @ Y2 ) ) )
= ( ( X = Y2 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1015_Bex__set__replicate,axiom,
! [N: nat,A2: tm,P3: tm > $o] :
( ( ? [X2: tm] :
( ( member_tm3 @ X2 @ ( set_tm2 @ ( replicate_tm @ N @ A2 ) ) )
& ( P3 @ X2 ) ) )
= ( ( P3 @ A2 )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_1016_Bex__set__replicate,axiom,
! [N: nat,A2: nat,P3: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat3 @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ A2 ) ) )
& ( P3 @ X2 ) ) )
= ( ( P3 @ A2 )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_1017_Bex__set__replicate,axiom,
! [N: nat,A2: set_nat,P3: set_nat > $o] :
( ( ? [X2: set_nat] :
( ( member_set_nat3 @ X2 @ ( set_set_nat2 @ ( replicate_set_nat @ N @ A2 ) ) )
& ( P3 @ X2 ) ) )
= ( ( P3 @ A2 )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_1018_Bex__set__replicate,axiom,
! [N: nat,A2: fm,P3: fm > $o] :
( ( ? [X2: fm] :
( ( member_fm3 @ X2 @ ( set_fm2 @ ( replicate_fm @ N @ A2 ) ) )
& ( P3 @ X2 ) ) )
= ( ( P3 @ A2 )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_1019_Ball__set__replicate,axiom,
! [N: nat,A2: tm,P3: tm > $o] :
( ( ! [X2: tm] :
( ( member_tm3 @ X2 @ ( set_tm2 @ ( replicate_tm @ N @ A2 ) ) )
=> ( P3 @ X2 ) ) )
= ( ( P3 @ A2 )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_1020_Ball__set__replicate,axiom,
! [N: nat,A2: nat,P3: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat3 @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ A2 ) ) )
=> ( P3 @ X2 ) ) )
= ( ( P3 @ A2 )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_1021_Ball__set__replicate,axiom,
! [N: nat,A2: set_nat,P3: set_nat > $o] :
( ( ! [X2: set_nat] :
( ( member_set_nat3 @ X2 @ ( set_set_nat2 @ ( replicate_set_nat @ N @ A2 ) ) )
=> ( P3 @ X2 ) ) )
= ( ( P3 @ A2 )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_1022_Ball__set__replicate,axiom,
! [N: nat,A2: fm,P3: fm > $o] :
( ( ! [X2: fm] :
( ( member_fm3 @ X2 @ ( set_fm2 @ ( replicate_fm @ N @ A2 ) ) )
=> ( P3 @ X2 ) ) )
= ( ( P3 @ A2 )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_1023_nths__empty,axiom,
! [Xs: list_tm] :
( ( nths_tm @ Xs @ bot_bot_set_nat )
= nil_tm ) ).
% nths_empty
thf(fact_1024_nths__empty,axiom,
! [Xs: list_nat] :
( ( nths_nat @ Xs @ bot_bot_set_nat )
= nil_nat ) ).
% nths_empty
thf(fact_1025_bind__simps_I2_J,axiom,
! [X: tm,Xs: list_tm,F: tm > list_nat] :
( ( bind_tm_nat @ ( cons_tm @ X @ Xs ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_tm_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_1026_bind__simps_I2_J,axiom,
! [X: nat,Xs: list_nat,F: nat > list_nat] :
( ( bind_nat_nat @ ( cons_nat @ X @ Xs ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_nat_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_1027_replicate__append__same,axiom,
! [I: nat,X: tm] :
( ( append_tm @ ( replicate_tm @ I @ X ) @ ( cons_tm @ X @ nil_tm ) )
= ( cons_tm @ X @ ( replicate_tm @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_1028_replicate__append__same,axiom,
! [I: nat,X: nat] :
( ( append_nat @ ( replicate_nat @ I @ X ) @ ( cons_nat @ X @ nil_nat ) )
= ( cons_nat @ X @ ( replicate_nat @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_1029_append__replicate__commute,axiom,
! [N: nat,X: nat,K: nat] :
( ( append_nat @ ( replicate_nat @ N @ X ) @ ( replicate_nat @ K @ X ) )
= ( append_nat @ ( replicate_nat @ K @ X ) @ ( replicate_nat @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_1030_append__eq__append__conv2,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us ) )
& ( ( append_nat @ Us @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us )
= Zs )
& ( Ys
= ( append_nat @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_1031_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us2: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us2 ) )
=> ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_1032_replicate__app__Cons__same,axiom,
! [N: nat,X: tm,Xs: list_tm] :
( ( append_tm @ ( replicate_tm @ N @ X ) @ ( cons_tm @ X @ Xs ) )
= ( cons_tm @ X @ ( append_tm @ ( replicate_tm @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_1033_replicate__app__Cons__same,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( append_nat @ ( replicate_nat @ N @ X ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( append_nat @ ( replicate_nat @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_1034_Cons__eq__appendI,axiom,
! [X: tm,Xs1: list_tm,Ys: list_tm,Xs: list_tm,Zs: list_tm] :
( ( ( cons_tm @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_tm @ Xs1 @ Zs ) )
=> ( ( cons_tm @ X @ Xs )
= ( append_tm @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_1035_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_1036_append__Cons,axiom,
! [X: tm,Xs: list_tm,Ys: list_tm] :
( ( append_tm @ ( cons_tm @ X @ Xs ) @ Ys )
= ( cons_tm @ X @ ( append_tm @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_1037_append__Cons,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys )
= ( cons_nat @ X @ ( append_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_1038_eq__Nil__appendI,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( Xs = Ys )
=> ( Xs
= ( append_tm @ nil_tm @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_1039_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_1040_append_Oleft__neutral,axiom,
! [A2: list_tm] :
( ( append_tm @ nil_tm @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_1041_append_Oleft__neutral,axiom,
! [A2: list_nat] :
( ( append_nat @ nil_nat @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_1042_append__Nil,axiom,
! [Ys: list_tm] :
( ( append_tm @ nil_tm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_1043_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_1044_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1045_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1046_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1047_diff__le__mono,axiom,
! [M: nat,N: nat,L3: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L3 ) @ ( minus_minus_nat @ N @ L3 ) ) ) ).
% diff_le_mono
thf(fact_1048_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1049_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1050_diff__le__mono2,axiom,
! [M: nat,N: nat,L3: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L3 @ N ) @ ( minus_minus_nat @ L3 @ M ) ) ) ).
