TPTP Problem File: SLH0165^1.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 : Equivalence_Relation_Enumeration/0007_Equivalence_Relation_Enumeration/prob_00389_014804__12107510_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1507 ( 595 unt; 235 typ; 0 def)
% Number of atoms : 3334 (1402 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 8924 ( 329 ~; 38 |; 134 &;7159 @)
% ( 0 <=>;1264 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 22 ( 21 usr)
% Number of type conns : 917 ( 917 >; 0 *; 0 +; 0 <<)
% Number of symbols : 215 ( 214 usr; 20 con; 0-8 aty)
% Number of variables : 3094 ( 138 ^;2906 !; 50 ?;3094 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:14:39.288
%------------------------------------------------------------------------------
% Could-be-implicit typings (21)
thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
list_l3822697302700470509at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_li2602615041181522887at_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
list_s1210847774152347623at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_se7855581050983116737at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_set_list_nat: $tType ).
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__List__Olist_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
list_set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (214)
thf(sy_c_Equiv__Relations_Oequiv_001t__Nat__Onat,type,
equiv_equiv_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Equivalence__Relation__Enumeration_Oenum__rgfs,type,
equiva7426478223624825838m_rgfs: nat > list_list_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Oequiv__rels,type,
equiva8721718519204927301v_rels: nat > list_s1210847774152347623at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__List__Olist_It__Nat__Onat_J,type,
equiva6490762433048536736st_nat: list_list_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__Nat__Onat,type,
equiva2048684438135499664of_nat: list_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__Set__Oset_It__Nat__Onat_J,type,
equiva50036288466687814et_nat: list_set_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
equiva1173177585473067681at_nat: list_s1210847774152347623at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Orgf,type,
equiva3371634703666331078on_rgf: list_nat > $o ).
thf(sy_c_Equivalence__Relation__Enumeration_Orgf__limit,type,
equiva5889994315859557365_limit: list_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Fun_Ofun__upd_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
fun_up5216017410020450626at_nat: ( list_nat > set_Pr1261947904930325089at_nat ) > list_nat > set_Pr1261947904930325089at_nat > list_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Nat__Onat,type,
fun_upd_nat_nat: ( nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
fun_upd_nat_set_nat: ( nat > set_nat ) > nat > set_nat > nat > set_nat ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
fun_up810672174005406034at_nat: ( nat > set_Pr1261947904930325089at_nat ) > nat > set_Pr1261947904930325089at_nat > nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Fun_Ofun__upd_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
fun_upd_set_nat_nat: ( set_nat > nat ) > set_nat > nat > set_nat > nat ).
thf(sy_c_Fun_Ofun__upd_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
fun_up2577977767889591691et_nat: ( set_nat > set_nat ) > set_nat > set_nat > set_nat > set_nat ).
thf(sy_c_Fun_Ofun__upd_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
fun_up8285502973659875868at_nat: ( set_nat > set_Pr1261947904930325089at_nat ) > set_nat > set_Pr1261947904930325089at_nat > set_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Int__Oint,type,
inj_on_int_int: ( int > int ) > set_int > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
inj_on8270031949885980868at_nat: ( list_list_nat > list_s1210847774152347623at_nat ) > set_list_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on3049792774292151987st_nat: ( list_nat > list_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
inj_on2669155312630230889et_nat: ( list_nat > list_set_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
inj_on1193113198435547220at_nat: ( list_nat > list_s1210847774152347623at_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
inj_on_list_nat_nat: ( list_nat > nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on1816901372521670873et_nat: ( list_nat > set_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inj_on7522185085906380110at_nat: ( list_nat > set_Pr1261947904930325089at_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on2090978070287468649st_nat: ( list_set_nat > list_nat ) > set_list_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
inj_on4452343878340264223et_nat: ( list_set_nat > list_set_nat ) > set_list_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
inj_on9019006866949024542at_nat: ( list_set_nat > list_s1210847774152347623at_nat ) > set_list_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Int__Oint,type,
inj_on_nat_int: ( nat > int ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
inj_on_nat_list_nat: ( nat > list_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on_nat_set_nat: ( nat > set_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inj_on7202621048659411806at_nat: ( nat > set_Pr1261947904930325089at_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on5467128325884351833st_nat: ( set_nat > list_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
inj_on_set_nat_nat: ( set_nat > nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on4604407203859583615et_nat: ( set_nat > set_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inj_on5888811482125702184at_nat: ( set_nat > set_Pr1261947904930325089at_nat ) > set_set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
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__List__Olist_It__Nat__Onat_J_J,type,
minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__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__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
inf_inf_int: int > int > int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
inf_inf_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
sup_sup_int: int > int > int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
sup_sup_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_List_Ocan__select_001t__List__Olist_It__Nat__Onat_J,type,
can_select_list_nat: ( list_nat > $o ) > set_list_nat > $o ).
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__Set__Oset_It__Nat__Onat_J,type,
can_select_set_nat: ( set_nat > $o ) > set_set_nat > $o ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Ocount__list_001t__List__Olist_It__Nat__Onat_J,type,
count_list_list_nat: list_list_nat > list_nat > nat ).
thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
count_list_nat: list_nat > nat > nat ).
thf(sy_c_List_Ocount__list_001t__Set__Oset_It__Nat__Onat_J,type,
count_list_set_nat: list_set_nat > set_nat > nat ).
thf(sy_c_List_Ocount__list_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
count_6440129622255701469at_nat: list_s1210847774152347623at_nat > set_Pr1261947904930325089at_nat > nat ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
distinct_list_nat: list_list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
distin4912746231890992349st_nat: list_set_list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Set__Oset_It__Nat__Onat_J,type,
distinct_set_nat: list_set_nat > $o ).
thf(sy_c_List_Odistinct_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
distin8719635941469336154at_nat: list_s1210847774152347623at_nat > $o ).
thf(sy_c_List_Odistinct_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
distinct_set_set_nat: list_set_set_nat > $o ).
thf(sy_c_List_Odistinct__adj_001t__List__Olist_It__Nat__Onat_J,type,
distin876741697294417026st_nat: list_list_nat > $o ).
thf(sy_c_List_Odistinct__adj_001t__Nat__Onat,type,
distinct_adj_nat: list_nat > $o ).
thf(sy_c_List_Odistinct__adj_001t__Set__Oset_It__Nat__Onat_J,type,
distinct_adj_set_nat: list_set_nat > $o ).
thf(sy_c_List_Odistinct__adj_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
distin3702590604212146495at_nat: list_s1210847774152347623at_nat > $o ).
thf(sy_c_List_Ofold_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
fold_l59423398878476163st_nat: ( list_nat > set_list_nat > set_list_nat ) > list_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
fold_nat_set_nat: ( nat > set_nat > set_nat ) > list_nat > set_nat > set_nat ).
thf(sy_c_List_Ofold_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
fold_s4794219702148550607et_nat: ( set_nat > set_set_nat > set_set_nat ) > list_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_List_Ofolding__insort__key_001t__Nat__Onat_001t__Nat__Onat,type,
foldin8133931898133206727at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > set_nat > ( nat > nat ) > $o ).
thf(sy_c_List_Ofolding__insort__key_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
foldin8172238680504804989et_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > set_set_nat > ( set_nat > nat ) > $o ).
thf(sy_c_List_Ofolding__insort__key_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
foldin4606603775849435773at_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > ( nat > set_nat ) > $o ).
thf(sy_c_List_Ofolding__insort__key_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
foldin2309727365711792691et_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_set_nat > ( set_nat > set_nat ) > $o ).
thf(sy_c_List_Ofolding__insort__key_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
foldin5963648469059051996st_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > set_list_nat > ( list_nat > set_Pr1261947904930325089at_nat ) > $o ).
thf(sy_c_List_Ofolding__insort__key_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
foldin5281791559409726668at_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > set_nat > ( nat > set_Pr1261947904930325089at_nat ) > $o ).
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thf(sy_c_List_Ofolding__insort__key__axioms_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
foldin6085089136383733792at_nat: set_set_nat > ( set_nat > nat ) > $o ).
thf(sy_c_List_Ogen__length_001t__List__Olist_It__Nat__Onat_J,type,
gen_length_list_nat: nat > list_list_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Set__Oset_It__Nat__Onat_J,type,
gen_length_set_nat: nat > list_set_nat > nat ).
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thf(sy_c_List_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
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_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__Nat__Onat_J,type,
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map_li2355978560338012748st_nat: ( list_list_nat > set_list_nat ) > list_list_list_nat > list_set_list_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__Nat__Onat_J_001t__Nat__Onat,type,
map_list_nat_nat: ( list_nat > nat ) > list_list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
map_list_nat_set_nat: ( list_nat > set_nat ) > list_list_nat > list_set_nat ).
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map_li6003994582982014139at_nat: ( list_nat > set_Pr1261947904930325089at_nat ) > list_list_nat > list_s1210847774152347623at_nat ).
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map_li3330360646112351000et_nat: ( list_set_nat > set_set_nat ) > list_list_set_nat > list_set_set_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
map_nat_list_nat: ( nat > list_nat ) > list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
map_nat_set_nat: ( nat > set_nat ) > list_nat > list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_set_nat_list_nat: ( set_nat > list_nat ) > list_set_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
map_set_nat_nat: ( set_nat > nat ) > list_set_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
map_set_nat_set_nat: ( set_nat > set_nat ) > list_set_nat > list_set_nat ).
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map_se1162299115588061717at_nat: ( set_nat > set_Pr1261947904930325089at_nat ) > list_set_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_list_nat2: list_list_list_nat > set_list_list_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__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__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_li2149138424243681404at_nat: list_l3822697302700470509at_nat > set_li2602615041181522887at_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
set_set_nat2: list_set_nat > set_set_nat ).
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thf(sy_c_List_Olist__ex1_001t__List__Olist_It__Nat__Onat_J,type,
list_ex1_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
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__Set__Oset_It__Nat__Onat_J,type,
list_ex1_set_nat: ( set_nat > $o ) > list_set_nat > $o ).
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map_ta8671482330076047857at_nat: ( list_nat > set_Pr1261947904930325089at_nat ) > list_list_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Nat__Onat,type,
map_tailrec_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
map_ta7283568089350952038et_nat: ( nat > set_nat ) > list_nat > list_set_nat ).
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thf(sy_c_List_Omap__tailrec_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
map_ta3717933184695582822at_nat: ( set_nat > nat ) > list_set_nat > list_nat ).
thf(sy_c_List_Omap__tailrec_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
map_ta8069416624492123164et_nat: ( set_nat > set_nat ) > list_set_nat > list_set_nat ).
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thf(sy_c_List_Omember_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_list_nat > list_nat > $o ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $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__List__Olist_It__Nat__Onat_J,type,
n_lists_list_nat: nat > list_list_nat > list_list_list_nat ).
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__Set__Oset_It__Nat__Onat_J,type,
n_lists_set_nat: nat > list_set_nat > list_list_set_nat ).
thf(sy_c_List_On__lists_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
n_list4589530985885001583at_nat: nat > list_s1210847774152347623at_nat > list_l3822697302700470509at_nat ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__Nat__Onat_J,type,
produc6783906451316923569st_nat: list_list_list_nat > list_list_list_nat ).
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__Set__Oset_It__Nat__Onat_J,type,
produc8109398739672286679et_nat: list_list_set_nat > list_list_set_nat ).
thf(sy_c_List_Oproduct__lists_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_List_OremoveAll_001t__List__Olist_It__Nat__Onat_J,type,
removeAll_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001t__Set__Oset_It__Nat__Onat_J,type,
removeAll_set_nat: set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_OremoveAll_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
remove5672899571770113645at_nat: set_Pr1261947904930325089at_nat > list_s1210847774152347623at_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Orotate1_001t__List__Olist_It__Nat__Onat_J,type,
rotate1_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
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_Orotate1_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
rotate4238613965387346100at_nat: list_s1210847774152347623at_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osubseqs_001t__List__Olist_It__Nat__Onat_J,type,
subseqs_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
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__List__Olist_It__Nat__Onat_J,type,
union_list_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
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_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
size_s6248950052170075156st_nat: list_list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
size_s8312130442339763642et_nat: list_list_set_nat > nat ).
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size_s3876073854528961369at_nat: list_l3822697302700470509at_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__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
size_s3254054031482475050et_nat: list_set_nat > nat ).
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thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
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__List__Olist_It__Nat__Onat_J_J,type,
bot_bot_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
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__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
top_top_list_nat_o: list_nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
top_to8617644770344506070st_nat: set_list_list_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
top_top_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
top_to667676211730484604et_nat: set_list_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat2: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat2: nat > set_nat > set_nat ).
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_Oremove_001t__List__Olist_It__Nat__Onat_J,type,
remove_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
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__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
member_list_list_nat: list_list_nat > set_list_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat2: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
member_list_set_nat: list_set_nat > set_list_set_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member2758946086501665296at_nat: list_s1210847774152347623at_nat > set_li2602615041181522887at_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat2: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).
thf(sy_v_f____,type,
f: set_nat > nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_r,type,
r: set_Pr1261947904930325089at_nat ).
thf(sy_v_y____,type,
y: list_set_nat ).
% Relevant facts (1271)
thf(fact_0_b_I2_J,axiom,
equiva3371634703666331078on_rgf @ ( map_set_nat_nat @ f @ y ) ).
% b(2)
thf(fact_1_a_I2_J,axiom,
( ( size_s3254054031482475050et_nat @ y )
= n ) ).
% a(2)
thf(fact_2_a_I1_J,axiom,
( ( equiva50036288466687814et_nat @ y )
= r ) ).
% a(1)
thf(fact_3_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_4_length__map,axiom,
! [F: set_nat > nat,Xs: list_set_nat] :
( ( size_size_list_nat @ ( map_set_nat_nat @ F @ Xs ) )
= ( size_s3254054031482475050et_nat @ Xs ) ) ).
% length_map
thf(fact_5_length__map,axiom,
! [F: nat > set_nat,Xs: list_nat] :
( ( size_s3254054031482475050et_nat @ ( map_nat_set_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_6_length__map,axiom,
! [F: set_nat > set_nat,Xs: list_set_nat] :
( ( size_s3254054031482475050et_nat @ ( map_set_nat_set_nat @ F @ Xs ) )
= ( size_s3254054031482475050et_nat @ Xs ) ) ).
% length_map
thf(fact_7_length__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( size_s8736152011456118867at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_8_length__map,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( size_size_list_nat @ ( map_list_nat_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_9_length__map,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( map_nat_list_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_10_length__map,axiom,
! [F: list_nat > set_nat,Xs: list_list_nat] :
( ( size_s3254054031482475050et_nat @ ( map_list_nat_set_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_11_length__map,axiom,
! [F: set_nat > list_nat,Xs: list_set_nat] :
( ( size_s3023201423986296836st_nat @ ( map_set_nat_list_nat @ F @ Xs ) )
= ( size_s3254054031482475050et_nat @ Xs ) ) ).
% length_map
thf(fact_12_length__map,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_13_calculation,axiom,
( ( equiva2048684438135499664of_nat @ ( map_set_nat_nat @ f @ y ) )
= r ) ).
% calculation
thf(fact_14_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: set_nat > nat,Ys: list_set_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_set_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3254054031482475050et_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_15_map__eq__imp__length__eq,axiom,
! [F: set_nat > nat,Xs: list_set_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_set_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_s3254054031482475050et_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_16_map__eq__imp__length__eq,axiom,
! [F: set_nat > nat,Xs: list_set_nat,G: set_nat > nat,Ys: list_set_nat] :
( ( ( map_set_nat_nat @ F @ Xs )
= ( map_set_nat_nat @ G @ Ys ) )
=> ( ( size_s3254054031482475050et_nat @ Xs )
= ( size_s3254054031482475050et_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_17_map__eq__imp__length__eq,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,G: nat > set_Pr1261947904930325089at_nat,Ys: list_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ Xs )
= ( map_na6577772983117884747at_nat @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_18_map__eq__imp__length__eq,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,Xs: list_nat,G: list_nat > set_Pr1261947904930325089at_nat,Ys: list_list_nat] :
( ( ( map_na6577772983117884747at_nat @ F @ Xs )
= ( map_li6003994582982014139at_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_19_map__eq__imp__length__eq,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,G: list_nat > set_Pr1261947904930325089at_nat,Ys: list_list_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ Xs )
= ( map_li6003994582982014139at_nat @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_20_map__eq__imp__length__eq,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,G: set_nat > set_Pr1261947904930325089at_nat,Ys: list_set_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ Xs )
= ( map_se1162299115588061717at_nat @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3254054031482475050et_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_21_map__eq__imp__length__eq,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,Xs: list_set_nat,G: list_nat > set_Pr1261947904930325089at_nat,Ys: list_list_nat] :
( ( ( map_se1162299115588061717at_nat @ F @ Xs )
= ( map_li6003994582982014139at_nat @ G @ Ys ) )
=> ( ( size_s3254054031482475050et_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_22_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_23_map__eq__imp__length__eq,axiom,
! [F: nat > set_nat,Xs: list_nat,G: nat > set_nat,Ys: list_nat] :
( ( ( map_nat_set_nat @ F @ Xs )
= ( map_nat_set_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_24_b_I1_J,axiom,
inj_on_set_nat_nat @ f @ ( set_set_nat2 @ y ) ).
% b(1)
thf(fact_25_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_s1210847774152347623at_nat] :
( ( size_s8736152011456118867at_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_26_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_nat] :
( ( size_s3023201423986296836st_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_27_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_28_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_set_nat] :
( ( size_s3254054031482475050et_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_29_neq__if__length__neq,axiom,
! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
( ( ( size_s8736152011456118867at_nat @ Xs )
!= ( size_s8736152011456118867at_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_30_neq__if__length__neq,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
!= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_31_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_32_neq__if__length__neq,axiom,
! [Xs: list_set_nat,Ys: list_set_nat] :
( ( ( size_s3254054031482475050et_nat @ Xs )
!= ( size_s3254054031482475050et_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_33_size__neq__size__imp__neq,axiom,
! [X: list_s1210847774152347623at_nat,Y: list_s1210847774152347623at_nat] :
( ( ( size_s8736152011456118867at_nat @ X )
!= ( size_s8736152011456118867at_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_34_size__neq__size__imp__neq,axiom,
! [X: list_list_nat,Y: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ X )
!= ( size_s3023201423986296836st_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_35_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_36_size__neq__size__imp__neq,axiom,
! [X: list_set_nat,Y: list_set_nat] :
( ( ( size_s3254054031482475050et_nat @ X )
!= ( size_s3254054031482475050et_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_37_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_38_kernel__of__eq__len,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_39_kernel__of__eq__len,axiom,
! [X: list_list_nat,Y: list_nat] :
( ( ( equiva6490762433048536736st_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( ( size_s3023201423986296836st_nat @ X )
= ( size_size_list_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_40_kernel__of__eq__len,axiom,
! [X: list_set_nat,Y: list_nat] :
( ( ( equiva50036288466687814et_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( ( size_s3254054031482475050et_nat @ X )
= ( size_size_list_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_41_kernel__of__eq__len,axiom,
! [X: list_nat,Y: list_list_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva6490762433048536736st_nat @ Y ) )
=> ( ( size_size_list_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_42_kernel__of__eq__len,axiom,
! [X: list_nat,Y: list_set_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva50036288466687814et_nat @ Y ) )
=> ( ( size_size_list_nat @ X )
= ( size_s3254054031482475050et_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_43_kernel__of__eq__len,axiom,
! [X: list_list_nat,Y: list_list_nat] :
( ( ( equiva6490762433048536736st_nat @ X )
= ( equiva6490762433048536736st_nat @ Y ) )
=> ( ( size_s3023201423986296836st_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_44_kernel__of__eq__len,axiom,
! [X: list_list_nat,Y: list_set_nat] :
( ( ( equiva6490762433048536736st_nat @ X )
= ( equiva50036288466687814et_nat @ Y ) )
=> ( ( size_s3023201423986296836st_nat @ X )
= ( size_s3254054031482475050et_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_45_kernel__of__eq__len,axiom,
! [X: list_set_nat,Y: list_list_nat] :
( ( ( equiva50036288466687814et_nat @ X )
= ( equiva6490762433048536736st_nat @ Y ) )
=> ( ( size_s3254054031482475050et_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_46_kernel__of__eq__len,axiom,
! [X: list_set_nat,Y: list_set_nat] :
( ( ( equiva50036288466687814et_nat @ X )
= ( equiva50036288466687814et_nat @ Y ) )
=> ( ( size_s3254054031482475050et_nat @ X )
= ( size_s3254054031482475050et_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_47_kernel__of__eq__len,axiom,
! [X: list_s1210847774152347623at_nat,Y: list_nat] :
( ( ( equiva1173177585473067681at_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( ( size_s8736152011456118867at_nat @ X )
= ( size_size_list_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_48__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062f_O_A_092_060lbrakk_062inj__on_Af_A_Iset_Ay_J_059_Argf_A_Imap_Af_Ay_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [F2: set_nat > nat] :
( ( inj_on_set_nat_nat @ F2 @ ( set_set_nat2 @ y ) )
=> ~ ( equiva3371634703666331078on_rgf @ ( map_set_nat_nat @ F2 @ y ) ) ) ).
% \<open>\<And>thesis. (\<And>f. \<lbrakk>inj_on f (set y); rgf (map f y)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_49_assms,axiom,
equiv_equiv_nat @ ( set_ord_lessThan_nat @ n ) @ r ).
% assms
thf(fact_50__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062y_O_A_092_060lbrakk_062kernel__of_Ay_A_061_Ar_059_Alength_Ay_A_061_An_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Y2: list_set_nat] :
( ( ( equiva50036288466687814et_nat @ Y2 )
= r )
=> ( ( size_s3254054031482475050et_nat @ Y2 )
!= n ) ) ).
% \<open>\<And>thesis. (\<And>y. \<lbrakk>kernel_of y = r; length y = n\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_51_map__eq__conv,axiom,
! [F: set_nat > nat,Xs: list_set_nat,G: set_nat > nat] :
( ( ( map_set_nat_nat @ F @ Xs )
= ( map_set_nat_nat @ G @ Xs ) )
= ( ! [X2: set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_52_map__eq__conv,axiom,
! [F: set_nat > set_nat,Xs: list_set_nat,G: set_nat > set_nat] :
( ( ( map_set_nat_set_nat @ F @ Xs )
= ( map_set_nat_set_nat @ G @ Xs ) )
= ( ! [X2: set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_53_map__eq__conv,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,Xs: list_set_nat,G: set_nat > set_Pr1261947904930325089at_nat] :
( ( ( map_se1162299115588061717at_nat @ F @ Xs )
= ( map_se1162299115588061717at_nat @ G @ Xs ) )
= ( ! [X2: set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_54_map__eq__conv,axiom,
! [F: nat > set_nat,Xs: list_nat,G: nat > set_nat] :
( ( ( map_nat_set_nat @ F @ Xs )
= ( map_nat_set_nat @ G @ Xs ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_55_map__eq__conv,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_56_map__eq__conv,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,Xs: list_nat,G: nat > set_Pr1261947904930325089at_nat] :
( ( ( map_na6577772983117884747at_nat @ F @ Xs )
= ( map_na6577772983117884747at_nat @ G @ Xs ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_57_map__eq__conv,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ Xs )
= ( map_li6003994582982014139at_nat @ G @ Xs ) )
= ( ! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_58_equiv__on__unique,axiom,
! [A: set_nat,P: set_Pr1261947904930325089at_nat,B: set_nat] :
( ( equiv_equiv_nat @ A @ P )
=> ( ( equiv_equiv_nat @ B @ P )
=> ( A = B ) ) ) ).
% equiv_on_unique
thf(fact_59_kernel__of__equiv,axiom,
! [Xs: list_s1210847774152347623at_nat] : ( equiv_equiv_nat @ ( set_ord_lessThan_nat @ ( size_s8736152011456118867at_nat @ Xs ) ) @ ( equiva1173177585473067681at_nat @ Xs ) ) ).
% kernel_of_equiv
thf(fact_60_kernel__of__equiv,axiom,
! [Xs: list_list_nat] : ( equiv_equiv_nat @ ( set_ord_lessThan_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) @ ( equiva6490762433048536736st_nat @ Xs ) ) ).
% kernel_of_equiv
thf(fact_61_kernel__of__equiv,axiom,
! [Xs: list_set_nat] : ( equiv_equiv_nat @ ( set_ord_lessThan_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) @ ( equiva50036288466687814et_nat @ Xs ) ) ).
