TPTP Problem File: SLH0161^1.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_00146_005846__11804992_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1437 ( 743 unt; 170 typ; 0 def)
% Number of atoms : 3316 (1634 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9341 ( 441 ~; 87 |; 298 &;7362 @)
% ( 0 <=>;1153 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 654 ( 654 >; 0 *; 0 +; 0 <<)
% Number of symbols : 159 ( 156 usr; 11 con; 0-3 aty)
% Number of variables : 3351 ( 152 ^;2911 !; 288 ?;3351 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:13:26.343
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
list_l5212752354702395664st_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__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__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_set_int: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $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__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (156)
thf(sy_c_Equivalence__Relation__Enumeration_Oenum__rgfs,type,
equiva7426478223624825838m_rgfs: nat > list_list_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Oenum__rgfs__rel,type,
equiva1432535406783100555fs_rel: nat > nat > $o ).
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_OFpow_001t__Int__Oint,type,
finite_Fpow_int: set_int > set_set_int ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: 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_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
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_If_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
if_list_list_nat: $o > list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_List_OListMem_001t__List__Olist_It__Nat__Onat_J,type,
listMem_list_nat: list_nat > list_list_nat > $o ).
thf(sy_c_List_OListMem_001t__Nat__Onat,type,
listMem_nat: nat > list_nat > $o ).
thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
append_list_list_nat: list_list_list_nat > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
append_list_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Obind_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
bind_l7796378977173581257st_nat: list_list_nat > ( list_nat > list_list_nat ) > list_list_nat ).
thf(sy_c_List_Obind_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
bind_list_nat_nat: list_list_nat > ( list_nat > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
bind_nat_list_nat: list_nat > ( nat > list_list_nat ) > list_list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Obutlast_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
butlas6429778205849610142st_nat: list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Obutlast_001t__List__Olist_It__Nat__Onat_J,type,
butlast_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
can_select_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
concat_list_list_nat: list_l5212752354702395664st_nat > list_list_list_nat ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__Nat__Onat_J,type,
concat_list_nat: list_list_list_nat > list_list_nat ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Ocoset_001t__List__Olist_It__Nat__Onat_J,type,
coset_list_nat: list_list_nat > set_list_nat ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
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_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_Olast_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
last_list_list_nat: list_list_list_nat > list_list_nat ).
thf(sy_c_List_Olast_001t__List__Olist_It__Nat__Onat_J,type,
last_list_nat: list_list_nat > list_nat ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord738340561235409698at_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Ostable__sort__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord227665693835759911at_nat: ( ( nat > nat ) > list_nat > list_nat ) > $o ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
cons_list_list_nat: list_list_nat > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
nil_list_list_nat: list_list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
map_li5769348595424326838st_nat: ( list_list_nat > list_list_list_nat ) > list_list_list_nat > list_l5212752354702395664st_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_li2855073862107769254st_nat: ( list_list_nat > list_list_nat ) > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_li960784813134754710st_nat: ( list_nat > list_list_nat ) > list_list_nat > list_list_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__Nat__Onat_001t__Int__Oint,type,
map_nat_int: ( nat > int ) > list_nat > list_int ).
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_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
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__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
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_Omaps_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
maps_l5785965478274863235st_nat: ( list_nat > list_list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Omaps_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
maps_list_nat_nat: ( list_nat > list_nat ) > list_list_nat > list_nat ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
maps_nat_list_nat: ( nat > list_list_nat ) > list_nat > list_list_nat ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_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_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_Onths_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
nths_list_list_nat: list_list_list_nat > set_nat > list_list_list_nat ).
thf(sy_c_List_Onths_001t__List__Olist_It__Nat__Onat_J,type,
nths_list_nat: list_list_nat > set_nat > list_list_nat ).
thf(sy_c_List_Onths_001t__Nat__Onat,type,
nths_nat: list_nat > set_nat > list_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_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_Orotate1_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
rotate6412633851404001245st_nat: list_list_list_nat > list_list_list_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_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_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
neg_numeral_dbl_int: int > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Oord_Omax_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
max_list_list_nat: ( list_list_nat > list_list_nat > $o ) > list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Orderings_Oord_Omax_001t__List__Olist_It__Nat__Onat_J,type,
max_list_nat: ( list_nat > list_nat > $o ) > list_nat > list_nat > list_nat ).
thf(sy_c_Orderings_Oord_Omin_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
min_list_list_nat: ( list_list_nat > list_list_nat > $o ) > list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Orderings_Oord_Omin_001t__List__Olist_It__Nat__Onat_J,type,
min_list_nat: ( list_nat > list_nat > $o ) > list_nat > list_nat > list_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__Num__Onum,type,
ord_less_num: num > num > $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__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__Num__Onum,type,
ord_less_eq_num: num > num > $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__Int__Oint_J_J,type,
ord_le4403425263959731960et_int: set_set_int > set_set_int > $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_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OPow_001t__Int__Oint,type,
pow_int: set_int > set_set_int ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
image_4042064729117200983st_nat: ( list_nat > list_list_nat ) > set_list_nat > set_list_list_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_1775855109352712557et_nat: ( list_nat > set_nat ) > set_list_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
image_nat_list_nat: ( nat > list_nat ) > set_nat > set_list_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Int__Oint_J,type,
image_3739036796817536367et_int: ( set_nat > set_int ) > set_set_nat > set_set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_It__Nat__Onat_J,type,
the_elem_list_nat: set_list_nat > list_nat ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
set_ord_atMost_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
set_or6656581121297822940st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
set_or5832277885323065728an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Int__Oint,type,
set_or1207661135979820486an_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > 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_Stirling_Ostirling,type,
stirling: nat > nat > nat ).
thf(sy_c_Stirling_Ostirling__row,type,
stirling_row: nat > list_nat ).
thf(sy_c_Stirling_Ostirling__row__aux_001t__Nat__Onat,type,
stirling_row_aux_nat: nat > nat > list_nat > list_nat ).
thf(sy_c_Sublist_Oprefix_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
prefix_list_list_nat: list_list_list_nat > list_list_list_nat > $o ).
thf(sy_c_Sublist_Oprefix_001t__List__Olist_It__Nat__Onat_J,type,
prefix_list_nat: list_list_nat > list_list_nat > $o ).
thf(sy_c_Sublist_Oprefix_001t__Nat__Onat,type,
prefix_nat: list_nat > list_nat > $o ).
thf(sy_c_Sublist_Oprefixes_001t__List__Olist_It__Nat__Onat_J,type,
prefixes_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_Sublist_Oprefixes_001t__Nat__Onat,type,
prefixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osublists_001t__List__Olist_It__Nat__Onat_J,type,
sublists_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_Sublist_Osublists_001t__Nat__Onat,type,
sublists_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osuffixes_001t__List__Olist_It__Nat__Onat_J,type,
suffixes_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_Sublist_Osuffixes_001t__Nat__Onat,type,
suffixes_nat: list_nat > list_list_nat ).
thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
accp_nat: ( nat > nat > $o ) > nat > $o ).
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__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_x____,type,
x: nat ).
thf(sy_v_xsa____,type,
xsa: list_nat ).
% Relevant facts (1261)
thf(fact_0_c,axiom,
equiva3371634703666331078on_rgf @ xsa ).
% c
thf(fact_1_snoc_OIH,axiom,
( ( equiva3371634703666331078on_rgf @ xsa )
=> ( ( set_nat2 @ xsa )
= ( set_ord_lessThan_nat @ ( equiva5889994315859557365_limit @ xsa ) ) ) ) ).
% snoc.IH
thf(fact_2_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_3_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_4_in__set__member,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_5_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_6_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P: nat > $o,Xs2: list_nat] :
? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs2 ) )
& ( P @ X2 )
& ! [Y2: nat] :
( ( ( member_nat2 @ Y2 @ ( set_nat2 @ Xs2 ) )
& ( P @ Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_7_snoc_Oprems,axiom,
equiva3371634703666331078on_rgf @ ( append_nat @ xsa @ ( cons_nat @ x @ nil_nat ) ) ).
% snoc.prems
thf(fact_8_ListMem__iff,axiom,
( listMem_nat
= ( ^ [X2: nat,Xs2: list_nat] : ( member_nat2 @ X2 @ ( set_nat2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_9_set__rotate1,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( rotate1_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_10_removeAll__id,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_11_set__sort,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( set_nat2 @ ( linord738340561235409698at_nat @ F @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_sort
thf(fact_12_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N ) ) ) ) ).
% atLeast_upt
thf(fact_13_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_14_list_Oinject,axiom,
! [X21: list_nat,X22: list_list_nat,Y21: list_nat,Y22: list_list_nat] :
( ( ( cons_list_nat @ X21 @ X22 )
= ( cons_list_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_15_append_Oassoc,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( append_nat @ ( append_nat @ A @ B ) @ C )
= ( append_nat @ A @ ( append_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_16_append__assoc,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
= ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_17_append__same__eq,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys @ Xs )
= ( append_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_18_same__append__eq,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_19_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_20_append_Oright__neutral,axiom,
! [A: list_list_nat] :
( ( append_list_nat @ A @ nil_list_nat )
= A ) ).
% append.right_neutral
thf(fact_21_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_22_append__Nil2,axiom,
! [Xs: list_list_nat] :
( ( append_list_nat @ Xs @ nil_list_nat )
= Xs ) ).
% append_Nil2
thf(fact_23_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_24_append__self__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_nat ) ) ).
% append_self_conv
thf(fact_25_self__append__conv,axiom,
! [Y: list_nat,Ys: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_26_self__append__conv,axiom,
! [Y: list_list_nat,Ys: list_list_nat] :
( ( Y
= ( append_list_nat @ Y @ Ys ) )
= ( Ys = nil_list_nat ) ) ).
% self_append_conv
thf(fact_27_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_28_append__self__conv2,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_nat ) ) ).
% append_self_conv2
thf(fact_29_self__append__conv2,axiom,
! [Y: list_nat,Xs: list_nat] :
( ( Y
= ( append_nat @ Xs @ Y ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_30_self__append__conv2,axiom,
! [Y: list_list_nat,Xs: list_list_nat] :
( ( Y
= ( append_list_nat @ Xs @ Y ) )
= ( Xs = nil_list_nat ) ) ).
% self_append_conv2
thf(fact_31_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_32_Nil__is__append__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( nil_list_nat
= ( append_list_nat @ Xs @ Ys ) )
= ( ( Xs = nil_list_nat )
& ( Ys = nil_list_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_33_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_34_append__is__Nil__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= nil_list_nat )
= ( ( Xs = nil_list_nat )
& ( Ys = nil_list_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_35_lessThan__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I @ K ) ) ).
% lessThan_iff
thf(fact_36_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat2 @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_37_sort__key__simps_I1_J,axiom,
! [F: nat > nat] :
( ( linord738340561235409698at_nat @ F @ nil_nat )
= nil_nat ) ).
% sort_key_simps(1)
thf(fact_38_rotate1__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rotate1_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_39_rotate1__is__Nil__conv,axiom,
! [Xs: list_list_nat] :
( ( ( rotate1_list_nat @ Xs )
= nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_40_removeAll__append,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( removeAll_nat @ X @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( removeAll_nat @ X @ Xs ) @ ( removeAll_nat @ X @ Ys ) ) ) ).
% removeAll_append
thf(fact_41_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_42_bind__simps_I1_J,axiom,
! [F: nat > list_list_nat] :
( ( bind_nat_list_nat @ nil_nat @ F )
= nil_list_nat ) ).
% bind_simps(1)
thf(fact_43_bind__simps_I1_J,axiom,
! [F: list_nat > list_nat] :
( ( bind_list_nat_nat @ nil_list_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_44_bind__simps_I1_J,axiom,
! [F: list_nat > list_list_nat] :
( ( bind_l7796378977173581257st_nat @ nil_list_nat @ F )
= nil_list_nat ) ).
% bind_simps(1)
thf(fact_45_list__ex1__simps_I1_J,axiom,
! [P2: list_nat > $o] :
~ ( list_ex1_list_nat @ P2 @ nil_list_nat ) ).
% list_ex1_simps(1)
thf(fact_46_list__ex1__simps_I1_J,axiom,
! [P2: nat > $o] :
~ ( list_ex1_nat @ P2 @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_47_append1__eq__conv,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_48_append1__eq__conv,axiom,
! [Xs: list_list_nat,X: list_nat,Ys: list_list_nat,Y: list_nat] :
( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) )
= ( append_list_nat @ Ys @ ( cons_list_nat @ Y @ nil_list_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_49_insert__Nil,axiom,
! [X: nat] :
( ( insert_nat @ X @ nil_nat )
= ( cons_nat @ X @ nil_nat ) ) ).
% insert_Nil
thf(fact_50_insert__Nil,axiom,
! [X: list_nat] :
( ( insert_list_nat @ X @ nil_list_nat )
= ( cons_list_nat @ X @ nil_list_nat ) ) ).
% insert_Nil
thf(fact_51_not__in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= ( cons_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_52_not__in__set__insert,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( insert_list_nat @ X @ Xs )
= ( cons_list_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_53_bind__simps_I2_J,axiom,
! [X: nat,Xs: list_nat,F: nat > list_nat] :
( ( bind_nat_nat @ ( cons_nat @ X @ Xs ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_nat_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_54_bind__simps_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat,F: list_nat > list_nat] :
( ( bind_list_nat_nat @ ( cons_list_nat @ X @ Xs ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_list_nat_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_55_member__rec_I2_J,axiom,
! [Y: nat] :
~ ( member_nat @ nil_nat @ Y ) ).
% member_rec(2)
thf(fact_56_member__rec_I2_J,axiom,
! [Y: list_nat] :
~ ( member_list_nat @ nil_list_nat @ Y ) ).
% member_rec(2)
thf(fact_57_member__rec_I1_J,axiom,
! [X: nat,Xs: list_nat,Y: nat] :
( ( member_nat @ ( cons_nat @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_nat @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_58_member__rec_I1_J,axiom,
! [X: list_nat,Xs: list_list_nat,Y: list_nat] :
( ( member_list_nat @ ( cons_list_nat @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_list_nat @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_59_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_60_list_Odistinct_I1_J,axiom,
! [X21: list_nat,X22: list_list_nat] :
( nil_list_nat
!= ( cons_list_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_61_rotate1_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( rotate1_nat @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_62_rotate1_Osimps_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( rotate1_list_nat @ ( cons_list_nat @ X @ Xs ) )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_63_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_64_rotate1_Osimps_I1_J,axiom,
( ( rotate1_list_nat @ nil_list_nat )
= nil_list_nat ) ).
% rotate1.simps(1)
thf(fact_65_removeAll_Osimps_I2_J,axiom,
! [X: nat,Y: nat,Xs: list_nat] :
( ( ( X = Y )
=> ( ( removeAll_nat @ X @ ( cons_nat @ Y @ Xs ) )
= ( removeAll_nat @ X @ Xs ) ) )
& ( ( X != Y )
=> ( ( removeAll_nat @ X @ ( cons_nat @ Y @ Xs ) )
= ( cons_nat @ Y @ ( removeAll_nat @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_66_removeAll_Osimps_I2_J,axiom,
! [X: list_nat,Y: list_nat,Xs: list_list_nat] :
( ( ( X = Y )
=> ( ( removeAll_list_nat @ X @ ( cons_list_nat @ Y @ Xs ) )
= ( removeAll_list_nat @ X @ Xs ) ) )
& ( ( X != Y )
=> ( ( removeAll_list_nat @ X @ ( cons_list_nat @ Y @ Xs ) )
= ( cons_list_nat @ Y @ ( removeAll_list_nat @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_67_removeAll_Osimps_I1_J,axiom,
! [X: nat] :
( ( removeAll_nat @ X @ nil_nat )
= nil_nat ) ).
% removeAll.simps(1)
thf(fact_68_removeAll_Osimps_I1_J,axiom,
! [X: list_nat] :
( ( removeAll_list_nat @ X @ nil_list_nat )
= nil_list_nat ) ).
% removeAll.simps(1)
thf(fact_69_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_70_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat2 @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_72_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_73_list_OdiscI,axiom,
! [List: list_list_nat,X21: list_nat,X22: list_list_nat] :
( ( List
= ( cons_list_nat @ X21 @ X22 ) )
=> ( List != nil_list_nat ) ) ).
% list.discI
thf(fact_74_elem,axiom,
! [X: nat,Xs: list_nat] : ( listMem_nat @ X @ ( cons_nat @ X @ Xs ) ) ).
% elem
thf(fact_75_elem,axiom,
! [X: list_nat,Xs: list_list_nat] : ( listMem_list_nat @ X @ ( cons_list_nat @ X @ Xs ) ) ).
% elem
thf(fact_76_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_77_list_Oexhaust,axiom,
! [Y: list_list_nat] :
( ( Y != nil_list_nat )
=> ~ ! [X212: list_nat,X222: list_list_nat] :
( Y
!= ( cons_list_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_78_ListMem_Ocases,axiom,
! [A1: nat,A22: list_nat] :
( ( listMem_nat @ A1 @ A22 )
=> ( ! [Xs3: list_nat] :
( A22
!= ( cons_nat @ A1 @ Xs3 ) )
=> ~ ! [Xs3: list_nat] :
( ? [Y3: nat] :
( A22
= ( cons_nat @ Y3 @ Xs3 ) )
=> ~ ( listMem_nat @ A1 @ Xs3 ) ) ) ) ).
% ListMem.cases
thf(fact_79_ListMem_Ocases,axiom,
! [A1: list_nat,A22: list_list_nat] :
( ( listMem_list_nat @ A1 @ A22 )
=> ( ! [Xs3: list_list_nat] :
( A22
!= ( cons_list_nat @ A1 @ Xs3 ) )
=> ~ ! [Xs3: list_list_nat] :
( ? [Y3: list_nat] :
( A22
= ( cons_list_nat @ Y3 @ Xs3 ) )
=> ~ ( listMem_list_nat @ A1 @ Xs3 ) ) ) ) ).
% ListMem.cases
thf(fact_80_ListMem_Osimps,axiom,
( listMem_nat
= ( ^ [A12: nat,A23: list_nat] :
( ? [X2: nat,Xs2: list_nat] :
( ( A12 = X2 )
& ( A23
= ( cons_nat @ X2 @ Xs2 ) ) )
| ? [X2: nat,Xs2: list_nat,Y2: nat] :
( ( A12 = X2 )
& ( A23
= ( cons_nat @ Y2 @ Xs2 ) )
& ( listMem_nat @ X2 @ Xs2 ) ) ) ) ) ).