% diff_le_mono2
thf(fact_1051_notin__set__nthsI,axiom,
! [X: tm,Xs: list_tm,I2: set_nat] :
( ~ ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ~ ( member_tm3 @ X @ ( set_tm2 @ ( nths_tm @ Xs @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_1052_notin__set__nthsI,axiom,
! [X: nat,Xs: list_nat,I2: set_nat] :
( ~ ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ~ ( member_nat3 @ X @ ( set_nat2 @ ( nths_nat @ Xs @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_1053_notin__set__nthsI,axiom,
! [X: set_nat,Xs: list_set_nat,I2: set_nat] :
( ~ ( member_set_nat3 @ X @ ( set_set_nat2 @ Xs ) )
=> ~ ( member_set_nat3 @ X @ ( set_set_nat2 @ ( nths_set_nat @ Xs @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_1054_notin__set__nthsI,axiom,
! [X: fm,Xs: list_fm,I2: set_nat] :
( ~ ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ~ ( member_fm3 @ X @ ( set_fm2 @ ( nths_fm @ Xs @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_1055_in__set__nthsD,axiom,
! [X: tm,Xs: list_tm,I2: set_nat] :
( ( member_tm3 @ X @ ( set_tm2 @ ( nths_tm @ Xs @ I2 ) ) )
=> ( member_tm3 @ X @ ( set_tm2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_1056_in__set__nthsD,axiom,
! [X: nat,Xs: list_nat,I2: set_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ ( nths_nat @ Xs @ I2 ) ) )
=> ( member_nat3 @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_1057_in__set__nthsD,axiom,
! [X: set_nat,Xs: list_set_nat,I2: set_nat] :
( ( member_set_nat3 @ X @ ( set_set_nat2 @ ( nths_set_nat @ Xs @ I2 ) ) )
=> ( member_set_nat3 @ X @ ( set_set_nat2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_1058_in__set__nthsD,axiom,
! [X: fm,Xs: list_fm,I2: set_nat] :
( ( member_fm3 @ X @ ( set_fm2 @ ( nths_fm @ Xs @ I2 ) ) )
=> ( member_fm3 @ X @ ( set_fm2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_1059_tm_Odistinct_I1_J,axiom,
! [X11: nat,X12: list_tm,X23: nat] :
( ( fun @ X11 @ X12 )
!= ( var @ X23 ) ) ).
% tm.distinct(1)
thf(fact_1060_tm_Oexhaust,axiom,
! [Y2: tm] :
( ! [X112: nat,X122: list_tm] :
( Y2
!= ( fun @ X112 @ X122 ) )
=> ~ ! [X24: nat] :
( Y2
!= ( var @ X24 ) ) ) ).
% tm.exhaust
thf(fact_1061_paramst_H_H_Ocases,axiom,
! [X: tm] :
( ! [N2: nat] :
( X
!= ( var @ N2 ) )
=> ~ ! [A5: nat,Ts2: list_tm] :
( X
!= ( fun @ A5 @ Ts2 ) ) ) ).
% paramst''.cases
thf(fact_1062_Cons__in__subseqsD,axiom,
! [Y2: tm,Ys: list_tm,Xs: list_tm] :
( ( member_list_tm @ ( cons_tm @ Y2 @ Ys ) @ ( set_list_tm2 @ ( subseqs_tm @ Xs ) ) )
=> ( member_list_tm @ Ys @ ( set_list_tm2 @ ( subseqs_tm @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_1063_Cons__in__subseqsD,axiom,
! [Y2: nat,Ys: list_nat,Xs: list_nat] :
( ( member_list_nat @ ( cons_nat @ Y2 @ Ys ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
=> ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_1064_rev__nonempty__induct,axiom,
! [Xs: list_tm,P3: list_tm > $o] :
( ( Xs != nil_tm )
=> ( ! [X3: tm] : ( P3 @ ( cons_tm @ X3 @ nil_tm ) )
=> ( ! [X3: tm,Xs3: list_tm] :
( ( Xs3 != nil_tm )
=> ( ( P3 @ Xs3 )
=> ( P3 @ ( append_tm @ Xs3 @ ( cons_tm @ X3 @ nil_tm ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_1065_rev__nonempty__induct,axiom,
! [Xs: list_nat,P3: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P3 @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P3 @ Xs3 )
=> ( P3 @ ( append_nat @ Xs3 @ ( cons_nat @ X3 @ nil_nat ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_1066_append__eq__Cons__conv,axiom,
! [Ys: list_tm,Zs: list_tm,X: tm,Xs: list_tm] :
( ( ( append_tm @ Ys @ Zs )
= ( cons_tm @ X @ Xs ) )
= ( ( ( Ys = nil_tm )
& ( Zs
= ( cons_tm @ X @ Xs ) ) )
| ? [Ys4: list_tm] :
( ( Ys
= ( cons_tm @ X @ Ys4 ) )
& ( ( append_tm @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_1067_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs: list_nat,X: nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( cons_nat @ X @ Xs ) )
= ( ( ( Ys = nil_nat )
& ( Zs
= ( cons_nat @ X @ Xs ) ) )
| ? [Ys4: list_nat] :
( ( Ys
= ( cons_nat @ X @ Ys4 ) )
& ( ( append_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_1068_Cons__eq__append__conv,axiom,
! [X: tm,Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( cons_tm @ X @ Xs )
= ( append_tm @ Ys @ Zs ) )
= ( ( ( Ys = nil_tm )
& ( ( cons_tm @ X @ Xs )
= Zs ) )
| ? [Ys4: list_tm] :
( ( ( cons_tm @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_tm @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_1069_Cons__eq__append__conv,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X @ Xs )
= Zs ) )
| ? [Ys4: list_nat] :
( ( ( cons_nat @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_1070_rev__exhaust,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
=> ~ ! [Ys2: list_tm,Y3: tm] :
( Xs
!= ( append_tm @ Ys2 @ ( cons_tm @ Y3 @ nil_tm ) ) ) ) ).
% rev_exhaust
thf(fact_1071_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys2: list_nat,Y3: nat] :
( Xs
!= ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_1072_rev__induct,axiom,
! [P3: list_tm > $o,Xs: list_tm] :
( ( P3 @ nil_tm )
=> ( ! [X3: tm,Xs3: list_tm] :
( ( P3 @ Xs3 )
=> ( P3 @ ( append_tm @ Xs3 @ ( cons_tm @ X3 @ nil_tm ) ) ) )
=> ( P3 @ Xs ) ) ) ).
% rev_induct
thf(fact_1073_rev__induct,axiom,
! [P3: list_nat > $o,Xs: list_nat] :
( ( P3 @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( P3 @ Xs3 )
=> ( P3 @ ( append_nat @ Xs3 @ ( cons_nat @ X3 @ nil_nat ) ) ) )
=> ( P3 @ Xs ) ) ) ).
% rev_induct
thf(fact_1074_replicate__0,axiom,
! [X: tm] :
( ( replicate_tm @ zero_zero_nat @ X )
= nil_tm ) ).
% replicate_0
thf(fact_1075_replicate__0,axiom,
! [X: nat] :
( ( replicate_nat @ zero_zero_nat @ X )
= nil_nat ) ).
% replicate_0
thf(fact_1076_split__list,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat3 @ X @ ( set_set_nat2 @ Xs ) )
=> ? [Ys2: list_set_nat,Zs2: list_set_nat] :
( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1077_split__list,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ? [Ys2: list_fm,Zs2: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1078_split__list,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ? [Ys2: list_tm,Zs2: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1079_split__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1080_split__list__last,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat3 @ X @ ( set_set_nat2 @ Xs ) )
=> ? [Ys2: list_set_nat,Zs2: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X @ Zs2 ) ) )
& ~ ( member_set_nat3 @ X @ ( set_set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1081_split__list__last,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ? [Ys2: list_fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs2 ) ) )
& ~ ( member_fm3 @ X @ ( set_fm2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1082_split__list__last,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ? [Ys2: list_tm,Zs2: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X @ Zs2 ) ) )
& ~ ( member_tm3 @ X @ ( set_tm2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1083_split__list__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat3 @ X @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1084_split__list__prop,axiom,
! [Xs: list_set_nat,P3: set_nat > $o] :
( ? [X4: set_nat] :
( ( member_set_nat3 @ X4 @ ( set_set_nat2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys2: list_set_nat,X3: set_nat] :
( ? [Zs2: list_set_nat] :
( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X3 @ Zs2 ) ) )
& ( P3 @ X3 ) ) ) ).
