TPTP Problem File: SLH0162^1.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_00154_006224__11828754_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1442 ( 642 unt; 157 typ; 0 def)
% Number of atoms : 3431 (1688 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10601 ( 371 ~; 60 |; 317 &;8576 @)
% ( 0 <=>;1277 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 624 ( 624 >; 0 *; 0 +; 0 <<)
% Number of symbols : 150 ( 147 usr; 14 con; 0-3 aty)
% Number of variables : 3647 ( 151 ^;3247 !; 249 ?;3647 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:13:30.061
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (147)
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_Equivalence__Relation__Enumeration_Orgf__limit__rel,type,
equiva5575797544161152836it_rel: list_nat > list_nat > $o ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
gcd_Gcd_nat: set_nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_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_If_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
if_list_set_nat: $o > list_set_nat > list_set_nat > list_set_nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
if_set_nat: $o > set_nat > set_nat > set_nat ).
thf(sy_c_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_Oappend_001t__Set__Oset_It__Nat__Onat_J,type,
append_set_nat: list_set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Obind_001t__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__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_Obutlast_001t__Set__Oset_It__Nat__Onat_J,type,
butlast_set_nat: list_set_nat > list_set_nat ).
thf(sy_c_List_Ocan__select_001t__List__Olist_It__Nat__Onat_J,type,
can_select_list_nat: ( list_nat > $o ) > set_list_nat > $o ).
thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
can_select_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_List_Ocan__select_001t__Set__Oset_It__Nat__Onat_J,type,
can_select_set_nat: ( set_nat > $o ) > set_set_nat > $o ).
thf(sy_c_List_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_Ocoset_001t__Set__Oset_It__Nat__Onat_J,type,
coset_set_nat: list_set_nat > set_set_nat ).
thf(sy_c_List_Ocount__list_001t__List__Olist_It__Nat__Onat_J,type,
count_list_list_nat: list_list_nat > list_nat > nat ).
thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
count_list_nat: list_nat > nat > nat ).
thf(sy_c_List_Ocount__list_001t__Set__Oset_It__Nat__Onat_J,type,
count_list_set_nat: list_set_nat > set_nat > nat ).
thf(sy_c_List_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_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_Olast_001t__Set__Oset_It__Nat__Onat_J,type,
last_set_nat: list_set_nat > set_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
cons_list_list_nat: list_list_nat > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__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_OCons_001t__Set__Oset_It__Nat__Onat_J,type,
cons_set_nat: set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__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_ONil_001t__Set__Oset_It__Nat__Onat_J,type,
nil_set_nat: list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__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__Nat__Onat_J,type,
map_li7225945977422193158st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
map_list_nat_nat: ( list_nat > nat ) > list_list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
map_list_nat_set_nat: ( list_nat > set_nat ) > list_list_nat > list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
map_nat_list_nat: ( nat > list_nat ) > list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
map_set_nat_set_nat: ( set_nat > set_nat ) > list_set_nat > list_set_nat ).
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_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
set_set_nat2: list_set_nat > set_set_nat ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_It__Nat__Onat_J,type,
list_ex1_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Set__Oset_It__Nat__Onat_J,type,
list_ex1_set_nat: ( set_nat > $o ) > list_set_nat > $o ).
thf(sy_c_List_Omap__tailrec_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_ta9145449198693458768st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Nat__Onat,type,
map_tailrec_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_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_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > 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__Nat__Onat,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oreplicate_001t__List__Olist_It__Nat__Onat_J,type,
replicate_list_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > 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_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bot_bot_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Oord_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__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
ord_max_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OPow_001t__List__Olist_It__Nat__Onat_J,type,
pow_list_nat: set_list_nat > set_set_list_nat ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
image_4375647060027540749st_nat: ( list_list_nat > set_list_nat ) > set_list_list_nat > set_set_list_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
image_7976474329151083847st_nat: ( list_nat > list_nat ) > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
image_list_nat_nat: ( list_nat > nat ) > set_list_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_1775855109352712557et_nat: ( list_nat > set_nat ) > set_list_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__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__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
image_5426082062715393517st_nat: ( set_nat > list_nat ) > set_set_nat > set_list_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat2: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat2: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat2: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Oremove_001t__List__Olist_It__Nat__Onat_J,type,
remove_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001t__Set__Oset_It__Nat__Onat_J,type,
remove_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_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_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or890127255671739683et_nat: set_nat > set_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__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__List__Olist_It__Nat__Onat_J,type,
accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $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_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
member_set_list_nat: set_list_nat > set_set_list_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 (1273)
thf(fact_0_snoc_OIH,axiom,
( ( equiva3371634703666331078on_rgf @ xsa )
=> ( ( set_nat2 @ xsa )
= ( set_ord_lessThan_nat @ ( equiva5889994315859557365_limit @ xsa ) ) ) ) ).