% ListMem.simps
thf(fact_81_ListMem_Osimps,axiom,
( listMem_list_nat
= ( ^ [A12: list_nat,A23: list_list_nat] :
( ? [X2: list_nat,Xs2: list_list_nat] :
( ( A12 = X2 )
& ( A23
= ( cons_list_nat @ X2 @ Xs2 ) ) )
| ? [X2: list_nat,Xs2: list_list_nat,Y2: list_nat] :
( ( A12 = X2 )
& ( A23
= ( cons_list_nat @ Y2 @ Xs2 ) )
& ( listMem_list_nat @ X2 @ Xs2 ) ) ) ) ) ).
% ListMem.simps
thf(fact_82_insert,axiom,
! [X: nat,Xs: list_nat,Y: nat] :
( ( listMem_nat @ X @ Xs )
=> ( listMem_nat @ X @ ( cons_nat @ Y @ Xs ) ) ) ).
% insert
thf(fact_83_insert,axiom,
! [X: list_nat,Xs: list_list_nat,Y: list_nat] :
( ( listMem_list_nat @ X @ Xs )
=> ( listMem_list_nat @ X @ ( cons_list_nat @ Y @ Xs ) ) ) ).
% insert
thf(fact_84_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X3: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs3 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_85_transpose_Ocases,axiom,
! [X: list_list_list_nat] :
( ( X != nil_list_list_nat )
=> ( ! [Xss: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ nil_list_nat @ Xss ) )
=> ~ ! [X3: list_nat,Xs3: list_list_nat,Xss: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_86_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X3: nat,Xs3: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_87_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_88_append__Nil,axiom,
! [Ys: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_89_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X3: nat] :
( X
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_90_remdups__adj_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [X3: list_nat] :
( X
!= ( cons_list_nat @ X3 @ nil_list_nat ) )
=> ~ ! [X3: list_nat,Y3: list_nat,Xs3: list_list_nat] :
( X
!= ( cons_list_nat @ X3 @ ( cons_list_nat @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_91_append__Cons,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys )
= ( cons_nat @ X @ ( append_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_92_append__Cons,axiom,
! [X: list_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( append_list_nat @ ( cons_list_nat @ X @ Xs ) @ Ys )
= ( cons_list_nat @ X @ ( append_list_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_93_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_94_append_Oleft__neutral,axiom,
! [A: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_95_rgf__limit_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X3: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs3 ) ) ) ).
% rgf_limit.cases
thf(fact_96_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X2: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat2 @ X2 @ ( set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_nat @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_97_List_Oinsert__def,axiom,
( insert_list_nat
= ( ^ [X2: list_nat,Xs2: list_list_nat] : ( if_list_list_nat @ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_list_nat @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_98_rev__induct,axiom,
! [P2: list_nat > $o,Xs: list_nat] :
( ( P2 @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( P2 @ Xs3 )
=> ( P2 @ ( append_nat @ Xs3 @ ( cons_nat @ X3 @ nil_nat ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_99_rev__induct,axiom,
! [P2: list_list_nat > $o,Xs: list_list_nat] :
( ( P2 @ nil_list_nat )
=> ( ! [X3: list_nat,Xs3: list_list_nat] :
( ( P2 @ Xs3 )
=> ( P2 @ ( append_list_nat @ Xs3 @ ( cons_list_nat @ X3 @ nil_list_nat ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_100_split__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_101_split__list,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ? [Ys2: list_list_nat,Zs2: list_list_nat] :
( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_102_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys2: list_nat,Y3: nat] :
( Xs
!= ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_103_rev__exhaust,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ~ ! [Ys2: list_list_nat,Y3: list_nat] :
( Xs
!= ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) ) ) ).
% rev_exhaust
thf(fact_104_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y2: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_105_neq__Nil__conv,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
= ( ? [Y2: list_nat,Ys3: list_list_nat] :
( Xs
= ( cons_list_nat @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_106_list__induct2_H,axiom,
! [P2: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P2 @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat] : ( P2 @ ( cons_nat @ X3 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys2: list_nat] : ( P2 @ nil_nat @ ( cons_nat @ Y3 @ Ys2 ) )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys2: list_nat] :
( ( P2 @ Xs3 @ Ys2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_107_list__induct2_H,axiom,
! [P2: list_nat > list_list_nat > $o,Xs: list_nat,Ys: list_list_nat] :
( ( P2 @ nil_nat @ nil_list_nat )
=> ( ! [X3: nat,Xs3: list_nat] : ( P2 @ ( cons_nat @ X3 @ Xs3 ) @ nil_list_nat )
=> ( ! [Y3: list_nat,Ys2: list_list_nat] : ( P2 @ nil_nat @ ( cons_list_nat @ Y3 @ Ys2 ) )
=> ( ! [X3: nat,Xs3: list_nat,Y3: list_nat,Ys2: list_list_nat] :
( ( P2 @ Xs3 @ Ys2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_list_nat @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_108_list__induct2_H,axiom,
! [P2: list_list_nat > list_nat > $o,Xs: list_list_nat,Ys: list_nat] :
( ( P2 @ nil_list_nat @ nil_nat )
=> ( ! [X3: list_nat,Xs3: list_list_nat] : ( P2 @ ( cons_list_nat @ X3 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys2: list_nat] : ( P2 @ nil_list_nat @ ( cons_nat @ Y3 @ Ys2 ) )
=> ( ! [X3: list_nat,Xs3: list_list_nat,Y3: nat,Ys2: list_nat] :
( ( P2 @ Xs3 @ Ys2 )
=> ( P2 @ ( cons_list_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_109_list__induct2_H,axiom,
! [P2: list_list_nat > list_list_nat > $o,Xs: list_list_nat,Ys: list_list_nat] :
( ( P2 @ nil_list_nat @ nil_list_nat )
=> ( ! [X3: list_nat,Xs3: list_list_nat] : ( P2 @ ( cons_list_nat @ X3 @ Xs3 ) @ nil_list_nat )
=> ( ! [Y3: list_nat,Ys2: list_list_nat] : ( P2 @ nil_list_nat @ ( cons_list_nat @ Y3 @ Ys2 ) )
=> ( ! [X3: list_nat,Xs3: list_list_nat,Y3: list_nat,Ys2: list_list_nat] :
( ( P2 @ Xs3 @ Ys2 )
=> ( P2 @ ( cons_list_nat @ X3 @ Xs3 ) @ ( cons_list_nat @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_110_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_111_eq__Nil__appendI,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_nat @ nil_list_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_112_not__Cons__self2,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_113_not__Cons__self2,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( cons_list_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_114_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_115_Cons__eq__appendI,axiom,
! [X: list_nat,Xs1: list_list_nat,Ys: list_list_nat,Xs: list_list_nat,Zs: list_list_nat] :
( ( ( cons_list_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_nat @ Xs1 @ Zs ) )
=> ( ( cons_list_nat @ X @ Xs )
= ( append_list_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_116_split__list__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat2 @ X @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_117_split__list__last,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ? [Ys2: list_list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X @ Zs2 ) ) )
& ~ ( member_list_nat2 @ X @ ( set_list_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_118_split__list__prop,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ? [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
& ( P2 @ X4 ) )
=> ? [Ys2: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P2 @ X3 ) ) ) ).
% split_list_prop
thf(fact_119_split__list__prop,axiom,
! [Xs: list_list_nat,P2: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat2 @ X4 @ ( set_list_nat2 @ Xs ) )
& ( P2 @ X4 ) )
=> ? [Ys2: list_list_nat,X3: list_nat] :
( ? [Zs2: list_list_nat] :
( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs2 ) ) )
& ( P2 @ X3 ) ) ) ).
% split_list_prop
thf(fact_120_split__list__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat2 @ X @ ( set_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_121_split__list__first,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ? [Ys2: list_list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X @ Zs2 ) ) )
& ~ ( member_list_nat2 @ X @ ( set_list_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_122_split__list__propE,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ? [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys2: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ~ ( P2 @ X3 ) ) ) ).
% split_list_propE
thf(fact_123_split__list__propE,axiom,
! [Xs: list_list_nat,P2: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat2 @ X4 @ ( set_list_nat2 @ Xs ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys2: list_list_nat,X3: list_nat] :
( ? [Zs2: list_list_nat] :
( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs2 ) ) )
=> ~ ( P2 @ X3 ) ) ) ).
% split_list_propE
thf(fact_124_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_125_append__Cons__eq__iff,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Xs4: list_nat,Ys4: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat2 @ X @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) )
= ( append_nat @ Xs4 @ ( cons_nat @ X @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_126_append__Cons__eq__iff,axiom,
! [X: list_nat,Xs: list_list_nat,Ys: list_list_nat,Xs4: list_list_nat,Ys4: list_list_nat] :
( ~ ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ~ ( member_list_nat2 @ X @ ( set_list_nat2 @ Ys ) )
=> ( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X @ Ys ) )
= ( append_list_nat @ Xs4 @ ( cons_list_nat @ X @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_127_in__set__conv__decomp,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_128_in__set__conv__decomp,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys3: list_list_nat,Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_129_Cons__eq__append__conv,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X @ Xs )
= Zs ) )
| ? [Ys5: list_nat] :
( ( ( cons_nat @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_nat @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_130_Cons__eq__append__conv,axiom,
! [X: list_nat,Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( ( cons_list_nat @ X @ Xs )
= ( append_list_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_nat )
& ( ( cons_list_nat @ X @ Xs )
= Zs ) )
| ? [Ys5: list_list_nat] :
( ( ( cons_list_nat @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_list_nat @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_131_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs: list_nat,X: nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( cons_nat @ X @ Xs ) )
= ( ( ( Ys = nil_nat )
& ( Zs
= ( cons_nat @ X @ Xs ) ) )
| ? [Ys5: list_nat] :
( ( Ys
= ( cons_nat @ X @ Ys5 ) )
& ( ( append_nat @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_132_append__eq__Cons__conv,axiom,
! [Ys: list_list_nat,Zs: list_list_nat,X: list_nat,Xs: list_list_nat] :
( ( ( append_list_nat @ Ys @ Zs )
= ( cons_list_nat @ X @ Xs ) )
= ( ( ( Ys = nil_list_nat )
& ( Zs
= ( cons_list_nat @ X @ Xs ) ) )
| ? [Ys5: list_list_nat] :
( ( Ys
= ( cons_list_nat @ X @ Ys5 ) )
& ( ( append_list_nat @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_133_rev__nonempty__induct,axiom,
! [Xs: list_nat,P2: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P2 @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( append_nat @ Xs3 @ ( cons_nat @ X3 @ nil_nat ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_134_rev__nonempty__induct,axiom,
! [Xs: list_list_nat,P2: list_list_nat > $o] :
( ( Xs != nil_list_nat )
=> ( ! [X3: list_nat] : ( P2 @ ( cons_list_nat @ X3 @ nil_list_nat ) )
=> ( ! [X3: list_nat,Xs3: list_list_nat] :
( ( Xs3 != nil_list_nat )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( append_list_nat @ Xs3 @ ( cons_list_nat @ X3 @ nil_list_nat ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_135_list__nonempty__induct,axiom,
! [Xs: list_nat,P2: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P2 @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( cons_nat @ X3 @ Xs3 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_136_list__nonempty__induct,axiom,
! [Xs: list_list_nat,P2: list_list_nat > $o] :
( ( Xs != nil_list_nat )
=> ( ! [X3: list_nat] : ( P2 @ ( cons_list_nat @ X3 @ nil_list_nat ) )
=> ( ! [X3: list_nat,Xs3: list_list_nat] :
( ( Xs3 != nil_list_nat )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( cons_list_nat @ X3 @ Xs3 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_137_split__list__last__prop,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ? [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
& ( P2 @ X4 ) )
=> ? [Ys2: list_nat,X3: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ ( set_nat2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_138_split__list__last__prop,axiom,
! [Xs: list_list_nat,P2: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat2 @ X4 @ ( set_list_nat2 @ Xs ) )
& ( P2 @ X4 ) )
=> ? [Ys2: list_list_nat,X3: list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Xa: list_nat] :
( ( member_list_nat2 @ Xa @ ( set_list_nat2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_139_split__list__first__prop,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ? [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
& ( P2 @ X4 ) )
=> ? [Ys2: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ ( set_nat2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_140_split__list__first__prop,axiom,
! [Xs: list_list_nat,P2: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat2 @ X4 @ ( set_list_nat2 @ Xs ) )
& ( P2 @ X4 ) )
=> ? [Ys2: list_list_nat,X3: list_nat] :
( ? [Zs2: list_list_nat] :
( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Xa: list_nat] :
( ( member_list_nat2 @ Xa @ ( set_list_nat2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_141_split__list__last__propE,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ? [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys2: list_nat,X3: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa: nat] :
( ( member_nat2 @ Xa @ ( set_nat2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_142_split__list__last__propE,axiom,
! [Xs: list_list_nat,P2: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat2 @ X4 @ ( set_list_nat2 @ Xs ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys2: list_list_nat,X3: list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs2 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa: list_nat] :
( ( member_list_nat2 @ Xa @ ( set_list_nat2 @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_143_append__eq__append__conv2,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us2: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us2 ) )
& ( ( append_nat @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_144_split__list__first__propE,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ? [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys2: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa: nat] :
( ( member_nat2 @ Xa @ ( set_nat2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_145_split__list__first__propE,axiom,
! [Xs: list_list_nat,P2: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat2 @ X4 @ ( set_list_nat2 @ Xs ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys2: list_list_nat,X3: list_nat] :
( ? [Zs2: list_list_nat] :
( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs2 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa: list_nat] :
( ( member_list_nat2 @ Xa @ ( set_list_nat2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_146_in__set__conv__decomp__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat2 @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_147_in__set__conv__decomp__last,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys3: list_list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs3 ) ) )
& ~ ( member_list_nat2 @ X @ ( set_list_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_148_in__set__conv__decomp__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat2 @ X @ ( set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_149_in__set__conv__decomp__first,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys3: list_list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs3 ) ) )
& ~ ( member_list_nat2 @ X @ ( set_list_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_150_split__list__last__prop__iff,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys3: list_nat,X2: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y2: nat] :
( ( member_nat2 @ Y2 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_151_split__list__last__prop__iff,axiom,
! [Xs: list_list_nat,P2: list_nat > $o] :
( ( ? [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys3: list_list_nat,X2: list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y2: list_nat] :
( ( member_list_nat2 @ Y2 @ ( set_list_nat2 @ Zs3 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_152_split__list__first__prop__iff,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys3: list_nat,X2: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y2: nat] :
( ( member_nat2 @ Y2 @ ( set_nat2 @ Ys3 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_153_split__list__first__prop__iff,axiom,
! [Xs: list_list_nat,P2: list_nat > $o] :
( ( ? [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys3: list_list_nat,X2: list_nat] :
( ? [Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y2: list_nat] :
( ( member_list_nat2 @ Y2 @ ( set_list_nat2 @ Ys3 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_154_lessThan__strict__subset__iff,axiom,
! [M: int,N2: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N2 ) )
= ( ord_less_int @ M @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_155_lessThan__strict__subset__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_156_set__ConsD,axiom,
! [Y: nat,X: nat,Xs: list_nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat2 @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_157_set__ConsD,axiom,
! [Y: list_nat,X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_list_nat2 @ Y @ ( set_list_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_158_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat2 @ E @ ( set_nat2 @ A ) )
=> ( ! [Z2: list_nat] :
( A
!= ( cons_nat @ E @ Z2 ) )
=> ~ ! [Z1: nat,Z2: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat2 @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_159_list_Oset__cases,axiom,
! [E: list_nat,A: list_list_nat] :
( ( member_list_nat2 @ E @ ( set_list_nat2 @ A ) )
=> ( ! [Z2: list_list_nat] :
( A
!= ( cons_list_nat @ E @ Z2 ) )
=> ~ ! [Z1: list_nat,Z2: list_list_nat] :
( ( A
= ( cons_list_nat @ Z1 @ Z2 ) )
=> ~ ( member_list_nat2 @ E @ ( set_list_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_160_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat2 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_161_list_Oset__intros_I1_J,axiom,
! [X21: list_nat,X22: list_list_nat] : ( member_list_nat2 @ X21 @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_162_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_163_list_Oset__intros_I2_J,axiom,
! [Y: list_nat,X22: list_list_nat,X21: list_nat] :
( ( member_list_nat2 @ Y @ ( set_list_nat2 @ X22 ) )
=> ( member_list_nat2 @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_164_rgf__limit_Osimps_I1_J,axiom,
( ( equiva5889994315859557365_limit @ nil_nat )
= zero_zero_nat ) ).
% rgf_limit.simps(1)
thf(fact_165_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_166_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_167_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_168_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_169_can__select__set__list__ex1,axiom,
! [P2: nat > $o,A2: list_nat] :
( ( can_select_nat @ P2 @ ( set_nat2 @ A2 ) )
= ( list_ex1_nat @ P2 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_170_the__elem__set,axiom,
! [X: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_171_the__elem__set,axiom,
! [X: list_nat] :
( ( the_elem_list_nat @ ( set_list_nat2 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_172_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_173_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_174_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_175_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_176_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_177_can__select__def,axiom,
( can_select_nat
= ( ^ [P: nat > $o,A3: set_nat] :
? [X2: nat] :
( ( member_nat2 @ X2 @ A3 )
& ( P @ X2 )
& ! [Y2: nat] :
( ( ( member_nat2 @ Y2 @ A3 )
& ( P @ Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_178_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_179_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_180_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_181_infinite__descent,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P2 @ M2 ) ) )
=> ( P2 @ N2 ) ) ).
% infinite_descent
thf(fact_182_nat__less__induct,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P2 @ M2 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N2 ) ) ).
% nat_less_induct
thf(fact_183_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_184_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_185_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_186_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_187_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less_nat @ M @ N2 )
| ( ord_less_nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_188_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_189_gr__implies__not__zero,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_190_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_191_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_192_infinite__descent0,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P2 @ M2 ) ) ) )
=> ( P2 @ N2 ) ) ) ).
% infinite_descent0
thf(fact_193_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_194_n__lists__Nil,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( n_lists_list_nat @ N2 @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) )
& ( ( N2 != zero_zero_nat )
=> ( ( n_lists_list_nat @ N2 @ nil_list_nat )
= nil_list_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_195_n__lists__Nil,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( n_lists_nat @ N2 @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N2 != zero_zero_nat )
=> ( ( n_lists_nat @ N2 @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_196_n__lists_Osimps_I1_J,axiom,
! [Xs: list_list_nat] :
( ( n_lists_list_nat @ zero_zero_nat @ Xs )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_197_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_198_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_199_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_200_product__lists_Osimps_I1_J,axiom,
( ( produc6783906451316923569st_nat @ nil_list_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_201_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_202_stable__sort__key__sort__key,axiom,
linord227665693835759911at_nat @ linord738340561235409698at_nat ).