% split_list_prop
thf(fact_1085_split__list__prop,axiom,
! [Xs: list_fm,P3: fm > $o] :
( ? [X4: fm] :
( ( member_fm3 @ X4 @ ( set_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys2: list_fm,X3: fm] :
( ? [Zs2: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X3 @ Zs2 ) ) )
& ( P3 @ X3 ) ) ) ).
% split_list_prop
thf(fact_1086_split__list__prop,axiom,
! [Xs: list_tm,P3: tm > $o] :
( ? [X4: tm] :
( ( member_tm3 @ X4 @ ( set_tm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys2: list_tm,X3: tm] :
( ? [Zs2: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X3 @ Zs2 ) ) )
& ( P3 @ X3 ) ) ) ).
% split_list_prop
thf(fact_1087_split__list__prop,axiom,
! [Xs: list_nat,P3: nat > $o] :
( ? [X4: nat] :
( ( member_nat3 @ X4 @ ( set_nat2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys2: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P3 @ X3 ) ) ) ).
% split_list_prop
thf(fact_1088_split__list__first,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat3 @ X @ ( set_set_nat2 @ Xs ) )
=> ? [Ys2: list_set_nat,Zs2: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X @ Zs2 ) ) )
& ~ ( member_set_nat3 @ X @ ( set_set_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_1089_split__list__first,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ? [Ys2: list_fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs2 ) ) )
& ~ ( member_fm3 @ X @ ( set_fm2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_1090_split__list__first,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ? [Ys2: list_tm,Zs2: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X @ Zs2 ) ) )
& ~ ( member_tm3 @ X @ ( set_tm2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_1091_split__list__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat3 @ X @ ( set_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_1092_split__list__propE,axiom,
! [Xs: list_set_nat,P3: set_nat > $o] :
( ? [X4: set_nat] :
( ( member_set_nat3 @ X4 @ ( set_set_nat2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys2: list_set_nat,X3: set_nat] :
( ? [Zs2: list_set_nat] :
( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X3 @ Zs2 ) ) )
=> ~ ( P3 @ X3 ) ) ) ).
% split_list_propE
thf(fact_1093_split__list__propE,axiom,
! [Xs: list_fm,P3: fm > $o] :
( ? [X4: fm] :
( ( member_fm3 @ X4 @ ( set_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys2: list_fm,X3: fm] :
( ? [Zs2: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X3 @ Zs2 ) ) )
=> ~ ( P3 @ X3 ) ) ) ).
% split_list_propE
thf(fact_1094_split__list__propE,axiom,
! [Xs: list_tm,P3: tm > $o] :
( ? [X4: tm] :
( ( member_tm3 @ X4 @ ( set_tm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys2: list_tm,X3: tm] :
( ? [Zs2: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X3 @ Zs2 ) ) )
=> ~ ( P3 @ X3 ) ) ) ).
% split_list_propE
thf(fact_1095_split__list__propE,axiom,
! [Xs: list_nat,P3: nat > $o] :
( ? [X4: nat] :
( ( member_nat3 @ X4 @ ( set_nat2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys2: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ~ ( P3 @ X3 ) ) ) ).
% split_list_propE
thf(fact_1096_append__Cons__eq__iff,axiom,
! [X: set_nat,Xs: list_set_nat,Ys: list_set_nat,Xs4: list_set_nat,Ys5: list_set_nat] :
( ~ ( member_set_nat3 @ X @ ( set_set_nat2 @ Xs ) )
=> ( ~ ( member_set_nat3 @ X @ ( set_set_nat2 @ Ys ) )
=> ( ( ( append_set_nat @ Xs @ ( cons_set_nat @ X @ Ys ) )
= ( append_set_nat @ Xs4 @ ( cons_set_nat @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1097_append__Cons__eq__iff,axiom,
! [X: fm,Xs: list_fm,Ys: list_fm,Xs4: list_fm,Ys5: list_fm] :
( ~ ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ( ~ ( member_fm3 @ X @ ( set_fm2 @ Ys ) )
=> ( ( ( append_fm @ Xs @ ( cons_fm @ X @ Ys ) )
= ( append_fm @ Xs4 @ ( cons_fm @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1098_append__Cons__eq__iff,axiom,
! [X: tm,Xs: list_tm,Ys: list_tm,Xs4: list_tm,Ys5: list_tm] :
( ~ ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ( ~ ( member_tm3 @ X @ ( set_tm2 @ Ys ) )
=> ( ( ( append_tm @ Xs @ ( cons_tm @ X @ Ys ) )
= ( append_tm @ Xs4 @ ( cons_tm @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1099_append__Cons__eq__iff,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Xs4: list_nat,Ys5: list_nat] :
( ~ ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat3 @ X @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) )
= ( append_nat @ Xs4 @ ( cons_nat @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1100_in__set__conv__decomp,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat3 @ X @ ( set_set_nat2 @ Xs ) )
= ( ? [Ys3: list_set_nat,Zs3: list_set_nat] :
( Xs
= ( append_set_nat @ Ys3 @ ( cons_set_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1101_in__set__conv__decomp,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
= ( ? [Ys3: list_fm,Zs3: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1102_in__set__conv__decomp,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
= ( ? [Ys3: list_tm,Zs3: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1103_in__set__conv__decomp,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1104_split__list__last__prop,axiom,
! [Xs: list_set_nat,P3: set_nat > $o] :
( ? [X4: set_nat] :
( ( member_set_nat3 @ X4 @ ( set_set_nat2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys2: list_set_nat,X3: set_nat,Zs2: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa2: set_nat] :
( ( member_set_nat3 @ Xa2 @ ( set_set_nat2 @ Zs2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_1105_split__list__last__prop,axiom,
! [Xs: list_fm,P3: fm > $o] :
( ? [X4: fm] :
( ( member_fm3 @ X4 @ ( set_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys2: list_fm,X3: fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa2: fm] :
( ( member_fm3 @ Xa2 @ ( set_fm2 @ Zs2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_1106_split__list__last__prop,axiom,
! [Xs: list_tm,P3: tm > $o] :
( ? [X4: tm] :
( ( member_tm3 @ X4 @ ( set_tm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys2: list_tm,X3: tm,Zs2: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa2: tm] :
( ( member_tm3 @ Xa2 @ ( set_tm2 @ Zs2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_1107_split__list__last__prop,axiom,
! [Xs: list_nat,P3: nat > $o] :
( ? [X4: nat] :
( ( member_nat3 @ X4 @ ( set_nat2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys2: list_nat,X3: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa2: nat] :