% snoc.IH
thf(fact_1_a,axiom,
( ( set_nat2 @ xsa )
= ( set_ord_lessThan_nat @ ( equiva5889994315859557365_limit @ xsa ) ) ) ).
% a
thf(fact_2_snoc_Oprems,axiom,
equiva3371634703666331078on_rgf @ ( append_nat @ xsa @ ( cons_nat @ x @ nil_nat ) ) ).
% snoc.prems
thf(fact_3_c,axiom,
equiva3371634703666331078on_rgf @ xsa ).
% c
thf(fact_4_rgf__limit_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X2: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs ) ) ) ).
% rgf_limit.cases
thf(fact_5__092_060open_062set_A_Ixs_A_064_A_091x_093_J_A_061_Ainsert_Ax_A_123_O_O_060rgf__limit_Axs_125_092_060close_062,axiom,
( ( set_nat2 @ ( append_nat @ xsa @ ( cons_nat @ x @ nil_nat ) ) )
= ( insert_nat2 @ x @ ( set_ord_lessThan_nat @ ( equiva5889994315859557365_limit @ xsa ) ) ) ) ).
% \<open>set (xs @ [x]) = insert x {..<rgf_limit xs}\<close>
thf(fact_6_append1__eq__conv,axiom,
! [Xs2: list_list_nat,X: list_nat,Ys: list_list_nat,Y: list_nat] :
( ( ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) )
= ( append_list_nat @ Ys @ ( cons_list_nat @ Y @ nil_list_nat ) ) )
= ( ( Xs2 = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_7_append1__eq__conv,axiom,
! [Xs2: list_nat,X: nat,Ys: list_nat,Y: nat] :
( ( ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs2 = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_8_append_Oright__neutral,axiom,
! [A: list_list_nat] :
( ( append_list_nat @ A @ nil_list_nat )
= A ) ).
% append.right_neutral
thf(fact_9_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_10_append__Nil2,axiom,
! [Xs2: list_list_nat] :
( ( append_list_nat @ Xs2 @ nil_list_nat )
= Xs2 ) ).
% append_Nil2
thf(fact_11_append__Nil2,axiom,
! [Xs2: list_nat] :
( ( append_nat @ Xs2 @ nil_nat )
= Xs2 ) ).
% append_Nil2
thf(fact_12_append__self__conv,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Ys )
= Xs2 )
= ( Ys = nil_list_nat ) ) ).
% append_self_conv
thf(fact_13_append__self__conv,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= Xs2 )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_14_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_15_self__append__conv,axiom,
! [Y: list_nat,Ys: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_16_append__self__conv2,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Ys )
= Ys )
= ( Xs2 = nil_list_nat ) ) ).
% append_self_conv2
thf(fact_17_append__self__conv2,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= Ys )
= ( Xs2 = nil_nat ) ) ).
% append_self_conv2
thf(fact_18_self__append__conv2,axiom,
! [Y: list_list_nat,Xs2: list_list_nat] :
( ( Y
= ( append_list_nat @ Xs2 @ Y ) )
= ( Xs2 = nil_list_nat ) ) ).
% self_append_conv2
thf(fact_19_self__append__conv2,axiom,
! [Y: list_nat,Xs2: list_nat] :
( ( Y
= ( append_nat @ Xs2 @ Y ) )
= ( Xs2 = nil_nat ) ) ).
% self_append_conv2
thf(fact_20_Nil__is__append__conv,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat] :
( ( nil_list_nat
= ( append_list_nat @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_list_nat )
& ( Ys = nil_list_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_21_Nil__is__append__conv,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_22_append__is__Nil__conv,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Ys )
= nil_list_nat )
= ( ( Xs2 = nil_list_nat )
& ( Ys = nil_list_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_23_append__is__Nil__conv,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= nil_nat )
= ( ( Xs2 = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_24_b,axiom,
ord_less_eq_nat @ x @ ( equiva5889994315859557365_limit @ xsa ) ).
% b
thf(fact_25_split__list,axiom,
! [X: set_nat,Xs2: list_set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
=> ? [Ys2: list_set_nat,Zs: list_set_nat] :
( Xs2
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X @ Zs ) ) ) ) ).
% split_list
thf(fact_26_split__list,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ? [Ys2: list_list_nat,Zs: list_list_nat] :
( Xs2
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X @ Zs ) ) ) ) ).
% split_list
thf(fact_27_split__list,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [Ys2: list_nat,Zs: list_nat] :
( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs ) ) ) ) ).
% split_list
thf(fact_28_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_29_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_30_same__append__eq,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat,Zs2: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Ys )
= ( append_list_nat @ Xs2 @ Zs2 ) )
= ( Ys = Zs2 ) ) ).