% stable_sort_key_sort_key
thf(fact_203_prefixes__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( prefixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( append_list_nat @ ( prefixes_nat @ Xs ) @ ( cons_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) @ nil_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_204_prefixes__snoc,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( prefixes_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( append_list_list_nat @ ( prefixes_list_nat @ Xs ) @ ( cons_list_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) @ nil_list_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_205_maps__simps_I1_J,axiom,
! [F: nat > list_nat,X: nat,Xs: list_nat] :
( ( maps_nat_nat @ F @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ ( F @ X ) @ ( maps_nat_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_206_maps__simps_I1_J,axiom,
! [F: list_nat > list_nat,X: list_nat,Xs: list_list_nat] :
( ( maps_list_nat_nat @ F @ ( cons_list_nat @ X @ Xs ) )
= ( append_nat @ ( F @ X ) @ ( maps_list_nat_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_207_subseqs_Osimps_I1_J,axiom,
( ( subseqs_list_nat @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_208_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_209_prefixes__not__Nil,axiom,
! [Xs: list_nat] :
( ( prefixes_nat @ Xs )
!= nil_list_nat ) ).
% prefixes_not_Nil
thf(fact_210_Cons__in__subseqsD,axiom,
! [Y: nat,Ys: list_nat,Xs: list_nat] :
( ( member_list_nat2 @ ( cons_nat @ Y @ Ys ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
=> ( member_list_nat2 @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_211_Cons__in__subseqsD,axiom,
! [Y: list_nat,Ys: list_list_nat,Xs: list_list_nat] :
( ( member_list_list_nat @ ( cons_list_nat @ Y @ Ys ) @ ( set_list_list_nat2 @ ( subseqs_list_nat @ Xs ) ) )
=> ( member_list_list_nat @ Ys @ ( set_list_list_nat2 @ ( subseqs_list_nat @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_212_prefixes_Osimps_I1_J,axiom,
( ( prefixes_list_nat @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_213_prefixes_Osimps_I1_J,axiom,
( ( prefixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_214_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_215_maps__simps_I2_J,axiom,
! [F: nat > list_list_nat] :
( ( maps_nat_list_nat @ F @ nil_nat )
= nil_list_nat ) ).
% maps_simps(2)
thf(fact_216_maps__simps_I2_J,axiom,
! [F: list_nat > list_nat] :
( ( maps_list_nat_nat @ F @ nil_list_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_217_maps__simps_I2_J,axiom,
! [F: list_nat > list_list_nat] :
( ( maps_l5785965478274863235st_nat @ F @ nil_list_nat )
= nil_list_nat ) ).
% maps_simps(2)
thf(fact_218_prefixes__eq__snoc,axiom,
! [Ys: list_list_nat,Xs: list_list_list_nat,X: list_list_nat] :
( ( ( prefixes_list_nat @ Ys )
= ( append_list_list_nat @ Xs @ ( cons_list_list_nat @ X @ nil_list_list_nat ) ) )
= ( ( ( ( Ys = nil_list_nat )
& ( Xs = nil_list_list_nat ) )
| ? [Z: list_nat,Zs3: list_list_nat] :
( ( Ys
= ( append_list_nat @ Zs3 @ ( cons_list_nat @ Z @ nil_list_nat ) ) )
& ( Xs
= ( prefixes_list_nat @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_219_prefixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( prefixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z: nat,Zs3: list_nat] :
( ( Ys
= ( append_nat @ Zs3 @ ( cons_nat @ Z @ nil_nat ) ) )
& ( Xs
= ( prefixes_nat @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_220_suffixes__eq__snoc,axiom,
! [Ys: list_list_nat,Xs: list_list_list_nat,X: list_list_nat] :
( ( ( suffixes_list_nat @ Ys )
= ( append_list_list_nat @ Xs @ ( cons_list_list_nat @ X @ nil_list_list_nat ) ) )
= ( ( ( ( Ys = nil_list_nat )
& ( Xs = nil_list_list_nat ) )
| ? [Z: list_nat,Zs3: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z @ Zs3 ) )
& ( Xs
= ( suffixes_list_nat @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_221_suffixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( suffixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs3 ) )
& ( Xs
= ( suffixes_nat @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_222_sublists_Osimps_I1_J,axiom,
( ( sublists_list_nat @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% sublists.simps(1)
thf(fact_223_sublists_Osimps_I1_J,axiom,
( ( sublists_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% sublists.simps(1)
thf(fact_224_concat__eq__append__conv,axiom,
! [Xss2: list_list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( ( concat_list_nat @ Xss2 )
= ( append_list_nat @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_list_nat )
=> ( ( Ys = nil_list_nat )
& ( Zs = nil_list_nat ) ) )
& ( ( Xss2 != nil_list_list_nat )
=> ? [Xss1: list_list_list_nat,Xs2: list_list_nat,Xs5: list_list_nat,Xss22: list_list_list_nat] :
( ( Xss2
= ( append_list_list_nat @ Xss1 @ ( cons_list_list_nat @ ( append_list_nat @ Xs2 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_list_nat @ ( concat_list_nat @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_list_nat @ Xs5 @ ( concat_list_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_225_concat__eq__append__conv,axiom,
! [Xss2: list_list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_nat )
=> ( ( Ys = nil_nat )
& ( Zs = nil_nat ) ) )
& ( ( Xss2 != nil_list_nat )
=> ? [Xss1: list_list_nat,Xs2: list_nat,Xs5: list_nat,Xss22: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs2 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_nat @ Xs5 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_226_suffixes_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( suffixes_nat @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( suffixes_nat @ Xs ) @ ( cons_list_nat @ ( cons_nat @ X @ Xs ) @ nil_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_227_suffixes_Osimps_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( suffixes_list_nat @ ( cons_list_nat @ X @ Xs ) )
= ( append_list_list_nat @ ( suffixes_list_nat @ Xs ) @ ( cons_list_list_nat @ ( cons_list_nat @ X @ Xs ) @ nil_list_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_228_enum__rgfs_Osimps_I1_J,axiom,
( ( equiva7426478223624825838m_rgfs @ zero_zero_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% enum_rgfs.simps(1)
thf(fact_229_concat__eq__appendD,axiom,
! [Xss2: list_list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( ( concat_list_nat @ Xss2 )
= ( append_list_nat @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_list_nat )
=> ? [Xss12: list_list_list_nat,Xs3: list_list_nat,Xs6: list_list_nat,Xss23: list_list_list_nat] :
( ( Xss2
= ( append_list_list_nat @ Xss12 @ ( cons_list_list_nat @ ( append_list_nat @ Xs3 @ Xs6 ) @ Xss23 ) ) )
& ( Ys
= ( append_list_nat @ ( concat_list_nat @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append_list_nat @ Xs6 @ ( concat_list_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_230_concat__eq__appendD,axiom,
! [Xss2: list_list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_nat )
=> ? [Xss12: list_list_nat,Xs3: list_nat,Xs6: list_nat,Xss23: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs3 @ Xs6 ) @ Xss23 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append_nat @ Xs6 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_231_upt__rec,axiom,
( upt
= ( ^ [I2: nat,J: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_232_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_233_nths__singleton,axiom,
! [A2: set_nat,X: nat] :
( ( ( member_nat2 @ zero_zero_nat @ A2 )
=> ( ( nths_nat @ ( cons_nat @ X @ nil_nat ) @ A2 )
= ( cons_nat @ X @ nil_nat ) ) )
& ( ~ ( member_nat2 @ zero_zero_nat @ A2 )
=> ( ( nths_nat @ ( cons_nat @ X @ nil_nat ) @ A2 )
= nil_nat ) ) ) ).
% nths_singleton
thf(fact_234_nths__singleton,axiom,
! [A2: set_nat,X: list_nat] :
( ( ( member_nat2 @ zero_zero_nat @ A2 )
=> ( ( nths_list_nat @ ( cons_list_nat @ X @ nil_list_nat ) @ A2 )
= ( cons_list_nat @ X @ nil_list_nat ) ) )
& ( ~ ( member_nat2 @ zero_zero_nat @ A2 )
=> ( ( nths_list_nat @ ( cons_list_nat @ X @ nil_list_nat ) @ A2 )
= nil_list_nat ) ) ) ).
% nths_singleton
thf(fact_235_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_236_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_237_Suc__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_eq
thf(fact_238_Suc__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_239_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_240_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_nat @ nil_nat @ A2 )
= nil_nat ) ).
% nths_nil
thf(fact_241_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_list_nat @ nil_list_nat @ A2 )
= nil_list_nat ) ).
% nths_nil
thf(fact_242_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_243_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_244_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_nat] :
( ( ( concat_nat @ Xss2 )
= nil_nat )
= ( ! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xss2 ) )
=> ( X2 = nil_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_245_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_list_nat] :
( ( ( concat_list_nat @ Xss2 )
= nil_list_nat )
= ( ! [X2: list_list_nat] :
( ( member_list_list_nat @ X2 @ ( set_list_list_nat2 @ Xss2 ) )
=> ( X2 = nil_list_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_246_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_nat] :
( ( nil_nat
= ( concat_nat @ Xss2 ) )
= ( ! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xss2 ) )
=> ( X2 = nil_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_247_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_list_nat] :
( ( nil_list_nat
= ( concat_list_nat @ Xss2 ) )
= ( ! [X2: list_list_nat] :
( ( member_list_list_nat @ X2 @ ( set_list_list_nat2 @ Xss2 ) )
=> ( X2 = nil_list_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_248_concat__append,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( concat_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( append_nat @ ( concat_nat @ Xs ) @ ( concat_nat @ Ys ) ) ) ).
% concat_append
thf(fact_249_concat__append,axiom,
! [Xs: list_list_list_nat,Ys: list_list_list_nat] :
( ( concat_list_nat @ ( append_list_list_nat @ Xs @ Ys ) )
= ( append_list_nat @ ( concat_list_nat @ Xs ) @ ( concat_list_nat @ Ys ) ) ) ).
% concat_append
thf(fact_250_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_251_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_252_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_253_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_254_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_255_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_256_zero__induct,axiom,
! [P2: nat > $o,K: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_257_diff__induct,axiom,
! [P2: nat > nat > $o,M: nat,N2: nat] :
( ! [X3: nat] : ( P2 @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P2 @ X3 @ Y3 )
=> ( P2 @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M @ N2 ) ) ) ) ).
% diff_induct
thf(fact_258_nat__induct,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) )
=> ( P2 @ N2 ) ) ) ).
% nat_induct
thf(fact_259_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_260_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_261_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_262_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_263_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_264_enum__rgfs_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% enum_rgfs.cases
thf(fact_265_not__less__less__Suc__eq,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_266_strict__inc__induct,axiom,
! [I: nat,J2: nat,P2: nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P2 @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( P2 @ ( suc @ I3 ) )
=> ( P2 @ I3 ) ) )
=> ( P2 @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_267_less__Suc__induct,axiom,
! [I: nat,J2: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ( P2 @ I3 @ J3 )
=> ( ( P2 @ J3 @ K2 )
=> ( P2 @ I3 @ K2 ) ) ) ) )
=> ( P2 @ I @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_268_less__trans__Suc,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_269_Suc__less__SucD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_SucD
thf(fact_270_less__antisym,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
=> ( M = N2 ) ) ) ).
% less_antisym
thf(fact_271_Suc__less__eq2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_272_All__less__Suc,axiom,
! [N2: nat,P2: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
=> ( P2 @ I2 ) ) )
= ( ( P2 @ N2 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( P2 @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_273_not__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_274_less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ).
% less_Suc_eq
thf(fact_275_Ex__less__Suc,axiom,
! [N2: nat,P2: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
& ( P2 @ I2 ) ) )
= ( ( P2 @ N2 )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
& ( P2 @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_276_less__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_277_less__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M @ N2 )
=> ( M = N2 ) ) ) ).
% less_SucE
thf(fact_278_Suc__lessI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ( suc @ M )
!= N2 )
=> ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_279_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_280_Suc__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_lessD
thf(fact_281_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_282_in__set__nthsD,axiom,
! [X: nat,Xs: list_nat,I4: set_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ ( nths_nat @ Xs @ I4 ) ) )
=> ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_283_notin__set__nthsI,axiom,
! [X: nat,Xs: list_nat,I4: set_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ~ ( member_nat2 @ X @ ( set_nat2 @ ( nths_nat @ Xs @ I4 ) ) ) ) ).
% notin_set_nthsI
thf(fact_284_suffixes__not__Nil,axiom,
! [Xs: list_nat] :
( ( suffixes_nat @ Xs )
!= nil_list_nat ) ).
% suffixes_not_Nil
thf(fact_285_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M ) )
= ( ord_less_nat @ N2 @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_286_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N2: nat,M: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M ) )
= ( ord_less_nat @ N2 @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_287_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_288_lift__Suc__mono__less,axiom,
! [F: nat > int,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ord_less_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_289_less__Suc__eq__0__disj,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( M = zero_zero_nat )
| ? [J: nat] :
( ( M
= ( suc @ J ) )
& ( ord_less_nat @ J @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_290_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_291_All__less__Suc2,axiom,
! [N2: nat,P2: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
=> ( P2 @ I2 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( P2 @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_292_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_293_Ex__less__Suc2,axiom,
! [N2: nat,P2: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
& ( P2 @ I2 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
& ( P2 @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_294_concat_Osimps_I2_J,axiom,
! [X: list_list_nat,Xs: list_list_list_nat] :
( ( concat_list_nat @ ( cons_list_list_nat @ X @ Xs ) )
= ( append_list_nat @ X @ ( concat_list_nat @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_295_concat_Osimps_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( concat_nat @ ( cons_list_nat @ X @ Xs ) )
= ( append_nat @ X @ ( concat_nat @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_296_concat_Osimps_I1_J,axiom,
( ( concat_list_nat @ nil_list_list_nat )
= nil_list_nat ) ).
% concat.simps(1)
thf(fact_297_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_298_upt__conv__Cons__Cons,axiom,
! [M: nat,N2: nat,Ns: list_nat,Q: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N2 @ Ns ) )
= ( upt @ M @ Q ) )
= ( ( cons_nat @ N2 @ Ns )
= ( upt @ ( suc @ M ) @ Q ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_299_upt__conv__Cons,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( upt @ I @ J2 )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J2 ) ) ) ) ).
% upt_conv_Cons
thf(fact_300_suffixes_Osimps_I1_J,axiom,
( ( suffixes_list_nat @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_301_suffixes_Osimps_I1_J,axiom,
( ( suffixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_302_member__remove,axiom,
! [X: nat,Y: nat,A2: set_nat] :
( ( member_nat2 @ X @ ( remove_nat @ Y @ A2 ) )
= ( ( member_nat2 @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_303_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_304_upt__rec__numeral,axiom,
! [M: num,N2: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
= ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_305_upt__Suc__append,axiom,
! [I: nat,J2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( upt @ I @ ( suc @ J2 ) )
= ( append_nat @ ( upt @ I @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_306_upt__Suc,axiom,
! [I: nat,J2: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
=> ( ( upt @ I @ ( suc @ J2 ) )
= ( append_nat @ ( upt @ I @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I @ J2 )
=> ( ( upt @ I @ ( suc @ J2 ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_307_gen__length__code_I2_J,axiom,
! [N2: nat,X: nat,Xs: list_nat] :
( ( gen_length_nat @ N2 @ ( cons_nat @ X @ Xs ) )
= ( gen_length_nat @ ( suc @ N2 ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_308_gen__length__code_I2_J,axiom,
! [N2: nat,X: list_nat,Xs: list_list_nat] :
( ( gen_length_list_nat @ N2 @ ( cons_list_nat @ X @ Xs ) )
= ( gen_length_list_nat @ ( suc @ N2 ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_309_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_310_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_311_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_312_numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% numeral_le_iff
thf(fact_313_numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% numeral_le_iff
thf(fact_314_numeral__less__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
= ( ord_less_num @ M @ N2 ) ) ).
% numeral_less_iff
thf(fact_315_numeral__less__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
= ( ord_less_num @ M @ N2 ) ) ).
% numeral_less_iff
thf(fact_316_Suc__le__mono,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N2 @ M ) ) ).
% Suc_le_mono
thf(fact_317_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_318_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_319_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_320_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_321_upt__conv__Nil,axiom,
! [J2: nat,I: nat] :
( ( ord_less_eq_nat @ J2 @ I )
=> ( ( upt @ I @ J2 )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_322_upt__eq__Nil__conv,axiom,
! [I: nat,J2: nat] :
( ( ( upt @ I @ J2 )
= nil_nat )
= ( ( J2 = zero_zero_nat )
| ( ord_less_eq_nat @ J2 @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_323_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_324_lift__Suc__mono__le,axiom,
! [F: nat > int,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_325_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_326_lift__Suc__antimono__le,axiom,
! [F: nat > int,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_327_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y4: nat] :
( ( P2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_328_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
| ( ord_less_eq_nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_329_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_330_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_331_le__trans,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_332_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_333_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M6: nat] :
( ( P2 @ X )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( ord_less_eq_nat @ X3 @ M6 ) )
=> ~ ! [M3: nat] :
( ( P2 @ M3 )
=> ~ ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_334_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_335_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_336_zero__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% zero_le_numeral
thf(fact_337_zero__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).
% zero_le_numeral
thf(fact_338_zero__neq__numeral,axiom,
! [N2: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N2 ) ) ).
% zero_neq_numeral
thf(fact_339_zero__neq__numeral,axiom,
! [N2: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N2 ) ) ).
% zero_neq_numeral
thf(fact_340_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_341_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_342_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_343_transitive__stepwise__le,axiom,
! [M: nat,N2: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_344_nat__induct__at__least,axiom,
! [M: nat,N2: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P2 @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_345_full__nat__induct,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( P2 @ M2 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N2 ) ) ).
% full_nat_induct
thf(fact_346_not__less__eq__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).
% not_less_eq_eq
thf(fact_347_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_348_le__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M @ N2 )
| ( M
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_349_Suc__le__D,axiom,
! [N2: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M7 )
=> ? [M3: nat] :
( M7
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_350_le__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_351_le__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M @ N2 )
=> ( M
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_352_Suc__leD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% Suc_leD
thf(fact_353_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_354_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_355_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_356_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_357_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
& ( M5 != N ) ) ) ) ).