% kernel_of_equiv
thf(fact_62_kernel__of__equiv,axiom,
! [Xs: list_nat] : ( equiv_equiv_nat @ ( set_ord_lessThan_nat @ ( size_size_list_nat @ Xs ) ) @ ( equiva2048684438135499664of_nat @ Xs ) ) ).
% kernel_of_equiv
thf(fact_63_all__rels__are__kernels,axiom,
! [N: nat,P: set_Pr1261947904930325089at_nat] :
( ( equiv_equiv_nat @ ( set_ord_lessThan_nat @ N ) @ P )
=> ? [X3: list_set_nat] :
( ( ( equiva50036288466687814et_nat @ X3 )
= P )
& ( ( size_s3254054031482475050et_nat @ X3 )
= N ) ) ) ).
% all_rels_are_kernels
thf(fact_64_kernel__of__under__inj__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,X: list_list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ ( set_list_nat2 @ X ) )
=> ( ( equiva6490762433048536736st_nat @ X )
= ( equiva1173177585473067681at_nat @ ( map_li6003994582982014139at_nat @ F @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_65_kernel__of__under__inj__map,axiom,
! [F: list_nat > set_nat,X: list_list_nat] :
( ( inj_on1816901372521670873et_nat @ F @ ( set_list_nat2 @ X ) )
=> ( ( equiva6490762433048536736st_nat @ X )
= ( equiva50036288466687814et_nat @ ( map_list_nat_set_nat @ F @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_66_kernel__of__under__inj__map,axiom,
! [F: list_nat > nat,X: list_list_nat] :
( ( inj_on_list_nat_nat @ F @ ( set_list_nat2 @ X ) )
=> ( ( equiva6490762433048536736st_nat @ X )
= ( equiva2048684438135499664of_nat @ ( map_list_nat_nat @ F @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_67_kernel__of__under__inj__map,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,X: list_set_nat] :
( ( inj_on5888811482125702184at_nat @ F @ ( set_set_nat2 @ X ) )
=> ( ( equiva50036288466687814et_nat @ X )
= ( equiva1173177585473067681at_nat @ ( map_se1162299115588061717at_nat @ F @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_68_kernel__of__under__inj__map,axiom,
! [F: set_nat > set_nat,X: list_set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ ( set_set_nat2 @ X ) )
=> ( ( equiva50036288466687814et_nat @ X )
= ( equiva50036288466687814et_nat @ ( map_set_nat_set_nat @ F @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_69_kernel__of__under__inj__map,axiom,
! [F: set_nat > nat,X: list_set_nat] :
( ( inj_on_set_nat_nat @ F @ ( set_set_nat2 @ X ) )
=> ( ( equiva50036288466687814et_nat @ X )
= ( equiva2048684438135499664of_nat @ ( map_set_nat_nat @ F @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_70_kernel__of__under__inj__map,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,X: list_nat] :
( ( inj_on7202621048659411806at_nat @ F @ ( set_nat2 @ X ) )
=> ( ( equiva2048684438135499664of_nat @ X )
= ( equiva1173177585473067681at_nat @ ( map_na6577772983117884747at_nat @ F @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_71_kernel__of__under__inj__map,axiom,
! [F: nat > set_nat,X: list_nat] :
( ( inj_on_nat_set_nat @ F @ ( set_nat2 @ X ) )
=> ( ( equiva2048684438135499664of_nat @ X )
= ( equiva50036288466687814et_nat @ ( map_nat_set_nat @ F @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_72_kernel__of__under__inj__map,axiom,
! [F: nat > nat,X: list_nat] :
( ( inj_on_nat_nat @ F @ ( set_nat2 @ X ) )
=> ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2048684438135499664of_nat @ ( map_nat_nat @ F @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_73_map__list__to__rgf,axiom,
! [X: list_set_nat] :
? [F2: set_nat > nat] :
( ( inj_on_set_nat_nat @ F2 @ ( set_set_nat2 @ X ) )
& ( equiva3371634703666331078on_rgf @ ( map_set_nat_nat @ F2 @ X ) ) ) ).
% map_list_to_rgf
thf(fact_74_map__list__to__rgf,axiom,
! [X: list_nat] :
? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ ( set_nat2 @ X ) )
& ( equiva3371634703666331078on_rgf @ ( map_nat_nat @ F2 @ X ) ) ) ).
% map_list_to_rgf
thf(fact_75_map__list__to__rgf,axiom,
! [X: list_list_nat] :
? [F2: list_nat > nat] :
( ( inj_on_list_nat_nat @ F2 @ ( set_list_nat2 @ X ) )
& ( equiva3371634703666331078on_rgf @ ( map_list_nat_nat @ F2 @ X ) ) ) ).
% map_list_to_rgf
thf(fact_76_ex__map__conv,axiom,
! [Ys: list_s1210847774152347623at_nat,F: list_nat > set_Pr1261947904930325089at_nat] :
( ( ? [Xs3: list_list_nat] :
( Ys
= ( map_li6003994582982014139at_nat @ F @ Xs3 ) ) )
= ( ! [X2: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X2 @ ( set_se5049602875457034614at_nat @ Ys ) )
=> ? [Y3: list_nat] :
( X2
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_77_ex__map__conv,axiom,
! [Ys: list_s1210847774152347623at_nat,F: set_nat > set_Pr1261947904930325089at_nat] :
( ( ? [Xs3: list_set_nat] :
( Ys
= ( map_se1162299115588061717at_nat @ F @ Xs3 ) ) )
= ( ! [X2: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X2 @ ( set_se5049602875457034614at_nat @ Ys ) )
=> ? [Y3: set_nat] :
( X2
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_78_ex__map__conv,axiom,
! [Ys: list_s1210847774152347623at_nat,F: nat > set_Pr1261947904930325089at_nat] :
( ( ? [Xs3: list_nat] :
( Ys
= ( map_na6577772983117884747at_nat @ F @ Xs3 ) ) )
= ( ! [X2: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X2 @ ( set_se5049602875457034614at_nat @ Ys ) )
=> ? [Y3: nat] :
( X2
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_79_ex__map__conv,axiom,
! [Ys: list_set_nat,F: set_nat > set_nat] :
( ( ? [Xs3: list_set_nat] :
( Ys
= ( map_set_nat_set_nat @ F @ Xs3 ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Ys ) )
=> ? [Y3: set_nat] :
( X2
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_80_ex__map__conv,axiom,
! [Ys: list_set_nat,F: nat > set_nat] :
( ( ? [Xs3: list_nat] :
( Ys
= ( map_nat_set_nat @ F @ Xs3 ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Ys ) )
=> ? [Y3: nat] :
( X2
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_81_ex__map__conv,axiom,
! [Ys: list_nat,F: set_nat > nat] :
( ( ? [Xs3: list_set_nat] :
( Ys
= ( map_set_nat_nat @ F @ Xs3 ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Ys ) )
=> ? [Y3: set_nat] :
( X2
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_82_ex__map__conv,axiom,
! [Ys: list_nat,F: nat > nat] :
( ( ? [Xs3: list_nat] :
( Ys
= ( map_nat_nat @ F @ Xs3 ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Ys ) )
=> ? [Y3: nat] :
( X2
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_83_map__cong,axiom,
! [Xs: list_set_nat,Ys: list_set_nat,F: set_nat > nat,G: set_nat > nat] :
( ( Xs = Ys )
=> ( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_set_nat_nat @ F @ Xs )
= ( map_set_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_84_map__cong,axiom,
! [Xs: list_set_nat,Ys: list_set_nat,F: set_nat > set_nat,G: set_nat > set_nat] :
( ( Xs = Ys )
=> ( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_set_nat_set_nat @ F @ Xs )
= ( map_set_nat_set_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_85_map__cong,axiom,
! [Xs: list_set_nat,Ys: list_set_nat,F: set_nat > set_Pr1261947904930325089at_nat,G: set_nat > set_Pr1261947904930325089at_nat] :
( ( Xs = Ys )
=> ( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_se1162299115588061717at_nat @ F @ Xs )
= ( map_se1162299115588061717at_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_86_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > set_nat,G: nat > set_nat] :
( ( Xs = Ys )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_set_nat @ F @ Xs )
= ( map_nat_set_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_87_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs = Ys )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_88_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > set_Pr1261947904930325089at_nat,G: nat > set_Pr1261947904930325089at_nat] :
( ( Xs = Ys )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_na6577772983117884747at_nat @ F @ Xs )
= ( map_na6577772983117884747at_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_89_map__cong,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ( Xs = Ys )
=> ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ Xs )
= ( map_li6003994582982014139at_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_90_map__idI,axiom,
! [Xs: list_set_nat,F: set_nat > set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_set_nat_set_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_91_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_92_map__idI,axiom,
! [Xs: list_list_nat,F: list_nat > list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_93_map__ext,axiom,
! [Xs: list_set_nat,F: set_nat > nat,G: set_nat > nat] :
( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_set_nat_nat @ F @ Xs )
= ( map_set_nat_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_94_map__ext,axiom,
! [Xs: list_set_nat,F: set_nat > set_nat,G: set_nat > set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_set_nat_set_nat @ F @ Xs )
= ( map_set_nat_set_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_95_map__ext,axiom,
! [Xs: list_set_nat,F: set_nat > set_Pr1261947904930325089at_nat,G: set_nat > set_Pr1261947904930325089at_nat] :
( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_se1162299115588061717at_nat @ F @ Xs )
= ( map_se1162299115588061717at_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_96_map__ext,axiom,
! [Xs: list_nat,F: nat > set_nat,G: nat > set_nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_set_nat @ F @ Xs )
= ( map_nat_set_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_97_map__ext,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_98_map__ext,axiom,
! [Xs: list_nat,F: nat > set_Pr1261947904930325089at_nat,G: nat > set_Pr1261947904930325089at_nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_na6577772983117884747at_nat @ F @ Xs )
= ( map_na6577772983117884747at_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_99_map__ext,axiom,
! [Xs: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ Xs )
= ( map_li6003994582982014139at_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_100_list_Omap__ident__strong,axiom,
! [T: list_set_nat,F: set_nat > set_nat] :
( ! [Z: set_nat] :
( ( member_set_nat2 @ Z @ ( set_set_nat2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_set_nat_set_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_101_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_102_list_Omap__ident__strong,axiom,
! [T: list_list_nat,F: list_nat > list_nat] :
( ! [Z: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_li7225945977422193158st_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_103_list_Oinj__map__strong,axiom,
! [X: list_set_nat,Xa: list_set_nat,F: set_nat > nat,Fa: set_nat > nat] :
( ! [Z: set_nat,Za: set_nat] :
( ( member_set_nat2 @ Z @ ( set_set_nat2 @ X ) )
=> ( ( member_set_nat2 @ Za @ ( set_set_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_set_nat_nat @ F @ X )
= ( map_set_nat_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_104_list_Oinj__map__strong,axiom,
! [X: list_set_nat,Xa: list_set_nat,F: set_nat > set_nat,Fa: set_nat > set_nat] :
( ! [Z: set_nat,Za: set_nat] :
( ( member_set_nat2 @ Z @ ( set_set_nat2 @ X ) )
=> ( ( member_set_nat2 @ Za @ ( set_set_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_set_nat_set_nat @ F @ X )
= ( map_set_nat_set_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_105_list_Oinj__map__strong,axiom,
! [X: list_set_nat,Xa: list_set_nat,F: set_nat > set_Pr1261947904930325089at_nat,Fa: set_nat > set_Pr1261947904930325089at_nat] :
( ! [Z: set_nat,Za: set_nat] :
( ( member_set_nat2 @ Z @ ( set_set_nat2 @ X ) )
=> ( ( member_set_nat2 @ Za @ ( set_set_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_se1162299115588061717at_nat @ F @ X )
= ( map_se1162299115588061717at_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_106_list_Oinj__map__strong,axiom,
! [X: list_nat,Xa: list_nat,F: nat > set_nat,Fa: nat > set_nat] :
( ! [Z: nat,Za: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ X ) )
=> ( ( member_nat2 @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_nat_set_nat @ F @ X )
= ( map_nat_set_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_107_list_Oinj__map__strong,axiom,
! [X: list_nat,Xa: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z: nat,Za: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ X ) )
=> ( ( member_nat2 @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_108_list_Oinj__map__strong,axiom,
! [X: list_nat,Xa: list_nat,F: nat > set_Pr1261947904930325089at_nat,Fa: nat > set_Pr1261947904930325089at_nat] :
( ! [Z: nat,Za: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ X ) )
=> ( ( member_nat2 @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_na6577772983117884747at_nat @ F @ X )
= ( map_na6577772983117884747at_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_109_list_Oinj__map__strong,axiom,
! [X: list_list_nat,Xa: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,Fa: list_nat > set_Pr1261947904930325089at_nat] :
( ! [Z: list_nat,Za: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ X ) )
=> ( ( member_list_nat2 @ Za @ ( set_list_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_li6003994582982014139at_nat @ F @ X )
= ( map_li6003994582982014139at_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_110_list_Omap__cong0,axiom,
! [X: list_set_nat,F: set_nat > nat,G: set_nat > nat] :
( ! [Z: set_nat] :
( ( member_set_nat2 @ Z @ ( set_set_nat2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_set_nat_nat @ F @ X )
= ( map_set_nat_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_111_list_Omap__cong0,axiom,
! [X: list_set_nat,F: set_nat > set_nat,G: set_nat > set_nat] :
( ! [Z: set_nat] :
( ( member_set_nat2 @ Z @ ( set_set_nat2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_set_nat_set_nat @ F @ X )
= ( map_set_nat_set_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_112_list_Omap__cong0,axiom,
! [X: list_set_nat,F: set_nat > set_Pr1261947904930325089at_nat,G: set_nat > set_Pr1261947904930325089at_nat] :
( ! [Z: set_nat] :
( ( member_set_nat2 @ Z @ ( set_set_nat2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_se1162299115588061717at_nat @ F @ X )
= ( map_se1162299115588061717at_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_113_list_Omap__cong0,axiom,
! [X: list_nat,F: nat > set_nat,G: nat > set_nat] :
( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_set_nat @ F @ X )
= ( map_nat_set_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_114_list_Omap__cong0,axiom,
! [X: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_115_list_Omap__cong0,axiom,
! [X: list_nat,F: nat > set_Pr1261947904930325089at_nat,G: nat > set_Pr1261947904930325089at_nat] :
( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_na6577772983117884747at_nat @ F @ X )
= ( map_na6577772983117884747at_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_116_list_Omap__cong0,axiom,
! [X: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ! [Z: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ X )
= ( map_li6003994582982014139at_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_117_list_Omap__cong,axiom,
! [X: list_set_nat,Ya: list_set_nat,F: set_nat > nat,G: set_nat > nat] :
( ( X = Ya )
=> ( ! [Z: set_nat] :
( ( member_set_nat2 @ Z @ ( set_set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_set_nat_nat @ F @ X )
= ( map_set_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_118_list_Omap__cong,axiom,
! [X: list_set_nat,Ya: list_set_nat,F: set_nat > set_nat,G: set_nat > set_nat] :
( ( X = Ya )
=> ( ! [Z: set_nat] :
( ( member_set_nat2 @ Z @ ( set_set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_set_nat_set_nat @ F @ X )
= ( map_set_nat_set_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_119_list_Omap__cong,axiom,
! [X: list_set_nat,Ya: list_set_nat,F: set_nat > set_Pr1261947904930325089at_nat,G: set_nat > set_Pr1261947904930325089at_nat] :
( ( X = Ya )
=> ( ! [Z: set_nat] :
( ( member_set_nat2 @ Z @ ( set_set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_se1162299115588061717at_nat @ F @ X )
= ( map_se1162299115588061717at_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_120_list_Omap__cong,axiom,
! [X: list_nat,Ya: list_nat,F: nat > set_nat,G: nat > set_nat] :
( ( X = Ya )
=> ( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_set_nat @ F @ X )
= ( map_nat_set_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_121_list_Omap__cong,axiom,
! [X: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X = Ya )
=> ( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_122_list_Omap__cong,axiom,
! [X: list_nat,Ya: list_nat,F: nat > set_Pr1261947904930325089at_nat,G: nat > set_Pr1261947904930325089at_nat] :
( ( X = Ya )
=> ( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_na6577772983117884747at_nat @ F @ X )
= ( map_na6577772983117884747at_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_123_list_Omap__cong,axiom,
! [X: list_list_nat,Ya: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ( X = Ya )
=> ( ! [Z: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ X )
= ( map_li6003994582982014139at_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_124_kernel__of__inj__on__rgfs__aux,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) )
=> ( ( equiva3371634703666331078on_rgf @ X )
=> ( ( equiva3371634703666331078on_rgf @ Y )
=> ( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( X = Y ) ) ) ) ) ).
% kernel_of_inj_on_rgfs_aux
thf(fact_125_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_126_map__eq__map__tailrec,axiom,
map_set_nat_nat = map_ta3717933184695582822at_nat ).
% map_eq_map_tailrec
thf(fact_127_map__eq__map__tailrec,axiom,
map_li6003994582982014139at_nat = map_ta8671482330076047857at_nat ).
% map_eq_map_tailrec
thf(fact_128_map__eq__map__tailrec,axiom,
map_set_nat_set_nat = map_ta8069416624492123164et_nat ).
% map_eq_map_tailrec
thf(fact_129_map__eq__map__tailrec,axiom,
map_nat_set_nat = map_ta7283568089350952038et_nat ).
% map_eq_map_tailrec
thf(fact_130_map__eq__map__tailrec,axiom,
map_nat_nat = map_tailrec_nat_nat ).
% map_eq_map_tailrec
thf(fact_131_map__eq__map__tailrec,axiom,
map_se1162299115588061717at_nat = map_ta5881658350000699979at_nat ).
% map_eq_map_tailrec
thf(fact_132_map__eq__map__tailrec,axiom,
map_na6577772983117884747at_nat = map_ta3239049050547484033at_nat ).
% map_eq_map_tailrec
thf(fact_133_List_Ocount__list__inj__map,axiom,
! [F: set_nat > nat,Xs: list_set_nat,X: set_nat] :
( ( inj_on_set_nat_nat @ F @ ( set_set_nat2 @ Xs ) )
=> ( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
=> ( ( count_list_nat @ ( map_set_nat_nat @ F @ Xs ) @ ( F @ X ) )
= ( count_list_set_nat @ Xs @ X ) ) ) ) ).
% List.count_list_inj_map
thf(fact_134_List_Ocount__list__inj__map,axiom,
! [F: set_nat > set_nat,Xs: list_set_nat,X: set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ ( set_set_nat2 @ Xs ) )
=> ( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
=> ( ( count_list_set_nat @ ( map_set_nat_set_nat @ F @ Xs ) @ ( F @ X ) )
= ( count_list_set_nat @ Xs @ X ) ) ) ) ).
% List.count_list_inj_map
thf(fact_135_List_Ocount__list__inj__map,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,Xs: list_set_nat,X: set_nat] :
( ( inj_on5888811482125702184at_nat @ F @ ( set_set_nat2 @ Xs ) )
=> ( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
=> ( ( count_6440129622255701469at_nat @ ( map_se1162299115588061717at_nat @ F @ Xs ) @ ( F @ X ) )
= ( count_list_set_nat @ Xs @ X ) ) ) ) ).
% List.count_list_inj_map
thf(fact_136_List_Ocount__list__inj__map,axiom,
! [F: nat > set_nat,Xs: list_nat,X: nat] :
( ( inj_on_nat_set_nat @ F @ ( set_nat2 @ Xs ) )
=> ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( count_list_set_nat @ ( map_nat_set_nat @ F @ Xs ) @ ( F @ X ) )
= ( count_list_nat @ Xs @ X ) ) ) ) ).
% List.count_list_inj_map
thf(fact_137_List_Ocount__list__inj__map,axiom,
! [F: nat > nat,Xs: list_nat,X: nat] :
( ( inj_on_nat_nat @ F @ ( set_nat2 @ Xs ) )
=> ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( count_list_nat @ ( map_nat_nat @ F @ Xs ) @ ( F @ X ) )
= ( count_list_nat @ Xs @ X ) ) ) ) ).
% List.count_list_inj_map
thf(fact_138_List_Ocount__list__inj__map,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,Xs: list_nat,X: nat] :
( ( inj_on7202621048659411806at_nat @ F @ ( set_nat2 @ Xs ) )
=> ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( count_6440129622255701469at_nat @ ( map_na6577772983117884747at_nat @ F @ Xs ) @ ( F @ X ) )
= ( count_list_nat @ Xs @ X ) ) ) ) ).
% List.count_list_inj_map
thf(fact_139_List_Ocount__list__inj__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,X: list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ ( set_list_nat2 @ Xs ) )
=> ( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( count_6440129622255701469at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) @ ( F @ X ) )
= ( count_list_list_nat @ Xs @ X ) ) ) ) ).
% List.count_list_inj_map
thf(fact_140_List_Ocount__list__inj__map,axiom,
! [F: set_nat > list_nat,Xs: list_set_nat,X: set_nat] :
( ( inj_on5467128325884351833st_nat @ F @ ( set_set_nat2 @ Xs ) )
=> ( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
=> ( ( count_list_list_nat @ ( map_set_nat_list_nat @ F @ Xs ) @ ( F @ X ) )
= ( count_list_set_nat @ Xs @ X ) ) ) ) ).
% List.count_list_inj_map
thf(fact_141_List_Ocount__list__inj__map,axiom,
! [F: nat > list_nat,Xs: list_nat,X: nat] :
( ( inj_on_nat_list_nat @ F @ ( set_nat2 @ Xs ) )
=> ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( count_list_list_nat @ ( map_nat_list_nat @ F @ Xs ) @ ( F @ X ) )
= ( count_list_nat @ Xs @ X ) ) ) ) ).
% List.count_list_inj_map
thf(fact_142_List_Ocount__list__inj__map,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,X: list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ ( set_list_nat2 @ Xs ) )
=> ( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( count_list_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) @ ( F @ X ) )
= ( count_list_list_nat @ Xs @ X ) ) ) ) ).