% same_append_eq
thf(fact_31_same__append__eq,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs2: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= ( append_nat @ Xs2 @ Zs2 ) )
= ( Ys = Zs2 ) ) ).
% same_append_eq
thf(fact_32_append__same__eq,axiom,
! [Ys: list_list_nat,Xs2: list_list_nat,Zs2: list_list_nat] :
( ( ( append_list_nat @ Ys @ Xs2 )
= ( append_list_nat @ Zs2 @ Xs2 ) )
= ( Ys = Zs2 ) ) ).
% append_same_eq
thf(fact_33_append__same__eq,axiom,
! [Ys: list_nat,Xs2: list_nat,Zs2: list_nat] :
( ( ( append_nat @ Ys @ Xs2 )
= ( append_nat @ Zs2 @ Xs2 ) )
= ( Ys = Zs2 ) ) ).
% append_same_eq
thf(fact_34_append__assoc,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat,Zs2: list_list_nat] :
( ( append_list_nat @ ( append_list_nat @ Xs2 @ Ys ) @ Zs2 )
= ( append_list_nat @ Xs2 @ ( append_list_nat @ Ys @ Zs2 ) ) ) ).
% append_assoc
thf(fact_35_append__assoc,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs2: list_nat] :
( ( append_nat @ ( append_nat @ Xs2 @ Ys ) @ Zs2 )
= ( append_nat @ Xs2 @ ( append_nat @ Ys @ Zs2 ) ) ) ).
% append_assoc
thf(fact_36_append_Oassoc,axiom,
! [A: list_list_nat,B: list_list_nat,C: list_list_nat] :
( ( append_list_nat @ ( append_list_nat @ A @ B ) @ C )
= ( append_list_nat @ A @ ( append_list_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_37_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_38_list_Osimps_I15_J,axiom,
! [X21: set_nat,X22: list_set_nat] :
( ( set_set_nat2 @ ( cons_set_nat @ X21 @ X22 ) )
= ( insert_set_nat2 @ X21 @ ( set_set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_39_list_Osimps_I15_J,axiom,
! [X21: list_nat,X22: list_list_nat] :
( ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) )
= ( insert_list_nat2 @ X21 @ ( set_list_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_40_list_Osimps_I15_J,axiom,
! [X21: nat,X22: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
= ( insert_nat2 @ X21 @ ( set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_41_not__Cons__self2,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( cons_list_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_42_not__Cons__self2,axiom,
! [X: nat,Xs2: list_nat] :
( ( cons_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_43_append__eq__append__conv2,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat,Zs2: list_list_nat,Ts: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Ys )
= ( append_list_nat @ Zs2 @ Ts ) )
= ( ? [Us: list_list_nat] :
( ( ( Xs2
= ( append_list_nat @ Zs2 @ Us ) )
& ( ( append_list_nat @ Us @ Ys )
= Ts ) )
| ( ( ( append_list_nat @ Xs2 @ Us )
= Zs2 )
& ( Ys
= ( append_list_nat @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_44_append__eq__append__conv2,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs2: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= ( append_nat @ Zs2 @ Ts ) )
= ( ? [Us: list_nat] :
( ( ( Xs2
= ( append_nat @ Zs2 @ Us ) )
& ( ( append_nat @ Us @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs2 @ Us )
= Zs2 )
& ( Ys
= ( append_nat @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_45_append__eq__appendI,axiom,
! [Xs2: list_list_nat,Xs1: list_list_nat,Zs2: list_list_nat,Ys: list_list_nat,Us2: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Xs1 )
= Zs2 )
=> ( ( Ys
= ( append_list_nat @ Xs1 @ Us2 ) )
=> ( ( append_list_nat @ Xs2 @ Ys )
= ( append_list_nat @ Zs2 @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_46_append__eq__appendI,axiom,
! [Xs2: list_nat,Xs1: list_nat,Zs2: list_nat,Ys: list_nat,Us2: list_nat] :
( ( ( append_nat @ Xs2 @ Xs1 )
= Zs2 )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us2 ) )
=> ( ( append_nat @ Xs2 @ Ys )
= ( append_nat @ Zs2 @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_47_list__nonempty__induct,axiom,
! [Xs2: list_list_nat,P: list_list_nat > $o] :
( ( Xs2 != nil_list_nat )
=> ( ! [X2: list_nat] : ( P @ ( cons_list_nat @ X2 @ nil_list_nat ) )
=> ( ! [X2: list_nat,Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_48_list__nonempty__induct,axiom,
! [Xs2: list_nat,P: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_49_list__induct2_H,axiom,
! [P: list_nat > list_list_nat > $o,Xs2: list_nat,Ys: list_list_nat] :
( ( P @ nil_nat @ nil_list_nat )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_list_nat )
=> ( ! [Y2: list_nat,Ys2: list_list_nat] : ( P @ nil_nat @ ( cons_list_nat @ Y2 @ Ys2 ) )
=> ( ! [X2: nat,Xs: list_nat,Y2: list_nat,Ys2: list_list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_50_list__induct2_H,axiom,
! [P: list_list_nat > list_nat > $o,Xs2: list_list_nat,Ys: list_nat] :
( ( P @ nil_list_nat @ nil_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_list_nat @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: nat,Ys2: list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_51_list__induct2_H,axiom,
! [P: list_list_nat > list_list_nat > $o,Xs2: list_list_nat,Ys: list_list_nat] :
( ( P @ nil_list_nat @ nil_list_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_list_nat )