( ( member_nat3 @ Xa2 @ ( set_nat2 @ Zs2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_1108_split__list__first__prop,axiom,
! [Xs: list_set_nat,P3: set_nat > $o] :
( ? [X4: set_nat] :
( ( member_set_nat3 @ X4 @ ( set_set_nat2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys2: list_set_nat,X3: set_nat] :
( ? [Zs2: list_set_nat] :
( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa2: set_nat] :
( ( member_set_nat3 @ Xa2 @ ( set_set_nat2 @ Ys2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_1109_split__list__first__prop,axiom,
! [Xs: list_fm,P3: fm > $o] :
( ? [X4: fm] :
( ( member_fm3 @ X4 @ ( set_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys2: list_fm,X3: fm] :
( ? [Zs2: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa2: fm] :
( ( member_fm3 @ Xa2 @ ( set_fm2 @ Ys2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_1110_split__list__first__prop,axiom,
! [Xs: list_tm,P3: tm > $o] :
( ? [X4: tm] :
( ( member_tm3 @ X4 @ ( set_tm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys2: list_tm,X3: tm] :
( ? [Zs2: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa2: tm] :
( ( member_tm3 @ Xa2 @ ( set_tm2 @ Ys2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_1111_split__list__first__prop,axiom,
! [Xs: list_nat,P3: nat > $o] :
( ? [X4: nat] :
( ( member_nat3 @ X4 @ ( set_nat2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys2: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa2: nat] :
( ( member_nat3 @ Xa2 @ ( set_nat2 @ Ys2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_1112_split__list__last__propE,axiom,
! [Xs: list_set_nat,P3: set_nat > $o] :
( ? [X4: set_nat] :
( ( member_set_nat3 @ X4 @ ( set_set_nat2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys2: list_set_nat,X3: set_nat,Zs2: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa2: set_nat] :
( ( member_set_nat3 @ Xa2 @ ( set_set_nat2 @ Zs2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1113_split__list__last__propE,axiom,
! [Xs: list_fm,P3: fm > $o] :
( ? [X4: fm] :
( ( member_fm3 @ X4 @ ( set_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys2: list_fm,X3: fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa2: fm] :
( ( member_fm3 @ Xa2 @ ( set_fm2 @ Zs2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1114_split__list__last__propE,axiom,
! [Xs: list_tm,P3: tm > $o] :
( ? [X4: tm] :
( ( member_tm3 @ X4 @ ( set_tm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys2: list_tm,X3: tm,Zs2: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa2: tm] :
( ( member_tm3 @ Xa2 @ ( set_tm2 @ Zs2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1115_split__list__last__propE,axiom,
! [Xs: list_nat,P3: nat > $o] :
( ? [X4: nat] :
( ( member_nat3 @ X4 @ ( set_nat2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys2: list_nat,X3: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa2: nat] :
( ( member_nat3 @ Xa2 @ ( set_nat2 @ Zs2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1116_split__list__first__propE,axiom,
! [Xs: list_set_nat,P3: set_nat > $o] :
( ? [X4: set_nat] :
( ( member_set_nat3 @ X4 @ ( set_set_nat2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys2: list_set_nat,X3: set_nat] :
( ? [Zs2: list_set_nat] :
( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa2: set_nat] :
( ( member_set_nat3 @ Xa2 @ ( set_set_nat2 @ Ys2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1117_split__list__first__propE,axiom,
! [Xs: list_fm,P3: fm > $o] :
( ? [X4: fm] :
( ( member_fm3 @ X4 @ ( set_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys2: list_fm,X3: fm] :
( ? [Zs2: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa2: fm] :
( ( member_fm3 @ Xa2 @ ( set_fm2 @ Ys2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1118_split__list__first__propE,axiom,
! [Xs: list_tm,P3: tm > $o] :
( ? [X4: tm] :
( ( member_tm3 @ X4 @ ( set_tm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys2: list_tm,X3: tm] :
( ? [Zs2: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa2: tm] :
( ( member_tm3 @ Xa2 @ ( set_tm2 @ Ys2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1119_split__list__first__propE,axiom,
! [Xs: list_nat,P3: nat > $o] :
( ? [X4: nat] :
( ( member_nat3 @ X4 @ ( set_nat2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys2: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa2: nat] :
( ( member_nat3 @ Xa2 @ ( set_nat2 @ Ys2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1120_in__set__conv__decomp__last,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat3 @ X @ ( set_set_nat2 @ Xs ) )
= ( ? [Ys3: list_set_nat,Zs3: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys3 @ ( cons_set_nat @ X @ Zs3 ) ) )
& ~ ( member_set_nat3 @ X @ ( set_set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1121_in__set__conv__decomp__last,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
= ( ? [Ys3: list_fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs3 ) ) )
& ~ ( member_fm3 @ X @ ( set_fm2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1122_in__set__conv__decomp__last,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
= ( ? [Ys3: list_tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X @ Zs3 ) ) )
& ~ ( member_tm3 @ X @ ( set_tm2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1123_in__set__conv__decomp__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat3 @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1124_in__set__conv__decomp__first,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat3 @ X @ ( set_set_nat2 @ Xs ) )
= ( ? [Ys3: list_set_nat,Zs3: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys3 @ ( cons_set_nat @ X @ Zs3 ) ) )
& ~ ( member_set_nat3 @ X @ ( set_set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1125_in__set__conv__decomp__first,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
= ( ? [Ys3: list_fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs3 ) ) )
& ~ ( member_fm3 @ X @ ( set_fm2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1126_in__set__conv__decomp__first,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
= ( ? [Ys3: list_tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X @ Zs3 ) ) )
& ~ ( member_tm3 @ X @ ( set_tm2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1127_in__set__conv__decomp__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat3 @ X @ ( set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1128_split__list__last__prop__iff,axiom,