% nat_less_le
thf(fact_358_less__imp__le__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_imp_le_nat
thf(fact_359_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N: nat] :
( ( ord_less_nat @ M5 @ N )
| ( M5 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_360_less__or__eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_361_le__neq__implies__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( M != N2 )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_362_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_363_zero__less__numeral,axiom,
! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% zero_less_numeral
thf(fact_364_zero__less__numeral,axiom,
! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).
% zero_less_numeral
thf(fact_365_not__numeral__less__zero,axiom,
! [N2: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_366_not__numeral__less__zero,axiom,
! [N2: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_367_Suc__leI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_leI
thf(fact_368_Suc__le__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_eq
thf(fact_369_dec__induct,axiom,
! [I: nat,J2: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P2 @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) ) )
=> ( P2 @ J2 ) ) ) ) ).
% dec_induct
thf(fact_370_inc__induct,axiom,
! [I: nat,J2: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P2 @ J2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% inc_induct
thf(fact_371_Suc__le__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_lessD
thf(fact_372_le__less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% le_less_Suc_eq
thf(fact_373_less__Suc__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_374_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_375_le__imp__less__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_376_ex__least__nat__le,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ N2 )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P2 @ I5 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_377_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_length_nat @ N2 @ nil_nat )
= N2 ) ).
% gen_length_code(1)
thf(fact_378_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_length_list_nat @ N2 @ nil_list_nat )
= N2 ) ).
% gen_length_code(1)
thf(fact_379_ex__least__nat__less,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ N2 )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N2 )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K2 )
=> ~ ( P2 @ I5 ) )
& ( P2 @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_380_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat2 @ C @ A2 )
=> ( member_nat2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_381_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_382_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_383_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_384_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_385_rgf__def,axiom,
( equiva3371634703666331078on_rgf
= ( ^ [X2: list_nat] :
! [Ys3: list_nat,Y2: nat] :
( ( prefix_nat @ ( append_nat @ Ys3 @ ( cons_nat @ Y2 @ nil_nat ) ) @ X2 )
=> ( ord_less_eq_nat @ Y2 @ ( equiva5889994315859557365_limit @ Ys3 ) ) ) ) ) ).
% rgf_def
thf(fact_386_nat__descend__induct,axiom,
! [N2: nat,P2: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P2 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P2 @ I5 ) )
=> ( P2 @ K2 ) ) )
=> ( P2 @ M ) ) ) ).
% nat_descend_induct
thf(fact_387_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_388_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_389_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_390_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
=> ( member_nat2 @ X3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_391_prefix__order_Oorder__refl,axiom,
! [X: list_nat] : ( prefix_nat @ X @ X ) ).
% prefix_order.order_refl
thf(fact_392_prefix__order_Odual__order_Orefl,axiom,
! [A: list_nat] : ( prefix_nat @ A @ A ) ).
% prefix_order.dual_order.refl
thf(fact_393_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_394_Cons__prefix__Cons,axiom,
! [X: list_nat,Xs: list_list_nat,Y: list_nat,Ys: list_list_nat] :
( ( prefix_list_nat @ ( cons_list_nat @ X @ Xs ) @ ( cons_list_nat @ Y @ Ys ) )
= ( ( X = Y )
& ( prefix_list_nat @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_395_Cons__prefix__Cons,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( prefix_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( ( X = Y )
& ( prefix_nat @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_396_prefix__Nil,axiom,
! [Xs: list_list_nat] :
( ( prefix_list_nat @ Xs @ nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% prefix_Nil
thf(fact_397_prefix__Nil,axiom,
! [Xs: list_nat] :
( ( prefix_nat @ Xs @ nil_nat )
= ( Xs = nil_nat ) ) ).
% prefix_Nil
thf(fact_398_prefix__bot_Oextremum__unique,axiom,
! [A: list_list_nat] :
( ( prefix_list_nat @ A @ nil_list_nat )
= ( A = nil_list_nat ) ) ).
% prefix_bot.extremum_unique
thf(fact_399_prefix__bot_Oextremum__unique,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
= ( A = nil_nat ) ) ).
% prefix_bot.extremum_unique
thf(fact_400_prefix__code_I1_J,axiom,
! [Xs: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ Xs ) ).
% prefix_code(1)
thf(fact_401_prefix__code_I1_J,axiom,
! [Xs: list_nat] : ( prefix_nat @ nil_nat @ Xs ) ).
% prefix_code(1)
thf(fact_402_atMost__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
= ( ord_less_eq_int @ I @ K ) ) ).
% atMost_iff
thf(fact_403_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat2 @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_404_same__prefix__prefix,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) )
= ( prefix_nat @ Ys @ Zs ) ) ).
% same_prefix_prefix
thf(fact_405_atMost__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_406_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_407_same__prefix__nil,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs @ Ys ) @ Xs )
= ( Ys = nil_list_nat ) ) ).
% same_prefix_nil
thf(fact_408_same__prefix__nil,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ Ys ) @ Xs )
= ( Ys = nil_nat ) ) ).
% same_prefix_nil
thf(fact_409_in__set__prefixes,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( member_list_nat2 @ Xs @ ( set_list_nat2 @ ( prefixes_nat @ Ys ) ) )
= ( prefix_nat @ Xs @ Ys ) ) ).
% in_set_prefixes
thf(fact_410_prefix__snoc,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,Y: list_nat] :
( ( prefix_list_nat @ Xs @ ( append_list_nat @ Ys @ ( cons_list_nat @ Y @ nil_list_nat ) ) )
= ( ( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ Y @ nil_list_nat ) ) )
| ( prefix_list_nat @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_411_prefix__snoc,axiom,
! [Xs: list_nat,Ys: list_nat,Y: nat] :
( ( prefix_nat @ Xs @ ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
| ( prefix_nat @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_412_set__mono__prefix,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_mono_prefix
thf(fact_413_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( member_nat2 @ X @ B2 ) ) ) ).
% in_mono
thf(fact_414_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat2 @ C @ A2 )
=> ( member_nat2 @ C @ B2 ) ) ) ).
% subsetD
thf(fact_415_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X2: nat] :
( ( member_nat2 @ X2 @ A3 )
=> ( member_nat2 @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_416_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T2: nat] :
( ( member_nat2 @ T2 @ A3 )
=> ( member_nat2 @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_417_prefix__order_Otrans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( prefix_nat @ A @ B )
=> ( ( prefix_nat @ B @ C )
=> ( prefix_nat @ A @ C ) ) ) ).
% prefix_order.trans
thf(fact_418_prefix__order_Oeq__iff,axiom,
( ( ^ [Y5: list_nat,Z4: list_nat] : ( Y5 = Z4 ) )
= ( ^ [A4: list_nat,B4: list_nat] :
( ( prefix_nat @ A4 @ B4 )
& ( prefix_nat @ B4 @ A4 ) ) ) ) ).
% prefix_order.eq_iff
thf(fact_419_prefix__order_Oantisym,axiom,
! [A: list_nat,B: list_nat] :
( ( prefix_nat @ A @ B )
=> ( ( prefix_nat @ B @ A )
=> ( A = B ) ) ) ).
% prefix_order.antisym
thf(fact_420_prefix__order_Oeq__refl,axiom,
! [X: list_nat,Y: list_nat] :
( ( X = Y )
=> ( prefix_nat @ X @ Y ) ) ).
% prefix_order.eq_refl
thf(fact_421_prefix__order_Oorder__trans,axiom,
! [X: list_nat,Y: list_nat,Z5: list_nat] :
( ( prefix_nat @ X @ Y )
=> ( ( prefix_nat @ Y @ Z5 )
=> ( prefix_nat @ X @ Z5 ) ) ) ).
% prefix_order.order_trans
thf(fact_422_prefix__order_Oantisym__conv,axiom,
! [Y: list_nat,X: list_nat] :
( ( prefix_nat @ Y @ X )
=> ( ( prefix_nat @ X @ Y )
= ( X = Y ) ) ) ).
% prefix_order.antisym_conv
thf(fact_423_prefix__order_Oorder__eq__iff,axiom,
( ( ^ [Y5: list_nat,Z4: list_nat] : ( Y5 = Z4 ) )
= ( ^ [X2: list_nat,Y2: list_nat] :
( ( prefix_nat @ X2 @ Y2 )
& ( prefix_nat @ Y2 @ X2 ) ) ) ) ).
% prefix_order.order_eq_iff
thf(fact_424_prefix__order_Oorder__antisym,axiom,
! [X: list_nat,Y: list_nat] :
( ( prefix_nat @ X @ Y )
=> ( ( prefix_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% prefix_order.order_antisym
thf(fact_425_prefix__order_Oord__eq__le__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( A = B )
=> ( ( prefix_nat @ B @ C )
=> ( prefix_nat @ A @ C ) ) ) ).
% prefix_order.ord_eq_le_trans
thf(fact_426_prefix__order_Oord__le__eq__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( prefix_nat @ A @ B )
=> ( ( B = C )
=> ( prefix_nat @ A @ C ) ) ) ).
% prefix_order.ord_le_eq_trans
thf(fact_427_prefix__order_Odual__order_Otrans,axiom,
! [B: list_nat,A: list_nat,C: list_nat] :
( ( prefix_nat @ B @ A )
=> ( ( prefix_nat @ C @ B )
=> ( prefix_nat @ C @ A ) ) ) ).
% prefix_order.dual_order.trans
thf(fact_428_prefix__order_Odual__order_Oeq__iff,axiom,
( ( ^ [Y5: list_nat,Z4: list_nat] : ( Y5 = Z4 ) )
= ( ^ [A4: list_nat,B4: list_nat] :
( ( prefix_nat @ B4 @ A4 )
& ( prefix_nat @ A4 @ B4 ) ) ) ) ).
% prefix_order.dual_order.eq_iff
thf(fact_429_prefix__order_Odual__order_Oantisym,axiom,
! [B: list_nat,A: list_nat] :
( ( prefix_nat @ B @ A )
=> ( ( prefix_nat @ A @ B )
=> ( A = B ) ) ) ).
% prefix_order.dual_order.antisym
thf(fact_430_prefix__same__cases,axiom,
! [Xs_1: list_nat,Ys: list_nat,Xs_2: list_nat] :
( ( prefix_nat @ Xs_1 @ Ys )
=> ( ( prefix_nat @ Xs_2 @ Ys )
=> ( ( prefix_nat @ Xs_1 @ Xs_2 )
| ( prefix_nat @ Xs_2 @ Xs_1 ) ) ) ) ).
% prefix_same_cases
thf(fact_431_Iic__subset__Iio__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_432_Iic__subset__Iio__iff,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_433_prefix__bot_Obot__least,axiom,
! [A: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_434_prefix__bot_Obot__least,axiom,
! [A: list_nat] : ( prefix_nat @ nil_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_435_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_list_nat] :
( ( prefix_list_nat @ A @ nil_list_nat )
=> ( A = nil_list_nat ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_436_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
=> ( A = nil_nat ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_437_Nil__prefix,axiom,
! [Xs: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ Xs ) ).
% Nil_prefix
thf(fact_438_Nil__prefix,axiom,
! [Xs: list_nat] : ( prefix_nat @ nil_nat @ Xs ) ).
% Nil_prefix
thf(fact_439_append__prefixD,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
=> ( prefix_nat @ Xs @ Zs ) ) ).
% append_prefixD
thf(fact_440_prefix__prefix,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ( prefix_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% prefix_prefix
thf(fact_441_prefix__append,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( prefix_nat @ Xs @ ( append_nat @ Ys @ Zs ) )
= ( ( prefix_nat @ Xs @ Ys )
| ? [Us2: list_nat] :
( ( Xs
= ( append_nat @ Ys @ Us2 ) )
& ( prefix_nat @ Us2 @ Zs ) ) ) ) ).
% prefix_append
thf(fact_442_prefix__def,axiom,
( prefix_nat
= ( ^ [Xs2: list_nat,Ys3: list_nat] :
? [Zs3: list_nat] :
( Ys3
= ( append_nat @ Xs2 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_443_prefixI,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( Ys
= ( append_nat @ Xs @ Zs ) )
=> ( prefix_nat @ Xs @ Ys ) ) ).
% prefixI
thf(fact_444_prefixE,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ~ ! [Zs2: list_nat] :
( Ys
!= ( append_nat @ Xs @ Zs2 ) ) ) ).
% prefixE
thf(fact_445_subset__code_I1_J,axiom,
! [Xs: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_446_not__prefix__induct,axiom,
! [Ps: list_list_nat,Ls: list_list_nat,P2: list_list_nat > list_list_nat > $o] :
( ~ ( prefix_list_nat @ Ps @ Ls )
=> ( ! [X3: list_nat,Xs3: list_list_nat] : ( P2 @ ( cons_list_nat @ X3 @ Xs3 ) @ nil_list_nat )
=> ( ! [X3: list_nat,Xs3: list_list_nat,Y3: list_nat,Ys2: list_list_nat] :
( ( X3 != Y3 )
=> ( P2 @ ( cons_list_nat @ X3 @ Xs3 ) @ ( cons_list_nat @ Y3 @ Ys2 ) ) )
=> ( ! [X3: list_nat,Xs3: list_list_nat,Y3: list_nat,Ys2: list_list_nat] :
( ( X3 = Y3 )
=> ( ~ ( prefix_list_nat @ Xs3 @ Ys2 )
=> ( ( P2 @ Xs3 @ Ys2 )
=> ( P2 @ ( cons_list_nat @ X3 @ Xs3 ) @ ( cons_list_nat @ Y3 @ Ys2 ) ) ) ) )
=> ( P2 @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_447_not__prefix__induct,axiom,
! [Ps: list_nat,Ls: list_nat,P2: list_nat > list_nat > $o] :
( ~ ( prefix_nat @ Ps @ Ls )
=> ( ! [X3: nat,Xs3: list_nat] : ( P2 @ ( cons_nat @ X3 @ Xs3 ) @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys2: list_nat] :
( ( X3 != Y3 )
=> ( P2 @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys2: list_nat] :
( ( X3 = Y3 )
=> ( ~ ( prefix_nat @ Xs3 @ Ys2 )
=> ( ( P2 @ Xs3 @ Ys2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) ) )
=> ( P2 @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_448_not__prefix__cases,axiom,
! [Ps: list_list_nat,Ls: list_list_nat] :
( ~ ( prefix_list_nat @ Ps @ Ls )
=> ( ( ( Ps != nil_list_nat )
=> ( Ls != nil_list_nat ) )
=> ( ! [A5: list_nat,As: list_list_nat] :
( ( Ps
= ( cons_list_nat @ A5 @ As ) )
=> ! [X3: list_nat,Xs3: list_list_nat] :
( ( Ls
= ( cons_list_nat @ X3 @ Xs3 ) )
=> ( ( X3 = A5 )
=> ( prefix_list_nat @ As @ Xs3 ) ) ) )
=> ~ ! [A5: list_nat] :
( ? [As: list_list_nat] :
( Ps
= ( cons_list_nat @ A5 @ As ) )
=> ! [X3: list_nat] :
( ? [Xs3: list_list_nat] :
( Ls
= ( cons_list_nat @ X3 @ Xs3 ) )
=> ( X3 = A5 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_449_not__prefix__cases,axiom,
! [Ps: list_nat,Ls: list_nat] :
( ~ ( prefix_nat @ Ps @ Ls )
=> ( ( ( Ps != nil_nat )
=> ( Ls != nil_nat ) )
=> ( ! [A5: nat,As: list_nat] :
( ( Ps
= ( cons_nat @ A5 @ As ) )
=> ! [X3: nat,Xs3: list_nat] :
( ( Ls
= ( cons_nat @ X3 @ Xs3 ) )
=> ( ( X3 = A5 )
=> ( prefix_nat @ As @ Xs3 ) ) ) )
=> ~ ! [A5: nat] :
( ? [As: list_nat] :
( Ps
= ( cons_nat @ A5 @ As ) )
=> ! [X3: nat] :
( ? [Xs3: list_nat] :
( Ls
= ( cons_nat @ X3 @ Xs3 ) )
=> ( X3 = A5 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_450_prefix__Cons,axiom,
! [Xs: list_list_nat,Y: list_nat,Ys: list_list_nat] :
( ( prefix_list_nat @ Xs @ ( cons_list_nat @ Y @ Ys ) )
= ( ( Xs = nil_list_nat )
| ? [Zs3: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Y @ Zs3 ) )
& ( prefix_list_nat @ Zs3 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_451_prefix__Cons,axiom,
! [Xs: list_nat,Y: nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ ( cons_nat @ Y @ Ys ) )
= ( ( Xs = nil_nat )
| ? [Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Y @ Zs3 ) )
& ( prefix_nat @ Zs3 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_452_prefix__code_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat] :
~ ( prefix_list_nat @ ( cons_list_nat @ X @ Xs ) @ nil_list_nat ) ).
% prefix_code(2)
thf(fact_453_prefix__code_I2_J,axiom,
! [X: nat,Xs: list_nat] :
~ ( prefix_nat @ ( cons_nat @ X @ Xs ) @ nil_nat ) ).
% prefix_code(2)
thf(fact_454_prefix__order_Olift__Suc__antimono__le,axiom,
! [F: nat > list_nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( prefix_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( prefix_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% prefix_order.lift_Suc_antimono_le
thf(fact_455_prefix__order_Olift__Suc__mono__le,axiom,
! [F: nat > list_nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( prefix_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( prefix_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% prefix_order.lift_Suc_mono_le
thf(fact_456_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_457_set__subset__Cons,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_458_set__subset__Cons,axiom,
! [Xs: list_list_nat,X: list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ ( cons_list_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_459_set__nths__subset,axiom,
! [Xs: list_nat,I4: set_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( nths_nat @ Xs @ I4 ) ) @ ( set_nat2 @ Xs ) ) ).
% set_nths_subset
thf(fact_460_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_461_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_462_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_463_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_464_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_465_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_466_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_467_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_468_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_469_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_470_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_471_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_472_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_473_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_474_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_475_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_476_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_477_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_478_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_479_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_480_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_481_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_482_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_483_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_484_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_485_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_486_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_487_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_488_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_489_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_490_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_491_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_492_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_493_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_494_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P2 @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P2 @ B5 @ A5 )
=> ( P2 @ A5 @ B5 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_495_linorder__wlog,axiom,
! [P2: int > int > $o,A: int,B: int] :
( ! [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
=> ( P2 @ A5 @ B5 ) )
=> ( ! [A5: int,B5: int] :
( ( P2 @ B5 @ A5 )
=> ( P2 @ A5 @ B5 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_496_order__trans,axiom,
! [X: nat,Y: nat,Z5: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z5 )
=> ( ord_less_eq_nat @ X @ Z5 ) ) ) ).