=> ( ! [Y2: list_nat,Ys2: list_list_nat] : ( P @ nil_list_nat @ ( cons_list_nat @ Y2 @ Ys2 ) )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: list_nat,Ys2: list_list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_52_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs2: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys2: list_nat] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_53_neq__Nil__conv,axiom,
! [Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
= ( ? [Y3: list_nat,Ys3: list_list_nat] :
( Xs2
= ( cons_list_nat @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_54_neq__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
= ( ? [Y3: nat,Ys3: list_nat] :
( Xs2
= ( cons_nat @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_55_remdups__adj_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [X2: list_nat] :
( X
!= ( cons_list_nat @ X2 @ nil_list_nat ) )
=> ~ ! [X2: list_nat,Y2: list_nat,Xs: list_list_nat] :
( X
!= ( cons_list_nat @ X2 @ ( cons_list_nat @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_56_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X2: nat] :
( X
!= ( cons_nat @ X2 @ nil_nat ) )
=> ~ ! [X2: nat,Y2: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_57_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 ) )
=> ~ ! [X2: list_nat,Xs: list_list_nat,Xss: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_58_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X2: nat,Xs: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_59_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X2: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_60_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_61_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_62_mem__Collect__eq,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_65_Collect__mem__eq,axiom,
! [A2: set_list_nat] :
( ( collect_list_nat
@ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_68_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_69_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_70_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_71_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_72_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_73_set__ConsD,axiom,
! [Y: set_nat,X: set_nat,Xs2: list_set_nat] :
( ( member_set_nat @ Y @ ( set_set_nat2 @ ( cons_set_nat @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_set_nat @ Y @ ( set_set_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_74_set__ConsD,axiom,
! [Y: list_nat,X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_list_nat @ Y @ ( set_list_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_75_set__ConsD,axiom,
! [Y: nat,X: nat,Xs2: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_nat @ Y @ ( set_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_76_list_Oset__cases,axiom,
! [E: set_nat,A: list_set_nat] :
( ( member_set_nat @ E @ ( set_set_nat2 @ A ) )
=> ( ! [Z2: list_set_nat] :
( A
!= ( cons_set_nat @ E @ Z2 ) )
=> ~ ! [Z1: set_nat,Z2: list_set_nat] :
( ( A
= ( cons_set_nat @ Z1 @ Z2 ) )
=> ~ ( member_set_nat @ E @ ( set_set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_77_list_Oset__cases,axiom,
! [E: list_nat,A: list_list_nat] :
( ( member_list_nat @ 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_nat @ E @ ( set_list_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_78_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A ) )
=> ( ! [Z2: list_nat] :
( A
!= ( cons_nat @ E @ Z2 ) )
=> ~ ! [Z1: nat,Z2: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_79_list_Oset__intros_I1_J,axiom,
! [X21: set_nat,X22: list_set_nat] : ( member_set_nat @ X21 @ ( set_set_nat2 @ ( cons_set_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_80_list_Oset__intros_I1_J,axiom,
! [X21: list_nat,X22: list_list_nat] : ( member_list_nat @ X21 @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_81_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_82_list_Oset__intros_I2_J,axiom,
! [Y: set_nat,X22: list_set_nat,X21: set_nat] :
( ( member_set_nat @ Y @ ( set_set_nat2 @ X22 ) )
=> ( member_set_nat @ Y @ ( set_set_nat2 @ ( cons_set_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_83_list_Oset__intros_I2_J,axiom,
! [Y: list_nat,X22: list_list_nat,X21: list_nat] :
( ( member_list_nat @ Y @ ( set_list_nat2 @ X22 ) )
=> ( member_list_nat @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_84_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_85_Cons__eq__appendI,axiom,
! [X: list_nat,Xs1: list_list_nat,Ys: list_list_nat,Xs2: list_list_nat,Zs2: list_list_nat] :
( ( ( cons_list_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs2
= ( append_list_nat @ Xs1 @ Zs2 ) )
=> ( ( cons_list_nat @ X @ Xs2 )
= ( append_list_nat @ Ys @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_86_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys: list_nat,Xs2: list_nat,Zs2: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs2
= ( append_nat @ Xs1 @ Zs2 ) )
=> ( ( cons_nat @ X @ Xs2 )
= ( append_nat @ Ys @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_87_append__Cons,axiom,
! [X: list_nat,Xs2: list_list_nat,Ys: list_list_nat] :
( ( append_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ Ys )
= ( cons_list_nat @ X @ ( append_list_nat @ Xs2 @ Ys ) ) ) ).