! [Xs: list_set_nat,P3: set_nat > $o] :
( ( ? [X2: set_nat] :
( ( member_set_nat3 @ X2 @ ( set_set_nat2 @ Xs ) )
& ( P3 @ X2 ) ) )
= ( ? [Ys3: list_set_nat,X2: set_nat,Zs3: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys3 @ ( cons_set_nat @ X2 @ Zs3 ) ) )
& ( P3 @ X2 )
& ! [Y: set_nat] :
( ( member_set_nat3 @ Y @ ( set_set_nat2 @ Zs3 ) )
=> ~ ( P3 @ Y ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1129_split__list__last__prop__iff,axiom,
! [Xs: list_fm,P3: fm > $o] :
( ( ? [X2: fm] :
( ( member_fm3 @ X2 @ ( set_fm2 @ Xs ) )
& ( P3 @ X2 ) ) )
= ( ? [Ys3: list_fm,X2: fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X2 @ Zs3 ) ) )
& ( P3 @ X2 )
& ! [Y: fm] :
( ( member_fm3 @ Y @ ( set_fm2 @ Zs3 ) )
=> ~ ( P3 @ Y ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1130_split__list__last__prop__iff,axiom,
! [Xs: list_tm,P3: tm > $o] :
( ( ? [X2: tm] :
( ( member_tm3 @ X2 @ ( set_tm2 @ Xs ) )
& ( P3 @ X2 ) ) )
= ( ? [Ys3: list_tm,X2: tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X2 @ Zs3 ) ) )
& ( P3 @ X2 )
& ! [Y: tm] :
( ( member_tm3 @ Y @ ( set_tm2 @ Zs3 ) )
=> ~ ( P3 @ Y ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1131_split__list__last__prop__iff,axiom,
! [Xs: list_nat,P3: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat3 @ X2 @ ( set_nat2 @ Xs ) )
& ( P3 @ X2 ) ) )
= ( ? [Ys3: list_nat,X2: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ( P3 @ X2 )
& ! [Y: nat] :
( ( member_nat3 @ Y @ ( set_nat2 @ Zs3 ) )
=> ~ ( P3 @ Y ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1132_split__list__first__prop__iff,axiom,
! [Xs: list_set_nat,P3: set_nat > $o] :
( ( ? [X2: set_nat] :
( ( member_set_nat3 @ X2 @ ( set_set_nat2 @ Xs ) )
& ( P3 @ X2 ) ) )
= ( ? [Ys3: list_set_nat,X2: set_nat] :
( ? [Zs3: list_set_nat] :
( Xs
= ( append_set_nat @ Ys3 @ ( cons_set_nat @ X2 @ Zs3 ) ) )
& ( P3 @ X2 )
& ! [Y: set_nat] :
( ( member_set_nat3 @ Y @ ( set_set_nat2 @ Ys3 ) )
=> ~ ( P3 @ Y ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1133_split__list__first__prop__iff,axiom,
! [Xs: list_fm,P3: fm > $o] :
( ( ? [X2: fm] :
( ( member_fm3 @ X2 @ ( set_fm2 @ Xs ) )
& ( P3 @ X2 ) ) )
= ( ? [Ys3: list_fm,X2: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X2 @ Zs3 ) ) )
& ( P3 @ X2 )
& ! [Y: fm] :
( ( member_fm3 @ Y @ ( set_fm2 @ Ys3 ) )
=> ~ ( P3 @ Y ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1134_split__list__first__prop__iff,axiom,
! [Xs: list_tm,P3: tm > $o] :
( ( ? [X2: tm] :
( ( member_tm3 @ X2 @ ( set_tm2 @ Xs ) )
& ( P3 @ X2 ) ) )
= ( ? [Ys3: list_tm,X2: tm] :
( ? [Zs3: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X2 @ Zs3 ) ) )
& ( P3 @ X2 )
& ! [Y: tm] :
( ( member_tm3 @ Y @ ( set_tm2 @ Ys3 ) )
=> ~ ( P3 @ Y ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1135_split__list__first__prop__iff,axiom,
! [Xs: list_nat,P3: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat3 @ X2 @ ( set_nat2 @ Xs ) )
& ( P3 @ X2 ) ) )
= ( ? [Ys3: list_nat,X2: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ( P3 @ X2 )
& ! [Y: nat] :
( ( member_nat3 @ Y @ ( set_nat2 @ Ys3 ) )
=> ~ ( P3 @ Y ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1136_new__term_Osimps_I1_J,axiom,
! [C: nat,N: nat] : ( new_term @ C @ ( var @ N ) ) ).
% new_term.simps(1)
thf(fact_1137_set__nths__subset,axiom,
! [Xs: list_set_nat,I2: set_nat] : ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( nths_set_nat @ Xs @ I2 ) ) @ ( set_set_nat2 @ Xs ) ) ).
% set_nths_subset
thf(fact_1138_set__nths__subset,axiom,
! [Xs: list_fm,I2: set_nat] : ( ord_less_eq_set_fm @ ( set_fm2 @ ( nths_fm @ Xs @ I2 ) ) @ ( set_fm2 @ Xs ) ) ).
% set_nths_subset
thf(fact_1139_set__nths__subset,axiom,
! [Xs: list_tm,I2: set_nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ ( nths_tm @ Xs @ I2 ) ) @ ( set_tm2 @ Xs ) ) ).
% set_nths_subset
thf(fact_1140_set__nths__subset,axiom,
! [Xs: list_nat,I2: set_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( nths_nat @ Xs @ I2 ) ) @ ( set_nat2 @ Xs ) ) ).
% set_nths_subset
thf(fact_1141_paramst_Osimps_I1_J,axiom,
! [N: nat] :
( ( paramst @ ( var @ N ) )
= bot_bot_set_nat ) ).
% paramst.simps(1)
thf(fact_1142_paramst_H_H_Osimps_I1_J,axiom,
! [N: nat] :
( ( paramst3 @ ( var @ N ) )
= bot_bot_set_nat ) ).
% paramst''.simps(1)
thf(fact_1143_rotate1_Osimps_I2_J,axiom,
! [X: tm,Xs: list_tm] :
( ( rotate1_tm @ ( cons_tm @ X @ Xs ) )
= ( append_tm @ Xs @ ( cons_tm @ X @ nil_tm ) ) ) ).
% rotate1.simps(2)
thf(fact_1144_rotate1_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( rotate1_nat @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_1145_paramst_H_Osimps_I1_J,axiom,
! [N: nat] :
( ( paramst2 @ ( var @ N ) )
= bot_bot_set_nat ) ).
% paramst'.simps(1)
thf(fact_1146_listFunTm_Osimps_I2_J,axiom,
! [N: nat] :
( ( listFunTm @ ( var @ N ) )
= nil_nat ) ).
% listFunTm.simps(2)
thf(fact_1147_set__replicate__conv__if,axiom,
! [N: nat,X: set_nat] :
( ( ( N = zero_zero_nat )
=> ( ( set_set_nat2 @ ( replicate_set_nat @ N @ X ) )
= bot_bot_set_set_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( set_set_nat2 @ ( replicate_set_nat @ N @ X ) )
= ( insert_set_nat2 @ X @ bot_bot_set_set_nat ) ) ) ) ).
% set_replicate_conv_if
thf(fact_1148_set__replicate__conv__if,axiom,
! [N: nat,X: tm] :
( ( ( N = zero_zero_nat )
=> ( ( set_tm2 @ ( replicate_tm @ N @ X ) )
= bot_bot_set_tm ) )
& ( ( N != zero_zero_nat )
=> ( ( set_tm2 @ ( replicate_tm @ N @ X ) )
= ( insert_tm2 @ X @ bot_bot_set_tm ) ) ) ) ).
% set_replicate_conv_if
thf(fact_1149_set__replicate__conv__if,axiom,
! [N: nat,X: nat] :
( ( ( N = zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
= bot_bot_set_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).
% set_replicate_conv_if
thf(fact_1150_set__replicate__conv__if,axiom,
! [N: nat,X: fm] :
( ( ( N = zero_zero_nat )
=> ( ( set_fm2 @ ( replicate_fm @ N @ X ) )
= bot_bot_set_fm ) )
& ( ( N != zero_zero_nat )
=> ( ( set_fm2 @ ( replicate_fm @ N @ X ) )
= ( insert_fm2 @ X @ bot_bot_set_fm ) ) ) ) ).
% set_replicate_conv_if
thf(fact_1151_prefixes__snoc,axiom,
! [Xs: list_tm,X: tm] :
( ( prefixes_tm @ ( append_tm @ Xs @ ( cons_tm @ X @ nil_tm ) ) )
= ( append_list_tm @ ( prefixes_tm @ Xs ) @ ( cons_list_tm @ ( append_tm @ Xs @ ( cons_tm @ X @ nil_tm ) ) @ nil_list_tm ) ) ) ).