% order_trans
thf(fact_497_order__trans,axiom,
! [X: int,Y: int,Z5: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z5 )
=> ( ord_less_eq_int @ X @ Z5 ) ) ) ).
% order_trans
thf(fact_498_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_499_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_500_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_501_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_502_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_503_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_504_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_505_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_506_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_507_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
& ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_508_le__cases3,axiom,
! [X: nat,Y: nat,Z5: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z5 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z5 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z5 )
=> ~ ( ord_less_eq_nat @ Z5 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z5 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z5 )
=> ~ ( ord_less_eq_nat @ Z5 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z5 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_509_le__cases3,axiom,
! [X: int,Y: int,Z5: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z5 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z5 ) )
=> ( ( ( ord_less_eq_int @ X @ Z5 )
=> ~ ( ord_less_eq_int @ Z5 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z5 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z5 )
=> ~ ( ord_less_eq_int @ Z5 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z5 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_510_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_511_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_512_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_513_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_514_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_515_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_516_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_517_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_518_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_519_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_520_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_521_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_522_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_523_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_524_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_525_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_526_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_527_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_528_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_529_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_530_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_531_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_532_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_533_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_534_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_535_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_536_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_537_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_538_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_539_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_540_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_541_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_542_order__less__trans,axiom,
! [X: nat,Y: nat,Z5: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z5 )
=> ( ord_less_nat @ X @ Z5 ) ) ) ).
% order_less_trans
thf(fact_543_order__less__trans,axiom,
! [X: int,Y: int,Z5: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z5 )
=> ( ord_less_int @ X @ Z5 ) ) ) ).
% order_less_trans
thf(fact_544_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_545_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_546_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_547_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_548_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_549_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_550_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_551_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_552_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_553_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_554_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_555_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_556_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_557_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_558_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_559_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_560_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_561_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_562_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P2 @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P2 @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P2 @ B5 @ A5 )
=> ( P2 @ A5 @ B5 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_563_linorder__less__wlog,axiom,
! [P2: int > int > $o,A: int,B: int] :
( ! [A5: int,B5: int] :
( ( ord_less_int @ A5 @ B5 )
=> ( P2 @ A5 @ B5 ) )
=> ( ! [A5: int] : ( P2 @ A5 @ A5 )
=> ( ! [A5: int,B5: int] :
( ( P2 @ B5 @ A5 )
=> ( P2 @ A5 @ B5 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_564_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P: nat > $o] :
? [N: nat] :
( ( P @ N )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N )
=> ~ ( P @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_565_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_566_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_567_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_568_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_569_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_570_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_571_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_572_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_573_less__induct,axiom,
! [P2: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X3 )
=> ( P2 @ Y4 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A ) ) ).
% less_induct
thf(fact_574_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_575_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_576_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_577_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_578_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_579_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_580_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_581_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_582_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_583_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_584_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_585_atMost__upto,axiom,
( set_ord_atMost_nat
= ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N ) ) ) ) ) ).
% atMost_upto
thf(fact_586_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_587_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_588_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_589_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_590_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_591_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_592_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_593_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_594_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_595_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_596_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_597_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
& ~ ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_598_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_599_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_600_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_601_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_602_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_603_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_604_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_605_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_606_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_607_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_608_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_609_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_610_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_611_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_int @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_612_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_613_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_614_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_615_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_616_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_617_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_618_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_619_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_620_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_621_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_622_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_623_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_624_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_625_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_626_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_627_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_628_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_629_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_630_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_631_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_632_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_633_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_634_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_635_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_636_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_637_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_638_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z5: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z5 )
=> ( ord_less_nat @ X @ Z5 ) ) ) ).
% order_le_less_trans
thf(fact_639_order__le__less__trans,axiom,
! [X: int,Y: int,Z5: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z5 )
=> ( ord_less_int @ X @ Z5 ) ) ) ).
% order_le_less_trans
thf(fact_640_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z5: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z5 )
=> ( ord_less_nat @ X @ Z5 ) ) ) ).
% order_less_le_trans
thf(fact_641_order__less__le__trans,axiom,
! [X: int,Y: int,Z5: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z5 )
=> ( ord_less_int @ X @ Z5 ) ) ) ).
% order_less_le_trans
thf(fact_642_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_643_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_644_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_645_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_646_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_647_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_648_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_649_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_650_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_651_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_652_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_653_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_654_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_655_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_656_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_657_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_658_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_659_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_660_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_661_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X4 ) ) ).
% minf(8)
thf(fact_662_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_663_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ord_less_eq_int @ X4 @ T ) ) ).
% minf(6)
thf(fact_664_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_665_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ord_less_eq_int @ T @ X4 ) ) ).
% pinf(8)
thf(fact_666_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_667_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ T ) ) ).
% pinf(6)
thf(fact_668_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_669_verit__comp__simplify1_I3_J,axiom,
! [B6: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
= ( ord_less_int @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_670_complete__interval,axiom,
! [A: nat,B: nat,P2: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C2 ) )
=> ( P2 @ X4 ) )
& ! [D: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D ) )
=> ( P2 @ X3 ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_671_complete__interval,axiom,
! [A: int,B: int,P2: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ C2 ) )
=> ( P2 @ X4 ) )
& ! [D: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D ) )
=> ( P2 @ X3 ) )
=> ( ord_less_eq_int @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_672_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_673_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_674_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_675_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_676_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ~ ( ord_less_nat @ T @ X4 ) ) ).
% minf(7)
thf(fact_677_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ~ ( ord_less_int @ T @ X4 ) ) ).
% minf(7)
thf(fact_678_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ord_less_nat @ X4 @ T ) ) ).
% minf(5)
thf(fact_679_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ord_less_int @ X4 @ T ) ) ).
% minf(5)
thf(fact_680_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_681_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_682_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_683_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_684_minf_I2_J,axiom,
! [P2: nat > $o,P4: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z6 )
=> ( ( P2 @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z6 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( ( P2 @ X4 )
| ( Q2 @ X4 ) )
= ( ( P4 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_685_minf_I2_J,axiom,
! [P2: int > $o,P4: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( P2 @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ( ( P2 @ X4 )
| ( Q2 @ X4 ) )
= ( ( P4 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_686_minf_I1_J,axiom,
! [P2: nat > $o,P4: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z6 )
=> ( ( P2 @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z6 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( ( P2 @ X4 )
& ( Q2 @ X4 ) )
= ( ( P4 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_687_minf_I1_J,axiom,
! [P2: int > $o,P4: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( P2 @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ( ( P2 @ X4 )
& ( Q2 @ X4 ) )
= ( ( P4 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_688_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ord_less_nat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_689_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ord_less_int @ T @ X4 ) ) ).
% pinf(7)
thf(fact_690_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_691_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ~ ( ord_less_int @ X4 @ T ) ) ).
% pinf(5)
thf(fact_692_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_693_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_694_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_695_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_696_pinf_I2_J,axiom,
! [P2: nat > $o,P4: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z6 @ X3 )
=> ( ( P2 @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z6 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( ( P2 @ X4 )
| ( Q2 @ X4 ) )
= ( ( P4 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_697_pinf_I2_J,axiom,
! [P2: int > $o,P4: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( P2 @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ( ( P2 @ X4 )
| ( Q2 @ X4 ) )
= ( ( P4 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_698_pinf_I1_J,axiom,
! [P2: nat > $o,P4: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z6 @ X3 )
=> ( ( P2 @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z6 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( ( P2 @ X4 )
& ( Q2 @ X4 ) )
= ( ( P4 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_699_pinf_I1_J,axiom,
! [P2: int > $o,P4: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( P2 @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ( ( P2 @ X4 )
& ( Q2 @ X4 ) )
= ( ( P4 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_700_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_701_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_702_subset__code_I3_J,axiom,
~ ( ord_le6045566169113846134st_nat @ ( coset_list_nat @ nil_list_nat ) @ ( set_list_nat2 @ nil_list_nat ) ) ).
% subset_code(3)
thf(fact_703_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_704_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ ( suc @ I ) @ J2 )
=> ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J2 ) )
= ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J2 ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_705_remove__code_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( remove_nat @ X @ ( coset_nat @ Xs ) )
= ( coset_nat @ ( insert_nat @ X @ Xs ) ) ) ).
% remove_code(2)
thf(fact_706_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J2: nat] :
( ( ord_less_eq_nat @ ( suc @ I ) @ J2 )
=> ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J2 ) )
= ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J2 ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_707_greaterThanLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat2 @ I @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_708_greaterThanLessThan__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or5832277885323065728an_int @ L @ U ) )
= ( ( ord_less_int @ L @ I )
& ( ord_less_int @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_709_greaterThanAtMost__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or6656581121297822940st_int @ L @ U ) )
= ( ( ord_less_int @ L @ I )
& ( ord_less_eq_int @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_710_greaterThanAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat2 @ I @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_711_Ioc__inj,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( set_or6656581121297822940st_int @ A @ B )
= ( set_or6656581121297822940st_int @ C @ D2 ) )
= ( ( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ D2 @ C ) )
| ( ( A = C )
& ( B = D2 ) ) ) ) ).
% Ioc_inj
thf(fact_712_Ioc__inj,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( set_or6659071591806873216st_nat @ A @ B )
= ( set_or6659071591806873216st_nat @ C @ D2 ) )
= ( ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ D2 @ C ) )
| ( ( A = C )
& ( B = D2 ) ) ) ) ).
% Ioc_inj
thf(fact_713_Ioc__subset__iff,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_set_int @ ( set_or6656581121297822940st_int @ A @ B ) @ ( set_or6656581121297822940st_int @ C @ D2 ) )
= ( ( ord_less_eq_int @ B @ A )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D2 ) ) ) ) ).
% Ioc_subset_iff
thf(fact_714_Ioc__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or6659071591806873216st_nat @ A @ B ) @ ( set_or6659071591806873216st_nat @ C @ D2 ) )
= ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).
% Ioc_subset_iff
thf(fact_715_subset__code_I2_J,axiom,
! [A2: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( coset_nat @ Ys ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat2 @ X2 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_716_greaterThanAtMost__upt,axiom,
( set_or6659071591806873216st_nat
= ( ^ [N: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N ) @ ( suc @ M5 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_717_greaterThanLessThan__upt,axiom,
( set_or5834768355832116004an_nat
= ( ^ [N: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N ) @ M5 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_718_prefix__bot_Omax__bot,axiom,
! [X: list_list_nat] :
( ( max_list_list_nat @ prefix_list_nat @ nil_list_nat @ X )
= X ) ).
% prefix_bot.max_bot
thf(fact_719_prefix__bot_Omax__bot,axiom,
! [X: list_nat] :
( ( max_list_nat @ prefix_nat @ nil_nat @ X )
= X ) ).
% prefix_bot.max_bot
thf(fact_720_prefix__bot_Omin__bot,axiom,
! [X: list_list_nat] :
( ( min_list_list_nat @ prefix_list_nat @ nil_list_nat @ X )
= nil_list_nat ) ).
% prefix_bot.min_bot
thf(fact_721_prefix__bot_Omin__bot,axiom,
! [X: list_nat] :
( ( min_list_nat @ prefix_nat @ nil_nat @ X )
= nil_nat ) ).
% prefix_bot.min_bot
thf(fact_722_prefix__bot_Omax__bot2,axiom,
! [X: list_list_nat] :
( ( max_list_list_nat @ prefix_list_nat @ X @ nil_list_nat )
= X ) ).
% prefix_bot.max_bot2
thf(fact_723_prefix__bot_Omax__bot2,axiom,
! [X: list_nat] :
( ( max_list_nat @ prefix_nat @ X @ nil_nat )
= X ) ).
% prefix_bot.max_bot2
thf(fact_724_prefix__bot_Omin__bot2,axiom,
! [X: list_list_nat] :
( ( min_list_list_nat @ prefix_list_nat @ X @ nil_list_nat )
= nil_list_nat ) ).
% prefix_bot.min_bot2
thf(fact_725_prefix__bot_Omin__bot2,axiom,
! [X: list_nat] :
( ( min_list_nat @ prefix_nat @ X @ nil_nat )
= nil_nat ) ).
% prefix_bot.min_bot2
thf(fact_726_prefix__order_Omin__def,axiom,
! [A: list_nat,B: list_nat] :
( ( ( prefix_nat @ A @ B )
=> ( ( min_list_nat @ prefix_nat @ A @ B )
= A ) )
& ( ~ ( prefix_nat @ A @ B )
=> ( ( min_list_nat @ prefix_nat @ A @ B )
= B ) ) ) ).
% prefix_order.min_def
thf(fact_727_prefix__order_Omax__def,axiom,
! [A: list_nat,B: list_nat] :
( ( ( prefix_nat @ A @ B )
=> ( ( max_list_nat @ prefix_nat @ A @ B )
= B ) )
& ( ~ ( prefix_nat @ A @ B )
=> ( ( max_list_nat @ prefix_nat @ A @ B )
= A ) ) ) ).
% prefix_order.max_def
thf(fact_728_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_729_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_730_butlast__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_731_butlast__snoc,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( butlast_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_732_sublists_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( sublists_nat @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( sublists_nat @ Xs ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_733_sublists_Osimps_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( sublists_list_nat @ ( cons_list_nat @ X @ Xs ) )
= ( append_list_list_nat @ ( sublists_list_nat @ Xs ) @ ( map_li2855073862107769254st_nat @ ( cons_list_nat @ X ) @ ( prefixes_list_nat @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_734_subset__subseqs,axiom,
! [X6: set_nat,Xs: list_nat] :
( ( ord_less_eq_set_nat @ X6 @ ( set_nat2 @ Xs ) )
=> ( member_set_nat @ X6 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_735_image__eqI,axiom,
! [B: int,F: nat > int,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat2 @ X @ A2 )
=> ( member_int @ B @ ( image_nat_int @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_736_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat2 @ X @ A2 )
=> ( member_nat2 @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_737_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_738_list_Omap__disc__iff,axiom,
! [F: list_nat > nat,A: list_list_nat] :
( ( ( map_list_nat_nat @ F @ A )
= nil_nat )
= ( A = nil_list_nat ) ) ).
% list.map_disc_iff
thf(fact_739_list_Omap__disc__iff,axiom,
! [F: list_nat > list_nat,A: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ A )
= nil_list_nat )
= ( A = nil_list_nat ) ) ).
% list.map_disc_iff
thf(fact_740_list_Omap__disc__iff,axiom,
! [F: nat > nat,A: list_nat] :
( ( ( map_nat_nat @ F @ A )
= nil_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_741_list_Omap__disc__iff,axiom,
! [F: list_nat > list_list_nat,A: list_list_nat] :
( ( ( map_li960784813134754710st_nat @ F @ A )
= nil_list_list_nat )
= ( A = nil_list_nat ) ) ).
% list.map_disc_iff
thf(fact_742_list_Omap__disc__iff,axiom,
! [F: nat > list_nat,A: list_nat] :
( ( ( map_nat_list_nat @ F @ A )
= nil_list_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_743_Nil__is__map__conv,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( nil_nat
= ( map_list_nat_nat @ F @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_744_Nil__is__map__conv,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat] :
( ( nil_list_nat
= ( map_li7225945977422193158st_nat @ F @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_745_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_746_Nil__is__map__conv,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( nil_list_list_nat
= ( map_li960784813134754710st_nat @ F @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_747_Nil__is__map__conv,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( nil_list_nat
= ( map_nat_list_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_748_map__is__Nil__conv,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_749_map__is__Nil__conv,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_750_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_751_map__is__Nil__conv,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( ( map_li960784813134754710st_nat @ F @ Xs )
= nil_list_list_nat )
= ( Xs = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_752_map__is__Nil__conv,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( ( map_nat_list_nat @ F @ Xs )
= nil_list_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_753_map__eq__conv,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat,G: list_nat > list_list_nat] :
( ( ( map_li960784813134754710st_nat @ F @ Xs )
= ( map_li960784813134754710st_nat @ G @ Xs ) )
= ( ! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_754_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_755_map__eq__conv,axiom,
! [F: nat > list_nat,Xs: list_nat,G: nat > list_nat] :
( ( ( map_nat_list_nat @ F @ Xs )
= ( map_nat_list_nat @ G @ Xs ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_756_map__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_757_map__append,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( map_li960784813134754710st_nat @ F @ ( append_list_nat @ Xs @ Ys ) )
= ( append_list_list_nat @ ( map_li960784813134754710st_nat @ F @ Xs ) @ ( map_li960784813134754710st_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_758_map__append,axiom,
! [F: nat > list_nat,Xs: list_nat,Ys: list_nat] :
( ( map_nat_list_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_list_nat @ ( map_nat_list_nat @ F @ Xs ) @ ( map_nat_list_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_759_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_760_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_761_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_762_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_763_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_764_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_765_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_766_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_767_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_768_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_769_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ N2 ) ) ).
% of_nat_numeral
thf(fact_770_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_int @ N2 ) ) ).
% of_nat_numeral
thf(fact_771_list_Oset__map,axiom,
! [F: list_nat > list_list_nat,V: list_list_nat] :
( ( set_list_list_nat2 @ ( map_li960784813134754710st_nat @ F @ V ) )
= ( image_4042064729117200983st_nat @ F @ ( set_list_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_772_list_Oset__map,axiom,
! [F: nat > int,V: list_nat] :
( ( set_int2 @ ( map_nat_int @ F @ V ) )
= ( image_nat_int @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_773_list_Oset__map,axiom,
! [F: nat > list_nat,V: list_nat] :
( ( set_list_nat2 @ ( map_nat_list_nat @ F @ V ) )
= ( image_nat_list_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_774_list_Oset__map,axiom,
! [F: nat > nat,V: list_nat] :
( ( set_nat2 @ ( map_nat_nat @ F @ V ) )
= ( image_nat_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_775_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_776_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_777_int__ops_I3_J,axiom,
! [N2: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_int @ N2 ) ) ).