% append_Cons
thf(fact_88_append__Cons,axiom,
! [X: nat,Xs2: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs2 ) @ Ys )
= ( cons_nat @ X @ ( append_nat @ Xs2 @ Ys ) ) ) ).
% append_Cons
thf(fact_89_eq__Nil__appendI,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append_list_nat @ nil_list_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_90_eq__Nil__appendI,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_91_append_Oleft__neutral,axiom,
! [A: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_92_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_93_append__Nil,axiom,
! [Ys: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_94_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_95_rev__nonempty__induct,axiom,
! [Xs2: list_list_nat,P: list_list_nat > $o] :
( ( Xs2 != nil_list_nat )
=> ( ! [X2: list_nat] : ( P @ ( cons_list_nat @ X2 @ nil_list_nat ) )
=> ( ! [X2: list_nat,Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( P @ Xs )
=> ( P @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_96_rev__nonempty__induct,axiom,
! [Xs2: list_nat,P: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( P @ Xs )
=> ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_97_append__eq__Cons__conv,axiom,
! [Ys: list_list_nat,Zs2: list_list_nat,X: list_nat,Xs2: list_list_nat] :
( ( ( append_list_nat @ Ys @ Zs2 )
= ( cons_list_nat @ X @ Xs2 ) )
= ( ( ( Ys = nil_list_nat )
& ( Zs2
= ( cons_list_nat @ X @ Xs2 ) ) )
| ? [Ys4: list_list_nat] :
( ( Ys
= ( cons_list_nat @ X @ Ys4 ) )
& ( ( append_list_nat @ Ys4 @ Zs2 )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_98_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs2: list_nat,X: nat,Xs2: list_nat] :
( ( ( append_nat @ Ys @ Zs2 )
= ( cons_nat @ X @ Xs2 ) )
= ( ( ( Ys = nil_nat )
& ( Zs2
= ( cons_nat @ X @ Xs2 ) ) )
| ? [Ys4: list_nat] :
( ( Ys
= ( cons_nat @ X @ Ys4 ) )
& ( ( append_nat @ Ys4 @ Zs2 )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_99_Cons__eq__append__conv,axiom,
! [X: list_nat,Xs2: list_list_nat,Ys: list_list_nat,Zs2: list_list_nat] :
( ( ( cons_list_nat @ X @ Xs2 )
= ( append_list_nat @ Ys @ Zs2 ) )
= ( ( ( Ys = nil_list_nat )
& ( ( cons_list_nat @ X @ Xs2 )
= Zs2 ) )
| ? [Ys4: list_list_nat] :
( ( ( cons_list_nat @ X @ Ys4 )
= Ys )
& ( Xs2
= ( append_list_nat @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_100_Cons__eq__append__conv,axiom,
! [X: nat,Xs2: list_nat,Ys: list_nat,Zs2: list_nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( append_nat @ Ys @ Zs2 ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X @ Xs2 )
= Zs2 ) )
| ? [Ys4: list_nat] :
( ( ( cons_nat @ X @ Ys4 )
= Ys )
& ( Xs2
= ( append_nat @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_101_rev__exhaust,axiom,
! [Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
=> ~ ! [Ys2: list_list_nat,Y2: list_nat] :
( Xs2
!= ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) ) ) ).