% prefixes_snoc
thf(fact_1152_prefixes__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( prefixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( append_list_nat @ ( prefixes_nat @ Xs ) @ ( cons_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) @ nil_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_1153_tm_Osize__gen_I2_J,axiom,
! [X23: nat] :
( ( size_tm @ ( var @ X23 ) )
= zero_zero_nat ) ).
% tm.size_gen(2)
thf(fact_1154_prefixes__eq__snoc,axiom,
! [Ys: list_tm,Xs: list_list_tm,X: list_tm] :
( ( ( prefixes_tm @ Ys )
= ( append_list_tm @ Xs @ ( cons_list_tm @ X @ nil_list_tm ) ) )
= ( ( ( ( Ys = nil_tm )
& ( Xs = nil_list_tm ) )
| ? [Z: tm,Zs3: list_tm] :
( ( Ys
= ( append_tm @ Zs3 @ ( cons_tm @ Z @ nil_tm ) ) )
& ( Xs
= ( prefixes_tm @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_1155_prefixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( prefixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z: nat,Zs3: list_nat] :
( ( Ys
= ( append_nat @ Zs3 @ ( cons_nat @ Z @ nil_nat ) ) )
& ( Xs
= ( prefixes_nat @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_1156_tm_Osize_I4_J,axiom,
! [X23: nat] :
( ( size_size_tm @ ( var @ X23 ) )
= zero_zero_nat ) ).
% tm.size(4)
thf(fact_1157_maps__simps_I1_J,axiom,
! [F: tm > list_nat,X: tm,Xs: list_tm] :
( ( maps_tm_nat @ F @ ( cons_tm @ X @ Xs ) )
= ( append_nat @ ( F @ X ) @ ( maps_tm_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_1158_maps__simps_I1_J,axiom,
! [F: nat > list_nat,X: nat,Xs: list_nat] :
( ( maps_nat_nat @ F @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ ( F @ X ) @ ( maps_nat_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_1159_maps__simps_I2_J,axiom,
! [F: tm > list_tm] :
( ( maps_tm_tm @ F @ nil_tm )
= nil_tm ) ).
% maps_simps(2)
thf(fact_1160_maps__simps_I2_J,axiom,
! [F: tm > list_nat] :
( ( maps_tm_nat @ F @ nil_tm )
= nil_nat ) ).
% maps_simps(2)
thf(fact_1161_maps__simps_I2_J,axiom,
! [F: nat > list_tm] :
( ( maps_nat_tm @ F @ nil_nat )
= nil_tm ) ).
% maps_simps(2)
thf(fact_1162_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_1163_prefixes_Osimps_I1_J,axiom,
( ( prefixes_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% prefixes.simps(1)
thf(fact_1164_prefixes_Osimps_I1_J,axiom,
( ( prefixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_1165_suffixes__eq__snoc,axiom,
! [Ys: list_tm,Xs: list_list_tm,X: list_tm] :
( ( ( suffixes_tm @ Ys )
= ( append_list_tm @ Xs @ ( cons_list_tm @ X @ nil_list_tm ) ) )
= ( ( ( ( Ys = nil_tm )
& ( Xs = nil_list_tm ) )
| ? [Z: tm,Zs3: list_tm] :
( ( Ys
= ( cons_tm @ Z @ Zs3 ) )
& ( Xs
= ( suffixes_tm @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_1166_suffixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( suffixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs3 ) )
& ( Xs
= ( suffixes_nat @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_1167_concat__eq__append__conv,axiom,
! [Xss2: list_list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( concat_tm @ Xss2 )
= ( append_tm @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_tm )
=> ( ( Ys = nil_tm )
& ( Zs = nil_tm ) ) )
& ( ( Xss2 != nil_list_tm )
=> ? [Xss1: list_list_tm,Xs2: list_tm,Xs5: list_tm,Xss22: list_list_tm] :
( ( Xss2
= ( append_list_tm @ Xss1 @ ( cons_list_tm @ ( append_tm @ Xs2 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_tm @ ( concat_tm @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_tm @ Xs5 @ ( concat_tm @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_1168_concat__eq__append__conv,axiom,
! [Xss2: list_list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_nat )
=> ( ( Ys = nil_nat )
& ( Zs = nil_nat ) ) )
& ( ( Xss2 != nil_list_nat )
=> ? [Xss1: list_list_nat,Xs2: list_nat,Xs5: list_nat,Xss22: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs2 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_nat @ Xs5 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_1169_suffixes_Osimps_I2_J,axiom,
! [X: tm,Xs: list_tm] :
( ( suffixes_tm @ ( cons_tm @ X @ Xs ) )
= ( append_list_tm @ ( suffixes_tm @ Xs ) @ ( cons_list_tm @ ( cons_tm @ X @ Xs ) @ nil_list_tm ) ) ) ).
% suffixes.simps(2)
thf(fact_1170_suffixes_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( suffixes_nat @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( suffixes_nat @ Xs ) @ ( cons_list_nat @ ( cons_nat @ X @ Xs ) @ nil_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_1171_concat__eq__appendD,axiom,
! [Xss2: list_list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_nat )
=> ? [Xss12: list_list_nat,Xs3: list_nat,Xs6: list_nat,Xss23: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs3 @ Xs6 ) @ Xss23 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append_nat @ Xs6 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_1172_concat__eq__appendD,axiom,
! [Xss2: list_list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( concat_tm @ Xss2 )
= ( append_tm @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_tm )
=> ? [Xss12: list_list_tm,Xs3: list_tm,Xs6: list_tm,Xss23: list_list_tm] :
( ( Xss2
= ( append_list_tm @ Xss12 @ ( cons_list_tm @ ( append_tm @ Xs3 @ Xs6 ) @ Xss23 ) ) )
& ( Ys
= ( append_tm @ ( concat_tm @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append_tm @ Xs6 @ ( concat_tm @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_1173_concat__replicate__trivial,axiom,
! [I: nat] :
( ( concat_tm @ ( replicate_list_tm @ I @ nil_tm ) )
= nil_tm ) ).
% concat_replicate_trivial
thf(fact_1174_concat__replicate__trivial,axiom,
! [I: nat] :
( ( concat_nat @ ( replicate_list_nat @ I @ nil_nat ) )
= nil_nat ) ).
% concat_replicate_trivial
thf(fact_1175_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_tm] :
( ( nil_tm
= ( concat_tm @ Xss2 ) )
= ( ! [X2: list_tm] :
( ( member_list_tm @ X2 @ ( set_list_tm2 @ Xss2 ) )
=> ( X2 = nil_tm ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1176_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_nat] :
( ( nil_nat
= ( concat_nat @ Xss2 ) )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xss2 ) )
=> ( X2 = nil_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1177_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_tm] :
( ( ( concat_tm @ Xss2 )
= nil_tm )
= ( ! [X2: list_tm] :
( ( member_list_tm @ X2 @ ( set_list_tm2 @ Xss2 ) )
=> ( X2 = nil_tm ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1178_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_nat] :
( ( ( concat_nat @ Xss2 )
= nil_nat )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xss2 ) )
=> ( X2 = nil_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1179_concat__append,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( concat_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( append_nat @ ( concat_nat @ Xs ) @ ( concat_nat @ Ys ) ) ) ).