% int_ops(3)
thf(fact_778_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_779_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_780_list_Osimps_I9_J,axiom,
! [F: nat > nat,X21: nat,X22: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_781_list_Osimps_I9_J,axiom,
! [F: nat > list_nat,X21: nat,X22: list_nat] :
( ( map_nat_list_nat @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_list_nat @ ( F @ X21 ) @ ( map_nat_list_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_782_list_Osimps_I9_J,axiom,
! [F: list_nat > list_list_nat,X21: list_nat,X22: list_list_nat] :
( ( map_li960784813134754710st_nat @ F @ ( cons_list_nat @ X21 @ X22 ) )
= ( cons_list_list_nat @ ( F @ X21 ) @ ( map_li960784813134754710st_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_783_list_Osimps_I9_J,axiom,
! [F: list_nat > nat,X21: list_nat,X22: list_list_nat] :
( ( map_list_nat_nat @ F @ ( cons_list_nat @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_list_nat_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_784_list_Osimps_I9_J,axiom,
! [F: list_nat > list_nat,X21: list_nat,X22: list_list_nat] :
( ( map_li7225945977422193158st_nat @ F @ ( cons_list_nat @ X21 @ X22 ) )
= ( cons_list_nat @ ( F @ X21 ) @ ( map_li7225945977422193158st_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_785_Cons__eq__map__D,axiom,
! [X: list_list_nat,Xs: list_list_list_nat,F: list_nat > list_list_nat,Ys: list_list_nat] :
( ( ( cons_list_list_nat @ X @ Xs )
= ( map_li960784813134754710st_nat @ F @ Ys ) )
=> ? [Z3: list_nat,Zs2: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z3 @ Zs2 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_li960784813134754710st_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_786_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z3: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs2 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_787_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F: list_nat > nat,Ys: list_list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_list_nat_nat @ F @ Ys ) )
=> ? [Z3: list_nat,Zs2: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z3 @ Zs2 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_list_nat_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_788_Cons__eq__map__D,axiom,
! [X: list_nat,Xs: list_list_nat,F: nat > list_nat,Ys: list_nat] :
( ( ( cons_list_nat @ X @ Xs )
= ( map_nat_list_nat @ F @ Ys ) )
=> ? [Z3: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs2 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_list_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_789_Cons__eq__map__D,axiom,
! [X: list_nat,Xs: list_list_nat,F: list_nat > list_nat,Ys: list_list_nat] :
( ( ( cons_list_nat @ X @ Xs )
= ( map_li7225945977422193158st_nat @ F @ Ys ) )
=> ? [Z3: list_nat,Zs2: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z3 @ Zs2 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_li7225945977422193158st_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_790_map__eq__Cons__D,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat,Y: list_list_nat,Ys: list_list_list_nat] :
( ( ( map_li960784813134754710st_nat @ F @ Xs )
= ( cons_list_list_nat @ Y @ Ys ) )
=> ? [Z3: list_nat,Zs2: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_li960784813134754710st_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_791_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z3: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_792_map__eq__Cons__D,axiom,
! [F: list_nat > nat,Xs: list_list_nat,Y: nat,Ys: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z3: list_nat,Zs2: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_list_nat_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_793_map__eq__Cons__D,axiom,
! [F: nat > list_nat,Xs: list_nat,Y: list_nat,Ys: list_list_nat] :
( ( ( map_nat_list_nat @ F @ Xs )
= ( cons_list_nat @ Y @ Ys ) )
=> ? [Z3: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_nat_list_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_794_map__eq__Cons__D,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,Y: list_nat,Ys: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= ( cons_list_nat @ Y @ Ys ) )
=> ? [Z3: list_nat,Zs2: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_li7225945977422193158st_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_795_Cons__eq__map__conv,axiom,
! [X: list_list_nat,Xs: list_list_list_nat,F: list_nat > list_list_nat,Ys: list_list_nat] :
( ( ( cons_list_list_nat @ X @ Xs )
= ( map_li960784813134754710st_nat @ F @ Ys ) )
= ( ? [Z: list_nat,Zs3: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z @ Zs3 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_li960784813134754710st_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_796_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs3 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_797_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F: list_nat > nat,Ys: list_list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_list_nat_nat @ F @ Ys ) )
= ( ? [Z: list_nat,Zs3: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z @ Zs3 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_list_nat_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_798_Cons__eq__map__conv,axiom,
! [X: list_nat,Xs: list_list_nat,F: nat > list_nat,Ys: list_nat] :
( ( ( cons_list_nat @ X @ Xs )
= ( map_nat_list_nat @ F @ Ys ) )
= ( ? [Z: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs3 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_nat_list_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_799_Cons__eq__map__conv,axiom,
! [X: list_nat,Xs: list_list_nat,F: list_nat > list_nat,Ys: list_list_nat] :
( ( ( cons_list_nat @ X @ Xs )
= ( map_li7225945977422193158st_nat @ F @ Ys ) )
= ( ? [Z: list_nat,Zs3: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z @ Zs3 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_li7225945977422193158st_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_800_map__eq__Cons__conv,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat,Y: list_list_nat,Ys: list_list_list_nat] :
( ( ( map_li960784813134754710st_nat @ F @ Xs )
= ( cons_list_list_nat @ Y @ Ys ) )
= ( ? [Z: list_nat,Zs3: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y )
& ( ( map_li960784813134754710st_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_801_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_802_map__eq__Cons__conv,axiom,
! [F: list_nat > nat,Xs: list_list_nat,Y: nat,Ys: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z: list_nat,Zs3: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y )
& ( ( map_list_nat_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_803_map__eq__Cons__conv,axiom,
! [F: nat > list_nat,Xs: list_nat,Y: list_nat,Ys: list_list_nat] :
( ( ( map_nat_list_nat @ F @ Xs )
= ( cons_list_nat @ Y @ Ys ) )
= ( ? [Z: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y )
& ( ( map_nat_list_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_804_map__eq__Cons__conv,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,Y: list_nat,Ys: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= ( cons_list_nat @ Y @ Ys ) )
= ( ? [Z: list_nat,Zs3: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y )
& ( ( map_li7225945977422193158st_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_805_list_Osimps_I8_J,axiom,
! [F: list_nat > nat] :
( ( map_list_nat_nat @ F @ nil_list_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_806_list_Osimps_I8_J,axiom,
! [F: list_nat > list_nat] :
( ( map_li7225945977422193158st_nat @ F @ nil_list_nat )
= nil_list_nat ) ).
% list.simps(8)
thf(fact_807_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_808_list_Osimps_I8_J,axiom,
! [F: list_nat > list_list_nat] :
( ( map_li960784813134754710st_nat @ F @ nil_list_nat )
= nil_list_list_nat ) ).
% list.simps(8)
thf(fact_809_list_Osimps_I8_J,axiom,
! [F: nat > list_nat] :
( ( map_nat_list_nat @ F @ nil_nat )
= nil_list_nat ) ).
% list.simps(8)
thf(fact_810_ex__map__conv,axiom,
! [Ys: list_list_list_nat,F: list_nat > list_list_nat] :
( ( ? [Xs2: list_list_nat] :
( Ys
= ( map_li960784813134754710st_nat @ F @ Xs2 ) ) )
= ( ! [X2: list_list_nat] :
( ( member_list_list_nat @ X2 @ ( set_list_list_nat2 @ Ys ) )
=> ? [Y2: list_nat] :
( X2
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_811_ex__map__conv,axiom,
! [Ys: list_list_nat,F: nat > list_nat] :
( ( ? [Xs2: list_nat] :
( Ys
= ( map_nat_list_nat @ F @ Xs2 ) ) )
= ( ! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Ys ) )
=> ? [Y2: nat] :
( X2
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_812_ex__map__conv,axiom,
! [Ys: list_nat,F: nat > nat] :
( ( ? [Xs2: list_nat] :
( Ys
= ( map_nat_nat @ F @ Xs2 ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Ys ) )
=> ? [Y2: nat] :
( X2
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_813_map__cong,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,F: list_nat > list_list_nat,G: list_nat > list_list_nat] :
( ( Xs = Ys )
=> ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li960784813134754710st_nat @ F @ Xs )
= ( map_li960784813134754710st_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_814_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_815_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > list_nat,G: nat > list_nat] :
( ( Xs = Ys )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_list_nat @ F @ Xs )
= ( map_nat_list_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_816_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_817_map__ext,axiom,
! [Xs: list_list_nat,F: list_nat > list_list_nat,G: list_nat > list_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li960784813134754710st_nat @ F @ Xs )
= ( map_li960784813134754710st_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_818_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_819_map__ext,axiom,
! [Xs: list_nat,F: nat > list_nat,G: nat > list_nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_list_nat @ F @ Xs )
= ( map_nat_list_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_820_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z3: nat] :
( ( member_nat2 @ Z3 @ ( set_nat2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_821_list_Oinj__map__strong,axiom,
! [X: list_list_nat,Xa2: list_list_nat,F: list_nat > list_list_nat,Fa: list_nat > list_list_nat] :
( ! [Z3: list_nat,Za: list_nat] :
( ( member_list_nat2 @ Z3 @ ( set_list_nat2 @ X ) )
=> ( ( member_list_nat2 @ Za @ ( set_list_nat2 @ Xa2 ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_li960784813134754710st_nat @ F @ X )
= ( map_li960784813134754710st_nat @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_822_list_Oinj__map__strong,axiom,
! [X: list_nat,Xa2: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z3: nat,Za: nat] :
( ( member_nat2 @ Z3 @ ( set_nat2 @ X ) )
=> ( ( member_nat2 @ Za @ ( set_nat2 @ Xa2 ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_823_list_Oinj__map__strong,axiom,
! [X: list_nat,Xa2: list_nat,F: nat > list_nat,Fa: nat > list_nat] :
( ! [Z3: nat,Za: nat] :
( ( member_nat2 @ Z3 @ ( set_nat2 @ X ) )
=> ( ( member_nat2 @ Za @ ( set_nat2 @ Xa2 ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_nat_list_nat @ F @ X )
= ( map_nat_list_nat @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_824_list_Omap__cong0,axiom,
! [X: list_list_nat,F: list_nat > list_list_nat,G: list_nat > list_list_nat] :
( ! [Z3: list_nat] :
( ( member_list_nat2 @ Z3 @ ( set_list_nat2 @ X ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_li960784813134754710st_nat @ F @ X )
= ( map_li960784813134754710st_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_825_list_Omap__cong0,axiom,
! [X: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z3: nat] :
( ( member_nat2 @ Z3 @ ( set_nat2 @ X ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_826_list_Omap__cong0,axiom,
! [X: list_nat,F: nat > list_nat,G: nat > list_nat] :
( ! [Z3: nat] :
( ( member_nat2 @ Z3 @ ( set_nat2 @ X ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_list_nat @ F @ X )
= ( map_nat_list_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_827_list_Omap__cong,axiom,
! [X: list_list_nat,Ya: list_list_nat,F: list_nat > list_list_nat,G: list_nat > list_list_nat] :
( ( X = Ya )
=> ( ! [Z3: list_nat] :
( ( member_list_nat2 @ Z3 @ ( set_list_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_li960784813134754710st_nat @ F @ X )
= ( map_li960784813134754710st_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_828_list_Omap__cong,axiom,
! [X: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X = Ya )
=> ( ! [Z3: nat] :
( ( member_nat2 @ Z3 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_829_list_Omap__cong,axiom,
! [X: list_nat,Ya: list_nat,F: nat > list_nat,G: nat > list_nat] :
( ( X = Ya )
=> ( ! [Z3: nat] :
( ( member_nat2 @ Z3 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_list_nat @ F @ X )
= ( map_nat_list_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_830_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: nat > nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( map_nat_nat @ F @ Xs ) )
= ( ? [Us2: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_831_append__eq__map__conv,axiom,
! [Ys: list_list_list_nat,Zs: list_list_list_nat,F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( ( append_list_list_nat @ Ys @ Zs )
= ( map_li960784813134754710st_nat @ F @ Xs ) )
= ( ? [Us2: list_list_nat,Vs: list_list_nat] :
( ( Xs
= ( append_list_nat @ Us2 @ Vs ) )
& ( Ys
= ( map_li960784813134754710st_nat @ F @ Us2 ) )
& ( Zs
= ( map_li960784813134754710st_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_832_append__eq__map__conv,axiom,
! [Ys: list_list_nat,Zs: list_list_nat,F: nat > list_nat,Xs: list_nat] :
( ( ( append_list_nat @ Ys @ Zs )
= ( map_nat_list_nat @ F @ Xs ) )
= ( ? [Us2: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs ) )
& ( Ys
= ( map_nat_list_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_list_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_833_map__eq__append__conv,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us2: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_834_map__eq__append__conv,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat,Ys: list_list_list_nat,Zs: list_list_list_nat] :
( ( ( map_li960784813134754710st_nat @ F @ Xs )
= ( append_list_list_nat @ Ys @ Zs ) )
= ( ? [Us2: list_list_nat,Vs: list_list_nat] :
( ( Xs
= ( append_list_nat @ Us2 @ Vs ) )
& ( Ys
= ( map_li960784813134754710st_nat @ F @ Us2 ) )
& ( Zs
= ( map_li960784813134754710st_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_835_map__eq__append__conv,axiom,
! [F: nat > list_nat,Xs: list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( ( map_nat_list_nat @ F @ Xs )
= ( append_list_nat @ Ys @ Zs ) )
= ( ? [Us2: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs ) )
& ( Ys
= ( map_nat_list_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_list_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_836_prefix__map__rightE,axiom,
! [Xs: list_list_list_nat,F: list_nat > list_list_nat,Ys: list_list_nat] :
( ( prefix_list_list_nat @ Xs @ ( map_li960784813134754710st_nat @ F @ Ys ) )
=> ? [Xs6: list_list_nat] :
( ( prefix_list_nat @ Xs6 @ Ys )
& ( Xs
= ( map_li960784813134754710st_nat @ F @ Xs6 ) ) ) ) ).
% prefix_map_rightE
thf(fact_837_prefix__map__rightE,axiom,
! [Xs: list_list_nat,F: nat > list_nat,Ys: list_nat] :
( ( prefix_list_nat @ Xs @ ( map_nat_list_nat @ F @ Ys ) )
=> ? [Xs6: list_nat] :
( ( prefix_nat @ Xs6 @ Ys )
& ( Xs
= ( map_nat_list_nat @ F @ Xs6 ) ) ) ) ).
% prefix_map_rightE
thf(fact_838_prefix__map__rightE,axiom,
! [Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ ( map_nat_nat @ F @ Ys ) )
=> ? [Xs6: list_nat] :
( ( prefix_nat @ Xs6 @ Ys )
& ( Xs
= ( map_nat_nat @ F @ Xs6 ) ) ) ) ).
% prefix_map_rightE
thf(fact_839_map__mono__prefix,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,F: list_nat > list_list_nat] :
( ( prefix_list_nat @ Xs @ Ys )
=> ( prefix_list_list_nat @ ( map_li960784813134754710st_nat @ F @ Xs ) @ ( map_li960784813134754710st_nat @ F @ Ys ) ) ) ).
% map_mono_prefix
thf(fact_840_map__mono__prefix,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ( prefix_list_nat @ ( map_nat_list_nat @ F @ Xs ) @ ( map_nat_list_nat @ F @ Ys ) ) ) ).
% map_mono_prefix
thf(fact_841_map__mono__prefix,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > nat] :
( ( prefix_nat @ Xs @ Ys )
=> ( prefix_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_mono_prefix
thf(fact_842_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_843_subset__image__iff,axiom,
! [B2: set_int,F: nat > int,A2: set_nat] :
( ( ord_less_eq_set_int @ B2 @ ( image_nat_int @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_844_image__subset__iff,axiom,
! [F: nat > int,A2: set_nat,B2: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( member_int @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_845_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( member_nat2 @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_846_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_847_subset__imageE,axiom,
! [B2: set_int,F: nat > int,A2: set_nat] :
( ( ord_less_eq_set_int @ B2 @ ( image_nat_int @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_nat_int @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_848_image__subsetI,axiom,
! [A2: set_nat,F: nat > int,B2: set_int] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
=> ( member_int @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_849_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
=> ( member_nat2 @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_850_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_851_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > int] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ ( image_nat_int @ F @ B2 ) ) ) ).
% image_mono
thf(fact_852_nths__map,axiom,
! [F: nat > nat,Xs: list_nat,I4: set_nat] :
( ( nths_nat @ ( map_nat_nat @ F @ Xs ) @ I4 )
= ( map_nat_nat @ F @ ( nths_nat @ Xs @ I4 ) ) ) ).
% nths_map
thf(fact_853_nths__map,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat,I4: set_nat] :
( ( nths_list_list_nat @ ( map_li960784813134754710st_nat @ F @ Xs ) @ I4 )
= ( map_li960784813134754710st_nat @ F @ ( nths_list_nat @ Xs @ I4 ) ) ) ).
% nths_map
thf(fact_854_nths__map,axiom,
! [F: nat > list_nat,Xs: list_nat,I4: set_nat] :
( ( nths_list_nat @ ( map_nat_list_nat @ F @ Xs ) @ I4 )
= ( map_nat_list_nat @ F @ ( nths_nat @ Xs @ I4 ) ) ) ).
% nths_map
thf(fact_855_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_856_rotate1__map,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( rotate6412633851404001245st_nat @ ( map_li960784813134754710st_nat @ F @ Xs ) )
= ( map_li960784813134754710st_nat @ F @ ( rotate1_list_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_857_rotate1__map,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( rotate1_list_nat @ ( map_nat_list_nat @ F @ Xs ) )
= ( map_nat_list_nat @ F @ ( rotate1_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_858_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_859_butlast_Osimps_I1_J,axiom,
( ( butlast_list_nat @ nil_list_nat )
= nil_list_nat ) ).
% butlast.simps(1)
thf(fact_860_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > int] :
( ( member_nat2 @ X @ A2 )
=> ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ A2 ) ) ) ).