% rev_exhaust
thf(fact_102_rev__exhaust,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ~ ! [Ys2: list_nat,Y2: nat] :
( Xs2
!= ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_103_rev__induct,axiom,
! [P: list_list_nat > $o,Xs2: list_list_nat] :
( ( P @ nil_list_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat] :
( ( P @ Xs )
=> ( P @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_104_rev__induct,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat] :
( ( P @ Xs )
=> ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_105_split__list__first__prop__iff,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ( ? [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_list_nat,X3: list_nat] :
( ? [Zs3: list_list_nat] :
( Xs2
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ ( set_list_nat2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_106_split__list__first__prop__iff,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_nat,X3: nat] :
( ? [Zs3: list_nat] :
( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_107_split__list__last__prop__iff,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ( ? [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_list_nat,X3: list_nat,Zs3: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ ( set_list_nat2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_108_split__list__last__prop__iff,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_nat,X3: nat,Zs3: list_nat] :
( ( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_109_in__set__conv__decomp__first,axiom,
! [X: set_nat,Xs2: list_set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_set_nat,Zs3: list_set_nat] :
( ( Xs2
= ( append_set_nat @ Ys3 @ ( cons_set_nat @ X @ Zs3 ) ) )
& ~ ( member_set_nat @ X @ ( set_set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_110_in__set__conv__decomp__first,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
= ( ? [Ys3: list_list_nat,Zs3: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs3 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_111_in__set__conv__decomp__first,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_112_in__set__conv__decomp__last,axiom,
! [X: set_nat,Xs2: list_set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_set_nat,Zs3: list_set_nat] :
( ( Xs2
= ( append_set_nat @ Ys3 @ ( cons_set_nat @ X @ Zs3 ) ) )
& ~ ( member_set_nat @ X @ ( set_set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_113_in__set__conv__decomp__last,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
= ( ? [Ys3: list_list_nat,Zs3: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs3 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_114_in__set__conv__decomp__last,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_115_split__list__first__propE,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_list_nat,X2: list_nat] :
( ? [Zs: list_list_nat] :
( Xs2
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X2 @ Zs ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ ( set_list_nat2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_116_split__list__first__propE,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_nat,X2: nat] :
( ? [Zs: list_nat] :
( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_117_split__list__last__propE,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_list_nat,X2: list_nat,Zs: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X2 @ Zs ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ ( set_list_nat2 @ Zs ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_118_split__list__last__propE,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_nat,X2: nat,Zs: list_nat] :
( ( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Zs ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_119_split__list__first__prop,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys2: list_list_nat,X2: list_nat] :
( ? [Zs: list_list_nat] :
( Xs2
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ ( set_list_nat2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_120_split__list__first__prop,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys2: list_nat,X2: nat] :
( ? [Zs: list_nat] :
( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_121_split__list__last__prop,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys2: list_list_nat,X2: list_nat,Zs: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ ( set_list_nat2 @ Zs ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_122_split__list__last__prop,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys2: list_nat,X2: nat,Zs: list_nat] :
( ( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Zs ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_123_in__set__conv__decomp,axiom,
! [X: set_nat,Xs2: list_set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_set_nat,Zs3: list_set_nat] :
( Xs2
= ( append_set_nat @ Ys3 @ ( cons_set_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_124_in__set__conv__decomp,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
= ( ? [Ys3: list_list_nat,Zs3: list_list_nat] :
( Xs2
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_125_in__set__conv__decomp,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_126_append__Cons__eq__iff,axiom,
! [X: set_nat,Xs2: list_set_nat,Ys: list_set_nat,Xs3: list_set_nat,Ys5: list_set_nat] :
( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
=> ( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Ys ) )
=> ( ( ( append_set_nat @ Xs2 @ ( cons_set_nat @ X @ Ys ) )
= ( append_set_nat @ Xs3 @ ( cons_set_nat @ X @ Ys5 ) ) )
= ( ( Xs2 = Xs3 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_127_append__Cons__eq__iff,axiom,
! [X: list_nat,Xs2: list_list_nat,Ys: list_list_nat,Xs3: list_list_nat,Ys5: list_list_nat] :
( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys ) )
=> ( ( ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ Ys ) )
= ( append_list_nat @ Xs3 @ ( cons_list_nat @ X @ Ys5 ) ) )
= ( ( Xs2 = Xs3 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_128_append__Cons__eq__iff,axiom,
! [X: nat,Xs2: list_nat,Ys: list_nat,Xs3: list_nat,Ys5: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ~ ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs2 @ ( cons_nat @ X @ Ys ) )
= ( append_nat @ Xs3 @ ( cons_nat @ X @ Ys5 ) ) )
= ( ( Xs2 = Xs3 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_129_split__list__propE,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_list_nat,X2: list_nat] :