% List.count_list_inj_map
thf(fact_143_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: set_nat > nat,X: list_set_nat,Y: set_nat] :
( ( inj_on_set_nat_nat @ F @ ( set_set_nat2 @ X ) )
=> ( ( member_set_nat2 @ Y @ ( set_set_nat2 @ X ) )
=> ( ( count_list_nat @ ( map_set_nat_nat @ F @ X ) @ ( F @ Y ) )
= ( count_list_set_nat @ X @ Y ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_144_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: set_nat > set_nat,X: list_set_nat,Y: set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ ( set_set_nat2 @ X ) )
=> ( ( member_set_nat2 @ Y @ ( set_set_nat2 @ X ) )
=> ( ( count_list_set_nat @ ( map_set_nat_set_nat @ F @ X ) @ ( F @ Y ) )
= ( count_list_set_nat @ X @ Y ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_145_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,X: list_set_nat,Y: set_nat] :
( ( inj_on5888811482125702184at_nat @ F @ ( set_set_nat2 @ X ) )
=> ( ( member_set_nat2 @ Y @ ( set_set_nat2 @ X ) )
=> ( ( count_6440129622255701469at_nat @ ( map_se1162299115588061717at_nat @ F @ X ) @ ( F @ Y ) )
= ( count_list_set_nat @ X @ Y ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_146_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: nat > set_nat,X: list_nat,Y: nat] :
( ( inj_on_nat_set_nat @ F @ ( set_nat2 @ X ) )
=> ( ( member_nat2 @ Y @ ( set_nat2 @ X ) )
=> ( ( count_list_set_nat @ ( map_nat_set_nat @ F @ X ) @ ( F @ Y ) )
= ( count_list_nat @ X @ Y ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_147_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: nat > nat,X: list_nat,Y: nat] :
( ( inj_on_nat_nat @ F @ ( set_nat2 @ X ) )
=> ( ( member_nat2 @ Y @ ( set_nat2 @ X ) )
=> ( ( count_list_nat @ ( map_nat_nat @ F @ X ) @ ( F @ Y ) )
= ( count_list_nat @ X @ Y ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_148_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,X: list_nat,Y: nat] :
( ( inj_on7202621048659411806at_nat @ F @ ( set_nat2 @ X ) )
=> ( ( member_nat2 @ Y @ ( set_nat2 @ X ) )
=> ( ( count_6440129622255701469at_nat @ ( map_na6577772983117884747at_nat @ F @ X ) @ ( F @ Y ) )
= ( count_list_nat @ X @ Y ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_149_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,X: list_list_nat,Y: list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ ( set_list_nat2 @ X ) )
=> ( ( member_list_nat2 @ Y @ ( set_list_nat2 @ X ) )
=> ( ( count_6440129622255701469at_nat @ ( map_li6003994582982014139at_nat @ F @ X ) @ ( F @ Y ) )
= ( count_list_list_nat @ X @ Y ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_150_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: set_nat > list_nat,X: list_set_nat,Y: set_nat] :
( ( inj_on5467128325884351833st_nat @ F @ ( set_set_nat2 @ X ) )
=> ( ( member_set_nat2 @ Y @ ( set_set_nat2 @ X ) )
=> ( ( count_list_list_nat @ ( map_set_nat_list_nat @ F @ X ) @ ( F @ Y ) )
= ( count_list_set_nat @ X @ Y ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_151_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: nat > list_nat,X: list_nat,Y: nat] :
( ( inj_on_nat_list_nat @ F @ ( set_nat2 @ X ) )
=> ( ( member_nat2 @ Y @ ( set_nat2 @ X ) )
=> ( ( count_list_list_nat @ ( map_nat_list_nat @ F @ X ) @ ( F @ Y ) )
= ( count_list_nat @ X @ Y ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_152_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: list_nat > list_nat,X: list_list_nat,Y: list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ ( set_list_nat2 @ X ) )
=> ( ( member_list_nat2 @ Y @ ( set_list_nat2 @ X ) )
=> ( ( count_list_list_nat @ ( map_li7225945977422193158st_nat @ F @ X ) @ ( F @ Y ) )
= ( count_list_list_nat @ X @ Y ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_153_distinct__adj__mapI,axiom,
! [Xs: list_set_nat,F: set_nat > nat] :
( ( distinct_adj_set_nat @ Xs )
=> ( ( inj_on_set_nat_nat @ F @ ( set_set_nat2 @ Xs ) )
=> ( distinct_adj_nat @ ( map_set_nat_nat @ F @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_154_distinct__adj__mapI,axiom,
! [Xs: list_set_nat,F: set_nat > set_nat] :
( ( distinct_adj_set_nat @ Xs )
=> ( ( inj_on4604407203859583615et_nat @ F @ ( set_set_nat2 @ Xs ) )
=> ( distinct_adj_set_nat @ ( map_set_nat_set_nat @ F @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_155_distinct__adj__mapI,axiom,
! [Xs: list_set_nat,F: set_nat > set_Pr1261947904930325089at_nat] :
( ( distinct_adj_set_nat @ Xs )
=> ( ( inj_on5888811482125702184at_nat @ F @ ( set_set_nat2 @ Xs ) )
=> ( distin3702590604212146495at_nat @ ( map_se1162299115588061717at_nat @ F @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_156_distinct__adj__mapI,axiom,
! [Xs: list_nat,F: nat > set_nat] :
( ( distinct_adj_nat @ Xs )
=> ( ( inj_on_nat_set_nat @ F @ ( set_nat2 @ Xs ) )
=> ( distinct_adj_set_nat @ ( map_nat_set_nat @ F @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_157_distinct__adj__mapI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ( distinct_adj_nat @ Xs )
=> ( ( inj_on_nat_nat @ F @ ( set_nat2 @ Xs ) )
=> ( distinct_adj_nat @ ( map_nat_nat @ F @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_158_distinct__adj__mapI,axiom,
! [Xs: list_nat,F: nat > set_Pr1261947904930325089at_nat] :
( ( distinct_adj_nat @ Xs )
=> ( ( inj_on7202621048659411806at_nat @ F @ ( set_nat2 @ Xs ) )
=> ( distin3702590604212146495at_nat @ ( map_na6577772983117884747at_nat @ F @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_159_distinct__adj__mapI,axiom,
! [Xs: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat] :
( ( distin876741697294417026st_nat @ Xs )
=> ( ( inj_on7522185085906380110at_nat @ F @ ( set_list_nat2 @ Xs ) )
=> ( distin3702590604212146495at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_160_distinct__adj__map__iff,axiom,
! [F: set_nat > nat,Xs: list_set_nat] :
( ( inj_on_set_nat_nat @ F @ ( set_set_nat2 @ Xs ) )
=> ( ( distinct_adj_nat @ ( map_set_nat_nat @ F @ Xs ) )
= ( distinct_adj_set_nat @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_161_distinct__adj__map__iff,axiom,
! [F: set_nat > set_nat,Xs: list_set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ ( set_set_nat2 @ Xs ) )
=> ( ( distinct_adj_set_nat @ ( map_set_nat_set_nat @ F @ Xs ) )
= ( distinct_adj_set_nat @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_162_distinct__adj__map__iff,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,Xs: list_set_nat] :
( ( inj_on5888811482125702184at_nat @ F @ ( set_set_nat2 @ Xs ) )
=> ( ( distin3702590604212146495at_nat @ ( map_se1162299115588061717at_nat @ F @ Xs ) )
= ( distinct_adj_set_nat @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_163_distinct__adj__map__iff,axiom,
! [F: nat > set_nat,Xs: list_nat] :
( ( inj_on_nat_set_nat @ F @ ( set_nat2 @ Xs ) )
=> ( ( distinct_adj_set_nat @ ( map_nat_set_nat @ F @ Xs ) )
= ( distinct_adj_nat @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_164_distinct__adj__map__iff,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( inj_on_nat_nat @ F @ ( set_nat2 @ Xs ) )
=> ( ( distinct_adj_nat @ ( map_nat_nat @ F @ Xs ) )
= ( distinct_adj_nat @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_165_distinct__adj__map__iff,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( inj_on7202621048659411806at_nat @ F @ ( set_nat2 @ Xs ) )
=> ( ( distin3702590604212146495at_nat @ ( map_na6577772983117884747at_nat @ F @ Xs ) )
= ( distinct_adj_nat @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_166_distinct__adj__map__iff,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ ( set_list_nat2 @ Xs ) )
=> ( ( distin3702590604212146495at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) )
= ( distin876741697294417026st_nat @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_167_folding__insort__key__axioms_Ointro,axiom,
! [F: set_nat > nat,S: set_set_nat] :
( ( inj_on_set_nat_nat @ F @ S )
=> ( foldin6085089136383733792at_nat @ S @ F ) ) ).
% folding_insort_key_axioms.intro
thf(fact_168_folding__insort__key__axioms__def,axiom,
( foldin6085089136383733792at_nat
= ( ^ [S2: set_set_nat,F3: set_nat > nat] : ( inj_on_set_nat_nat @ F3 @ S2 ) ) ) ).
% folding_insort_key_axioms_def
thf(fact_169_inj__onD,axiom,
! [F: set_nat > nat,A: set_set_nat,X: set_nat,Y: set_nat] :
( ( inj_on_set_nat_nat @ F @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_set_nat2 @ X @ A )
=> ( ( member_set_nat2 @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_170_inj__onI,axiom,
! [A: set_set_nat,F: set_nat > nat] :
( ! [X3: set_nat,Y2: set_nat] :
( ( member_set_nat2 @ X3 @ A )
=> ( ( member_set_nat2 @ Y2 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) )
=> ( inj_on_set_nat_nat @ F @ A ) ) ).
% inj_onI
thf(fact_171_inj__on__def,axiom,
( inj_on_set_nat_nat
= ( ^ [F3: set_nat > nat,A2: set_set_nat] :
! [X2: set_nat] :
( ( member_set_nat2 @ X2 @ A2 )
=> ! [Y3: set_nat] :
( ( member_set_nat2 @ Y3 @ A2 )
=> ( ( ( F3 @ X2 )
= ( F3 @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_172_inj__on__cong,axiom,
! [A: set_set_nat,F: set_nat > nat,G: set_nat > nat] :
( ! [A3: set_nat] :
( ( member_set_nat2 @ A3 @ A )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on_set_nat_nat @ F @ A )
= ( inj_on_set_nat_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_173_distinct__adj__mapD,axiom,
! [F: set_nat > nat,Xs: list_set_nat] :
( ( distinct_adj_nat @ ( map_set_nat_nat @ F @ Xs ) )
=> ( distinct_adj_set_nat @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_174_distinct__adj__mapD,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( distin3702590604212146495at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) )
=> ( distin876741697294417026st_nat @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_175_distinct__adj__mapD,axiom,
! [F: set_nat > set_nat,Xs: list_set_nat] :
( ( distinct_adj_set_nat @ ( map_set_nat_set_nat @ F @ Xs ) )
=> ( distinct_adj_set_nat @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_176_distinct__adj__mapD,axiom,
! [F: nat > set_nat,Xs: list_nat] :
( ( distinct_adj_set_nat @ ( map_nat_set_nat @ F @ Xs ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_177_distinct__adj__mapD,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( map_nat_nat @ F @ Xs ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_178_distinct__adj__mapD,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,Xs: list_set_nat] :
( ( distin3702590604212146495at_nat @ ( map_se1162299115588061717at_nat @ F @ Xs ) )
=> ( distinct_adj_set_nat @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_179_distinct__adj__mapD,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( distin3702590604212146495at_nat @ ( map_na6577772983117884747at_nat @ F @ Xs ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_180_inj__on__inverseI,axiom,
! [A: set_set_nat,G: nat > set_nat,F: set_nat > nat] :
( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ A )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_set_nat_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_181_inj__on__contraD,axiom,
! [F: set_nat > nat,A: set_set_nat,X: set_nat,Y: set_nat] :
( ( inj_on_set_nat_nat @ F @ A )
=> ( ( X != Y )
=> ( ( member_set_nat2 @ X @ A )
=> ( ( member_set_nat2 @ Y @ A )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_182_inj__on__eq__iff,axiom,
! [F: set_nat > nat,A: set_set_nat,X: set_nat,Y: set_nat] :
( ( inj_on_set_nat_nat @ F @ A )
=> ( ( member_set_nat2 @ X @ A )
=> ( ( member_set_nat2 @ Y @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_183_mem__Collect__eq,axiom,
! [A4: nat,P2: nat > $o] :
( ( member_nat2 @ A4 @ ( collect_nat @ P2 ) )
= ( P2 @ A4 ) ) ).
% mem_Collect_eq
thf(fact_184_mem__Collect__eq,axiom,
! [A4: list_nat,P2: list_nat > $o] :
( ( member_list_nat2 @ A4 @ ( collect_list_nat @ P2 ) )
= ( P2 @ A4 ) ) ).
% mem_Collect_eq
thf(fact_185_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_186_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X2: list_nat] : ( member_list_nat2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_187_rgf__imp__initial__segment,axiom,
! [Xs: list_nat] :
( ( equiva3371634703666331078on_rgf @ Xs )
=> ( ( set_nat2 @ Xs )
= ( set_ord_lessThan_nat @ ( equiva5889994315859557365_limit @ Xs ) ) ) ) ).
% rgf_imp_initial_segment
thf(fact_188_in__set__member,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
= ( member_set_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_189_in__set__member,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_190_in__set__member,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
= ( member_list_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_191_count__notin,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ~ ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
=> ( ( count_list_set_nat @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_192_count__notin,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( count_list_nat @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_193_count__notin,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( count_list_list_nat @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_194_inj__on__map__eq__map,axiom,
! [F: set_nat > nat,Xs: list_set_nat,Ys: list_set_nat] :
( ( inj_on_set_nat_nat @ F @ ( sup_sup_set_set_nat @ ( set_set_nat2 @ Xs ) @ ( set_set_nat2 @ Ys ) ) )
=> ( ( ( map_set_nat_nat @ F @ Xs )
= ( map_set_nat_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_195_inj__on__map__eq__map,axiom,
! [F: set_nat > set_nat,Xs: list_set_nat,Ys: list_set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ ( sup_sup_set_set_nat @ ( set_set_nat2 @ Xs ) @ ( set_set_nat2 @ Ys ) ) )
=> ( ( ( map_set_nat_set_nat @ F @ Xs )
= ( map_set_nat_set_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_196_inj__on__map__eq__map,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,Xs: list_set_nat,Ys: list_set_nat] :
( ( inj_on5888811482125702184at_nat @ F @ ( sup_sup_set_set_nat @ ( set_set_nat2 @ Xs ) @ ( set_set_nat2 @ Ys ) ) )
=> ( ( ( map_se1162299115588061717at_nat @ F @ Xs )
= ( map_se1162299115588061717at_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_197_inj__on__map__eq__map,axiom,
! [F: nat > set_nat,Xs: list_nat,Ys: list_nat] :
( ( inj_on_nat_set_nat @ F @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( ( ( map_nat_set_nat @ F @ Xs )
= ( map_nat_set_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_198_inj__on__map__eq__map,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_199_inj__on__map__eq__map,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
( ( inj_on7202621048659411806at_nat @ F @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( ( ( map_na6577772983117884747at_nat @ F @ Xs )
= ( map_na6577772983117884747at_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_200_inj__on__map__eq__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys ) ) )
=> ( ( ( map_li6003994582982014139at_nat @ F @ Xs )
= ( map_li6003994582982014139at_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_201_map__inj__on,axiom,
! [F: set_nat > nat,Xs: list_set_nat,Ys: list_set_nat] :
( ( ( map_set_nat_nat @ F @ Xs )
= ( map_set_nat_nat @ F @ Ys ) )
=> ( ( inj_on_set_nat_nat @ F @ ( sup_sup_set_set_nat @ ( set_set_nat2 @ Xs ) @ ( set_set_nat2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_202_map__inj__on,axiom,
! [F: set_nat > set_nat,Xs: list_set_nat,Ys: list_set_nat] :
( ( ( map_set_nat_set_nat @ F @ Xs )
= ( map_set_nat_set_nat @ F @ Ys ) )
=> ( ( inj_on4604407203859583615et_nat @ F @ ( sup_sup_set_set_nat @ ( set_set_nat2 @ Xs ) @ ( set_set_nat2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_203_map__inj__on,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,Xs: list_set_nat,Ys: list_set_nat] :
( ( ( map_se1162299115588061717at_nat @ F @ Xs )
= ( map_se1162299115588061717at_nat @ F @ Ys ) )
=> ( ( inj_on5888811482125702184at_nat @ F @ ( sup_sup_set_set_nat @ ( set_set_nat2 @ Xs ) @ ( set_set_nat2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_204_map__inj__on,axiom,
! [F: nat > set_nat,Xs: list_nat,Ys: list_nat] :
( ( ( map_nat_set_nat @ F @ Xs )
= ( map_nat_set_nat @ F @ Ys ) )
=> ( ( inj_on_nat_set_nat @ F @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_205_map__inj__on,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_206_map__inj__on,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
( ( ( map_na6577772983117884747at_nat @ F @ Xs )
= ( map_na6577772983117884747at_nat @ F @ Ys ) )
=> ( ( inj_on7202621048659411806at_nat @ F @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_207_map__inj__on,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ Xs )
= ( map_li6003994582982014139at_nat @ F @ Ys ) )
=> ( ( inj_on7522185085906380110at_nat @ F @ ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_208_distinct__map,axiom,
! [F: set_nat > nat,Xs: list_set_nat] :
( ( distinct_nat @ ( map_set_nat_nat @ F @ Xs ) )
= ( ( distinct_set_nat @ Xs )
& ( inj_on_set_nat_nat @ F @ ( set_set_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_209_distinct__map,axiom,
! [F: set_nat > set_nat,Xs: list_set_nat] :
( ( distinct_set_nat @ ( map_set_nat_set_nat @ F @ Xs ) )
= ( ( distinct_set_nat @ Xs )
& ( inj_on4604407203859583615et_nat @ F @ ( set_set_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_210_distinct__map,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,Xs: list_set_nat] :
( ( distin8719635941469336154at_nat @ ( map_se1162299115588061717at_nat @ F @ Xs ) )
= ( ( distinct_set_nat @ Xs )
& ( inj_on5888811482125702184at_nat @ F @ ( set_set_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_211_distinct__map,axiom,
! [F: nat > set_nat,Xs: list_nat] :
( ( distinct_set_nat @ ( map_nat_set_nat @ F @ Xs ) )
= ( ( distinct_nat @ Xs )
& ( inj_on_nat_set_nat @ F @ ( set_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_212_distinct__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( distinct_nat @ ( map_nat_nat @ F @ Xs ) )
= ( ( distinct_nat @ Xs )
& ( inj_on_nat_nat @ F @ ( set_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_213_distinct__map,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( distin8719635941469336154at_nat @ ( map_na6577772983117884747at_nat @ F @ Xs ) )
= ( ( distinct_nat @ Xs )
& ( inj_on7202621048659411806at_nat @ F @ ( set_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_214_distinct__map,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( distinct_nat @ ( map_list_nat_nat @ F @ Xs ) )
= ( ( distinct_list_nat @ Xs )
& ( inj_on_list_nat_nat @ F @ ( set_list_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_215_distinct__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( distin8719635941469336154at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) )
= ( ( distinct_list_nat @ Xs )
& ( inj_on7522185085906380110at_nat @ F @ ( set_list_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_216_inj__map__eq__map,axiom,
! [F: set_nat > nat,Xs: list_set_nat,Ys: list_set_nat] :
( ( inj_on_set_nat_nat @ F @ top_top_set_set_nat )
=> ( ( ( map_set_nat_nat @ F @ Xs )
= ( map_set_nat_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_217_inj__map__eq__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ top_top_set_list_nat )
=> ( ( ( map_li6003994582982014139at_nat @ F @ Xs )
= ( map_li6003994582982014139at_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_218_inj__map__eq__map,axiom,
! [F: set_nat > set_nat,Xs: list_set_nat,Ys: list_set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ top_top_set_set_nat )
=> ( ( ( map_set_nat_set_nat @ F @ Xs )
= ( map_set_nat_set_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_219_inj__map__eq__map,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,Xs: list_set_nat,Ys: list_set_nat] :
( ( inj_on5888811482125702184at_nat @ F @ top_top_set_set_nat )
=> ( ( ( map_se1162299115588061717at_nat @ F @ Xs )
= ( map_se1162299115588061717at_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_220_inj__map__eq__map,axiom,
! [F: nat > set_nat,Xs: list_nat,Ys: list_nat] :
( ( inj_on_nat_set_nat @ F @ top_top_set_nat )
=> ( ( ( map_nat_set_nat @ F @ Xs )
= ( map_nat_set_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_221_inj__map__eq__map,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_222_inj__map__eq__map,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
( ( inj_on7202621048659411806at_nat @ F @ top_top_set_nat )
=> ( ( ( map_na6577772983117884747at_nat @ F @ Xs )
= ( map_na6577772983117884747at_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_223_map__fun__upd,axiom,
! [Y: set_nat,Xs: list_set_nat,F: set_nat > nat,V: nat] :
( ~ ( member_set_nat2 @ Y @ ( set_set_nat2 @ Xs ) )
=> ( ( map_set_nat_nat @ ( fun_upd_set_nat_nat @ F @ Y @ V ) @ Xs )
= ( map_set_nat_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_224_map__fun__upd,axiom,
! [Y: set_nat,Xs: list_set_nat,F: set_nat > set_nat,V: set_nat] :
( ~ ( member_set_nat2 @ Y @ ( set_set_nat2 @ Xs ) )
=> ( ( map_set_nat_set_nat @ ( fun_up2577977767889591691et_nat @ F @ Y @ V ) @ Xs )
= ( map_set_nat_set_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_225_map__fun__upd,axiom,
! [Y: set_nat,Xs: list_set_nat,F: set_nat > set_Pr1261947904930325089at_nat,V: set_Pr1261947904930325089at_nat] :
( ~ ( member_set_nat2 @ Y @ ( set_set_nat2 @ Xs ) )
=> ( ( map_se1162299115588061717at_nat @ ( fun_up8285502973659875868at_nat @ F @ Y @ V ) @ Xs )
= ( map_se1162299115588061717at_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_226_map__fun__upd,axiom,
! [Y: nat,Xs: list_nat,F: nat > set_nat,V: set_nat] :
( ~ ( member_nat2 @ Y @ ( set_nat2 @ Xs ) )
=> ( ( map_nat_set_nat @ ( fun_upd_nat_set_nat @ F @ Y @ V ) @ Xs )
= ( map_nat_set_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_227_map__fun__upd,axiom,
! [Y: nat,Xs: list_nat,F: nat > nat,V: nat] :
( ~ ( member_nat2 @ Y @ ( set_nat2 @ Xs ) )
=> ( ( map_nat_nat @ ( fun_upd_nat_nat @ F @ Y @ V ) @ Xs )
= ( map_nat_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_228_map__fun__upd,axiom,
! [Y: nat,Xs: list_nat,F: nat > set_Pr1261947904930325089at_nat,V: set_Pr1261947904930325089at_nat] :
( ~ ( member_nat2 @ Y @ ( set_nat2 @ Xs ) )
=> ( ( map_na6577772983117884747at_nat @ ( fun_up810672174005406034at_nat @ F @ Y @ V ) @ Xs )
= ( map_na6577772983117884747at_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_229_map__fun__upd,axiom,
! [Y: list_nat,Xs: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,V: set_Pr1261947904930325089at_nat] :
( ~ ( member_list_nat2 @ Y @ ( set_list_nat2 @ Xs ) )
=> ( ( map_li6003994582982014139at_nat @ ( fun_up5216017410020450626at_nat @ F @ Y @ V ) @ Xs )
= ( map_li6003994582982014139at_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_230_in__set__insert,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
=> ( ( insert_set_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_231_in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_232_in__set__insert,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( insert_list_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_233_list__ex1__iff,axiom,
( list_ex1_set_nat
= ( ^ [P3: set_nat > $o,Xs3: list_set_nat] :
? [X2: set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs3 ) )
& ( P3 @ X2 )
& ! [Y3: set_nat] :
( ( ( member_set_nat2 @ Y3 @ ( set_set_nat2 @ Xs3 ) )
& ( P3 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_234_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P3: nat > $o,Xs3: list_nat] :
? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs3 ) )
& ( P3 @ X2 )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ ( set_nat2 @ Xs3 ) )
& ( P3 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_235_list__ex1__iff,axiom,
( list_ex1_list_nat
= ( ^ [P3: list_nat > $o,Xs3: list_list_nat] :
? [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs3 ) )
& ( P3 @ X2 )
& ! [Y3: list_nat] :
( ( ( member_list_nat2 @ Y3 @ ( set_list_nat2 @ Xs3 ) )
& ( P3 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_236_distinct__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( distinct_nat @ ( insert_nat @ X @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct_insert
thf(fact_237_inj__map,axiom,
! [F: set_nat > nat] :
( ( inj_on2090978070287468649st_nat @ ( map_set_nat_nat @ F ) @ top_to667676211730484604et_nat )
= ( inj_on_set_nat_nat @ F @ top_top_set_set_nat ) ) ).
% inj_map
thf(fact_238_inj__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat] :
( ( inj_on8270031949885980868at_nat @ ( map_li6003994582982014139at_nat @ F ) @ top_to8617644770344506070st_nat )
= ( inj_on7522185085906380110at_nat @ F @ top_top_set_list_nat ) ) ).
% inj_map
thf(fact_239_inj__map,axiom,
! [F: set_nat > set_nat] :
( ( inj_on4452343878340264223et_nat @ ( map_set_nat_set_nat @ F ) @ top_to667676211730484604et_nat )
= ( inj_on4604407203859583615et_nat @ F @ top_top_set_set_nat ) ) ).
% inj_map
thf(fact_240_inj__map,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat] :
( ( inj_on9019006866949024542at_nat @ ( map_se1162299115588061717at_nat @ F ) @ top_to667676211730484604et_nat )
= ( inj_on5888811482125702184at_nat @ F @ top_top_set_set_nat ) ) ).
% inj_map
thf(fact_241_inj__map,axiom,
! [F: nat > set_nat] :
( ( inj_on2669155312630230889et_nat @ ( map_nat_set_nat @ F ) @ top_top_set_list_nat )
= ( inj_on_nat_set_nat @ F @ top_top_set_nat ) ) ).
% inj_map
thf(fact_242_inj__map,axiom,
! [F: nat > nat] :
( ( inj_on3049792774292151987st_nat @ ( map_nat_nat @ F ) @ top_top_set_list_nat )
= ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% inj_map
thf(fact_243_inj__map,axiom,
! [F: nat > set_Pr1261947904930325089at_nat] :
( ( inj_on1193113198435547220at_nat @ ( map_na6577772983117884747at_nat @ F ) @ top_top_set_list_nat )
= ( inj_on7202621048659411806at_nat @ F @ top_top_set_nat ) ) ).