% concat_append
thf(fact_1180_concat__append,axiom,
! [Xs: list_list_tm,Ys: list_list_tm] :
( ( concat_tm @ ( append_list_tm @ Xs @ Ys ) )
= ( append_tm @ ( concat_tm @ Xs ) @ ( concat_tm @ Ys ) ) ) ).
% concat_append
thf(fact_1181_comm__append__are__replicate,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Ys @ Xs ) )
=> ? [M2: nat,N2: nat,Zs2: list_nat] :
( ( ( concat_nat @ ( replicate_list_nat @ M2 @ Zs2 ) )
= Xs )
& ( ( concat_nat @ ( replicate_list_nat @ N2 @ Zs2 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_1182_comm__append__are__replicate,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= ( append_tm @ Ys @ Xs ) )
=> ? [M2: nat,N2: nat,Zs2: list_tm] :
( ( ( concat_tm @ ( replicate_list_tm @ M2 @ Zs2 ) )
= Xs )
& ( ( concat_tm @ ( replicate_list_tm @ N2 @ Zs2 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_1183_concat_Osimps_I1_J,axiom,
( ( concat_tm @ nil_list_tm )
= nil_tm ) ).
% concat.simps(1)
thf(fact_1184_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_1185_concat_Osimps_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( concat_nat @ ( cons_list_nat @ X @ Xs ) )
= ( append_nat @ X @ ( concat_nat @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_1186_concat_Osimps_I2_J,axiom,
! [X: list_tm,Xs: list_list_tm] :
( ( concat_tm @ ( cons_list_tm @ X @ Xs ) )
= ( append_tm @ X @ ( concat_tm @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_1187_suffixes_Osimps_I1_J,axiom,
( ( suffixes_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% suffixes.simps(1)
thf(fact_1188_suffixes_Osimps_I1_J,axiom,
( ( suffixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_1189_sublists_Osimps_I2_J,axiom,
! [X: tm,Xs: list_tm] :
( ( sublists_tm @ ( cons_tm @ X @ Xs ) )
= ( append_list_tm @ ( sublists_tm @ Xs ) @ ( map_list_tm_list_tm @ ( cons_tm @ X ) @ ( prefixes_tm @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_1190_sublists_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( sublists_nat @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( sublists_nat @ Xs ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_1191_butlast__snoc,axiom,
! [Xs: list_tm,X: tm] :
( ( butlast_tm @ ( append_tm @ Xs @ ( cons_tm @ X @ nil_tm ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_1192_butlast__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_1193_subset__subseqs,axiom,
! [X6: set_set_nat,Xs: list_set_nat] :
( ( ord_le6893508408891458716et_nat @ X6 @ ( set_set_nat2 @ Xs ) )
=> ( member_set_set_nat @ X6 @ ( image_8726355809080528601et_nat @ set_set_nat2 @ ( set_list_set_nat2 @ ( subseqs_set_nat @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_1194_subset__subseqs,axiom,
! [X6: set_fm,Xs: list_fm] :
( ( ord_less_eq_set_fm @ X6 @ ( set_fm2 @ Xs ) )
=> ( member_set_fm @ X6 @ ( image_list_fm_set_fm @ set_fm2 @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_1195_subset__subseqs,axiom,
! [X6: set_tm,Xs: list_tm] :
( ( ord_less_eq_set_tm @ X6 @ ( set_tm2 @ Xs ) )
=> ( member_set_tm @ X6 @ ( image_list_tm_set_tm @ set_tm2 @ ( set_list_tm2 @ ( subseqs_tm @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_1196_subset__subseqs,axiom,
! [X6: set_nat,Xs: list_nat] :
( ( ord_less_eq_set_nat @ X6 @ ( set_nat2 @ Xs ) )
=> ( member_set_nat3 @ X6 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_1197_last__snoc,axiom,
! [Xs: list_tm,X: tm] :
( ( last_tm @ ( append_tm @ Xs @ ( cons_tm @ X @ nil_tm ) ) )
= X ) ).
% last_snoc
thf(fact_1198_last__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% last_snoc
thf(fact_1199_image__eqI,axiom,
! [B2: nat,F: set_nat > nat,X: set_nat,A: set_set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_set_nat3 @ X @ A )
=> ( member_nat3 @ B2 @ ( image_set_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1200_image__eqI,axiom,
! [B2: fm,F: set_nat > fm,X: set_nat,A: set_set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_set_nat3 @ X @ A )
=> ( member_fm3 @ B2 @ ( image_set_nat_fm @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1201_image__eqI,axiom,
! [B2: set_nat,F: set_nat > set_nat,X: set_nat,A: set_set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_set_nat3 @ X @ A )
=> ( member_set_nat3 @ B2 @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1202_subtermFm_Osimps_I1_J,axiom,
! [Uu: nat,Ts: list_tm] :
( ( subtermFm @ ( pre @ Uu @ Ts ) )
= ( concat_tm @ ( map_tm_list_tm @ subtermTm @ Ts ) ) ) ).
% subtermFm.simps(1)
thf(fact_1203_paramst_H_H_Oelims,axiom,
! [X: tm,Y2: set_nat] :
( ( ( paramst3 @ X )
= Y2 )
=> ( ( ? [N2: nat] :
( X
= ( var @ N2 ) )
=> ( Y2 != bot_bot_set_nat ) )
=> ~ ! [A5: nat,Ts2: list_tm] :
( ( X
= ( fun @ A5 @ Ts2 ) )
=> ( Y2
!= ( sup_sup_set_nat @ ( insert_nat2 @ A5 @ bot_bot_set_nat ) @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ paramst3 @ ( set_tm2 @ Ts2 ) ) ) ) ) ) ) ) ).
% paramst''.elims
thf(fact_1204_paramst_H_H_Osimps_I2_J,axiom,
! [A2: nat,Ts: list_tm] :
( ( paramst3 @ ( fun @ A2 @ Ts ) )
= ( sup_sup_set_nat @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ paramst3 @ ( set_tm2 @ Ts ) ) ) ) ) ).
% paramst''.simps(2)
thf(fact_1205_p0,axiom,
( paramsts
= ( ^ [Ts3: list_tm] : ( comple7399068483239264473et_nat @ ( set_set_nat2 @ ( map_tm_set_nat @ paramst @ Ts3 ) ) ) ) ) ).
% p0
thf(fact_1206_paramst_H_Osimps_I2_J,axiom,
! [A2: nat,Ts: list_tm] :
( ( paramst2 @ ( fun @ A2 @ Ts ) )
= ( sup_sup_set_nat @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) @ ( comple7399068483239264473et_nat @ ( set_set_nat2 @ ( map_tm_set_nat @ paramst2 @ Ts ) ) ) ) ) ).
% paramst'.simps(2)
thf(fact_1207_params_H_Osimps_I1_J,axiom,
! [B2: nat,Ts: list_tm] :
( ( params2 @ ( pre @ B2 @ Ts ) )
= ( comple7399068483239264473et_nat @ ( set_set_nat2 @ ( map_tm_set_nat @ paramst2 @ Ts ) ) ) ) ).