( ? [Zs: list_list_nat] :
( Xs2
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X2 @ Zs ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_130_split__list__propE,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_nat,X2: nat] :
( ? [Zs: list_nat] :
( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_131_split__list__first,axiom,
! [X: set_nat,Xs2: list_set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
=> ? [Ys2: list_set_nat,Zs: list_set_nat] :
( ( Xs2
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X @ Zs ) ) )
& ~ ( member_set_nat @ X @ ( set_set_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_132_split__list__first,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ? [Ys2: list_list_nat,Zs: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X @ Zs ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_133_split__list__first,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [Ys2: list_nat,Zs: list_nat] :
( ( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_134_split__list__prop,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys2: list_list_nat,X2: list_nat] :
( ? [Zs: list_list_nat] :
( Xs2
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X2 @ Zs ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_135_split__list__prop,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys2: list_nat,X2: nat] :
( ? [Zs: list_nat] :
( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_136_split__list__last,axiom,
! [X: set_nat,Xs2: list_set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
=> ? [Ys2: list_set_nat,Zs: list_set_nat] :
( ( Xs2
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X @ Zs ) ) )
& ~ ( member_set_nat @ X @ ( set_set_nat2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_137_split__list__last,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ? [Ys2: list_list_nat,Zs: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X @ Zs ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_138_split__list__last,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [Ys2: list_nat,Zs: list_nat] :
( ( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_139_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_140_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_141_insertCI,axiom,
! [A: list_nat,B2: set_list_nat,B: list_nat] :
( ( ~ ( member_list_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_list_nat @ A @ ( insert_list_nat2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_142_insertCI,axiom,
! [A: set_nat,B2: set_set_nat,B: set_nat] :
( ( ~ ( member_set_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_set_nat @ A @ ( insert_set_nat2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_143_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_144_insert__iff,axiom,
! [A: list_nat,B: list_nat,A2: set_list_nat] :
( ( member_list_nat @ A @ ( insert_list_nat2 @ B @ A2 ) )
= ( ( A = B )
| ( member_list_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_145_insert__iff,axiom,
! [A: set_nat,B: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ ( insert_set_nat2 @ B @ A2 ) )
= ( ( A = B )
| ( member_set_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_146_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat2 @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_147_insert__absorb2,axiom,
! [X: set_nat,A2: set_set_nat] :
( ( insert_set_nat2 @ X @ ( insert_set_nat2 @ X @ A2 ) )
= ( insert_set_nat2 @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_148_insert__absorb2,axiom,
! [X: nat,A2: set_nat] :
( ( insert_nat2 @ X @ ( insert_nat2 @ X @ A2 ) )
= ( insert_nat2 @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_149_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_150_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_151_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_152_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_153_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_154_the__elem__set,axiom,
! [X: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_155_rgf__def,axiom,
( equiva3371634703666331078on_rgf
= ( ^ [X3: list_nat] :
! [Ys3: list_nat,Y3: nat] :
( ( prefix_nat @ ( append_nat @ Ys3 @ ( cons_nat @ Y3 @ nil_nat ) ) @ X3 )
=> ( ord_less_eq_nat @ Y3 @ ( equiva5889994315859557365_limit @ Ys3 ) ) ) ) ) ).
% rgf_def
thf(fact_156_bind__simps_I2_J,axiom,
! [X: nat,Xs2: list_nat,F: nat > list_list_nat] :
( ( bind_nat_list_nat @ ( cons_nat @ X @ Xs2 ) @ F )
= ( append_list_nat @ ( F @ X ) @ ( bind_nat_list_nat @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_157_bind__simps_I2_J,axiom,
! [X: list_nat,Xs2: list_list_nat,F: list_nat > list_nat] :
( ( bind_list_nat_nat @ ( cons_list_nat @ X @ Xs2 ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_list_nat_nat @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_158_bind__simps_I2_J,axiom,
! [X: list_nat,Xs2: list_list_nat,F: list_nat > list_list_nat] :
( ( bind_l7796378977173581257st_nat @ ( cons_list_nat @ X @ Xs2 ) @ F )
= ( append_list_nat @ ( F @ X ) @ ( bind_l7796378977173581257st_nat @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_159_bind__simps_I2_J,axiom,
! [X: nat,Xs2: list_nat,F: nat > list_nat] :
( ( bind_nat_nat @ ( cons_nat @ X @ Xs2 ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_nat_nat @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_160_List_Oset__insert,axiom,
! [X: set_nat,Xs2: list_set_nat] :
( ( set_set_nat2 @ ( insert_set_nat @ X @ Xs2 ) )
= ( insert_set_nat2 @ X @ ( set_set_nat2 @ Xs2 ) ) ) ).
% List.set_insert
thf(fact_161_List_Oset__insert,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( set_list_nat2 @ ( insert_list_nat @ X @ Xs2 ) )
= ( insert_list_nat2 @ X @ ( set_list_nat2 @ Xs2 ) ) ) ).
% List.set_insert
thf(fact_162_List_Oset__insert,axiom,
! [X: nat,Xs2: list_nat] :
( ( set_nat2 @ ( insert_nat @ X @ Xs2 ) )
= ( insert_nat2 @ X @ ( set_nat2 @ Xs2 ) ) ) ).
% List.set_insert
thf(fact_163_subsetI,axiom,
! [A2: set_list_nat,B2: set_list_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A2 )
=> ( member_list_nat @ X2 @ B2 ) )
=> ( ord_le6045566169113846134st_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_164_subsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ X2 @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_165_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_166_subset__antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_167_prefix__order_Odual__order_Orefl,axiom,
! [A: list_nat] : ( prefix_nat @ A @ A ) ).