% inj_map
thf(fact_244_inj__mapI,axiom,
! [F: set_nat > nat] :
( ( inj_on_set_nat_nat @ F @ top_top_set_set_nat )
=> ( inj_on2090978070287468649st_nat @ ( map_set_nat_nat @ F ) @ top_to667676211730484604et_nat ) ) ).
% inj_mapI
thf(fact_245_inj__mapI,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat] :
( ( inj_on7522185085906380110at_nat @ F @ top_top_set_list_nat )
=> ( inj_on8270031949885980868at_nat @ ( map_li6003994582982014139at_nat @ F ) @ top_to8617644770344506070st_nat ) ) ).
% inj_mapI
thf(fact_246_inj__mapI,axiom,
! [F: set_nat > set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ top_top_set_set_nat )
=> ( inj_on4452343878340264223et_nat @ ( map_set_nat_set_nat @ F ) @ top_to667676211730484604et_nat ) ) ).
% inj_mapI
thf(fact_247_inj__mapI,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat] :
( ( inj_on5888811482125702184at_nat @ F @ top_top_set_set_nat )
=> ( inj_on9019006866949024542at_nat @ ( map_se1162299115588061717at_nat @ F ) @ top_to667676211730484604et_nat ) ) ).
% inj_mapI
thf(fact_248_inj__mapI,axiom,
! [F: nat > set_nat] :
( ( inj_on_nat_set_nat @ F @ top_top_set_nat )
=> ( inj_on2669155312630230889et_nat @ ( map_nat_set_nat @ F ) @ top_top_set_list_nat ) ) ).
% inj_mapI
thf(fact_249_inj__mapI,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( inj_on3049792774292151987st_nat @ ( map_nat_nat @ F ) @ top_top_set_list_nat ) ) ).
% inj_mapI
thf(fact_250_inj__mapI,axiom,
! [F: nat > set_Pr1261947904930325089at_nat] :
( ( inj_on7202621048659411806at_nat @ F @ top_top_set_nat )
=> ( inj_on1193113198435547220at_nat @ ( map_na6577772983117884747at_nat @ F ) @ top_top_set_list_nat ) ) ).
% inj_mapI
thf(fact_251_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ Xs ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_252_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_set_nat,F: set_nat > nat,Xs: list_set_nat] :
( ( foldin8172238680504804989et_nat @ Less_eq @ Less @ S @ F )
=> ( ( distinct_nat @ ( map_set_nat_nat @ F @ Xs ) )
=> ( distinct_set_nat @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_253_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [Less_eq: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Less: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,S: set_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( foldin5963648469059051996st_nat @ Less_eq @ Less @ S @ F )
=> ( ( distin8719635941469336154at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) )
=> ( distinct_list_nat @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_254_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [Less_eq: set_nat > set_nat > $o,Less: set_nat > set_nat > $o,S: set_set_nat,F: set_nat > set_nat,Xs: list_set_nat] :
( ( foldin2309727365711792691et_nat @ Less_eq @ Less @ S @ F )
=> ( ( distinct_set_nat @ ( map_set_nat_set_nat @ F @ Xs ) )
=> ( distinct_set_nat @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_255_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [Less_eq: set_nat > set_nat > $o,Less: set_nat > set_nat > $o,S: set_nat,F: nat > set_nat,Xs: list_nat] :
( ( foldin4606603775849435773at_nat @ Less_eq @ Less @ S @ F )
=> ( ( distinct_set_nat @ ( map_nat_set_nat @ F @ Xs ) )
=> ( distinct_nat @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_256_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_nat,F: nat > nat,Xs: list_nat] :
( ( foldin8133931898133206727at_nat @ Less_eq @ Less @ S @ F )
=> ( ( distinct_nat @ ( map_nat_nat @ F @ Xs ) )
=> ( distinct_nat @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_257_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [Less_eq: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Less: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,S: set_set_nat,F: set_nat > set_Pr1261947904930325089at_nat,Xs: list_set_nat] :
( ( foldin3406279397012058242et_nat @ Less_eq @ Less @ S @ F )
=> ( ( distin8719635941469336154at_nat @ ( map_se1162299115588061717at_nat @ F @ Xs ) )
=> ( distinct_set_nat @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_258_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [Less_eq: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Less: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,S: set_nat,F: nat > set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( foldin5281791559409726668at_nat @ Less_eq @ Less @ S @ F )
=> ( ( distin8719635941469336154at_nat @ ( map_na6577772983117884747at_nat @ F @ Xs ) )
=> ( distinct_nat @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_259_inj__mapD,axiom,
! [F: set_nat > nat] :
( ( inj_on2090978070287468649st_nat @ ( map_set_nat_nat @ F ) @ top_to667676211730484604et_nat )
=> ( inj_on_set_nat_nat @ F @ top_top_set_set_nat ) ) ).
% inj_mapD
thf(fact_260_inj__mapD,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat] :
( ( inj_on8270031949885980868at_nat @ ( map_li6003994582982014139at_nat @ F ) @ top_to8617644770344506070st_nat )
=> ( inj_on7522185085906380110at_nat @ F @ top_top_set_list_nat ) ) ).
% inj_mapD
thf(fact_261_inj__mapD,axiom,
! [F: set_nat > set_nat] :
( ( inj_on4452343878340264223et_nat @ ( map_set_nat_set_nat @ F ) @ top_to667676211730484604et_nat )
=> ( inj_on4604407203859583615et_nat @ F @ top_top_set_set_nat ) ) ).
% inj_mapD
thf(fact_262_inj__mapD,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat] :
( ( inj_on9019006866949024542at_nat @ ( map_se1162299115588061717at_nat @ F ) @ top_to667676211730484604et_nat )
=> ( inj_on5888811482125702184at_nat @ F @ top_top_set_set_nat ) ) ).
% inj_mapD
thf(fact_263_inj__mapD,axiom,
! [F: nat > set_nat] :
( ( inj_on2669155312630230889et_nat @ ( map_nat_set_nat @ F ) @ top_top_set_list_nat )
=> ( inj_on_nat_set_nat @ F @ top_top_set_nat ) ) ).
% inj_mapD
thf(fact_264_inj__mapD,axiom,
! [F: nat > nat] :
( ( inj_on3049792774292151987st_nat @ ( map_nat_nat @ F ) @ top_top_set_list_nat )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% inj_mapD
thf(fact_265_inj__mapD,axiom,
! [F: nat > set_Pr1261947904930325089at_nat] :
( ( inj_on1193113198435547220at_nat @ ( map_na6577772983117884747at_nat @ F ) @ top_top_set_list_nat )
=> ( inj_on7202621048659411806at_nat @ F @ top_top_set_nat ) ) ).
% inj_mapD
thf(fact_266_injD,axiom,
! [F: set_nat > nat,X: set_nat,Y: set_nat] :
( ( inj_on_set_nat_nat @ F @ top_top_set_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_267_injI,axiom,
! [F: set_nat > nat] :
( ! [X3: set_nat,Y2: set_nat] :
( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) )
=> ( inj_on_set_nat_nat @ F @ top_top_set_set_nat ) ) ).
% injI
thf(fact_268_inj__eq,axiom,
! [F: set_nat > nat,X: set_nat,Y: set_nat] :
( ( inj_on_set_nat_nat @ F @ top_top_set_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_269_inj__def,axiom,
! [F: set_nat > nat] :
( ( inj_on_set_nat_nat @ F @ top_top_set_set_nat )
= ( ! [X2: set_nat,Y3: set_nat] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ).
% inj_def
thf(fact_270_folding__insort__key_Oinj__on,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_set_nat,F: set_nat > nat] :
( ( foldin8172238680504804989et_nat @ Less_eq @ Less @ S @ F )
=> ( inj_on_set_nat_nat @ F @ S ) ) ).
% folding_insort_key.inj_on
thf(fact_271_map__injective,axiom,
! [F: set_nat > nat,Xs: list_set_nat,Ys: list_set_nat] :
( ( ( map_set_nat_nat @ F @ Xs )
= ( map_set_nat_nat @ F @ Ys ) )
=> ( ( inj_on_set_nat_nat @ F @ top_top_set_set_nat )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_272_map__injective,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ Xs )
= ( map_li6003994582982014139at_nat @ F @ Ys ) )
=> ( ( inj_on7522185085906380110at_nat @ F @ top_top_set_list_nat )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_273_map__injective,axiom,
! [F: set_nat > set_nat,Xs: list_set_nat,Ys: list_set_nat] :
( ( ( map_set_nat_set_nat @ F @ Xs )
= ( map_set_nat_set_nat @ F @ Ys ) )
=> ( ( inj_on4604407203859583615et_nat @ F @ top_top_set_set_nat )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_274_map__injective,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,Xs: list_set_nat,Ys: list_set_nat] :
( ( ( map_se1162299115588061717at_nat @ F @ Xs )
= ( map_se1162299115588061717at_nat @ F @ Ys ) )
=> ( ( inj_on5888811482125702184at_nat @ F @ top_top_set_set_nat )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_275_map__injective,axiom,
! [F: nat > set_nat,Xs: list_nat,Ys: list_nat] :
( ( ( map_nat_set_nat @ F @ Xs )
= ( map_nat_set_nat @ F @ Ys ) )
=> ( ( inj_on_nat_set_nat @ F @ top_top_set_nat )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_276_map__injective,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_277_map__injective,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
( ( ( map_na6577772983117884747at_nat @ F @ Xs )
= ( map_na6577772983117884747at_nat @ F @ Ys ) )
=> ( ( inj_on7202621048659411806at_nat @ F @ top_top_set_nat )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_278_count__list__0__iff,axiom,
! [Xs: list_set_nat,X: set_nat] :
( ( ( count_list_set_nat @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_279_count__list__0__iff,axiom,
! [Xs: list_nat,X: nat] :
( ( ( count_list_nat @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_280_count__list__0__iff,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( ( count_list_list_nat @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_281_sup__top__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% sup_top_left
thf(fact_282_sup__top__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% sup_top_right
thf(fact_283_boolean__algebra_Odisj__one__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_284_boolean__algebra_Odisj__one__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_285_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_286_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_287_set__union,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( set_list_nat2 @ ( union_list_nat @ Xs @ Ys ) )
= ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_288_can__select__set__list__ex1,axiom,
! [P2: set_nat > $o,A: list_set_nat] :
( ( can_select_set_nat @ P2 @ ( set_set_nat2 @ A ) )
= ( list_ex1_set_nat @ P2 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_289_can__select__set__list__ex1,axiom,
! [P2: nat > $o,A: list_nat] :
( ( can_select_nat @ P2 @ ( set_nat2 @ A ) )
= ( list_ex1_nat @ P2 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_290_can__select__set__list__ex1,axiom,
! [P2: list_nat > $o,A: list_list_nat] :
( ( can_select_list_nat @ P2 @ ( set_list_nat2 @ A ) )
= ( list_ex1_list_nat @ P2 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_291_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat2 @ C @ B )
=> ( member_nat2 @ C @ A ) )
=> ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_292_UnCI,axiom,
! [C: list_nat,B: set_list_nat,A: set_list_nat] :
( ( ~ ( member_list_nat2 @ C @ B )
=> ( member_list_nat2 @ C @ A ) )
=> ( member_list_nat2 @ C @ ( sup_sup_set_list_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_293_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat2 @ C @ A )
| ( member_nat2 @ C @ B ) ) ) ).
% Un_iff
thf(fact_294_Un__iff,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ ( sup_sup_set_list_nat @ A @ B ) )
= ( ( member_list_nat2 @ C @ A )
| ( member_list_nat2 @ C @ B ) ) ) ).
% Un_iff
thf(fact_295_UNIV__I,axiom,
! [X: list_nat] : ( member_list_nat2 @ X @ top_top_set_list_nat ) ).
% UNIV_I
thf(fact_296_UNIV__I,axiom,
! [X: nat] : ( member_nat2 @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_297_distinct__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
= ( distinct_nat @ Ys ) ) ).
% distinct_union
thf(fact_298_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_299_can__select__def,axiom,
( can_select_nat
= ( ^ [P3: nat > $o,A2: set_nat] :
? [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
& ( P3 @ X2 )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ A2 )
& ( P3 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_300_can__select__def,axiom,
( can_select_list_nat
= ( ^ [P3: list_nat > $o,A2: set_list_nat] :
? [X2: list_nat] :
( ( member_list_nat2 @ X2 @ A2 )
& ( P3 @ X2 )
& ! [Y3: list_nat] :
( ( ( member_list_nat2 @ Y3 @ A2 )
& ( P3 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_301_UNIV__witness,axiom,
? [X3: list_nat] : ( member_list_nat2 @ X3 @ top_top_set_list_nat ) ).
% UNIV_witness
thf(fact_302_UNIV__witness,axiom,
? [X3: nat] : ( member_nat2 @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_303_UNIV__eq__I,axiom,
! [A: set_list_nat] :
( ! [X3: list_nat] : ( member_list_nat2 @ X3 @ A )
=> ( top_top_set_list_nat = A ) ) ).
% UNIV_eq_I
thf(fact_304_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X3: nat] : ( member_nat2 @ X3 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_305_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat2 @ C @ B )
=> ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_306_UnI2,axiom,
! [C: list_nat,B: set_list_nat,A: set_list_nat] :
( ( member_list_nat2 @ C @ B )
=> ( member_list_nat2 @ C @ ( sup_sup_set_list_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_307_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_308_UnI1,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ A )
=> ( member_list_nat2 @ C @ ( sup_sup_set_list_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_309_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ B ) ) ) ).
% UnE
thf(fact_310_UnE,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ ( sup_sup_set_list_nat @ A @ B ) )
=> ( ~ ( member_list_nat2 @ C @ A )
=> ( member_list_nat2 @ C @ B ) ) ) ).
% UnE
thf(fact_311_Un__UNIV__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_312_Un__UNIV__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_313_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_314_iso__tuple__UNIV__I,axiom,
! [X: list_nat] : ( member_list_nat2 @ X @ top_top_set_list_nat ) ).
% iso_tuple_UNIV_I
thf(fact_315_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat2 @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_316_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_317_length__code,axiom,
( size_s3254054031482475050et_nat
= ( gen_length_set_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_318_length__code,axiom,
( size_s8736152011456118867at_nat
= ( gen_le5092146751752969972at_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_319_length__code,axiom,
( size_s3023201423986296836st_nat
= ( gen_length_list_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_320_map__removeAll__inj,axiom,
! [F: set_nat > nat,X: set_nat,Xs: list_set_nat] :
( ( inj_on_set_nat_nat @ F @ top_top_set_set_nat )
=> ( ( map_set_nat_nat @ F @ ( removeAll_set_nat @ X @ Xs ) )
= ( removeAll_nat @ ( F @ X ) @ ( map_set_nat_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_321_map__removeAll__inj,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,X: list_nat,Xs: list_list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ top_top_set_list_nat )
=> ( ( map_li6003994582982014139at_nat @ F @ ( removeAll_list_nat @ X @ Xs ) )
= ( remove5672899571770113645at_nat @ ( F @ X ) @ ( map_li6003994582982014139at_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_322_map__removeAll__inj,axiom,
! [F: set_nat > set_nat,X: set_nat,Xs: list_set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ top_top_set_set_nat )
=> ( ( map_set_nat_set_nat @ F @ ( removeAll_set_nat @ X @ Xs ) )
= ( removeAll_set_nat @ ( F @ X ) @ ( map_set_nat_set_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_323_map__removeAll__inj,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,X: set_nat,Xs: list_set_nat] :
( ( inj_on5888811482125702184at_nat @ F @ top_top_set_set_nat )
=> ( ( map_se1162299115588061717at_nat @ F @ ( removeAll_set_nat @ X @ Xs ) )
= ( remove5672899571770113645at_nat @ ( F @ X ) @ ( map_se1162299115588061717at_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_324_map__removeAll__inj,axiom,
! [F: nat > set_nat,X: nat,Xs: list_nat] :
( ( inj_on_nat_set_nat @ F @ top_top_set_nat )
=> ( ( map_nat_set_nat @ F @ ( removeAll_nat @ X @ Xs ) )
= ( removeAll_set_nat @ ( F @ X ) @ ( map_nat_set_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_325_map__removeAll__inj,axiom,
! [F: nat > nat,X: nat,Xs: list_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( map_nat_nat @ F @ ( removeAll_nat @ X @ Xs ) )
= ( removeAll_nat @ ( F @ X ) @ ( map_nat_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_326_map__removeAll__inj,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,X: nat,Xs: list_nat] :
( ( inj_on7202621048659411806at_nat @ F @ top_top_set_nat )
=> ( ( map_na6577772983117884747at_nat @ F @ ( removeAll_nat @ X @ Xs ) )
= ( remove5672899571770113645at_nat @ ( F @ X ) @ ( map_na6577772983117884747at_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_327_distinct__set__subseqs,axiom,
! [Xs: list_set_nat] :
( ( distinct_set_nat @ Xs )
=> ( distinct_set_set_nat @ ( map_li3330360646112351000et_nat @ set_set_nat2 @ ( subseqs_set_nat @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_328_distinct__set__subseqs,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_set_nat @ ( map_list_nat_set_nat @ set_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_329_distinct__set__subseqs,axiom,
! [Xs: list_list_nat] :
( ( distinct_list_nat @ Xs )
=> ( distin4912746231890992349st_nat @ ( map_li2355978560338012748st_nat @ set_list_nat2 @ ( subseqs_list_nat @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_330_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).
% atLeast_upt
thf(fact_331_removeAll__id,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ~ ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
=> ( ( removeAll_set_nat @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_332_removeAll__id,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_333_removeAll__id,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( removeAll_list_nat @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_334_set__rotate1,axiom,
! [Xs: list_set_nat] :
( ( set_set_nat2 @ ( rotate1_set_nat @ Xs ) )
= ( set_set_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_335_set__rotate1,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( rotate1_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_336_set__rotate1,axiom,
! [Xs: list_list_nat] :
( ( set_list_nat2 @ ( rotate1_list_nat @ Xs ) )
= ( set_list_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_337_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_338_length__rotate1,axiom,
! [Xs: list_set_nat] :
( ( size_s3254054031482475050et_nat @ ( rotate1_set_nat @ Xs ) )
= ( size_s3254054031482475050et_nat @ Xs ) ) ).
% length_rotate1
thf(fact_339_length__rotate1,axiom,
! [Xs: list_s1210847774152347623at_nat] :
( ( size_s8736152011456118867at_nat @ ( rotate4238613965387346100at_nat @ Xs ) )
= ( size_s8736152011456118867at_nat @ Xs ) ) ).
% length_rotate1
thf(fact_340_length__rotate1,axiom,
! [Xs: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( rotate1_list_nat @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_rotate1
thf(fact_341_distinct1__rotate,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ ( rotate1_nat @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct1_rotate
thf(fact_342_distinct__upt,axiom,
! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).
% distinct_upt
thf(fact_343_subseqs__refl,axiom,
! [Xs: list_nat] : ( member_list_nat2 @ Xs @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ).
% subseqs_refl
thf(fact_344_rotate1__map,axiom,
! [F: set_nat > nat,Xs: list_set_nat] :
( ( rotate1_nat @ ( map_set_nat_nat @ F @ Xs ) )
= ( map_set_nat_nat @ F @ ( rotate1_set_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_345_rotate1__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( rotate4238613965387346100at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) )
= ( map_li6003994582982014139at_nat @ F @ ( rotate1_list_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_346_rotate1__map,axiom,
! [F: set_nat > set_nat,Xs: list_set_nat] :
( ( rotate1_set_nat @ ( map_set_nat_set_nat @ F @ Xs ) )
= ( map_set_nat_set_nat @ F @ ( rotate1_set_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_347_rotate1__map,axiom,
! [F: nat > set_nat,Xs: list_nat] :
( ( rotate1_set_nat @ ( map_nat_set_nat @ F @ Xs ) )
= ( map_nat_set_nat @ F @ ( rotate1_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_348_rotate1__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( rotate1_nat @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( rotate1_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_349_rotate1__map,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,Xs: list_set_nat] :
( ( rotate4238613965387346100at_nat @ ( map_se1162299115588061717at_nat @ F @ Xs ) )
= ( map_se1162299115588061717at_nat @ F @ ( rotate1_set_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_350_rotate1__map,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( rotate4238613965387346100at_nat @ ( map_na6577772983117884747at_nat @ F @ Xs ) )
= ( map_na6577772983117884747at_nat @ F @ ( rotate1_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_351_distinct__removeAll,axiom,
! [Xs: list_nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( removeAll_nat @ X @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_352_subseqs__distinctD,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( member_list_nat2 @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
=> ( ( distinct_nat @ Xs )
=> ( distinct_nat @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_353_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_354_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_355_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_356_top__empty__eq,axiom,
( top_top_list_nat_o
= ( ^ [X2: list_nat] : ( member_list_nat2 @ X2 @ top_top_set_list_nat ) ) ) ).
% top_empty_eq
thf(fact_357_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X2: nat] : ( member_nat2 @ X2 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_358_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_359_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_360_remove__code_I1_J,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( remove_list_nat @ X @ ( set_list_nat2 @ Xs ) )
= ( set_list_nat2 @ ( removeAll_list_nat @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_361_map__removeAll__inj__on,axiom,
! [F: set_nat > nat,X: set_nat,Xs: list_set_nat] :
( ( inj_on_set_nat_nat @ F @ ( insert_set_nat2 @ X @ ( set_set_nat2 @ Xs ) ) )
=> ( ( map_set_nat_nat @ F @ ( removeAll_set_nat @ X @ Xs ) )
= ( removeAll_nat @ ( F @ X ) @ ( map_set_nat_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_362_map__removeAll__inj__on,axiom,
! [F: set_nat > set_nat,X: set_nat,Xs: list_set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ ( insert_set_nat2 @ X @ ( set_set_nat2 @ Xs ) ) )
=> ( ( map_set_nat_set_nat @ F @ ( removeAll_set_nat @ X @ Xs ) )
= ( removeAll_set_nat @ ( F @ X ) @ ( map_set_nat_set_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_363_map__removeAll__inj__on,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,X: set_nat,Xs: list_set_nat] :
( ( inj_on5888811482125702184at_nat @ F @ ( insert_set_nat2 @ X @ ( set_set_nat2 @ Xs ) ) )
=> ( ( map_se1162299115588061717at_nat @ F @ ( removeAll_set_nat @ X @ Xs ) )
= ( remove5672899571770113645at_nat @ ( F @ X ) @ ( map_se1162299115588061717at_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_364_map__removeAll__inj__on,axiom,
! [F: nat > set_nat,X: nat,Xs: list_nat] :
( ( inj_on_nat_set_nat @ F @ ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) )
=> ( ( map_nat_set_nat @ F @ ( removeAll_nat @ X @ Xs ) )
= ( removeAll_set_nat @ ( F @ X ) @ ( map_nat_set_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_365_map__removeAll__inj__on,axiom,
! [F: nat > nat,X: nat,Xs: list_nat] :
( ( inj_on_nat_nat @ F @ ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) )
=> ( ( map_nat_nat @ F @ ( removeAll_nat @ X @ Xs ) )
= ( removeAll_nat @ ( F @ X ) @ ( map_nat_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_366_map__removeAll__inj__on,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,X: nat,Xs: list_nat] :
( ( inj_on7202621048659411806at_nat @ F @ ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) )
=> ( ( map_na6577772983117884747at_nat @ F @ ( removeAll_nat @ X @ Xs ) )
= ( remove5672899571770113645at_nat @ ( F @ X ) @ ( map_na6577772983117884747at_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_367_map__removeAll__inj__on,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,X: list_nat,Xs: list_list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ ( insert_list_nat2 @ X @ ( set_list_nat2 @ Xs ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ ( removeAll_list_nat @ X @ Xs ) )
= ( remove5672899571770113645at_nat @ ( F @ X ) @ ( map_li6003994582982014139at_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_368_char_Osize_I2_J,axiom,
! [X1: $o,X22: $o,X32: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o] :
( ( size_size_char @ ( char2 @ X1 @ X22 @ X32 @ X4 @ X5 @ X6 @ X7 @ X8 ) )
= zero_zero_nat ) ).