% imageI
thf(fact_861_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat2 @ X @ A2 )
=> ( member_nat2 @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_862_image__iff,axiom,
! [Z5: int,F: nat > int,A2: set_nat] :
( ( member_int @ Z5 @ ( image_nat_int @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
& ( Z5
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_863_image__iff,axiom,
! [Z5: nat,F: nat > nat,A2: set_nat] :
( ( member_nat2 @ Z5 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
& ( Z5
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_864_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P2: nat > $o] :
( ? [X4: nat] :
( ( member_nat2 @ X4 @ ( image_nat_nat @ F @ A2 ) )
& ( P2 @ X4 ) )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_865_bex__imageD,axiom,
! [F: nat > int,A2: set_nat,P2: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( image_nat_int @ F @ A2 ) )
& ( P2 @ X4 ) )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_866_image__cong,axiom,
! [M6: set_nat,N5: set_nat,F: nat > nat,G: nat > nat] :
( ( M6 = N5 )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ N5 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M6 )
= ( image_nat_nat @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_867_image__cong,axiom,
! [M6: set_nat,N5: set_nat,F: nat > int,G: nat > int] :
( ( M6 = N5 )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ N5 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_int @ F @ M6 )
= ( image_nat_int @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_868_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P2: nat > $o] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( image_nat_nat @ F @ A2 ) )
=> ( P2 @ X3 ) )
=> ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_869_ball__imageD,axiom,
! [F: nat > int,A2: set_nat,P2: int > $o] :
( ! [X3: int] :
( ( member_int @ X3 @ ( image_nat_int @ F @ A2 ) )
=> ( P2 @ X3 ) )
=> ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_870_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: int,F: nat > int] :
( ( member_nat2 @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_nat_int @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_871_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat2 @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat2 @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_872_map__butlast,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( map_nat_nat @ F @ ( butlast_nat @ Xs ) )
= ( butlast_nat @ ( map_nat_nat @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_873_map__butlast,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( map_li960784813134754710st_nat @ F @ ( butlast_list_nat @ Xs ) )
= ( butlas6429778205849610142st_nat @ ( map_li960784813134754710st_nat @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_874_map__butlast,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( map_nat_list_nat @ F @ ( butlast_nat @ Xs ) )
= ( butlast_list_nat @ ( map_nat_list_nat @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_875_image__set,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( image_4042064729117200983st_nat @ F @ ( set_list_nat2 @ Xs ) )
= ( set_list_list_nat2 @ ( map_li960784813134754710st_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_876_image__set,axiom,
! [F: nat > int,Xs: list_nat] :
( ( image_nat_int @ F @ ( set_nat2 @ Xs ) )
= ( set_int2 @ ( map_nat_int @ F @ Xs ) ) ) ).
% image_set
thf(fact_877_image__set,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( image_nat_list_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_list_nat2 @ ( map_nat_list_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_878_image__set,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( image_nat_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( map_nat_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_879_map__concat,axiom,
! [F: list_nat > list_nat,Xs: list_list_list_nat] :
( ( map_li7225945977422193158st_nat @ F @ ( concat_list_nat @ Xs ) )
= ( concat_list_nat @ ( map_li2855073862107769254st_nat @ ( map_li7225945977422193158st_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_880_map__concat,axiom,
! [F: nat > nat,Xs: list_list_nat] :
( ( map_nat_nat @ F @ ( concat_nat @ Xs ) )
= ( concat_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_881_map__concat,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_list_nat] :
( ( map_li960784813134754710st_nat @ F @ ( concat_list_nat @ Xs ) )
= ( concat_list_list_nat @ ( map_li5769348595424326838st_nat @ ( map_li960784813134754710st_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_882_map__concat,axiom,
! [F: nat > list_nat,Xs: list_list_nat] :
( ( map_nat_list_nat @ F @ ( concat_nat @ Xs ) )
= ( concat_list_nat @ ( map_li960784813134754710st_nat @ ( map_nat_list_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_883_prefixeq__butlast,axiom,
! [Xs: list_nat] : ( prefix_nat @ ( butlast_nat @ Xs ) @ Xs ) ).
% prefixeq_butlast
thf(fact_884_in__set__butlastD,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
=> ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_885_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_886_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_887_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_888_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_889_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_890_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_891_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_892_of__nat__mono,axiom,
! [I: nat,J2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J2 ) ) ) ).
% of_nat_mono
thf(fact_893_of__nat__mono,axiom,
! [I: nat,J2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ).
% of_nat_mono
thf(fact_894_of__nat__less__imp__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_895_of__nat__less__imp__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_896_less__imp__of__nat__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_897_less__imp__of__nat__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_898_butlast_Osimps_I2_J,axiom,
! [Xs: list_nat,X: nat] :
( ( ( Xs = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
= nil_nat ) )
& ( ( Xs != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( butlast_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_899_butlast_Osimps_I2_J,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( ( Xs = nil_list_nat )
=> ( ( butlast_list_nat @ ( cons_list_nat @ X @ Xs ) )
= nil_list_nat ) )
& ( ( Xs != nil_list_nat )
=> ( ( butlast_list_nat @ ( cons_list_nat @ X @ Xs ) )
= ( cons_list_nat @ X @ ( butlast_list_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_900_List_Obind__def,axiom,
( bind_l7796378977173581257st_nat
= ( ^ [Xs2: list_list_nat,F2: list_nat > list_list_nat] : ( concat_list_nat @ ( map_li960784813134754710st_nat @ F2 @ Xs2 ) ) ) ) ).
% List.bind_def
thf(fact_901_List_Obind__def,axiom,
( bind_nat_nat
= ( ^ [Xs2: list_nat,F2: nat > list_nat] : ( concat_nat @ ( map_nat_list_nat @ F2 @ Xs2 ) ) ) ) ).
% List.bind_def
thf(fact_902_butlast__append,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( ( Ys = nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
= ( butlast_nat @ Xs ) ) )
& ( ( Ys != nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ Xs @ ( butlast_nat @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_903_butlast__append,axiom,
! [Ys: list_list_nat,Xs: list_list_nat] :
( ( ( Ys = nil_list_nat )
=> ( ( butlast_list_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( butlast_list_nat @ Xs ) ) )
& ( ( Ys != nil_list_nat )
=> ( ( butlast_list_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( append_list_nat @ Xs @ ( butlast_list_nat @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_904_in__set__butlast__appendI,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( ( member_nat2 @ X @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
| ( member_nat2 @ X @ ( set_nat2 @ ( butlast_nat @ Ys ) ) ) )
=> ( member_nat2 @ X @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_905_maps__def,axiom,
( maps_l5785965478274863235st_nat
= ( ^ [F2: list_nat > list_list_nat,Xs2: list_list_nat] : ( concat_list_nat @ ( map_li960784813134754710st_nat @ F2 @ Xs2 ) ) ) ) ).
% maps_def
thf(fact_906_maps__def,axiom,
( maps_nat_nat
= ( ^ [F2: nat > list_nat,Xs2: list_nat] : ( concat_nat @ ( map_nat_list_nat @ F2 @ Xs2 ) ) ) ) ).
% maps_def
thf(fact_907_concat__map__maps,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( concat_list_nat @ ( map_li960784813134754710st_nat @ F @ Xs ) )
= ( maps_l5785965478274863235st_nat @ F @ Xs ) ) ).
% concat_map_maps
thf(fact_908_concat__map__maps,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( concat_nat @ ( map_nat_list_nat @ F @ Xs ) )
= ( maps_nat_nat @ F @ Xs ) ) ).
% concat_map_maps
thf(fact_909_prefixes_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( prefixes_nat @ ( cons_nat @ X @ Xs ) )
= ( cons_list_nat @ nil_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_910_prefixes_Osimps_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( prefixes_list_nat @ ( cons_list_nat @ X @ Xs ) )
= ( cons_list_list_nat @ nil_list_nat @ ( map_li2855073862107769254st_nat @ ( cons_list_nat @ X ) @ ( prefixes_list_nat @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_911_int__eq__iff__numeral,axiom,
! [M: nat,V: num] :
( ( ( semiri1314217659103216013at_int @ M )
= ( numeral_numeral_int @ V ) )
= ( M
= ( numeral_numeral_nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_912_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_913_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_914_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P2: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P2 @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_915_all__subset__image,axiom,
! [F: nat > int,A2: set_nat,P2: set_int > $o] :
( ( ! [B3: set_int] :
( ( ord_less_eq_set_int @ B3 @ ( image_nat_int @ F @ A2 ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P2 @ ( image_nat_int @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_916_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_917_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_918_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_919_conj__le__cong,axiom,
! [X: int,X7: int,P2: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P2 = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_920_imp__le__cong,axiom,
! [X: int,X7: int,P2: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P2 = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_921_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_922_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_923_map__Suc__upt,axiom,
! [M: nat,N2: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N2 ) )
= ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% map_Suc_upt
thf(fact_924_zle__int,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% zle_int
thf(fact_925_append__butlast__last__id,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( append_nat @ ( butlast_nat @ Xs ) @ ( cons_nat @ ( last_nat @ Xs ) @ nil_nat ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_926_append__butlast__last__id,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( append_list_nat @ ( butlast_list_nat @ Xs ) @ ( cons_list_nat @ ( last_list_nat @ Xs ) @ nil_list_nat ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_927_subseqs__powset,axiom,
! [Xs: list_nat] :
( ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
= ( pow_nat @ ( set_nat2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_928_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_929_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_930_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_931_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_932_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_933_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_934_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_935_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_936_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_937_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_938_neg__numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( M = N2 ) ) ).
% neg_numeral_eq_iff
thf(fact_939_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_940_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_941_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_942_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_943_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_944_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_945_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_946_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_947_negative__eq__positive,axiom,
! [N2: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N2 = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_948_image__uminus__greaterThanLessThan,axiom,
! [X: int,Y: int] :
( ( image_int_int @ uminus_uminus_int @ ( set_or5832277885323065728an_int @ X @ Y ) )
= ( set_or5832277885323065728an_int @ ( uminus_uminus_int @ Y ) @ ( uminus_uminus_int @ X ) ) ) ).
% image_uminus_greaterThanLessThan
thf(fact_949_negative__zle,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_950_last__appendR,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( Ys != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Ys ) ) ) ).
% last_appendR
thf(fact_951_last__appendR,axiom,
! [Ys: list_list_nat,Xs: list_list_nat] :
( ( Ys != nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( last_list_nat @ Ys ) ) ) ).
% last_appendR
thf(fact_952_last__appendL,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( Ys = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Xs ) ) ) ).
% last_appendL
thf(fact_953_last__appendL,axiom,
! [Ys: list_list_nat,Xs: list_list_nat] :
( ( Ys = nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( last_list_nat @ Xs ) ) ) ).
% last_appendL
thf(fact_954_negative__zless,axiom,
! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_955_last__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% last_snoc
thf(fact_956_last__snoc,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( last_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= X ) ).
% last_snoc
thf(fact_957_neg__numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( ord_less_eq_num @ N2 @ M ) ) ).
% neg_numeral_le_iff
thf(fact_958_neg__numeral__less__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
= ( ord_less_num @ N2 @ M ) ) ).
% neg_numeral_less_iff
thf(fact_959_image__Pow__surj,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= B2 )
=> ( ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A2 ) )
= ( pow_nat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_960_image__Pow__surj,axiom,
! [F: nat > int,A2: set_nat,B2: set_int] :
( ( ( image_nat_int @ F @ A2 )
= B2 )
=> ( ( image_3739036796817536367et_int @ ( image_nat_int @ F ) @ ( pow_nat @ A2 ) )
= ( pow_int @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_961_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_962_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_963_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_964_neg__numeral__neq__numeral,axiom,
! [M: num,N2: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
!= ( numeral_numeral_int @ N2 ) ) ).
% neg_numeral_neq_numeral
thf(fact_965_numeral__neq__neg__numeral,axiom,
! [M: num,N2: num] :
( ( numeral_numeral_int @ M )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_966_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_967_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_968_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_969_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_970_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_971_neg__numeral__le__numeral,axiom,
! [M: num,N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).
% neg_numeral_le_numeral
thf(fact_972_not__numeral__le__neg__numeral,axiom,
! [M: num,N2: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_973_zero__neq__neg__numeral,axiom,
! [N2: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% zero_neq_neg_numeral
thf(fact_974_neg__numeral__less__numeral,axiom,
! [M: num,N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).
% neg_numeral_less_numeral
thf(fact_975_not__numeral__less__neg__numeral,axiom,
! [M: num,N2: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_976_int__of__nat__induct,axiom,
! [P2: int > $o,Z5: int] :
( ! [N3: nat] : ( P2 @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P2 @ Z5 ) ) ) ).
% int_of_nat_induct
thf(fact_977_int__cases,axiom,
! [Z5: int] :
( ! [N3: nat] :
( Z5
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z5
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_978_image__Pow__mono,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A2 ) ) @ ( pow_nat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_979_image__Pow__mono,axiom,
! [F: nat > int,A2: set_nat,B2: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ B2 )
=> ( ord_le4403425263959731960et_int @ ( image_3739036796817536367et_int @ ( image_nat_int @ F ) @ ( pow_nat @ A2 ) ) @ ( pow_int @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_980_last__ConsR,axiom,
! [Xs: list_nat,X: nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_981_last__ConsR,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( Xs != nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs ) )
= ( last_list_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_982_last__ConsL,axiom,
! [Xs: list_nat,X: nat] :
( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_983_last__ConsL,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( Xs = nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_984_last_Osimps,axiom,
! [Xs: list_nat,X: nat] :
( ( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= ( last_nat @ Xs ) ) ) ) ).
% last.simps
thf(fact_985_last_Osimps,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( ( Xs = nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs ) )
= ( last_list_nat @ Xs ) ) ) ) ).
% last.simps
thf(fact_986_last__in__set,axiom,
! [As2: list_list_nat] :
( ( As2 != nil_list_nat )
=> ( member_list_nat2 @ ( last_list_nat @ As2 ) @ ( set_list_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_987_last__in__set,axiom,
! [As2: list_nat] :
( ( As2 != nil_nat )
=> ( member_nat2 @ ( last_nat @ As2 ) @ ( set_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_988_longest__common__suffix,axiom,
! [Xs: list_nat,Ys: list_nat] :
? [Ss: list_nat,Xs6: list_nat,Ys6: list_nat] :
( ( Xs
= ( append_nat @ Xs6 @ Ss ) )
& ( Ys
= ( append_nat @ Ys6 @ Ss ) )
& ( ( Xs6 = nil_nat )
| ( Ys6 = nil_nat )
| ( ( last_nat @ Xs6 )
!= ( last_nat @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_989_longest__common__suffix,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
? [Ss: list_list_nat,Xs6: list_list_nat,Ys6: list_list_nat] :
( ( Xs
= ( append_list_nat @ Xs6 @ Ss ) )
& ( Ys
= ( append_list_nat @ Ys6 @ Ss ) )
& ( ( Xs6 = nil_list_nat )
| ( Ys6 = nil_list_nat )
| ( ( last_list_nat @ Xs6 )
!= ( last_list_nat @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_990_last__append,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( ( Ys = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Xs ) ) )
& ( ( Ys != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Ys ) ) ) ) ).
% last_append
thf(fact_991_last__append,axiom,
! [Ys: list_list_nat,Xs: list_list_nat] :
( ( ( Ys = nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( last_list_nat @ Xs ) ) )
& ( ( Ys != nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( last_list_nat @ Ys ) ) ) ) ).
% last_append
thf(fact_992_last__map,axiom,
! [Xs: list_nat,F: nat > nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( map_nat_nat @ F @ Xs ) )
= ( F @ ( last_nat @ Xs ) ) ) ) ).
% last_map
thf(fact_993_last__map,axiom,
! [Xs: list_list_nat,F: list_nat > list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( last_list_list_nat @ ( map_li960784813134754710st_nat @ F @ Xs ) )
= ( F @ ( last_list_nat @ Xs ) ) ) ) ).
% last_map
thf(fact_994_last__map,axiom,
! [Xs: list_nat,F: nat > list_nat] :
( ( Xs != nil_nat )
=> ( ( last_list_nat @ ( map_nat_list_nat @ F @ Xs ) )
= ( F @ ( last_nat @ Xs ) ) ) ) ).
% last_map
thf(fact_995_neg__numeral__le__zero,axiom,
! [N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).
% neg_numeral_le_zero
thf(fact_996_not__zero__le__neg__numeral,axiom,
! [N2: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_997_not__zero__less__neg__numeral,axiom,
! [N2: num] :
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_998_neg__numeral__less__zero,axiom,
! [N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).
% neg_numeral_less_zero
thf(fact_999_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1000_int__zle__neg,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N2 = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1001_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_1002_negative__zle__0,axiom,
! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1003_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_1004_not__zle__0__negative,axiom,
! [N2: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ).
% not_zle_0_negative
thf(fact_1005_negative__zless__0,axiom,
! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1006_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N3: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_1007_snoc__eq__iff__butlast,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= Ys )
= ( ( Ys != nil_nat )
& ( ( butlast_nat @ Ys )
= Xs )
& ( ( last_nat @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_1008_snoc__eq__iff__butlast,axiom,
! [Xs: list_list_nat,X: list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) )
= Ys )
= ( ( Ys != nil_list_nat )
& ( ( butlast_list_nat @ Ys )
= Xs )
& ( ( last_list_nat @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_1009_compl__coset,axiom,
! [Xs: list_nat] :
( ( uminus5710092332889474511et_nat @ ( coset_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% compl_coset
thf(fact_1010_coset__def,axiom,
( coset_nat
= ( ^ [Xs2: list_nat] : ( uminus5710092332889474511et_nat @ ( set_nat2 @ Xs2 ) ) ) ) ).
% coset_def
thf(fact_1011_image__Fpow__mono,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A2 ) ) @ ( finite_Fpow_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_1012_image__Fpow__mono,axiom,
! [F: nat > int,A2: set_nat,B2: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A2 ) @ B2 )
=> ( ord_le4403425263959731960et_int @ ( image_3739036796817536367et_int @ ( image_nat_int @ F ) @ ( finite_Fpow_nat @ A2 ) ) @ ( finite_Fpow_int @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_1013_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_1014_nat__less__iff,axiom,
! [W: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_1015_ComplI,axiom,
! [C: nat,A2: set_nat] :
( ~ ( member_nat2 @ C @ A2 )
=> ( member_nat2 @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% ComplI
thf(fact_1016_Compl__iff,axiom,
! [C: nat,A2: set_nat] :
( ( member_nat2 @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( ~ ( member_nat2 @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_1017_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_1018_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% nat_numeral
thf(fact_1019_nat__le__0,axiom,
! [Z5: int] :
( ( ord_less_eq_int @ Z5 @ zero_zero_int )
=> ( ( nat2 @ Z5 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1020_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1021_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= zero_zero_nat ) ).
% nat_neg_numeral
thf(fact_1022_zless__nat__conj,axiom,
! [W: int,Z5: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z5 ) )
= ( ( ord_less_int @ zero_zero_int @ Z5 )
& ( ord_less_int @ W @ Z5 ) ) ) ).
% zless_nat_conj
thf(fact_1023_nat__zminus__int,axiom,
! [N2: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_1024_int__nat__eq,axiom,
! [Z5: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z5 ) )
= Z5 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z5 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_1025_zero__less__nat__eq,axiom,
! [Z5: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z5 ) )
= ( ord_less_int @ zero_zero_int @ Z5 ) ) ).
% zero_less_nat_eq
thf(fact_1026_ComplD,axiom,
! [C: nat,A2: set_nat] :
( ( member_nat2 @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
=> ~ ( member_nat2 @ C @ A2 ) ) ).