% params'.simps(1)
thf(fact_1208_p2,axiom,
params2 = params ).
% p2
thf(fact_1209_paramst_H_H_Opelims,axiom,
! [X: tm,Y2: set_nat] :
( ( ( paramst3 @ X )
= Y2 )
=> ( ( accp_tm @ paramst_rel @ X )
=> ( ! [N2: nat] :
( ( X
= ( var @ N2 ) )
=> ( ( Y2 = bot_bot_set_nat )
=> ~ ( accp_tm @ paramst_rel @ ( var @ N2 ) ) ) )
=> ~ ! [A5: nat,Ts2: list_tm] :
( ( X
= ( fun @ A5 @ Ts2 ) )
=> ( ( Y2
= ( sup_sup_set_nat @ ( insert_nat2 @ A5 @ bot_bot_set_nat ) @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ paramst3 @ ( set_tm2 @ Ts2 ) ) ) ) )
=> ~ ( accp_tm @ paramst_rel @ ( fun @ A5 @ Ts2 ) ) ) ) ) ) ) ).
% paramst''.pelims
thf(fact_1210_subtermTm_Osimps_I1_J,axiom,
! [N: nat,Ts: list_tm] :
( ( subtermTm @ ( fun @ N @ Ts ) )
= ( cons_tm @ ( fun @ N @ Ts ) @ ( remdups_tm @ ( concat_tm @ ( map_tm_list_tm @ subtermTm @ Ts ) ) ) ) ) ).
% subtermTm.simps(1)
thf(fact_1211_params_H_H_Osimps_I1_J,axiom,
! [B2: nat,Ts: list_tm] :
( ( params3 @ ( pre @ B2 @ Ts ) )
= ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ paramst3 @ ( set_tm2 @ Ts ) ) ) ) ).
% params''.simps(1)
thf(fact_1212_p2_H,axiom,
params3 = params ).
% p2'
thf(fact_1213_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1214_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z4: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z4 )
=> ( R @ X3 @ Z4 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1215_nat__induct__at__least,axiom,
! [M: nat,N: nat,P3: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P3 @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P3 @ N2 )
=> ( P3 @ ( suc @ N2 ) ) ) )
=> ( P3 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1216_full__nat__induct,axiom,
! [P3: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( P3 @ M3 ) )
=> ( P3 @ N2 ) )
=> ( P3 @ N ) ) ).
% full_nat_induct
thf(fact_1217_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1218_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1219_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1220_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M2: nat] :
( M4
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_1221_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1222_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1223_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1224_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1225_liftt_Osimps_I1_J,axiom,
! [I: nat] :
( ( liftt @ ( var @ I ) )
= ( var @ ( suc @ I ) ) ) ).
% liftt.simps(1)
thf(fact_1226_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_1227_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_1228_remdups__upt,axiom,
! [M: nat,N: nat] :
( ( remdups_nat @ ( upt @ M @ N ) )
= ( upt @ M @ N ) ) ).
% remdups_upt
thf(fact_1229_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_1230_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_1231_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
= ( upt @ M @ Q2 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q2 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_1232_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_1233_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_1234_upt__rec,axiom,
( upt
= ( ^ [I3: nat,J2: nat] : ( if_list_nat @ ( ord_less_nat @ I3 @ J2 ) @ ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J2 ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_1235_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( M5 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1236_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1237_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
| ( M5 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1238_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1239_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1240_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1241_diff__less__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1242_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1243_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1244_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1245_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1246_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1247_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1248_inc__induct,axiom,
! [I: nat,J: nat,P3: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P3 @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P3 @ ( suc @ N2 ) )
=> ( P3 @ N2 ) ) ) )
=> ( P3 @ I ) ) ) ) ).
% inc_induct
thf(fact_1249_dec__induct,axiom,
! [I: nat,J: nat,P3: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P3 @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P3 @ N2 )
=> ( P3 @ ( suc @ N2 ) ) ) ) )
=> ( P3 @ J ) ) ) ) ).
% dec_induct
thf(fact_1250_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1251_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1252_ex__least__nat__le,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ N )
=> ( ~ ( P3 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P3 @ I5 ) )
& ( P3 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1253_ex__least__nat__less,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ N )
=> ( ~ ( P3 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K2 )
=> ~ ( P3 @ I5 ) )
& ( P3 @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1254_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_1255_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_1256_nat__descend__induct,axiom,
! [N: nat,P3: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P3 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P3 @ I5 ) )
=> ( P3 @ K2 ) ) )
=> ( P3 @ M ) ) ) ).
% nat_descend_induct
thf(fact_1257_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_1258_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% ile0_eq
thf(fact_1259_i0__lb,axiom,
! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% i0_lb
thf(fact_1260_last__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( last_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% last_upt
thf(fact_1261_sub__term_Osimps_I1_J,axiom,
! [N: nat,V: nat,S2: tm] :
( ( ( ord_less_nat @ N @ V )
=> ( ( sub_term @ V @ S2 @ ( var @ N ) )
= ( var @ N ) ) )
& ( ~ ( ord_less_nat @ N @ V )
=> ( ( ( N = V )
=> ( ( sub_term @ V @ S2 @ ( var @ N ) )
= S2 ) )
& ( ( N != V )
=> ( ( sub_term @ V @ S2 @ ( var @ N ) )
= ( var @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% sub_term.simps(1)
thf(fact_1262_substt_Osimps_I1_J,axiom,
! [K: nat,I: nat,S2: tm] :
( ( ( ord_less_nat @ K @ I )
=> ( ( substt @ ( var @ I ) @ S2 @ K )
= ( var @ ( minus_minus_nat @ I @ one_one_nat ) ) ) )
& ( ~ ( ord_less_nat @ K @ I )
=> ( ( ( I = K )
=> ( ( substt @ ( var @ I ) @ S2 @ K )
= S2 ) )
& ( ( I != K )
=> ( ( substt @ ( var @ I ) @ S2 @ K )
= ( var @ I ) ) ) ) ) ) ).
% substt.simps(1)
thf(fact_1263_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1264_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_1265_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1266_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1267_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1268_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
=> ( ( nth_nat @ ( upt @ I @ J ) @ K )
= ( plus_plus_nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_1269_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1270_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1271_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( F @ M2 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1272_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
% Helper facts (9)
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y2: list_nat] :
( ( if_list_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y2: list_nat] :
( ( if_list_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
! [X: list_fm,Y2: list_fm] :
( ( if_list_fm @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
! [X: list_fm,Y2: list_fm] :
( ( if_list_fm @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X: list_tm,Y2: list_tm] :
( ( if_list_tm @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X: list_tm,Y2: list_tm] :
( ( if_list_tm @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [X: list_set_nat,Y2: list_set_nat] :
( ( if_list_set_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [X: list_set_nat,Y2: list_set_nat] :
( ( if_list_set_nat @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [X3: tm] :
( ( member_tm3 @ X3 @ ( set_tm2 @ ts ) )
=> ( member_tm3 @ X3 @ ( terms @ s ) ) ) ).
%------------------------------------------------------------------------------