% char.size(2)
thf(fact_369_length__removeAll__less,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_370_length__removeAll__less,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_s3254054031482475050et_nat @ ( removeAll_set_nat @ X @ Xs ) ) @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_371_length__removeAll__less,axiom,
! [X: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat] :
( ( member2643936169264416010at_nat @ X @ ( set_se5049602875457034614at_nat @ Xs ) )
=> ( ord_less_nat @ ( size_s8736152011456118867at_nat @ ( remove5672899571770113645at_nat @ X @ Xs ) ) @ ( size_s8736152011456118867at_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_372_length__removeAll__less,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_s3023201423986296836st_nat @ ( removeAll_list_nat @ X @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_373_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_374_insertCI,axiom,
! [A4: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat2 @ A4 @ B )
=> ( A4 = B2 ) )
=> ( member_nat2 @ A4 @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_375_insertCI,axiom,
! [A4: list_nat,B: set_list_nat,B2: list_nat] :
( ( ~ ( member_list_nat2 @ A4 @ B )
=> ( A4 = B2 ) )
=> ( member_list_nat2 @ A4 @ ( insert_list_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_376_insert__iff,axiom,
! [A4: nat,B2: nat,A: set_nat] :
( ( member_nat2 @ A4 @ ( insert_nat2 @ B2 @ A ) )
= ( ( A4 = B2 )
| ( member_nat2 @ A4 @ A ) ) ) ).
% insert_iff
thf(fact_377_insert__iff,axiom,
! [A4: list_nat,B2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A4 @ ( insert_list_nat2 @ B2 @ A ) )
= ( ( A4 = B2 )
| ( member_list_nat2 @ A4 @ A ) ) ) ).
% insert_iff
thf(fact_378_char_Oinject,axiom,
! [X1: $o,X22: $o,X32: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o,Y1: $o,Y22: $o,Y32: $o,Y4: $o,Y5: $o,Y6: $o,Y7: $o,Y8: $o] :
( ( ( char2 @ X1 @ X22 @ X32 @ X4 @ X5 @ X6 @ X7 @ X8 )
= ( char2 @ Y1 @ Y22 @ Y32 @ Y4 @ Y5 @ Y6 @ Y7 @ Y8 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 )
& ( X32 = Y32 )
& ( X4 = Y4 )
& ( X5 = Y5 )
& ( X6 = Y6 )
& ( X7 = Y7 )
& ( X8 = Y8 ) ) ) ).
% char.inject
thf(fact_379_member__remove,axiom,
! [X: nat,Y: nat,A: set_nat] :
( ( member_nat2 @ X @ ( remove_nat @ Y @ A ) )
= ( ( member_nat2 @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_380_member__remove,axiom,
! [X: list_nat,Y: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ X @ ( remove_list_nat @ Y @ A ) )
= ( ( member_list_nat2 @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_381_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_382_bot__nat__0_Onot__eq__extremum,axiom,
! [A4: nat] :
( ( A4 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A4 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_383_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_384_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_385_lessThan__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I @ K ) ) ).
% lessThan_iff
thf(fact_386_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat2 @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_387_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_388_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_389_List_Oset__insert,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( set_list_nat2 @ ( insert_list_nat @ X @ Xs ) )
= ( insert_list_nat2 @ X @ ( set_list_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_390_char_Oexhaust,axiom,
! [Y: char] :
~ ! [X12: $o,X23: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( Y
!= ( char2 @ X12 @ X23 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) ) ).
% char.exhaust
thf(fact_391_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_392_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_393_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_394_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_395_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_396_order_Oasym,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A4 ) ) ).
% order.asym
thf(fact_397_order_Oasym,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ B2 )
=> ~ ( ord_less_int @ B2 @ A4 ) ) ).
% order.asym
thf(fact_398_ord__eq__less__trans,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( A4 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_399_ord__eq__less__trans,axiom,
! [A4: int,B2: int,C: int] :
( ( A4 = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_400_ord__less__eq__trans,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_401_ord__less__eq__trans,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_402_less__induct,axiom,
! [P2: nat > $o,A4: nat] :
( ! [X3: nat] :
( ! [Y9: nat] :
( ( ord_less_nat @ Y9 @ X3 )
=> ( P2 @ Y9 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A4 ) ) ).
% less_induct
thf(fact_403_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_404_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_405_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_406_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_407_dual__order_Oasym,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ~ ( ord_less_nat @ A4 @ B2 ) ) ).
% dual_order.asym
thf(fact_408_dual__order_Oasym,axiom,
! [B2: int,A4: int] :
( ( ord_less_int @ B2 @ A4 )
=> ~ ( ord_less_int @ A4 @ B2 ) ) ).
% dual_order.asym
thf(fact_409_dual__order_Oirrefl,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_410_dual__order_Oirrefl,axiom,
! [A4: int] :
~ ( ord_less_int @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_411_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X9: nat] : ( P4 @ X9 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_412_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A4: nat,B2: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P2 @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A4 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_413_linorder__less__wlog,axiom,
! [P2: int > int > $o,A4: int,B2: int] :
( ! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: int] : ( P2 @ A3 @ A3 )
=> ( ! [A3: int,B3: int] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A4 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_414_order_Ostrict__trans,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% order.strict_trans
thf(fact_415_order_Ostrict__trans,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% order.strict_trans
thf(fact_416_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_417_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_418_dual__order_Ostrict__trans,axiom,
! [B2: nat,A4: nat,C: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_419_dual__order_Ostrict__trans,axiom,
! [B2: int,A4: int,C: int] :
( ( ord_less_int @ B2 @ A4 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_420_order_Ostrict__implies__not__eq,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( A4 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_421_order_Ostrict__implies__not__eq,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( A4 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_422_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ( A4 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_423_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A4: int] :
( ( ord_less_int @ B2 @ A4 )
=> ( A4 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_424_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_425_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_426_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_427_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_428_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_429_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_430_order__less__asym_H,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A4 ) ) ).
% order_less_asym'
thf(fact_431_order__less__asym_H,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ B2 )
=> ~ ( ord_less_int @ B2 @ A4 ) ) ).
% order_less_asym'
thf(fact_432_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_433_order__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_434_ord__eq__less__subst,axiom,
! [A4: nat,F: nat > nat,B2: nat,C: nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_435_ord__eq__less__subst,axiom,
! [A4: int,F: nat > int,B2: nat,C: nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_436_ord__eq__less__subst,axiom,
! [A4: nat,F: int > nat,B2: int,C: int] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_437_ord__eq__less__subst,axiom,
! [A4: int,F: int > int,B2: int,C: int] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_438_ord__less__eq__subst,axiom,
! [A4: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_439_ord__less__eq__subst,axiom,
! [A4: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_440_ord__less__eq__subst,axiom,
! [A4: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_441_ord__less__eq__subst,axiom,
! [A4: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_442_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_443_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_444_order__less__subst1,axiom,
! [A4: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_445_order__less__subst1,axiom,
! [A4: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_446_order__less__subst1,axiom,
! [A4: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_447_order__less__subst1,axiom,
! [A4: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_448_order__less__subst2,axiom,
! [A4: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_449_order__less__subst2,axiom,
! [A4: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_450_order__less__subst2,axiom,
! [A4: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_451_order__less__subst2,axiom,
! [A4: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_452_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_453_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_454_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P2: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_455_order__less__imp__triv,axiom,
! [X: int,Y: int,P2: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_456_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_457_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_458_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_459_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_460_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_461_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_462_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_463_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_464_lessThan__strict__subset__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M2 ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_465_lessThan__strict__subset__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_466_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_467_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_468_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_469_less__not__refl3,axiom,
! [S3: nat,T: nat] :
( ( ord_less_nat @ S3 @ T )
=> ( S3 != T ) ) ).
% less_not_refl3
thf(fact_470_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_471_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_472_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_473_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_474_insertE,axiom,
! [A4: nat,B2: nat,A: set_nat] :
( ( member_nat2 @ A4 @ ( insert_nat2 @ B2 @ A ) )
=> ( ( A4 != B2 )
=> ( member_nat2 @ A4 @ A ) ) ) ).
% insertE
thf(fact_475_insertE,axiom,
! [A4: list_nat,B2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A4 @ ( insert_list_nat2 @ B2 @ A ) )
=> ( ( A4 != B2 )
=> ( member_list_nat2 @ A4 @ A ) ) ) ).
% insertE
thf(fact_476_insertI1,axiom,
! [A4: nat,B: set_nat] : ( member_nat2 @ A4 @ ( insert_nat2 @ A4 @ B ) ) ).
% insertI1
thf(fact_477_insertI1,axiom,
! [A4: list_nat,B: set_list_nat] : ( member_list_nat2 @ A4 @ ( insert_list_nat2 @ A4 @ B ) ) ).
% insertI1
thf(fact_478_insertI2,axiom,
! [A4: nat,B: set_nat,B2: nat] :
( ( member_nat2 @ A4 @ B )
=> ( member_nat2 @ A4 @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_479_insertI2,axiom,
! [A4: list_nat,B: set_list_nat,B2: list_nat] :
( ( member_list_nat2 @ A4 @ B )
=> ( member_list_nat2 @ A4 @ ( insert_list_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_480_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat2 @ X @ A )
=> ~ ! [B4: set_nat] :
( ( A
= ( insert_nat2 @ X @ B4 ) )
=> ( member_nat2 @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_481_Set_Oset__insert,axiom,
! [X: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ X @ A )
=> ~ ! [B4: set_list_nat] :
( ( A
= ( insert_list_nat2 @ X @ B4 ) )
=> ( member_list_nat2 @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_482_insert__ident,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat2 @ X @ A )
=> ( ~ ( member_nat2 @ X @ B )
=> ( ( ( insert_nat2 @ X @ A )
= ( insert_nat2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_483_insert__ident,axiom,
! [X: list_nat,A: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ X @ A )
=> ( ~ ( member_list_nat2 @ X @ B )
=> ( ( ( insert_list_nat2 @ X @ A )
= ( insert_list_nat2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_484_insert__absorb,axiom,
! [A4: nat,A: set_nat] :
( ( member_nat2 @ A4 @ A )
=> ( ( insert_nat2 @ A4 @ A )
= A ) ) ).
% insert_absorb
thf(fact_485_insert__absorb,axiom,
! [A4: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A4 @ A )
=> ( ( insert_list_nat2 @ A4 @ A )
= A ) ) ).
% insert_absorb
thf(fact_486_insert__eq__iff,axiom,
! [A4: nat,A: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat2 @ A4 @ A )
=> ( ~ ( member_nat2 @ B2 @ B )
=> ( ( ( insert_nat2 @ A4 @ A )
= ( insert_nat2 @ B2 @ B ) )
= ( ( ( A4 = B2 )
=> ( A = B ) )
& ( ( A4 != B2 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat2 @ B2 @ C3 ) )
& ~ ( member_nat2 @ B2 @ C3 )
& ( B
= ( insert_nat2 @ A4 @ C3 ) )
& ~ ( member_nat2 @ A4 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_487_insert__eq__iff,axiom,
! [A4: list_nat,A: set_list_nat,B2: list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ A4 @ A )
=> ( ~ ( member_list_nat2 @ B2 @ B )
=> ( ( ( insert_list_nat2 @ A4 @ A )
= ( insert_list_nat2 @ B2 @ B ) )
= ( ( ( A4 = B2 )
=> ( A = B ) )
& ( ( A4 != B2 )
=> ? [C3: set_list_nat] :
( ( A
= ( insert_list_nat2 @ B2 @ C3 ) )
& ~ ( member_list_nat2 @ B2 @ C3 )
& ( B
= ( insert_list_nat2 @ A4 @ C3 ) )
& ~ ( member_list_nat2 @ A4 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_488_mk__disjoint__insert,axiom,
! [A4: nat,A: set_nat] :
( ( member_nat2 @ A4 @ A )
=> ? [B4: set_nat] :
( ( A
= ( insert_nat2 @ A4 @ B4 ) )
& ~ ( member_nat2 @ A4 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_489_mk__disjoint__insert,axiom,
! [A4: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A4 @ A )
=> ? [B4: set_list_nat] :
( ( A
= ( insert_list_nat2 @ A4 @ B4 ) )
& ~ ( member_list_nat2 @ A4 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_490_union__set__fold,axiom,
! [Xs: list_set_nat,A: set_set_nat] :
( ( sup_sup_set_set_nat @ ( set_set_nat2 @ Xs ) @ A )
= ( fold_s4794219702148550607et_nat @ insert_set_nat2 @ Xs @ A ) ) ).
% union_set_fold
thf(fact_491_union__set__fold,axiom,
! [Xs: list_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ A )
= ( fold_nat_set_nat @ insert_nat2 @ Xs @ A ) ) ).
% union_set_fold
thf(fact_492_union__set__fold,axiom,
! [Xs: list_list_nat,A: set_list_nat] :
( ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ A )
= ( fold_l59423398878476163st_nat @ insert_list_nat2 @ Xs @ A ) ) ).
% union_set_fold
thf(fact_493_char_Osize__gen,axiom,
! [X1: $o,X22: $o,X32: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o] :
( ( size_char @ ( char2 @ X1 @ X22 @ X32 @ X4 @ X5 @ X6 @ X7 @ X8 ) )
= zero_zero_nat ) ).
% char.size_gen
thf(fact_494_insert__UNIV,axiom,
! [X: nat] :
( ( insert_nat2 @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_495_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_496_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_497_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_498_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_499_top_Onot__eq__extremum,axiom,
! [A4: set_nat] :
( ( A4 != top_top_set_nat )
= ( ord_less_set_nat @ A4 @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_500_top_Oextremum__strict,axiom,
! [A4: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A4 ) ).
% top.extremum_strict
thf(fact_501_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B2: nat,A4: nat] :
( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_502_sup_Ostrict__coboundedI2,axiom,
! [C: int,B2: int,A4: int] :
( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ ( sup_sup_int @ A4 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_503_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A4: nat,B2: nat] :
( ( ord_less_nat @ C @ A4 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_504_sup_Ostrict__coboundedI1,axiom,
! [C: int,A4: int,B2: int] :
( ( ord_less_int @ C @ A4 )
=> ( ord_less_int @ C @ ( sup_sup_int @ A4 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_505_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( A5
= ( sup_sup_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_506_sup_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B5: int,A5: int] :
( ( A5
= ( sup_sup_int @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_507_sup_Ostrict__boundedE,axiom,
! [B2: nat,C: nat,A4: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C ) @ A4 )
=> ~ ( ( ord_less_nat @ B2 @ A4 )
=> ~ ( ord_less_nat @ C @ A4 ) ) ) ).
% sup.strict_boundedE
thf(fact_508_sup_Ostrict__boundedE,axiom,
! [B2: int,C: int,A4: int] :
( ( ord_less_int @ ( sup_sup_int @ B2 @ C ) @ A4 )
=> ~ ( ( ord_less_int @ B2 @ A4 )
=> ~ ( ord_less_int @ C @ A4 ) ) ) ).
% sup.strict_boundedE
thf(fact_509_sup_Oabsorb4,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( sup_sup_nat @ A4 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_510_sup_Oabsorb4,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( sup_sup_int @ A4 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_511_sup_Oabsorb3,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ( ( sup_sup_nat @ A4 @ B2 )
= A4 ) ) ).
% sup.absorb3
thf(fact_512_sup_Oabsorb3,axiom,
! [B2: int,A4: int] :
( ( ord_less_int @ B2 @ A4 )
=> ( ( sup_sup_int @ A4 @ B2 )
= A4 ) ) ).
% sup.absorb3
thf(fact_513_less__supI2,axiom,
! [X: nat,B2: nat,A4: nat] :
( ( ord_less_nat @ X @ B2 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% less_supI2
thf(fact_514_less__supI2,axiom,
! [X: int,B2: int,A4: int] :
( ( ord_less_int @ X @ B2 )
=> ( ord_less_int @ X @ ( sup_sup_int @ A4 @ B2 ) ) ) ).
% less_supI2
thf(fact_515_less__supI1,axiom,
! [X: nat,A4: nat,B2: nat] :
( ( ord_less_nat @ X @ A4 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% less_supI1
thf(fact_516_less__supI1,axiom,
! [X: int,A4: int,B2: int] :
( ( ord_less_int @ X @ A4 )
=> ( ord_less_int @ X @ ( sup_sup_int @ A4 @ B2 ) ) ) ).
% less_supI1
thf(fact_517_bot__nat__0_Oextremum__strict,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_518_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_519_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_520_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_521_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_522_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_523_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_524_length__induct,axiom,
! [P2: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_525_length__induct,axiom,
! [P2: list_set_nat > $o,Xs: list_set_nat] :
( ! [Xs2: list_set_nat] :
( ! [Ys2: list_set_nat] :
( ( ord_less_nat @ ( size_s3254054031482475050et_nat @ Ys2 ) @ ( size_s3254054031482475050et_nat @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_526_length__induct,axiom,
! [P2: list_s1210847774152347623at_nat > $o,Xs: list_s1210847774152347623at_nat] :
( ! [Xs2: list_s1210847774152347623at_nat] :
( ! [Ys2: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ ( size_s8736152011456118867at_nat @ Ys2 ) @ ( size_s8736152011456118867at_nat @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_527_length__induct,axiom,
! [P2: list_list_nat > $o,Xs: list_list_nat] :
( ! [Xs2: list_list_nat] :
( ! [Ys2: list_list_nat] :
( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ Ys2 ) @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_528_rgf__limit__ge,axiom,
! [Y: nat,Xs: list_nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ Y @ ( equiva5889994315859557365_limit @ Xs ) ) ) ).
% rgf_limit_ge
thf(fact_529_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_530_length__pos__if__in__set,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_531_length__pos__if__in__set,axiom,
! [X: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat] :
( ( member2643936169264416010at_nat @ X @ ( set_se5049602875457034614at_nat @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s8736152011456118867at_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_532_length__pos__if__in__set,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_533_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_534_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_535_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_536_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_537_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs: list_nat] :
( ( member_list_nat2 @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_538_length__n__lists__elem,axiom,
! [Ys: list_set_nat,N: nat,Xs: list_set_nat] :
( ( member_list_set_nat @ Ys @ ( set_list_set_nat2 @ ( n_lists_set_nat @ N @ Xs ) ) )
=> ( ( size_s3254054031482475050et_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_539_length__n__lists__elem,axiom,
! [Ys: list_s1210847774152347623at_nat,N: nat,Xs: list_s1210847774152347623at_nat] :
( ( member2758946086501665296at_nat @ Ys @ ( set_li2149138424243681404at_nat @ ( n_list4589530985885001583at_nat @ N @ Xs ) ) )
=> ( ( size_s8736152011456118867at_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_540_length__n__lists__elem,axiom,
! [Ys: list_list_nat,N: nat,Xs: list_list_nat] :
( ( member_list_list_nat @ Ys @ ( set_list_list_nat2 @ ( n_lists_list_nat @ N @ Xs ) ) )
=> ( ( size_s3023201423986296836st_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_541_distinct__product__lists,axiom,
! [Xss: list_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xss ) )
=> ( distinct_nat @ X3 ) )
=> ( distinct_list_nat @ ( product_lists_nat @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_542_in__set__product__lists__length,axiom,
! [Xs: list_nat,Xss: list_list_nat] :
( ( member_list_nat2 @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_543_in__set__product__lists__length,axiom,
! [Xs: list_set_nat,Xss: list_list_set_nat] :
( ( member_list_set_nat @ Xs @ ( set_list_set_nat2 @ ( produc8109398739672286679et_nat @ Xss ) ) )
=> ( ( size_s3254054031482475050et_nat @ Xs )
= ( size_s8312130442339763642et_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_544_in__set__product__lists__length,axiom,
! [Xs: list_s1210847774152347623at_nat,Xss: list_l3822697302700470509at_nat] :
( ( member2758946086501665296at_nat @ Xs @ ( set_li2149138424243681404at_nat @ ( produc5476187143726666448at_nat @ Xss ) ) )
=> ( ( size_s8736152011456118867at_nat @ Xs )
= ( size_s3876073854528961369at_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_545_in__set__product__lists__length,axiom,
! [Xs: list_list_nat,Xss: list_list_list_nat] :
( ( member_list_list_nat @ Xs @ ( set_list_list_nat2 @ ( produc6783906451316923569st_nat @ Xss ) ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s6248950052170075156st_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_546_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_547_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_548_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_549_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_550_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_551_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_552_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_553_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_554_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_555_psubsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ B ) ) ) ).
% psubsetD
thf(fact_556_psubsetD,axiom,
! [A: set_list_nat,B: set_list_nat,C: list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B )
=> ( ( member_list_nat2 @ C @ A )
=> ( member_list_nat2 @ C @ B ) ) ) ).
% psubsetD
thf(fact_557_inj__on__of__nat,axiom,
! [N4: set_nat] : ( inj_on_nat_int @ semiri1314217659103216013at_int @ N4 ) ).
% inj_on_of_nat
thf(fact_558_inj__of__nat,axiom,
inj_on_nat_int @ semiri1314217659103216013at_int @ top_top_set_nat ).
% inj_of_nat
thf(fact_559_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_560_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_561_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_562_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_563_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_564_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_565_distinct__n__lists,axiom,
! [Xs: list_nat,N: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_list_nat @ ( n_lists_nat @ N @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_566_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_567_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_568_minus__set__fold,axiom,
! [A: set_set_nat,Xs: list_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ ( set_set_nat2 @ Xs ) )
= ( fold_s4794219702148550607et_nat @ remove_set_nat @ Xs @ A ) ) ).
% minus_set_fold
thf(fact_569_minus__set__fold,axiom,
! [A: set_nat,Xs: list_nat] :
( ( minus_minus_set_nat @ A @ ( set_nat2 @ Xs ) )
= ( fold_nat_set_nat @ remove_nat @ Xs @ A ) ) ).
% minus_set_fold
thf(fact_570_minus__set__fold,axiom,
! [A: set_list_nat,Xs: list_list_nat] :
( ( minus_7954133019191499631st_nat @ A @ ( set_list_nat2 @ Xs ) )
= ( fold_l59423398878476163st_nat @ remove_list_nat @ Xs @ A ) ) ).
% minus_set_fold
thf(fact_571_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_572_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_573_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_574_length__greater__0__conv,axiom,
! [Xs: list_set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) )
= ( Xs != nil_set_nat ) ) ).
% length_greater_0_conv
thf(fact_575_length__greater__0__conv,axiom,
! [Xs: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s8736152011456118867at_nat @ Xs ) )
= ( Xs != nil_se357566008730718055at_nat ) ) ).
% length_greater_0_conv
thf(fact_576_length__greater__0__conv,axiom,
! [Xs: list_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) )
= ( Xs != nil_list_nat ) ) ).
% length_greater_0_conv
thf(fact_577_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ A )
=> ( ( member_nat2 @ C @ B )
=> ( member_nat2 @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_578_IntI,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ A )
=> ( ( member_list_nat2 @ C @ B )
=> ( member_list_nat2 @ C @ ( inf_inf_set_list_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_579_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat2 @ C @ A )
& ( member_nat2 @ C @ B ) ) ) ).
% Int_iff
thf(fact_580_Int__iff,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ ( inf_inf_set_list_nat @ A @ B ) )
= ( ( member_list_nat2 @ C @ A )
& ( member_list_nat2 @ C @ B ) ) ) ).
% Int_iff
thf(fact_581_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ A )
=> ( ~ ( member_nat2 @ C @ B )
=> ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_582_DiffI,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ A )
=> ( ~ ( member_list_nat2 @ C @ B )
=> ( member_list_nat2 @ C @ ( minus_7954133019191499631st_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_583_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat2 @ C @ A )
& ~ ( member_nat2 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_584_Diff__iff,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ ( minus_7954133019191499631st_nat @ A @ B ) )
= ( ( member_list_nat2 @ C @ A )
& ~ ( member_list_nat2 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_585_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A4: nat] :
( ( minus_minus_nat @ A4 @ A4 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_586_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A4: int] :
( ( minus_minus_int @ A4 @ A4 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_587_diff__zero,axiom,
! [A4: nat] :
( ( minus_minus_nat @ A4 @ zero_zero_nat )
= A4 ) ).
% diff_zero
thf(fact_588_diff__zero,axiom,
! [A4: int] :
( ( minus_minus_int @ A4 @ zero_zero_int )
= A4 ) ).
% diff_zero
thf(fact_589_zero__diff,axiom,
! [A4: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A4 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_590_diff__0__right,axiom,
! [A4: int] :
( ( minus_minus_int @ A4 @ zero_zero_int )
= A4 ) ).
% diff_0_right
thf(fact_591_diff__self,axiom,
! [A4: int] :
( ( minus_minus_int @ A4 @ A4 )
= zero_zero_int ) ).