% ComplD
thf(fact_1027_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1028_nat__numeral__as__int,axiom,
( numeral_numeral_nat
= ( ^ [I2: num] : ( nat2 @ ( numeral_numeral_int @ I2 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_1029_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_1030_ex__nat,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_1031_all__nat,axiom,
( ( ^ [P3: nat > $o] :
! [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_1032_eq__nat__nat__iff,axiom,
! [Z5: int,Z7: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
=> ( ( ( nat2 @ Z5 )
= ( nat2 @ Z7 ) )
= ( Z5 = Z7 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1033_nat__mono__iff,axiom,
! [Z5: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z5 )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z5 ) )
= ( ord_less_int @ W @ Z5 ) ) ) ).
% nat_mono_iff
thf(fact_1034_nat__le__iff,axiom,
! [X: int,N2: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N2 )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nat_le_iff
thf(fact_1035_zless__nat__eq__int__zless,axiom,
! [M: nat,Z5: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z5 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z5 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1036_int__eq__iff,axiom,
! [M: nat,Z5: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z5 )
= ( ( M
= ( nat2 @ Z5 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z5 ) ) ) ).
% int_eq_iff
thf(fact_1037_nat__0__le,axiom,
! [Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z5 ) )
= Z5 ) ) ).
% nat_0_le
thf(fact_1038_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1039_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat2 @ W )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1040_nat__less__eq__zless,axiom,
! [W: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z5 ) )
= ( ord_less_int @ W @ Z5 ) ) ) ).
% nat_less_eq_zless
thf(fact_1041_split__nat,axiom,
! [P2: nat > $o,I: int] :
( ( P2 @ ( nat2 @ I ) )
= ( ! [N: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N ) )
=> ( P2 @ N ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P2 @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1042_nat__le__eq__zle,axiom,
! [W: int,Z5: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z5 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z5 ) )
= ( ord_less_eq_int @ W @ Z5 ) ) ) ).
% nat_le_eq_zle
thf(fact_1043_le__nat__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1044_one__less__nat__eq,axiom,
! [Z5: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z5 ) )
= ( ord_less_int @ one_one_int @ Z5 ) ) ).
% one_less_nat_eq
thf(fact_1045_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
= ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_1046_image__uminus__lessThan,axiom,
! [X: int] :
( ( image_int_int @ uminus_uminus_int @ ( set_ord_lessThan_int @ X ) )
= ( set_or1207661135979820486an_int @ ( uminus_uminus_int @ X ) ) ) ).
% image_uminus_lessThan
thf(fact_1047_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1316708129612266289at_nat @ N2 )
= one_one_nat )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1048_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1314217659103216013at_int @ N2 )
= one_one_int )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1049_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1050_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1051_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_1052_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_1053_greaterThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat2 @ I @ ( set_or1210151606488870762an_nat @ K ) )
= ( ord_less_nat @ K @ I ) ) ).
% greaterThan_iff
thf(fact_1054_greaterThan__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_or1207661135979820486an_int @ K ) )
= ( ord_less_int @ K @ I ) ) ).
% greaterThan_iff
thf(fact_1055_atLeastLessThan__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_int @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1056_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat2 @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1057_ivl__subset,axiom,
! [I: int,J2: int,M: int,N2: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J2 ) @ ( set_or4662586982721622107an_int @ M @ N2 ) )
= ( ( ord_less_eq_int @ J2 @ I )
| ( ( ord_less_eq_int @ M @ I )
& ( ord_less_eq_int @ J2 @ N2 ) ) ) ) ).
% ivl_subset
thf(fact_1058_ivl__subset,axiom,
! [I: nat,J2: nat,M: nat,N2: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J2 ) @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
= ( ( ord_less_eq_nat @ J2 @ I )
| ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ J2 @ N2 ) ) ) ) ).
% ivl_subset
thf(fact_1059_greaterThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_or1210151606488870762an_nat @ X ) @ ( set_or1210151606488870762an_nat @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% greaterThan_subset_iff
thf(fact_1060_greaterThan__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_or1207661135979820486an_int @ X ) @ ( set_or1207661135979820486an_int @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% greaterThan_subset_iff
thf(fact_1061_Compl__greaterThan,axiom,
! [K: nat] :
( ( uminus5710092332889474511et_nat @ ( set_or1210151606488870762an_nat @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% Compl_greaterThan
thf(fact_1062_Compl__atMost,axiom,
! [K: nat] :
( ( uminus5710092332889474511et_nat @ ( set_ord_atMost_nat @ K ) )
= ( set_or1210151606488870762an_nat @ K ) ) ).
% Compl_atMost
thf(fact_1063_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1064_image__uminus__greaterThanAtMost,axiom,
! [X: int,Y: int] :
( ( image_int_int @ uminus_uminus_int @ ( set_or6656581121297822940st_int @ X @ Y ) )
= ( set_or4662586982721622107an_int @ ( uminus_uminus_int @ Y ) @ ( uminus_uminus_int @ X ) ) ) ).
% image_uminus_greaterThanAtMost
thf(fact_1065_image__uminus__atLeastLessThan,axiom,
! [X: int,Y: int] :
( ( image_int_int @ uminus_uminus_int @ ( set_or4662586982721622107an_int @ X @ Y ) )
= ( set_or6656581121297822940st_int @ ( uminus_uminus_int @ Y ) @ ( uminus_uminus_int @ X ) ) ) ).
% image_uminus_atLeastLessThan
thf(fact_1066_image__uminus__greaterThan,axiom,
! [X: int] :
( ( image_int_int @ uminus_uminus_int @ ( set_or1207661135979820486an_int @ X ) )
= ( set_ord_lessThan_int @ ( uminus_uminus_int @ X ) ) ) ).
% image_uminus_greaterThan
thf(fact_1067_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_1068_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_1069_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_1070_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_1071_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_1072_atLeastLessThan__eq__iff,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D2 ) )
= ( ( A = C )
& ( B = D2 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_1073_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
= ( ( A = C )
& ( B = D2 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_1074_Ico__eq__Ico,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( ( set_or4662586982721622107an_int @ L @ H )
= ( set_or4662586982721622107an_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_int @ L @ H )
& ~ ( ord_less_int @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_1075_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_1076_atLeastLessThan__inj_I1_J,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D2 ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_1077_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_1078_atLeastLessThan__inj_I2_J,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D2 ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( B = D2 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_1079_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( B = D2 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_1080_atLeastLessThan__subset__iff,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ B @ A )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_1081_atLeastLessThan__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_1082_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1083_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1084_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1085_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_1086_one__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% one_le_numeral
thf(fact_1087_one__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).
% one_le_numeral
thf(fact_1088_not__numeral__less__one,axiom,
! [N2: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_1089_not__numeral__less__one,axiom,
! [N2: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_1090_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_1091_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_1092_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_1093_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_1094_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_1095_numeral__neq__neg__one,axiom,
! [N2: num] :
( ( numeral_numeral_int @ N2 )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_1096_one__neq__neg__numeral,axiom,
! [N2: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% one_neq_neg_numeral
thf(fact_1097_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_1098_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_1099_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_1100_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_1101_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_1102_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_le_neg_one
thf(fact_1103_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% neg_numeral_le_neg_one
thf(fact_1104_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_le_numeral
thf(fact_1105_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_le_one
thf(fact_1106_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_less_one
thf(fact_1107_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_less_numeral
thf(fact_1108_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_less_neg_one
thf(fact_1109_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_1110_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_1111_int__one__le__iff__zero__less,axiom,
! [Z5: int] :
( ( ord_less_eq_int @ one_one_int @ Z5 )
= ( ord_less_int @ zero_zero_int @ Z5 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1112_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1113_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_1114_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1115_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_1116_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1117_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1118_image__Suc__atLeastLessThan,axiom,
! [I: nat,J2: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J2 ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J2 ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_1119_sorted__list__of__set__range,axiom,
! [M: nat,N2: nat] :
( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
= ( upt @ M @ N2 ) ) ).
% sorted_list_of_set_range
thf(fact_1120_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1121_all__nat__less__eq,axiom,
! [N2: nat,P2: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( P2 @ M5 ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
=> ( P2 @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_1122_ex__nat__less__eq,axiom,
! [N2: nat,P2: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ( P2 @ M5 ) ) )
= ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
& ( P2 @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1123_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_1124_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1125_atLeastLessThan__upt,axiom,
( set_or4665077453230672383an_nat
= ( ^ [I2: nat,J: nat] : ( set_nat2 @ ( upt @ I2 @ J ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_1126_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_1127_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1128_nat__induct__non__zero,axiom,
! [N2: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P2 @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1129_image__int__atLeastLessThan,axiom,
! [A: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
= ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastLessThan
thf(fact_1130_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1131_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_1132_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1133_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_1134_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1135_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1136_rgf__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( equiva3371634703666331078on_rgf @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( ( equiva3371634703666331078on_rgf @ Xs )
& ( ord_less_nat @ X @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ Xs ) @ one_one_nat ) ) ) ) ).
% rgf_snoc
thf(fact_1137_stirling__row__code_I1_J,axiom,
( ( stirling_row @ zero_zero_nat )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% stirling_row_code(1)
thf(fact_1138_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1139_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1140_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1141_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1142_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1143_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1144_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1145_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1146_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_1147_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1148_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_1149_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_1150_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1151_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_1152_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1153_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_1154_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1155_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_1156_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1157_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_1158_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1159_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1160_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1161_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_1162_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1163_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1164_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1165_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1166_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_1167_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_1168_add__numeral__left,axiom,
! [V: num,W: num,Z5: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z5 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z5 ) ) ).
% add_numeral_left
thf(fact_1169_add__numeral__left,axiom,
! [V: num,W: num,Z5: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z5 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z5 ) ) ).
% add_numeral_left
thf(fact_1170_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_1171_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_1172_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_1173_add__Suc__right,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ M @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% add_Suc_right
thf(fact_1174_add__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1175_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1176_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1177_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1178_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1179_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_1180_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1181_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_1182_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_1183_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_1184_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_1185_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_1186_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1187_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1188_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1189_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_1190_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1191_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_1192_add__gr__0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1193_Suc__eq__plus1,axiom,
( suc
= ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1194_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1195_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1196_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1197_add__Suc,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N2 )
= ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% add_Suc
thf(fact_1198_add__Suc__shift,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N2 )
= ( plus_plus_nat @ M @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_1199_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1200_add__eq__self__zero,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= M )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1201_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1202_trans__less__add2,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_1203_trans__less__add1,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_1204_add__less__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_1205_not__add__less2,axiom,
! [J2: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I ) @ I ) ).
% not_add_less2
thf(fact_1206_not__add__less1,axiom,
! [I: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ I ) ).
% not_add_less1
thf(fact_1207_add__less__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_1208_add__lessD1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1209_add__leE,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1210_le__add1,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).
% le_add1
thf(fact_1211_le__add2,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).
% le_add2
thf(fact_1212_add__leD1,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% add_leD1
thf(fact_1213_add__leD2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1214_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1215_add__le__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_1216_add__le__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_1217_trans__le__add1,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_1218_trans__le__add2,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1219_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N: nat] :
? [K3: nat] :
( N
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1220_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1221_less__imp__add__positive,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_1222_less__imp__Suc__add,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ? [K2: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1223_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N: nat] :
? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1224_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1225_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1226_less__natE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ! [Q4: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1227_one__is__add,axiom,
! [M: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N2 ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1228_add__is__1,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1229_stirling__row__nonempty,axiom,
! [N2: nat] :
( ( stirling_row @ N2 )
!= nil_nat ) ).
% stirling_row_nonempty
thf(fact_1230_upt__add__eq__append,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( upt @ I @ ( plus_plus_nat @ J2 @ K ) )
= ( append_nat @ ( upt @ I @ J2 ) @ ( upt @ J2 @ ( plus_plus_nat @ J2 @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_1231_upt__eq__Cons__conv,axiom,
! [I: nat,J2: nat,X: nat,Xs: list_nat] :
( ( ( upt @ I @ J2 )
= ( cons_nat @ X @ Xs ) )
= ( ( ord_less_nat @ I @ J2 )
& ( I = X )
& ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J2 )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_1232_zle__add1__eq__le,axiom,
! [W: int,Z5: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z5 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z5 ) ) ).
% zle_add1_eq_le
thf(fact_1233_int__ge__induct,axiom,
! [K: int,I: int,P2: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P2 @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1234_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z: int] :
? [N: nat] :
( Z
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1235_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
= ( set_or5832277885323065728an_int @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_1236_int__Suc,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1237_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1238_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z: int] :
? [N: nat] :
( Z
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1239_zless__imp__add1__zle,axiom,
! [W: int,Z5: int] :
( ( ord_less_int @ W @ Z5 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z5 ) ) ).
% zless_imp_add1_zle
thf(fact_1240_add1__zle__eq,axiom,
! [W: int,Z5: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z5 )
= ( ord_less_int @ W @ Z5 ) ) ).
% add1_zle_eq
thf(fact_1241_nat__add__distrib,axiom,
! [Z5: int,Z7: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
=> ( ( nat2 @ ( plus_plus_int @ Z5 @ Z7 ) )
= ( plus_plus_nat @ ( nat2 @ Z5 ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1242_le__imp__0__less,axiom,
! [Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z5 ) ) ) ).
% le_imp_0_less
thf(fact_1243_Suc__as__int,axiom,
( suc
= ( ^ [A4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_1244_Suc__nat__eq__nat__zadd1,axiom,
! [Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( suc @ ( nat2 @ Z5 ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z5 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1245_stirling__row__code_I2_J,axiom,
! [N2: nat] :
( ( stirling_row @ ( suc @ N2 ) )
= ( stirling_row_aux_nat @ N2 @ zero_zero_nat @ ( stirling_row @ N2 ) ) ) ).
% stirling_row_code(2)
thf(fact_1246_Euclid__induct,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( P2 @ A5 @ B5 )
= ( P2 @ B5 @ A5 ) )
=> ( ! [A5: nat] : ( P2 @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B5: nat] :
( ( P2 @ A5 @ B5 )
=> ( P2 @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
=> ( P2 @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1247_enum__rgfs_Oelims,axiom,
! [X: nat,Y: list_list_nat] :
( ( ( equiva7426478223624825838m_rgfs @ X )
= Y )
=> ( ( ( X = zero_zero_nat )
=> ( Y
!= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ( Y
!= ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ^ [X2: list_nat] :
( map_nat_list_nat
@ ^ [Y2: nat] : ( append_nat @ X2 @ ( cons_nat @ Y2 @ nil_nat ) )
@ ( upt @ zero_zero_nat @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ X2 ) @ one_one_nat ) ) )
@ ( equiva7426478223624825838m_rgfs @ N3 ) ) ) ) ) ) ) ).
% enum_rgfs.elims
thf(fact_1248_sort__upt,axiom,
! [M: nat,N2: nat] :
( ( linord738340561235409698at_nat
@ ^ [X2: nat] : X2
@ ( upt @ M @ N2 ) )
= ( upt @ M @ N2 ) ) ).
% sort_upt
thf(fact_1249_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_1250_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1251_map__add__upt,axiom,
! [N2: nat,M: nat] :
( ( map_nat_nat
@ ^ [I2: nat] : ( plus_plus_nat @ I2 @ N2 )
@ ( upt @ zero_zero_nat @ M ) )
= ( upt @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% map_add_upt
thf(fact_1252_enum__rgfs_Osimps_I2_J,axiom,
! [N2: nat] :
( ( equiva7426478223624825838m_rgfs @ ( suc @ N2 ) )
= ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ^ [X2: list_nat] :
( map_nat_list_nat
@ ^ [Y2: nat] : ( append_nat @ X2 @ ( cons_nat @ Y2 @ nil_nat ) )
@ ( upt @ zero_zero_nat @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ X2 ) @ one_one_nat ) ) )
@ ( equiva7426478223624825838m_rgfs @ N2 ) ) ) ) ).
% enum_rgfs.simps(2)
thf(fact_1253_enum__rgfs_Opelims,axiom,
! [X: nat,Y: list_list_nat] :
( ( ( equiva7426478223624825838m_rgfs @ X )
= Y )
=> ( ( accp_nat @ equiva1432535406783100555fs_rel @ X )
=> ( ( ( X = zero_zero_nat )
=> ( ( Y
= ( cons_list_nat @ nil_nat @ nil_list_nat ) )
=> ~ ( accp_nat @ equiva1432535406783100555fs_rel @ zero_zero_nat ) ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ( ( Y
= ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ^ [X2: list_nat] :
( map_nat_list_nat
@ ^ [Y2: nat] : ( append_nat @ X2 @ ( cons_nat @ Y2 @ nil_nat ) )
@ ( upt @ zero_zero_nat @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ X2 ) @ one_one_nat ) ) )
@ ( equiva7426478223624825838m_rgfs @ N3 ) ) ) )
=> ~ ( accp_nat @ equiva1432535406783100555fs_rel @ ( suc @ N3 ) ) ) ) ) ) ) ).
% enum_rgfs.pelims
thf(fact_1254_stirling__row__def,axiom,
( stirling_row
= ( ^ [N: nat] : ( map_nat_nat @ ( stirling @ N ) @ ( upt @ zero_zero_nat @ ( suc @ N ) ) ) ) ) ).
% stirling_row_def
thf(fact_1255_stirling__same,axiom,
! [N2: nat] :
( ( stirling @ N2 @ N2 )
= one_one_nat ) ).
% stirling_same
thf(fact_1256_stirling__0,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( stirling @ N2 @ zero_zero_nat )
= zero_zero_nat ) ) ).
% stirling_0
thf(fact_1257_stirling__less,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ N2 @ K )
=> ( ( stirling @ N2 @ K )
= zero_zero_nat ) ) ).
% stirling_less
thf(fact_1258_stirling_Osimps_I2_J,axiom,
! [K: nat] :
( ( stirling @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% stirling.simps(2)
thf(fact_1259_stirling_Osimps_I3_J,axiom,
! [N2: nat] :
( ( stirling @ ( suc @ N2 ) @ zero_zero_nat )
= zero_zero_nat ) ).
% stirling.simps(3)
thf(fact_1260_stirling_Osimps_I1_J,axiom,
( ( stirling @ zero_zero_nat @ zero_zero_nat )
= one_one_nat ) ).
% stirling.simps(1)
% Helper facts (5)
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
! [X: list_list_nat,Y: list_list_nat] :
( ( if_list_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
! [X: list_list_nat,Y: list_list_nat] :
( ( if_list_list_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( set_nat2 @ xsa )
= ( set_ord_lessThan_nat @ ( equiva5889994315859557365_limit @ xsa ) ) ) ).
%------------------------------------------------------------------------------