% diff_self
thf(fact_592_inf__top_Oright__neutral,axiom,
! [A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ top_top_set_nat )
= A4 ) ).
% inf_top.right_neutral
thf(fact_593_inf__top_Oneutr__eq__iff,axiom,
! [A4: set_nat,B2: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A4 @ B2 ) )
= ( ( A4 = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_594_inf__top_Oleft__neutral,axiom,
! [A4: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A4 )
= A4 ) ).
% inf_top.left_neutral
thf(fact_595_inf__top_Oeq__neutr__iff,axiom,
! [A4: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A4 @ B2 )
= top_top_set_nat )
= ( ( A4 = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_596_top__eq__inf__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X @ Y ) )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_597_inf__eq__top__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= top_top_set_nat )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_598_inf__top__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% inf_top_right
thf(fact_599_inf__top__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X )
= X ) ).
% inf_top_left
thf(fact_600_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_601_Int__UNIV,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_602_Int__insert__left__if0,axiom,
! [A4: nat,C4: set_nat,B: set_nat] :
( ~ ( member_nat2 @ A4 @ C4 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A4 @ B ) @ C4 )
= ( inf_inf_set_nat @ B @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_603_Int__insert__left__if0,axiom,
! [A4: list_nat,C4: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ A4 @ C4 )
=> ( ( inf_inf_set_list_nat @ ( insert_list_nat2 @ A4 @ B ) @ C4 )
= ( inf_inf_set_list_nat @ B @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_604_Int__insert__left__if1,axiom,
! [A4: nat,C4: set_nat,B: set_nat] :
( ( member_nat2 @ A4 @ C4 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A4 @ B ) @ C4 )
= ( insert_nat2 @ A4 @ ( inf_inf_set_nat @ B @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_605_Int__insert__left__if1,axiom,
! [A4: list_nat,C4: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ A4 @ C4 )
=> ( ( inf_inf_set_list_nat @ ( insert_list_nat2 @ A4 @ B ) @ C4 )
= ( insert_list_nat2 @ A4 @ ( inf_inf_set_list_nat @ B @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_606_Int__insert__right__if0,axiom,
! [A4: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat2 @ A4 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat2 @ A4 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_607_Int__insert__right__if0,axiom,
! [A4: list_nat,A: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ A4 @ A )
=> ( ( inf_inf_set_list_nat @ A @ ( insert_list_nat2 @ A4 @ B ) )
= ( inf_inf_set_list_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_608_Int__insert__right__if1,axiom,
! [A4: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ A4 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat2 @ A4 @ B ) )
= ( insert_nat2 @ A4 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_609_Int__insert__right__if1,axiom,
! [A4: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ A4 @ A )
=> ( ( inf_inf_set_list_nat @ A @ ( insert_list_nat2 @ A4 @ B ) )
= ( insert_list_nat2 @ A4 @ ( inf_inf_set_list_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_610_list_Omap__disc__iff,axiom,
! [F: set_nat > nat,A4: list_set_nat] :
( ( ( map_set_nat_nat @ F @ A4 )
= nil_nat )
= ( A4 = nil_set_nat ) ) ).
% list.map_disc_iff
thf(fact_611_list_Omap__disc__iff,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,A4: list_list_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ A4 )
= nil_se357566008730718055at_nat )
= ( A4 = nil_list_nat ) ) ).
% list.map_disc_iff
thf(fact_612_list_Omap__disc__iff,axiom,
! [F: set_nat > set_nat,A4: list_set_nat] :
( ( ( map_set_nat_set_nat @ F @ A4 )
= nil_set_nat )
= ( A4 = nil_set_nat ) ) ).
% list.map_disc_iff
thf(fact_613_list_Omap__disc__iff,axiom,
! [F: nat > set_nat,A4: list_nat] :
( ( ( map_nat_set_nat @ F @ A4 )
= nil_set_nat )
= ( A4 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_614_list_Omap__disc__iff,axiom,
! [F: nat > nat,A4: list_nat] :
( ( ( map_nat_nat @ F @ A4 )
= nil_nat )
= ( A4 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_615_list_Omap__disc__iff,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,A4: list_set_nat] :
( ( ( map_se1162299115588061717at_nat @ F @ A4 )
= nil_se357566008730718055at_nat )
= ( A4 = nil_set_nat ) ) ).
% list.map_disc_iff
thf(fact_616_list_Omap__disc__iff,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,A4: list_nat] :
( ( ( map_na6577772983117884747at_nat @ F @ A4 )
= nil_se357566008730718055at_nat )
= ( A4 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_617_Nil__is__map__conv,axiom,
! [F: set_nat > nat,Xs: list_set_nat] :
( ( nil_nat
= ( map_set_nat_nat @ F @ Xs ) )
= ( Xs = nil_set_nat ) ) ).
% Nil_is_map_conv
thf(fact_618_Nil__is__map__conv,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( nil_se357566008730718055at_nat
= ( map_li6003994582982014139at_nat @ F @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_619_Nil__is__map__conv,axiom,
! [F: set_nat > set_nat,Xs: list_set_nat] :
( ( nil_set_nat
= ( map_set_nat_set_nat @ F @ Xs ) )
= ( Xs = nil_set_nat ) ) ).
% Nil_is_map_conv
thf(fact_620_Nil__is__map__conv,axiom,
! [F: nat > set_nat,Xs: list_nat] :
( ( nil_set_nat
= ( map_nat_set_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_621_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_622_Nil__is__map__conv,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,Xs: list_set_nat] :
( ( nil_se357566008730718055at_nat
= ( map_se1162299115588061717at_nat @ F @ Xs ) )
= ( Xs = nil_set_nat ) ) ).
% Nil_is_map_conv
thf(fact_623_Nil__is__map__conv,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( nil_se357566008730718055at_nat
= ( map_na6577772983117884747at_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_624_map__is__Nil__conv,axiom,
! [F: set_nat > nat,Xs: list_set_nat] :
( ( ( map_set_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_set_nat ) ) ).
% map_is_Nil_conv
thf(fact_625_map__is__Nil__conv,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ Xs )
= nil_se357566008730718055at_nat )
= ( Xs = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_626_map__is__Nil__conv,axiom,
! [F: set_nat > set_nat,Xs: list_set_nat] :
( ( ( map_set_nat_set_nat @ F @ Xs )
= nil_set_nat )
= ( Xs = nil_set_nat ) ) ).
% map_is_Nil_conv
thf(fact_627_map__is__Nil__conv,axiom,
! [F: nat > set_nat,Xs: list_nat] :
( ( ( map_nat_set_nat @ F @ Xs )
= nil_set_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_628_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_629_map__is__Nil__conv,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,Xs: list_set_nat] :
( ( ( map_se1162299115588061717at_nat @ F @ Xs )
= nil_se357566008730718055at_nat )
= ( Xs = nil_set_nat ) ) ).
% map_is_Nil_conv
thf(fact_630_map__is__Nil__conv,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( ( map_na6577772983117884747at_nat @ F @ Xs )
= nil_se357566008730718055at_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_631_Diff__insert0,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat2 @ X @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ B ) )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_632_Diff__insert0,axiom,
! [X: list_nat,A: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ X @ A )
=> ( ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat2 @ X @ B ) )
= ( minus_7954133019191499631st_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_633_insert__Diff1,axiom,
! [X: nat,B: set_nat,A: set_nat] :
( ( member_nat2 @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_634_insert__Diff1,axiom,
! [X: list_nat,B: set_list_nat,A: set_list_nat] :
( ( member_list_nat2 @ X @ B )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat2 @ X @ A ) @ B )
= ( minus_7954133019191499631st_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_635_rotate1__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rotate1_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_636_list__ex1__simps_I1_J,axiom,
! [P2: nat > $o] :
~ ( list_ex1_nat @ P2 @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_637_diff__gt__0__iff__gt,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A4 @ B2 ) )
= ( ord_less_int @ B2 @ A4 ) ) ).
% diff_gt_0_iff_gt
thf(fact_638_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_639_length__0__conv,axiom,
! [Xs: list_set_nat] :
( ( ( size_s3254054031482475050et_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_set_nat ) ) ).
% length_0_conv
thf(fact_640_length__0__conv,axiom,
! [Xs: list_s1210847774152347623at_nat] :
( ( ( size_s8736152011456118867at_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_se357566008730718055at_nat ) ) ).
% length_0_conv
thf(fact_641_length__0__conv,axiom,
! [Xs: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_nat ) ) ).
% length_0_conv
thf(fact_642_power__eq__0__iff,axiom,
! [A4: int,N: nat] :
( ( ( power_power_int @ A4 @ N )
= zero_zero_int )
= ( ( A4 = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_643_power__eq__0__iff,axiom,
! [A4: nat,N: nat] :
( ( ( power_power_nat @ A4 @ N )
= zero_zero_nat )
= ( ( A4 = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_644_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_645_diff__eq__diff__eq,axiom,
! [A4: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A4 @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( A4 = B2 )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_646_diff__right__commute,axiom,
! [A4: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A4 @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A4 @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_647_diff__right__commute,axiom,
! [A4: int,C: int,B2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A4 @ C ) @ B2 )
= ( minus_minus_int @ ( minus_minus_int @ A4 @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_648_power__not__zero,axiom,
! [A4: int,N: nat] :
( ( A4 != zero_zero_int )
=> ( ( power_power_int @ A4 @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_649_power__not__zero,axiom,
! [A4: nat,N: nat] :
( ( A4 != zero_zero_nat )
=> ( ( power_power_nat @ A4 @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_650_diff__left__imp__eq,axiom,
! [A4: int,B2: int,C: int] :
( ( ( minus_minus_int @ A4 @ B2 )
= ( minus_minus_int @ A4 @ C ) )
=> ( B2 = C ) ) ).
% diff_left_imp_eq
thf(fact_651_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat2 @ C @ A )
=> ~ ( member_nat2 @ C @ B ) ) ) ).
% IntE
thf(fact_652_IntE,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ ( inf_inf_set_list_nat @ A @ B ) )
=> ~ ( ( member_list_nat2 @ C @ A )
=> ~ ( member_list_nat2 @ C @ B ) ) ) ).
% IntE
thf(fact_653_DiffE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ B ) ) ) ).
% DiffE
thf(fact_654_DiffE,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ ( minus_7954133019191499631st_nat @ A @ B ) )
=> ~ ( ( member_list_nat2 @ C @ A )
=> ( member_list_nat2 @ C @ B ) ) ) ).
% DiffE
thf(fact_655_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat2 @ C @ A ) ) ).
% IntD1
thf(fact_656_IntD1,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ ( inf_inf_set_list_nat @ A @ B ) )
=> ( member_list_nat2 @ C @ A ) ) ).
% IntD1
thf(fact_657_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat2 @ C @ B ) ) ).
% IntD2
thf(fact_658_IntD2,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ ( inf_inf_set_list_nat @ A @ B ) )
=> ( member_list_nat2 @ C @ B ) ) ).
% IntD2
thf(fact_659_DiffD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat2 @ C @ A ) ) ).
% DiffD1
thf(fact_660_DiffD1,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ ( minus_7954133019191499631st_nat @ A @ B ) )
=> ( member_list_nat2 @ C @ A ) ) ).
% DiffD1
thf(fact_661_DiffD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat2 @ C @ B ) ) ).
% DiffD2
thf(fact_662_DiffD2,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ ( minus_7954133019191499631st_nat @ A @ B ) )
=> ~ ( member_list_nat2 @ C @ B ) ) ).
% DiffD2
thf(fact_663_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_664_less__infI1,axiom,
! [A4: nat,X: nat,B2: nat] :
( ( ord_less_nat @ A4 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A4 @ B2 ) @ X ) ) ).
% less_infI1
thf(fact_665_less__infI1,axiom,
! [A4: int,X: int,B2: int] :
( ( ord_less_int @ A4 @ X )
=> ( ord_less_int @ ( inf_inf_int @ A4 @ B2 ) @ X ) ) ).
% less_infI1
thf(fact_666_less__infI2,axiom,
! [B2: nat,X: nat,A4: nat] :
( ( ord_less_nat @ B2 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A4 @ B2 ) @ X ) ) ).
% less_infI2
thf(fact_667_less__infI2,axiom,
! [B2: int,X: int,A4: int] :
( ( ord_less_int @ B2 @ X )
=> ( ord_less_int @ ( inf_inf_int @ A4 @ B2 ) @ X ) ) ).
% less_infI2
thf(fact_668_inf_Oabsorb3,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( inf_inf_nat @ A4 @ B2 )
= A4 ) ) ).
% inf.absorb3
thf(fact_669_inf_Oabsorb3,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( inf_inf_int @ A4 @ B2 )
= A4 ) ) ).
% inf.absorb3
thf(fact_670_inf_Oabsorb4,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ( ( inf_inf_nat @ A4 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_671_inf_Oabsorb4,axiom,
! [B2: int,A4: int] :
( ( ord_less_int @ B2 @ A4 )
=> ( ( inf_inf_int @ A4 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_672_inf_Ostrict__boundedE,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A4 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_nat @ A4 @ B2 )
=> ~ ( ord_less_nat @ A4 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_673_inf_Ostrict__boundedE,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_int @ A4 @ ( inf_inf_int @ B2 @ C ) )
=> ~ ( ( ord_less_int @ A4 @ B2 )
=> ~ ( ord_less_int @ A4 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_674_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( A5
= ( inf_inf_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_675_inf_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] :
( ( A5
= ( inf_inf_int @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_676_inf_Ostrict__coboundedI1,axiom,
! [A4: nat,C: nat,B2: nat] :
( ( ord_less_nat @ A4 @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A4 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_677_inf_Ostrict__coboundedI1,axiom,
! [A4: int,C: int,B2: int] :
( ( ord_less_int @ A4 @ C )
=> ( ord_less_int @ ( inf_inf_int @ A4 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_678_inf_Ostrict__coboundedI2,axiom,
! [B2: nat,C: nat,A4: nat] :
( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A4 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_679_inf_Ostrict__coboundedI2,axiom,
! [B2: int,C: int,A4: int] :
( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ ( inf_inf_int @ A4 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_680_boolean__algebra_Oconj__one__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_681_eq__iff__diff__eq__0,axiom,
( ( ^ [Y10: int,Z3: int] : ( Y10 = Z3 ) )
= ( ^ [A5: int,B5: int] :
( ( minus_minus_int @ A5 @ B5 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_682_diff__strict__mono,axiom,
! [A4: int,B2: int,D: int,C: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A4 @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_683_diff__eq__diff__less,axiom,
! [A4: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A4 @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A4 @ B2 )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_684_diff__strict__left__mono,axiom,
! [B2: int,A4: int,C: int] :
( ( ord_less_int @ B2 @ A4 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A4 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_685_diff__strict__right__mono,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ord_less_int @ ( minus_minus_int @ A4 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_686_Int__UNIV__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% Int_UNIV_right
thf(fact_687_Int__UNIV__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B )
= B ) ).
% Int_UNIV_left
thf(fact_688_Int__insert__left,axiom,
! [A4: nat,C4: set_nat,B: set_nat] :
( ( ( member_nat2 @ A4 @ C4 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A4 @ B ) @ C4 )
= ( insert_nat2 @ A4 @ ( inf_inf_set_nat @ B @ C4 ) ) ) )
& ( ~ ( member_nat2 @ A4 @ C4 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A4 @ B ) @ C4 )
= ( inf_inf_set_nat @ B @ C4 ) ) ) ) ).
% Int_insert_left
thf(fact_689_Int__insert__left,axiom,
! [A4: list_nat,C4: set_list_nat,B: set_list_nat] :
( ( ( member_list_nat2 @ A4 @ C4 )
=> ( ( inf_inf_set_list_nat @ ( insert_list_nat2 @ A4 @ B ) @ C4 )
= ( insert_list_nat2 @ A4 @ ( inf_inf_set_list_nat @ B @ C4 ) ) ) )
& ( ~ ( member_list_nat2 @ A4 @ C4 )
=> ( ( inf_inf_set_list_nat @ ( insert_list_nat2 @ A4 @ B ) @ C4 )
= ( inf_inf_set_list_nat @ B @ C4 ) ) ) ) ).
% Int_insert_left
thf(fact_690_Int__insert__right,axiom,
! [A4: nat,A: set_nat,B: set_nat] :
( ( ( member_nat2 @ A4 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat2 @ A4 @ B ) )
= ( insert_nat2 @ A4 @ ( inf_inf_set_nat @ A @ B ) ) ) )
& ( ~ ( member_nat2 @ A4 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat2 @ A4 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_691_Int__insert__right,axiom,
! [A4: list_nat,A: set_list_nat,B: set_list_nat] :
( ( ( member_list_nat2 @ A4 @ A )
=> ( ( inf_inf_set_list_nat @ A @ ( insert_list_nat2 @ A4 @ B ) )
= ( insert_list_nat2 @ A4 @ ( inf_inf_set_list_nat @ A @ B ) ) ) )
& ( ~ ( member_list_nat2 @ A4 @ A )
=> ( ( inf_inf_set_list_nat @ A @ ( insert_list_nat2 @ A4 @ B ) )
= ( inf_inf_set_list_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_692_insert__Diff__if,axiom,
! [X: nat,B: set_nat,A: set_nat] :
( ( ( member_nat2 @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) )
& ( ~ ( member_nat2 @ 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_693_insert__Diff__if,axiom,
! [X: list_nat,B: set_list_nat,A: set_list_nat] :
( ( ( member_list_nat2 @ X @ B )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat2 @ X @ A ) @ B )
= ( minus_7954133019191499631st_nat @ A @ B ) ) )
& ( ~ ( member_list_nat2 @ X @ B )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat2 @ X @ A ) @ B )
= ( insert_list_nat2 @ X @ ( minus_7954133019191499631st_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_694_inj__on__diff__left,axiom,
! [A4: int,A: set_int] : ( inj_on_int_int @ ( minus_minus_int @ A4 ) @ A ) ).
% inj_on_diff_left
thf(fact_695_inj__on__Int,axiom,
! [F: set_nat > nat,A: set_set_nat,B: set_set_nat] :
( ( ( inj_on_set_nat_nat @ F @ A )
| ( inj_on_set_nat_nat @ F @ B ) )
=> ( inj_on_set_nat_nat @ F @ ( inf_inf_set_set_nat @ A @ B ) ) ) ).
% inj_on_Int
thf(fact_696_inj__on__diff,axiom,
! [F: set_nat > nat,A: set_set_nat,B: set_set_nat] :
( ( inj_on_set_nat_nat @ F @ A )
=> ( inj_on_set_nat_nat @ F @ ( minus_2163939370556025621et_nat @ A @ B ) ) ) ).
% inj_on_diff
thf(fact_697_psubset__imp__ex__mem,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ? [B3: nat] : ( member_nat2 @ B3 @ ( minus_minus_set_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_698_psubset__imp__ex__mem,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B )
=> ? [B3: list_nat] : ( member_list_nat2 @ B3 @ ( minus_7954133019191499631st_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_699_list_Osimps_I8_J,axiom,
! [F: set_nat > nat] :
( ( map_set_nat_nat @ F @ nil_set_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_700_list_Osimps_I8_J,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat] :
( ( map_li6003994582982014139at_nat @ F @ nil_list_nat )
= nil_se357566008730718055at_nat ) ).
% list.simps(8)
thf(fact_701_list_Osimps_I8_J,axiom,
! [F: set_nat > set_nat] :
( ( map_set_nat_set_nat @ F @ nil_set_nat )
= nil_set_nat ) ).
% list.simps(8)
thf(fact_702_list_Osimps_I8_J,axiom,
! [F: nat > set_nat] :
( ( map_nat_set_nat @ F @ nil_nat )
= nil_set_nat ) ).
% list.simps(8)
thf(fact_703_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_704_list_Osimps_I8_J,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat] :
( ( map_se1162299115588061717at_nat @ F @ nil_set_nat )
= nil_se357566008730718055at_nat ) ).
% list.simps(8)
thf(fact_705_list_Osimps_I8_J,axiom,
! [F: nat > set_Pr1261947904930325089at_nat] :
( ( map_na6577772983117884747at_nat @ F @ nil_nat )
= nil_se357566008730718055at_nat ) ).
% list.simps(8)
thf(fact_706_zero__less__power,axiom,
! [A4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A4 @ N ) ) ) ).
% zero_less_power
thf(fact_707_zero__less__power,axiom,
! [A4: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A4 )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A4 @ N ) ) ) ).
% zero_less_power
thf(fact_708_distinct_Osimps_I1_J,axiom,
distinct_nat @ nil_nat ).
% distinct.simps(1)
thf(fact_709_nat__power__less__imp__less,axiom,
! [I: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_710_distinct__adj__Nil,axiom,
distinct_adj_nat @ nil_nat ).
% distinct_adj_Nil
thf(fact_711_removeAll_Osimps_I1_J,axiom,
! [X: nat] :
( ( removeAll_nat @ X @ nil_nat )
= nil_nat ) ).
% removeAll.simps(1)
thf(fact_712_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_713_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_714_member__rec_I2_J,axiom,
! [Y: nat] :
~ ( member_nat @ nil_nat @ Y ) ).
% member_rec(2)
thf(fact_715_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B5 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_716_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_717_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_718_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_719_list_Osize_I3_J,axiom,
( ( size_s3254054031482475050et_nat @ nil_set_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_720_list_Osize_I3_J,axiom,
( ( size_s8736152011456118867at_nat @ nil_se357566008730718055at_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_721_list_Osize_I3_J,axiom,
( ( size_s3023201423986296836st_nat @ nil_list_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_722_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_723_count__list_Osimps_I1_J,axiom,
! [Y: nat] :
( ( count_list_nat @ nil_nat @ Y )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_724_count__list_Osimps_I1_J,axiom,
! [Y: list_nat] :
( ( count_list_list_nat @ nil_list_nat @ Y )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_725_rgf__limit_Osimps_I1_J,axiom,
( ( equiva5889994315859557365_limit @ nil_nat )
= zero_zero_nat ) ).
% rgf_limit.simps(1)
thf(fact_726_UNIV__coset,axiom,
( top_top_set_nat
= ( coset_nat @ nil_nat ) ) ).
% UNIV_coset
thf(fact_727_length__n__lists,axiom,
! [N: nat,Xs: list_set_nat] :
( ( size_s8312130442339763642et_nat @ ( n_lists_set_nat @ N @ Xs ) )
= ( power_power_nat @ ( size_s3254054031482475050et_nat @ Xs ) @ N ) ) ).
% length_n_lists
thf(fact_728_length__n__lists,axiom,
! [N: nat,Xs: list_s1210847774152347623at_nat] :
( ( size_s3876073854528961369at_nat @ ( n_list4589530985885001583at_nat @ N @ Xs ) )
= ( power_power_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ N ) ) ).
% length_n_lists
thf(fact_729_length__n__lists,axiom,
! [N: nat,Xs: list_list_nat] :
( ( size_s6248950052170075156st_nat @ ( n_lists_list_nat @ N @ Xs ) )
= ( power_power_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) ) ).
% length_n_lists
thf(fact_730_length__n__lists,axiom,
! [N: nat,Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( n_lists_nat @ N @ Xs ) )
= ( power_power_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% length_n_lists
thf(fact_731_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_732_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_733_set__removeAll,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( set_list_nat2 @ ( removeAll_list_nat @ X @ Xs ) )
= ( minus_7954133019191499631st_nat @ ( set_list_nat2 @ Xs ) @ ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) ) ) ).
% set_removeAll
thf(fact_734_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_735_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_736_power__strict__decreasing__iff,axiom,
! [B2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ B2 @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B2 @ M2 ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_737_power__strict__decreasing__iff,axiom,
! [B2: int,M2: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ B2 @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B2 @ M2 ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_738_power__strict__mono,axiom,
! [A4: nat,B2: nat,N: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A4 @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_739_power__strict__mono,axiom,
! [A4: int,B2: int,N: nat] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A4 @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_740_power__mono__iff,axiom,
! [A4: nat,B2: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A4 @ N ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_eq_nat @ A4 @ B2 ) ) ) ) ) ).
% power_mono_iff
thf(fact_741_power__mono__iff,axiom,
! [A4: int,B2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A4 @ N ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_eq_int @ A4 @ B2 ) ) ) ) ) ).
% power_mono_iff
thf(fact_742_dual__order_Orefl,axiom,
! [A4: nat] : ( ord_less_eq_nat @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_743_dual__order_Orefl,axiom,
! [A4: int] : ( ord_less_eq_int @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_744_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_745_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_746_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X2: nat] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_747_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_748_all__not__in__conv,axiom,
! [A: set_list_nat] :
( ( ! [X2: list_nat] :
~ ( member_list_nat2 @ X2 @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% all_not_in_conv
thf(fact_749_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat2 @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_750_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_751_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_752_empty__iff,axiom,
! [C: list_nat] :
~ ( member_list_nat2 @ C @ bot_bot_set_list_nat ) ).
% empty_iff
thf(fact_753_empty__iff,axiom,
! [C: nat] :
~ ( member_nat2 @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_754_neg__equal__iff__equal,axiom,
! [A4: int,B2: int] :
( ( ( uminus_uminus_int @ A4 )
= ( uminus_uminus_int @ B2 ) )
= ( A4 = B2 ) ) ).
% neg_equal_iff_equal
thf(fact_755_add_Oinverse__inverse,axiom,
! [A4: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A4 ) )
= A4 ) ).
% add.inverse_inverse
thf(fact_756_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_757_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_758_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_759_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_760_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_761_bot__nat__0_Oextremum,axiom,
! [A4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A4 ) ).
% bot_nat_0.extremum
thf(fact_762_insert__subset,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ A ) @ B )
= ( ( member_nat2 @ X @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_763_insert__subset,axiom,
! [X: list_nat,A: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( insert_list_nat2 @ X @ A ) @ B )
= ( ( member_list_nat2 @ X @ B )
& ( ord_le6045566169113846134st_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_764_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_765_neg__le__iff__le,axiom,
! [B2: int,A4: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_eq_int @ A4 @ B2 ) ) ).
% neg_le_iff_le
thf(fact_766_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_767_neg__0__equal__iff__equal,axiom,
! [A4: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A4 ) )
= ( zero_zero_int = A4 ) ) ).
% neg_0_equal_iff_equal
thf(fact_768_neg__equal__0__iff__equal,axiom,
! [A4: int] :
( ( ( uminus_uminus_int @ A4 )
= zero_zero_int )
= ( A4 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_769_equal__neg__zero,axiom,
! [A4: int] :
( ( A4
= ( uminus_uminus_int @ A4 ) )
= ( A4 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_770_neg__equal__zero,axiom,
! [A4: int] :
( ( ( uminus_uminus_int @ A4 )
= A4 )
= ( A4 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_771_inf_Obounded__iff,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A4 @ B2 )
& ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_772_inf_Obounded__iff,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_eq_int @ A4 @ ( inf_inf_int @ B2 @ C ) )
= ( ( ord_less_eq_int @ A4 @ B2 )
& ( ord_less_eq_int @ A4 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_773_le__inf__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_774_le__inf__iff,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z2 ) )
= ( ( ord_less_eq_int @ X @ Y )
& ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_775_neg__less__iff__less,axiom,
! [B2: int,A4: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_int @ A4 @ B2 ) ) ).
% neg_less_iff_less
thf(fact_776_le__sup__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_777_le__sup__iff,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_int @ X @ Z2 )
& ( ord_less_eq_int @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_778_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A4: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A4 )
= ( ( ord_less_eq_nat @ B2 @ A4 )
& ( ord_less_eq_nat @ C @ A4 ) ) ) ).
% sup.bounded_iff
thf(fact_779_sup_Obounded__iff,axiom,
! [B2: int,C: int,A4: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A4 )
= ( ( ord_less_eq_int @ B2 @ A4 )
& ( ord_less_eq_int @ C @ A4 ) ) ) ).
% sup.bounded_iff
thf(fact_780_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_781_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_782_minus__diff__eq,axiom,
! [A4: int,B2: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A4 @ B2 ) )
= ( minus_minus_int @ B2 @ A4 ) ) ).
% minus_diff_eq
thf(fact_783_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_784_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_785_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_786_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_787_singleton__insert__inj__eq,axiom,
! [B2: nat,A4: nat,A: set_nat] :
( ( ( insert_nat2 @ B2 @ bot_bot_set_nat )
= ( insert_nat2 @ A4 @ A ) )
= ( ( A4 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_788_singleton__insert__inj__eq_H,axiom,
! [A4: nat,A: set_nat,B2: nat] :
( ( ( insert_nat2 @ A4 @ A )
= ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
= ( ( A4 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_789_singletonI,axiom,
! [A4: list_nat] : ( member_list_nat2 @ A4 @ ( insert_list_nat2 @ A4 @ bot_bot_set_list_nat ) ) ).
% singletonI
thf(fact_790_singletonI,axiom,
! [A4: nat] : ( member_nat2 @ A4 @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_791_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_792_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_793_bot__eq__sup__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y ) )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_794_sup__eq__bot__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_795_sup__bot_Oeq__neutr__iff,axiom,
! [A4: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A4 @ B2 )
= bot_bot_set_nat )
= ( ( A4 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_796_sup__bot_Oleft__neutral,axiom,
! [A4: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A4 )
= A4 ) ).
% sup_bot.left_neutral
thf(fact_797_sup__bot_Oneutr__eq__iff,axiom,
! [A4: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A4 @ B2 ) )
= ( ( A4 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_798_sup__bot_Oright__neutral,axiom,
! [A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ bot_bot_set_nat )
= A4 ) ).
% sup_bot.right_neutral
thf(fact_799_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_800_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_801_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_802_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_803_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_804_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_805_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_806_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_807_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_808_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_809_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_810_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_811_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_812_inj__on__empty,axiom,
! [F: set_nat > nat] : ( inj_on_set_nat_nat @ F @ bot_bot_set_set_nat ) ).
% inj_on_empty
thf(fact_813_inj__uminus,axiom,
! [A: set_int] : ( inj_on_int_int @ uminus_uminus_int @ A ) ).
% inj_uminus
thf(fact_814_lessThan__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_815_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_816_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_817_rotate1__length01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_818_rotate1__length01,axiom,
! [Xs: list_set_nat] :
( ( ord_less_eq_nat @ ( size_s3254054031482475050et_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_set_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_819_rotate1__length01,axiom,
! [Xs: list_s1210847774152347623at_nat] :
( ( ord_less_eq_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ one_one_nat )
=> ( ( rotate4238613965387346100at_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_820_rotate1__length01,axiom,
! [Xs: list_list_nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_list_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_821_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_822_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_823_diff__ge__0__iff__ge,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A4 @ B2 ) )
= ( ord_less_eq_int @ B2 @ A4 ) ) ).
% diff_ge_0_iff_ge
thf(fact_824_neg__less__eq__nonneg,axiom,
! [A4: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A4 ) @ A4 )
= ( ord_less_eq_int @ zero_zero_int @ A4 ) ) ).
% neg_less_eq_nonneg
thf(fact_825_less__eq__neg__nonpos,axiom,
! [A4: int] :
( ( ord_less_eq_int @ A4 @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_eq_int @ A4 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_826_neg__le__0__iff__le,axiom,
! [A4: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A4 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A4 ) ) ).
% neg_le_0_iff_le
thf(fact_827_neg__0__le__iff__le,axiom,
! [A4: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_eq_int @ A4 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_828_less__neg__neg,axiom,
! [A4: int] :
( ( ord_less_int @ A4 @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_int @ A4 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_829_neg__less__pos,axiom,
! [A4: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A4 ) @ A4 )
= ( ord_less_int @ zero_zero_int @ A4 ) ) ).
% neg_less_pos
thf(fact_830_neg__0__less__iff__less,axiom,
! [A4: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_int @ A4 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_831_neg__less__0__iff__less,axiom,
! [A4: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A4 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A4 ) ) ).
% neg_less_0_iff_less
thf(fact_832_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_833_verit__minus__simplify_I3_J,axiom,
! [B2: int] :
( ( minus_minus_int @ zero_zero_int @ B2 )
= ( uminus_uminus_int @ B2 ) ) ).
% verit_minus_simplify(3)
thf(fact_834_diff__0,axiom,
! [A4: int] :
( ( minus_minus_int @ zero_zero_int @ A4 )
= ( uminus_uminus_int @ A4 ) ) ).
% diff_0
thf(fact_835_boolean__algebra_Ocompl__zero,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.compl_zero
thf(fact_836_boolean__algebra_Ocompl__one,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.compl_one
thf(fact_837_inf__compl__bot__left1,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( inf_inf_set_nat @ X @ Y ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left1
thf(fact_838_inf__compl__bot__left2,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left2
thf(fact_839_inf__compl__bot__right,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_right
thf(fact_840_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_left
thf(fact_841_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ X ) )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_right
thf(fact_842_sup__compl__top__left1,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( sup_sup_set_nat @ X @ Y ) )
= top_top_set_nat ) ).
% sup_compl_top_left1
thf(fact_843_sup__compl__top__left2,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) )
= top_top_set_nat ) ).
% sup_compl_top_left2
thf(fact_844_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ X )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_left
thf(fact_845_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ ( uminus5710092332889474511et_nat @ X ) )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_right
thf(fact_846_set__empty2,axiom,
! [Xs: list_set_nat] :
( ( bot_bot_set_set_nat
= ( set_set_nat2 @ Xs ) )
= ( Xs = nil_set_nat ) ) ).
% set_empty2
thf(fact_847_set__empty2,axiom,
! [Xs: list_list_nat] :
( ( bot_bot_set_list_nat
= ( set_list_nat2 @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% set_empty2
thf(fact_848_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_849_set__empty,axiom,
! [Xs: list_set_nat] :
( ( ( set_set_nat2 @ Xs )
= bot_bot_set_set_nat )
= ( Xs = nil_set_nat ) ) ).
% set_empty
thf(fact_850_set__empty,axiom,
! [Xs: list_list_nat] :
( ( ( set_list_nat2 @ Xs )
= bot_bot_set_list_nat )
= ( Xs = nil_list_nat ) ) ).
% set_empty
thf(fact_851_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_852_Diff__UNIV,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_853_disjoint__insert_I2_J,axiom,
! [A: set_list_nat,B2: list_nat,B: set_list_nat] :
( ( bot_bot_set_list_nat
= ( inf_inf_set_list_nat @ A @ ( insert_list_nat2 @ B2 @ B ) ) )
= ( ~ ( member_list_nat2 @ B2 @ A )
& ( bot_bot_set_list_nat
= ( inf_inf_set_list_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_854_disjoint__insert_I2_J,axiom,
! [A: set_nat,B2: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat2 @ B2 @ B ) ) )
= ( ~ ( member_nat2 @ B2 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_855_disjoint__insert_I1_J,axiom,
! [B: set_list_nat,A4: list_nat,A: set_list_nat] :
( ( ( inf_inf_set_list_nat @ B @ ( insert_list_nat2 @ A4 @ A ) )
= bot_bot_set_list_nat )
= ( ~ ( member_list_nat2 @ A4 @ B )
& ( ( inf_inf_set_list_nat @ B @ A )
= bot_bot_set_list_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_856_disjoint__insert_I1_J,axiom,
! [B: set_nat,A4: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat2 @ A4 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat2 @ A4 @ B )
& ( ( inf_inf_set_nat @ B @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_857_insert__disjoint_I2_J,axiom,
! [A4: list_nat,A: set_list_nat,B: set_list_nat] :
( ( bot_bot_set_list_nat
= ( inf_inf_set_list_nat @ ( insert_list_nat2 @ A4 @ A ) @ B ) )
= ( ~ ( member_list_nat2 @ A4 @ B )
& ( bot_bot_set_list_nat
= ( inf_inf_set_list_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_858_insert__disjoint_I2_J,axiom,
! [A4: nat,A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat2 @ A4 @ A ) @ B ) )
= ( ~ ( member_nat2 @ A4 @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_859_insert__disjoint_I1_J,axiom,
! [A4: list_nat,A: set_list_nat,B: set_list_nat] :
( ( ( inf_inf_set_list_nat @ ( insert_list_nat2 @ A4 @ A ) @ B )
= bot_bot_set_list_nat )
= ( ~ ( member_list_nat2 @ A4 @ B )
& ( ( inf_inf_set_list_nat @ A @ B )
= bot_bot_set_list_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_860_insert__disjoint_I1_J,axiom,
! [A4: nat,A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat2 @ A4 @ A ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat2 @ A4 @ B )
& ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_861_insert__Diff__single,axiom,
! [A4: nat,A: set_nat] :
( ( insert_nat2 @ A4 @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) ) )
= ( insert_nat2 @ A4 @ A ) ) ).
% insert_Diff_single
thf(fact_862_Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_863_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_864_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_865_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_866_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_867_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_868_single__Diff__lessThan,axiom,
! [K: nat] :
( ( minus_minus_set_nat @ ( insert_nat2 @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
= ( insert_nat2 @ K @ bot_bot_set_nat ) ) ).
% single_Diff_lessThan
thf(fact_869_power__decreasing__iff,axiom,
! [B2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ B2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ M2 ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_870_power__decreasing__iff,axiom,
! [B2: int,M2: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ B2 @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B2 @ M2 ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_871_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_872_diff__less__mono,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A4 )
=> ( ord_less_nat @ ( minus_minus_nat @ A4 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_873_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_874_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_875_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_876_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_877_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_878_power__decreasing,axiom,
! [N: nat,N4: nat,A4: nat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_eq_nat @ A4 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A4 @ N4 ) @ ( power_power_nat @ A4 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_879_power__decreasing,axiom,
! [N: nat,N4: nat,A4: int] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A4 )
=> ( ( ord_less_eq_int @ A4 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A4 @ N4 ) @ ( power_power_int @ A4 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_880_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_881_inf__shunt,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% inf_shunt
thf(fact_882_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_883_inf__cancel__left1,axiom,
! [X: set_nat,A4: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ A4 ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ B2 ) )
= bot_bot_set_nat ) ).
% inf_cancel_left1
thf(fact_884_inf__cancel__left2,axiom,
! [X: set_nat,A4: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ A4 ) @ ( inf_inf_set_nat @ X @ B2 ) )
= bot_bot_set_nat ) ).
% inf_cancel_left2
thf(fact_885_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_886_subset__singleton__iff,axiom,
! [X10: set_nat,A4: nat] :
( ( ord_less_eq_set_nat @ X10 @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) )
= ( ( X10 = bot_bot_set_nat )
| ( X10
= ( insert_nat2 @ A4 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_887_ex__in__conv,axiom,
! [A: set_list_nat] :
( ( ? [X2: list_nat] : ( member_list_nat2 @ X2 @ A ) )
= ( A != bot_bot_set_list_nat ) ) ).
% ex_in_conv
thf(fact_888_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat2 @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_889_equals0I,axiom,
! [A: set_list_nat] :
( ! [Y2: list_nat] :
~ ( member_list_nat2 @ Y2 @ A )
=> ( A = bot_bot_set_list_nat ) ) ).
% equals0I
thf(fact_890_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat2 @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_891_equals0D,axiom,
! [A: set_list_nat,A4: list_nat] :
( ( A = bot_bot_set_list_nat )
=> ~ ( member_list_nat2 @ A4 @ A ) ) ).
% equals0D
thf(fact_892_equals0D,axiom,
! [A: set_nat,A4: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat2 @ A4 @ A ) ) ).
% equals0D
thf(fact_893_emptyE,axiom,
! [A4: list_nat] :
~ ( member_list_nat2 @ A4 @ bot_bot_set_list_nat ) ).
% emptyE
thf(fact_894_emptyE,axiom,
! [A4: nat] :
~ ( member_nat2 @ A4 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_895_Iio__eq__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_896_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_897_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_898_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_899_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_900_ord__le__eq__subst,axiom,
! [A4: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_901_ord__le__eq__subst,axiom,
! [A4: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_902_ord__le__eq__subst,axiom,
! [A4: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_903_ord__le__eq__subst,axiom,
! [A4: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_904_ord__eq__le__subst,axiom,
! [A4: nat,F: nat > nat,B2: nat,C: nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_905_ord__eq__le__subst,axiom,
! [A4: int,F: nat > int,B2: nat,C: nat] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_906_ord__eq__le__subst,axiom,
! [A4: nat,F: int > nat,B2: int,C: int] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_907_ord__eq__le__subst,axiom,
! [A4: int,F: int > int,B2: int,C: int] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_908_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_909_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_910_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_911_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_912_order__subst2,axiom,
! [A4: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_913_order__subst2,axiom,
! [A4: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_914_order__subst2,axiom,
! [A4: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_915_order__subst2,axiom,
! [A4: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_916_order__subst1,axiom,
! [A4: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_917_order__subst1,axiom,
! [A4: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_918_order__subst1,axiom,
! [A4: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_919_order__subst1,axiom,
! [A4: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_920_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y10: nat,Z3: nat] : ( Y10 = Z3 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_921_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y10: int,Z3: int] : ( Y10 = Z3 ) )
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_922_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_923_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_924_antisym,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_925_antisym,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_926_le__imp__neg__le,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A4 ) ) ) ).
% le_imp_neg_le
thf(fact_927_bot_Oextremum__uniqueI,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
=> ( A4 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_928_bot_Oextremum__uniqueI,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ bot_bot_nat )
=> ( A4 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_929_minus__le__iff,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A4 ) @ B2 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A4 ) ) ).
% minus_le_iff
thf(fact_930_le__minus__iff,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A4 ) ) ) ).
% le_minus_iff
thf(fact_931_bot_Oextremum__unique,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
= ( A4 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_932_bot_Oextremum__unique,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ bot_bot_nat )
= ( A4 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_933_dual__order_Otrans,axiom,
! [B2: nat,A4: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_934_dual__order_Otrans,axiom,
! [B2: int,A4: int,C: int] :
( ( ord_less_eq_int @ B2 @ A4 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_935_minus__equation__iff,axiom,
! [A4: int,B2: int] :
( ( ( uminus_uminus_int @ A4 )
= B2 )
= ( ( uminus_uminus_int @ B2 )
= A4 ) ) ).
% minus_equation_iff
thf(fact_936_equation__minus__iff,axiom,
! [A4: int,B2: int] :
( ( A4
= ( uminus_uminus_int @ B2 ) )
= ( B2
= ( uminus_uminus_int @ A4 ) ) ) ).
% equation_minus_iff
thf(fact_937_dual__order_Oantisym,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ( ord_less_eq_nat @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_938_dual__order_Oantisym,axiom,
! [B2: int,A4: int] :
( ( ord_less_eq_int @ B2 @ A4 )
=> ( ( ord_less_eq_int @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_939_dual__order_Oeq__iff,axiom,
( ( ^ [Y10: nat,Z3: nat] : ( Y10 = Z3 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_940_dual__order_Oeq__iff,axiom,
( ( ^ [Y10: int,Z3: int] : ( Y10 = Z3 ) )
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_941_bot_Oextremum,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).
% bot.extremum
thf(fact_942_bot_Oextremum,axiom,
! [A4: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A4 ) ).
% bot.extremum
thf(fact_943_linorder__wlog,axiom,
! [P2: nat > nat > $o,A4: nat,B2: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A4 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_944_linorder__wlog,axiom,
! [P2: int > int > $o,A4: int,B2: int] :
( ! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: int,B3: int] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A4 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_945_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_946_order__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_947_order_Otrans,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% order.trans
thf(fact_948_order_Otrans,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A4 @ C ) ) ) ).
% order.trans
thf(fact_949_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_950_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_951_ord__le__eq__trans,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_952_ord__le__eq__trans,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_int @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_953_ord__eq__le__trans,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( A4 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_954_ord__eq__le__trans,axiom,
! [A4: int,B2: int,C: int] :
( ( A4 = B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_955_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y10: nat,Z3: nat] : ( Y10 = Z3 ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_956_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y10: int,Z3: int] : ( Y10 = Z3 ) )
= ( ^ [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
& ( ord_less_eq_int @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_957_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_958_le__cases3,axiom,
! [X: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_959_nle__le,axiom,
! [A4: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A4 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A4 )
& ( B2 != A4 ) ) ) ).
% nle_le
thf(fact_960_nle__le,axiom,
! [A4: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A4 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A4 )
& ( B2 != A4 ) ) ) ).
% nle_le
thf(fact_961_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_962_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_963_sup__shunt,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= top_top_set_nat )
= ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_964_power__le__one,axiom,
! [A4: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ( ord_less_eq_nat @ A4 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A4 @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_965_power__le__one,axiom,
! [A4: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A4 )
=> ( ( ord_less_eq_int @ A4 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A4 @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_966_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_967_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_968_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_969_subset__insert__iff,axiom,
! [A: set_list_nat,X: list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( insert_list_nat2 @ X @ B ) )
= ( ( ( member_list_nat2 @ X @ A )
=> ( ord_le6045566169113846134st_nat @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) ) @ B ) )
& ( ~ ( member_list_nat2 @ X @ A )
=> ( ord_le6045566169113846134st_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_970_subset__insert__iff,axiom,
! [A: set_nat,X: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ B ) )
= ( ( ( member_nat2 @ X @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat2 @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_971_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_972_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_973_bot_Onot__eq__extremum,axiom,
! [A4: set_nat] :
( ( A4 != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A4 ) ) ).
% bot.not_eq_extremum
thf(fact_974_bot_Onot__eq__extremum,axiom,
! [A4: nat] :
( ( A4 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A4 ) ) ).
% bot.not_eq_extremum
thf(fact_975_bot_Oextremum__strict,axiom,
! [A4: set_nat] :
~ ( ord_less_set_nat @ A4 @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_976_bot_Oextremum__strict,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_977_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_978_minus__less__iff,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A4 ) @ B2 )
= ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A4 ) ) ).
% minus_less_iff
thf(fact_979_less__minus__iff,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_int @ B2 @ ( uminus_uminus_int @ A4 ) ) ) ).
% less_minus_iff
thf(fact_980_int__ops_I6_J,axiom,
! [A4: nat,B2: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A4 @ B2 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A4 @ B2 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_981_minus__diff__commute,axiom,
! [B2: int,A4: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B2 ) @ A4 )
= ( minus_minus_int @ ( uminus_uminus_int @ A4 ) @ B2 ) ) ).
% minus_diff_commute
thf(fact_982_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_983_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_984_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_985_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_986_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_987_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_988_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_989_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_990_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_991_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_992_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_993_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_994_order__less__le__subst2,axiom,
! [A4: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_995_order__less__le__subst2,axiom,
! [A4: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_996_order__less__le__subst2,axiom,
! [A4: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_997_order__less__le__subst2,axiom,
! [A4: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_998_order__less__le__subst1,axiom,
! [A4: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_999_order__less__le__subst1,axiom,
! [A4: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1000_order__less__le__subst1,axiom,
! [A4: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1001_order__less__le__subst1,axiom,
! [A4: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1002_order__le__less__subst2,axiom,
! [A4: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1003_order__le__less__subst2,axiom,
! [A4: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1004_order__le__less__subst2,axiom,
! [A4: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1005_order__le__less__subst2,axiom,
! [A4: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1006_order__le__less__subst1,axiom,
! [A4: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1007_order__le__less__subst1,axiom,
! [A4: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1008_order__le__less__subst1,axiom,
! [A4: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1009_order__le__less__subst1,axiom,
! [A4: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1010_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1011_order__less__le__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1012_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1013_order__le__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1014_order__neq__le__trans,axiom,
! [A4: nat,B2: nat] :
( ( A4 != B2 )
=> ( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ord_less_nat @ A4 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1015_order__neq__le__trans,axiom,
! [A4: int,B2: int] :
( ( A4 != B2 )
=> ( ( ord_less_eq_int @ A4 @ B2 )
=> ( ord_less_int @ A4 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1016_order__le__neq__trans,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( A4 != B2 )
=> ( ord_less_nat @ A4 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1017_order__le__neq__trans,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( A4 != B2 )
=> ( ord_less_int @ A4 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1018_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1019_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1020_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1021_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1022_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1023_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1024_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1025_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1026_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1027_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1028_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ( ord_less_eq_nat @ B2 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_1029_dual__order_Ostrict__implies__order,axiom,
! [B2: int,A4: int] :
( ( ord_less_int @ B2 @ A4 )
=> ( ord_less_eq_int @ B2 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_1030_order_Ostrict__implies__order,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ord_less_eq_nat @ A4 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1031_order_Ostrict__implies__order,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ord_less_eq_int @ A4 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1032_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1033_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B5: int,A5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1034_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A4: nat,C: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1035_dual__order_Ostrict__trans2,axiom,
! [B2: int,A4: int,C: int] :
( ( ord_less_int @ B2 @ A4 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_int @ C @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1036_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A4: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1037_dual__order_Ostrict__trans1,axiom,
! [B2: int,A4: int,C: int] :
( ( ord_less_eq_int @ B2 @ A4 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1038_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1039_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B5: int,A5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1040_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1041_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A5: int] :
( ( ord_less_int @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1042_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1043_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ~ ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1044_order_Ostrict__trans2,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% order.strict_trans2
thf(fact_1045_order_Ostrict__trans2,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% order.strict_trans2
thf(fact_1046_order_Ostrict__trans1,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% order.strict_trans1
thf(fact_1047_order_Ostrict__trans1,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% order.strict_trans1
thf(fact_1048_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1049_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1050_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1051_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_int @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1052_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1053_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1054_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1055_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
& ~ ( ord_less_eq_int @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1056_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1057_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1058_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1059_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1060_nless__le,axiom,
! [A4: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A4 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A4 @ B2 )
| ( A4 = B2 ) ) ) ).
% nless_le
thf(fact_1061_nless__le,axiom,
! [A4: int,B2: int] :
( ( ~ ( ord_less_int @ A4 @ B2 ) )
= ( ~ ( ord_less_eq_int @ A4 @ B2 )
| ( A4 = B2 ) ) ) ).
% nless_le
thf(fact_1062_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_1063_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_1064_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_1065_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_1066_diff__eq__diff__less__eq,axiom,
! [A4: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A4 @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A4 @ B2 )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1067_diff__right__mono,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A4 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_1068_diff__left__mono,axiom,
! [B2: int,A4: int,C: int] :
( ( ord_less_eq_int @ B2 @ A4 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A4 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_1069_diff__mono,axiom,
! [A4: int,B2: int,D: int,C: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A4 @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% diff_mono
thf(fact_1070_top__greatest,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ top_top_set_nat ) ).
% top_greatest
thf(fact_1071_top_Oextremum__unique,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A4 )
= ( A4 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_1072_top_Oextremum__uniqueI,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A4 )
=> ( A4 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_1073_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1074_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_1075_inf_OcoboundedI2,axiom,
! [B2: nat,C: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1076_inf_OcoboundedI2,axiom,
! [B2: int,C: int,A4: int] :
( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ ( inf_inf_int @ A4 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1077_inf_OcoboundedI1,axiom,
! [A4: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1078_inf_OcoboundedI1,axiom,
! [A4: int,C: int,B2: int] :
( ( ord_less_eq_int @ A4 @ C )
=> ( ord_less_eq_int @ ( inf_inf_int @ A4 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1079_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_1080_inf_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A5: int] :
( ( inf_inf_int @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_1081_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_1082_inf_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] :
( ( inf_inf_int @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_1083_inf_Ocobounded2,axiom,
! [A4: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_1084_inf_Ocobounded2,axiom,
! [A4: int,B2: int] : ( ord_less_eq_int @ ( inf_inf_int @ A4 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_1085_inf_Ocobounded1,axiom,
! [A4: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B2 ) @ A4 ) ).
% inf.cobounded1
thf(fact_1086_inf_Ocobounded1,axiom,
! [A4: int,B2: int] : ( ord_less_eq_int @ ( inf_inf_int @ A4 @ B2 ) @ A4 ) ).
% inf.cobounded1
thf(fact_1087_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( A5
= ( inf_inf_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_1088_inf_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] :
( A5
= ( inf_inf_int @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_1089_inf__greatest,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1090_inf__greatest,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Z2 )
=> ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1091_inf_OboundedI,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ A4 @ C )
=> ( ord_less_eq_nat @ A4 @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1092_inf_OboundedI,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_eq_int @ A4 @ C )
=> ( ord_less_eq_int @ A4 @ ( inf_inf_int @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1093_inf_OboundedE,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A4 @ B2 )
=> ~ ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% inf.boundedE
thf(fact_1094_inf_OboundedE,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_eq_int @ A4 @ ( inf_inf_int @ B2 @ C ) )
=> ~ ( ( ord_less_eq_int @ A4 @ B2 )
=> ~ ( ord_less_eq_int @ A4 @ C ) ) ) ).
% inf.boundedE
thf(fact_1095_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1096_inf__absorb2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( inf_inf_int @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1097_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1098_inf__absorb1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( inf_inf_int @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1099_inf_Oabsorb2,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ( inf_inf_nat @ A4 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1100_inf_Oabsorb2,axiom,
! [B2: int,A4: int] :
( ( ord_less_eq_int @ B2 @ A4 )
=> ( ( inf_inf_int @ A4 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1101_inf_Oabsorb1,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( inf_inf_nat @ A4 @ B2 )
= A4 ) ) ).
% inf.absorb1
thf(fact_1102_inf_Oabsorb1,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( inf_inf_int @ A4 @ B2 )
= A4 ) ) ).
% inf.absorb1
thf(fact_1103_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( inf_inf_nat @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1104_le__iff__inf,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y3: int] :
( ( inf_inf_int @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1105_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Z )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y2 @ Z ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1106_inf__unique,axiom,
! [F: int > int > int,X: int,Y: int] :
( ! [X3: int,Y2: int] : ( ord_less_eq_int @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: int,Y2: int] : ( ord_less_eq_int @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: int,Y2: int,Z: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ( ord_less_eq_int @ X3 @ Z )
=> ( ord_less_eq_int @ X3 @ ( F @ Y2 @ Z ) ) ) )
=> ( ( inf_inf_int @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1107_inf_OorderI,axiom,
! [A4: nat,B2: nat] :
( ( A4
= ( inf_inf_nat @ A4 @ B2 ) )
=> ( ord_less_eq_nat @ A4 @ B2 ) ) ).
% inf.orderI
thf(fact_1108_inf_OorderI,axiom,
! [A4: int,B2: int] :
( ( A4
= ( inf_inf_int @ A4 @ B2 ) )
=> ( ord_less_eq_int @ A4 @ B2 ) ) ).
% inf.orderI
thf(fact_1109_inf_OorderE,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( A4
= ( inf_inf_nat @ A4 @ B2 ) ) ) ).
% inf.orderE
thf(fact_1110_inf_OorderE,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( A4
= ( inf_inf_int @ A4 @ B2 ) ) ) ).
% inf.orderE
thf(fact_1111_le__infI2,axiom,
! [B2: nat,X: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_1112_le__infI2,axiom,
! [B2: int,X: int,A4: int] :
( ( ord_less_eq_int @ B2 @ X )
=> ( ord_less_eq_int @ ( inf_inf_int @ A4 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_1113_le__infI1,axiom,
! [A4: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_1114_le__infI1,axiom,
! [A4: int,X: int,B2: int] :
( ( ord_less_eq_int @ A4 @ X )
=> ( ord_less_eq_int @ ( inf_inf_int @ A4 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_1115_inf__mono,axiom,
! [A4: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A4 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B2 ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_1116_inf__mono,axiom,
! [A4: int,C: int,B2: int,D: int] :
( ( ord_less_eq_int @ A4 @ C )
=> ( ( ord_less_eq_int @ B2 @ D )
=> ( ord_less_eq_int @ ( inf_inf_int @ A4 @ B2 ) @ ( inf_inf_int @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_1117_le__infI,axiom,
! [X: nat,A4: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A4 )
=> ( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A4 @ B2 ) ) ) ) ).
% le_infI
thf(fact_1118_le__infI,axiom,
! [X: int,A4: int,B2: int] :
( ( ord_less_eq_int @ X @ A4 )
=> ( ( ord_less_eq_int @ X @ B2 )
=> ( ord_less_eq_int @ X @ ( inf_inf_int @ A4 @ B2 ) ) ) ) ).
% le_infI
thf(fact_1119_le__infE,axiom,
! [X: nat,A4: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A4 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A4 )
=> ~ ( ord_less_eq_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_1120_le__infE,axiom,
! [X: int,A4: int,B2: int] :
( ( ord_less_eq_int @ X @ ( inf_inf_int @ A4 @ B2 ) )
=> ~ ( ( ord_less_eq_int @ X @ A4 )
=> ~ ( ord_less_eq_int @ X @ B2 ) ) ) ).
% le_infE
thf(fact_1121_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_1122_inf__le2,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_1123_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_1124_inf__le1,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_1125_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1126_inf__sup__ord_I1_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1127_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1128_inf__sup__ord_I2_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1129_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1130_bot__nat__0_Oextremum__uniqueI,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
=> ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1131_bot__nat__0_Oextremum__unique,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
= ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1132_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1133_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1134_inf__sup__ord_I4_J,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1135_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1136_inf__sup__ord_I3_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1137_le__supE,axiom,
! [A4: nat,B2: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A4 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A4 @ X )
=> ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_1138_le__supE,axiom,
! [A4: int,B2: int,X: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ A4 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_int @ A4 @ X )
=> ~ ( ord_less_eq_int @ B2 @ X ) ) ) ).
% le_supE
thf(fact_1139_le__supI,axiom,
! [A4: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ X )
=> ( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A4 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_1140_le__supI,axiom,
! [A4: int,X: int,B2: int] :
( ( ord_less_eq_int @ A4 @ X )
=> ( ( ord_less_eq_int @ B2 @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ A4 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_1141_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_1142_sup__ge1,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge1
thf(fact_1143_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_1144_sup__ge2,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge2
thf(fact_1145_le__supI1,axiom,
! [X: nat,A4: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A4 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% le_supI1
thf(fact_1146_le__supI1,axiom,
! [X: int,A4: int,B2: int] :
( ( ord_less_eq_int @ X @ A4 )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A4 @ B2 ) ) ) ).
% le_supI1
thf(fact_1147_le__supI2,axiom,
! [X: nat,B2: nat,A4: nat] :
( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% le_supI2
thf(fact_1148_le__supI2,axiom,
! [X: int,B2: int,A4: int] :
( ( ord_less_eq_int @ X @ B2 )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A4 @ B2 ) ) ) ).
% le_supI2
thf(fact_1149_sup_Omono,axiom,
! [C: nat,A4: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A4 )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A4 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_1150_sup_Omono,axiom,
! [C: int,A4: int,D: int,B2: int] :
( ( ord_less_eq_int @ C @ A4 )
=> ( ( ord_less_eq_int @ D @ B2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ C @ D ) @ ( sup_sup_int @ A4 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_1151_sup__mono,axiom,
! [A4: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A4 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A4 @ B2 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_1152_sup__mono,axiom,
! [A4: int,C: int,B2: int,D: int] :
( ( ord_less_eq_int @ A4 @ C )
=> ( ( ord_less_eq_int @ B2 @ D )
=> ( ord_less_eq_int @ ( sup_sup_int @ A4 @ B2 ) @ ( sup_sup_int @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_1153_sup__least,axiom,
! [Y: nat,X: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1154_sup__least,axiom,
! [Y: int,X: int,Z2: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ Z2 @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1155_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( sup_sup_nat @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_1156_le__iff__sup,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y3: int] :
( ( sup_sup_int @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_1157_sup_OorderE,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( A4
= ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% sup.orderE
thf(fact_1158_sup_OorderE,axiom,
! [B2: int,A4: int] :
( ( ord_less_eq_int @ B2 @ A4 )
=> ( A4
= ( sup_sup_int @ A4 @ B2 ) ) ) ).
% sup.orderE
thf(fact_1159_sup_OorderI,axiom,
! [A4: nat,B2: nat] :
( ( A4
= ( sup_sup_nat @ A4 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A4 ) ) ).
% sup.orderI
thf(fact_1160_sup_OorderI,axiom,
! [A4: int,B2: int] :
( ( A4
= ( sup_sup_int @ A4 @ B2 ) )
=> ( ord_less_eq_int @ B2 @ A4 ) ) ).
% sup.orderI
thf(fact_1161_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ Z @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1162_sup__unique,axiom,
! [F: int > int > int,X: int,Y: int] :
( ! [X3: int,Y2: int] : ( ord_less_eq_int @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: int,Y2: int] : ( ord_less_eq_int @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: int,Y2: int,Z: int] :
( ( ord_less_eq_int @ Y2 @ X3 )
=> ( ( ord_less_eq_int @ Z @ X3 )
=> ( ord_less_eq_int @ ( F @ Y2 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_int @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1163_sup_Oabsorb1,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ( sup_sup_nat @ A4 @ B2 )
= A4 ) ) ).
% sup.absorb1
thf(fact_1164_sup_Oabsorb1,axiom,
! [B2: int,A4: int] :
( ( ord_less_eq_int @ B2 @ A4 )
=> ( ( sup_sup_int @ A4 @ B2 )
= A4 ) ) ).
% sup.absorb1
thf(fact_1165_sup_Oabsorb2,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( sup_sup_nat @ A4 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_1166_sup_Oabsorb2,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( sup_sup_int @ A4 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_1167_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1168_sup__absorb1,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( sup_sup_int @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1169_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1170_sup__absorb2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( sup_sup_int @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1171_sup_OboundedE,axiom,
! [B2: nat,C: nat,A4: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A4 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A4 )
=> ~ ( ord_less_eq_nat @ C @ A4 ) ) ) ).
% sup.boundedE
thf(fact_1172_sup_OboundedE,axiom,
! [B2: int,C: int,A4: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A4 )
=> ~ ( ( ord_less_eq_int @ B2 @ A4 )
=> ~ ( ord_less_eq_int @ C @ A4 ) ) ) ).
% sup.boundedE
thf(fact_1173_sup_OboundedI,axiom,
! [B2: nat,A4: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ( ord_less_eq_nat @ C @ A4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A4 ) ) ) ).
% sup.boundedI
thf(fact_1174_sup_OboundedI,axiom,
! [B2: int,A4: int,C: int] :
( ( ord_less_eq_int @ B2 @ A4 )
=> ( ( ord_less_eq_int @ C @ A4 )
=> ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A4 ) ) ) ).
% sup.boundedI
thf(fact_1175_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_1176_sup_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A5: int] :
( A5
= ( sup_sup_int @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_1177_sup_Ocobounded1,axiom,
! [A4: nat,B2: nat] : ( ord_less_eq_nat @ A4 @ ( sup_sup_nat @ A4 @ B2 ) ) ).
% sup.cobounded1
thf(fact_1178_sup_Ocobounded1,axiom,
! [A4: int,B2: int] : ( ord_less_eq_int @ A4 @ ( sup_sup_int @ A4 @ B2 ) ) ).
% sup.cobounded1
thf(fact_1179_sup_Ocobounded2,axiom,
! [B2: nat,A4: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A4 @ B2 ) ) ).
% sup.cobounded2
thf(fact_1180_sup_Ocobounded2,axiom,
! [B2: int,A4: int] : ( ord_less_eq_int @ B2 @ ( sup_sup_int @ A4 @ B2 ) ) ).
% sup.cobounded2
thf(fact_1181_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_1182_sup_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A5: int] :
( ( sup_sup_int @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_1183_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_1184_sup_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] :
( ( sup_sup_int @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_1185_sup_OcoboundedI1,axiom,
! [C: nat,A4: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A4 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1186_sup_OcoboundedI1,axiom,
! [C: int,A4: int,B2: int] :
( ( ord_less_eq_int @ C @ A4 )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A4 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1187_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A4: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1188_sup_OcoboundedI2,axiom,
! [C: int,B2: int,A4: int] :
( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A4 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1189_subset__insert,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat2 @ X @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_1190_subset__insert,axiom,
! [X: list_nat,A: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ X @ A )
=> ( ( ord_le6045566169113846134st_nat @ A @ ( insert_list_nat2 @ X @ B ) )
= ( ord_le6045566169113846134st_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_1191_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1192_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1193_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1194_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1195_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_1196_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1197_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_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( member_set_nat2 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_1198_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_1199_subset__code_I1_J,axiom,
! [Xs: list_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ B )
= ( ! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( member_list_nat2 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_1200_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1201_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P2 ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1202_Int__Collect__mono,axiom,
! [A: set_list_nat,B: set_list_nat,P2: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ A )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( inf_inf_set_list_nat @ A @ ( collect_list_nat @ P2 ) ) @ ( inf_inf_set_list_nat @ B @ ( collect_list_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1203_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_1204_singleton__inject,axiom,
! [A4: nat,B2: nat] :
( ( ( insert_nat2 @ A4 @ bot_bot_set_nat )
= ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
=> ( A4 = B2 ) ) ).
% singleton_inject
thf(fact_1205_insert__not__empty,axiom,
! [A4: nat,A: set_nat] :
( ( insert_nat2 @ A4 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_1206_doubleton__eq__iff,axiom,
! [A4: nat,B2: nat,C: nat,D: nat] :
( ( ( insert_nat2 @ A4 @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
= ( insert_nat2 @ C @ ( insert_nat2 @ D @ bot_bot_set_nat ) ) )
= ( ( ( A4 = C )
& ( B2 = D ) )
| ( ( A4 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1207_singleton__iff,axiom,
! [B2: list_nat,A4: list_nat] :
( ( member_list_nat2 @ B2 @ ( insert_list_nat2 @ A4 @ bot_bot_set_list_nat ) )
= ( B2 = A4 ) ) ).
% singleton_iff
thf(fact_1208_singleton__iff,axiom,
! [B2: nat,A4: nat] :
( ( member_nat2 @ B2 @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) )
= ( B2 = A4 ) ) ).
% singleton_iff
thf(fact_1209_singletonD,axiom,
! [B2: list_nat,A4: list_nat] :
( ( member_list_nat2 @ B2 @ ( insert_list_nat2 @ A4 @ bot_bot_set_list_nat ) )
=> ( B2 = A4 ) ) ).
% singletonD
thf(fact_1210_singletonD,axiom,
! [B2: nat,A4: nat] :
( ( member_nat2 @ B2 @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) )
=> ( B2 = A4 ) ) ).
% singletonD
thf(fact_1211_inj__on__subset,axiom,
! [F: set_nat > nat,A: set_set_nat,B: set_set_nat] :
( ( inj_on_set_nat_nat @ F @ A )
=> ( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( inj_on_set_nat_nat @ F @ B ) ) ) ).
% inj_on_subset
thf(fact_1212_subset__inj__on,axiom,
! [F: set_nat > nat,B: set_set_nat,A: set_set_nat] :
( ( inj_on_set_nat_nat @ F @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( inj_on_set_nat_nat @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_1213_disjoint__iff__not__equal,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ! [Y3: nat] :
( ( member_nat2 @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1214_Int__empty__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_1215_Int__empty__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_1216_disjoint__iff,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ( inf_inf_set_list_nat @ A @ B )
= bot_bot_set_list_nat )
= ( ! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ A )
=> ~ ( member_list_nat2 @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1217_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ~ ( member_nat2 @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1218_Int__emptyI,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ A )
=> ~ ( member_list_nat2 @ X3 @ B ) )
=> ( ( inf_inf_set_list_nat @ A @ B )
= bot_bot_set_list_nat ) ) ).
% Int_emptyI
thf(fact_1219_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ~ ( member_nat2 @ X3 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_1220_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_1221_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_1222_not__psubset__empty,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_1223_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ X @ Y )
= top_top_set_nat )
=> ( ( uminus5710092332889474511et_nat @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1224_self__le__power,axiom,
! [A4: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A4 @ ( power_power_nat @ A4 @ N ) ) ) ) ).
% self_le_power
thf(fact_1225_self__le__power,axiom,
! [A4: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A4 @ ( power_power_int @ A4 @ N ) ) ) ) ).
% self_le_power
thf(fact_1226_psubset__insert__iff,axiom,
! [A: set_list_nat,X: list_nat,B: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ ( insert_list_nat2 @ X @ B ) )
= ( ( ( member_list_nat2 @ X @ B )
=> ( ord_le1190675801316882794st_nat @ A @ B ) )
& ( ~ ( member_list_nat2 @ X @ B )
=> ( ( ( member_list_nat2 @ X @ A )
=> ( ord_le1190675801316882794st_nat @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) ) @ B ) )
& ( ~ ( member_list_nat2 @ X @ A )
=> ( ord_le6045566169113846134st_nat @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1227_psubset__insert__iff,axiom,
! [A: set_nat,X: nat,B: set_nat] :
( ( ord_less_set_nat @ A @ ( insert_nat2 @ X @ B ) )
= ( ( ( member_nat2 @ X @ B )
=> ( ord_less_set_nat @ A @ B ) )
& ( ~ ( member_nat2 @ X @ B )
=> ( ( ( member_nat2 @ X @ A )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat2 @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1228_sup__cancel__left1,axiom,
! [X: set_nat,A4: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ A4 ) @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ B2 ) )
= top_top_set_nat ) ).
% sup_cancel_left1
thf(fact_1229_sup__cancel__left2,axiom,
! [X: set_nat,A4: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ A4 ) @ ( sup_sup_set_nat @ X @ B2 ) )
= top_top_set_nat ) ).
% sup_cancel_left2
thf(fact_1230_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_1231_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B5 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1232_power__mono,axiom,
! [A4: nat,B2: nat,N: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A4 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A4 @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).
% power_mono
thf(fact_1233_power__mono,axiom,
! [A4: int,B2: int,N: nat] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A4 )
=> ( ord_less_eq_int @ ( power_power_int @ A4 @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% power_mono
thf(fact_1234_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P2 @ I3 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1235_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1236_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_1237_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1238_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1239_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_1240_zdiff__int__split,axiom,
! [P2: int > $o,X: nat,Y: nat] :
( ( P2 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P2 @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1241_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_1242_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1243_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1244_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1245_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1246_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1247_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1248_le__diff__iff_H,axiom,
! [A4: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A4 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A4 ) ) ) ) ).
% le_diff_iff'
thf(fact_1249_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1250_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M4: nat] :
( ( P2 @ X )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) )
=> ~ ! [M5: nat] :
( ( P2 @ M5 )
=> ~ ! [X11: nat] :
( ( P2 @ X11 )
=> ( ord_less_eq_nat @ X11 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1251_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B2: nat] :
( ( P2 @ K )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y9: nat] :
( ( P2 @ Y9 )
=> ( ord_less_eq_nat @ Y9 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1252_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_1253_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_1254_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_1255_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_1256_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1257_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1258_imp__le__cong,axiom,
! [X: int,X13: int,P2: $o,P5: $o] :
( ( X = X13 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X13 )
=> ( P2 = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X13 )
=> P5 ) ) ) ) ).
% imp_le_cong
thf(fact_1259_conj__le__cong,axiom,
! [X: int,X13: int,P2: $o,P5: $o] :
( ( X = X13 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X13 )
=> ( P2 = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X13 )
& P5 ) ) ) ) ).
% conj_le_cong
thf(fact_1260_equiv__rels__enum,axiom,
! [X: list_nat] :
( ( equiva3371634703666331078on_rgf @ X )
=> ( ( count_list_list_nat @ ( equiva7426478223624825838m_rgfs @ ( size_size_list_nat @ X ) ) @ X )
= one_one_nat ) ) ).
% equiv_rels_enum
thf(fact_1261_enum__rgfs__returns__rgfs,axiom,
! [X: list_nat,N: nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ ( equiva7426478223624825838m_rgfs @ N ) ) )
=> ( equiva3371634703666331078on_rgf @ X ) ) ).
% enum_rgfs_returns_rgfs
thf(fact_1262_enum__rgfs__len,axiom,
! [X: list_nat,N: nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ ( equiva7426478223624825838m_rgfs @ N ) ) )
=> ( ( size_size_list_nat @ X )
= N ) ) ).
% enum_rgfs_len
thf(fact_1263_equiv__rels__def,axiom,
( equiva8721718519204927301v_rels
= ( ^ [N2: nat] : ( map_li6003994582982014139at_nat @ equiva2048684438135499664of_nat @ ( equiva7426478223624825838m_rgfs @ N2 ) ) ) ) ).
% equiv_rels_def
thf(fact_1264_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% finite_lessThan
thf(fact_1265_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M: nat] :
! [X2: nat] :
( ( member_nat2 @ X2 @ N5 )
=> ( ord_less_eq_nat @ X2 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1266_sorted__upt,axiom,
! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M2 @ N ) ) ).
% sorted_upt
thf(fact_1267_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M: nat] :
! [X2: nat] :
( ( member_nat2 @ X2 @ N5 )
=> ( ord_less_nat @ X2 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1268_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ N4 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1269_sorted__wrt__upt,axiom,
! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M2 @ N ) ) ).
% sorted_wrt_upt
thf(fact_1270_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
% Conjectures (1)
thf(conj_0,conjecture,
( ( size_size_list_nat @ ( map_set_nat_nat @ f @ y ) )
= n ) ).
%------------------------------------------------------------------------------