% prefix_order.dual_order.refl
thf(fact_168_prefix__order_Oorder__refl,axiom,
! [X: list_nat] : ( prefix_nat @ X @ X ) ).
% prefix_order.order_refl
thf(fact_169_insert__subset,axiom,
! [X: list_nat,A2: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( insert_list_nat2 @ X @ A2 ) @ B2 )
= ( ( member_list_nat @ X @ B2 )
& ( ord_le6045566169113846134st_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_170_insert__subset,axiom,
! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( insert_set_nat2 @ X @ A2 ) @ B2 )
= ( ( member_set_nat @ X @ B2 )
& ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_171_insert__subset,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ A2 ) @ B2 )
= ( ( member_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_172_Cons__prefix__Cons,axiom,
! [X: list_nat,Xs2: list_list_nat,Y: list_nat,Ys: list_list_nat] :
( ( prefix_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ ( cons_list_nat @ Y @ Ys ) )
= ( ( X = Y )
& ( prefix_list_nat @ Xs2 @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_173_Cons__prefix__Cons,axiom,
! [X: nat,Xs2: list_nat,Y: nat,Ys: list_nat] :
( ( prefix_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys ) )
= ( ( X = Y )
& ( prefix_nat @ Xs2 @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_174_prefix__code_I1_J,axiom,
! [Xs2: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ Xs2 ) ).
% prefix_code(1)
thf(fact_175_prefix__code_I1_J,axiom,
! [Xs2: list_nat] : ( prefix_nat @ nil_nat @ Xs2 ) ).
% prefix_code(1)
thf(fact_176_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_177_prefix__bot_Oextremum__unique,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
= ( A = nil_nat ) ) ).
% prefix_bot.extremum_unique
thf(fact_178_prefix__Nil,axiom,
! [Xs2: list_list_nat] :
( ( prefix_list_nat @ Xs2 @ nil_list_nat )
= ( Xs2 = nil_list_nat ) ) ).
% prefix_Nil
thf(fact_179_prefix__Nil,axiom,
! [Xs2: list_nat] :
( ( prefix_nat @ Xs2 @ nil_nat )
= ( Xs2 = nil_nat ) ) ).
% prefix_Nil
thf(fact_180_same__prefix__prefix,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat,Zs2: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs2 @ Ys ) @ ( append_list_nat @ Xs2 @ Zs2 ) )
= ( prefix_list_nat @ Ys @ Zs2 ) ) ).
% same_prefix_prefix
thf(fact_181_same__prefix__prefix,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs2: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs2 @ Ys ) @ ( append_nat @ Xs2 @ Zs2 ) )
= ( prefix_nat @ Ys @ Zs2 ) ) ).
% same_prefix_prefix
thf(fact_182_in__set__insert,axiom,
! [X: set_nat,Xs2: list_set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
=> ( ( insert_set_nat @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_183_in__set__insert,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( ( insert_list_nat @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_184_in__set__insert,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( insert_nat @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_185_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_186_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_187_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_188_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_189_same__prefix__nil,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs2 @ Ys ) @ Xs2 )
= ( Ys = nil_list_nat ) ) ).
% same_prefix_nil
thf(fact_190_same__prefix__nil,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs2 @ Ys ) @ Xs2 )
= ( Ys = nil_nat ) ) ).
% same_prefix_nil
thf(fact_191_insert__Nil,axiom,
! [X: list_nat] :
( ( insert_list_nat @ X @ nil_list_nat )
= ( cons_list_nat @ X @ nil_list_nat ) ) ).
% insert_Nil
thf(fact_192_insert__Nil,axiom,
! [X: nat] :
( ( insert_nat @ X @ nil_nat )
= ( cons_nat @ X @ nil_nat ) ) ).
% insert_Nil
thf(fact_193_not__in__set__insert,axiom,
! [X: set_nat,Xs2: list_set_nat] :
( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
=> ( ( insert_set_nat @ X @ Xs2 )
= ( cons_set_nat @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_194_not__in__set__insert,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( ( insert_list_nat @ X @ Xs2 )
= ( cons_list_nat @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_195_not__in__set__insert,axiom,
! [X: nat,Xs2: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( insert_nat @ X @ Xs2 )
= ( cons_nat @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_196_prefix__snoc,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat,Y: list_nat] :
( ( prefix_list_nat @ Xs2 @ ( append_list_nat @ Ys @ ( cons_list_nat @ Y @ nil_list_nat ) ) )
= ( ( Xs2
= ( append_list_nat @ Ys @ ( cons_list_nat @ Y @ nil_list_nat ) ) )
| ( prefix_list_nat @ Xs2 @ Ys ) ) ) ).
% prefix_snoc
thf(fact_197_prefix__snoc,axiom,
! [Xs2: list_nat,Ys: list_nat,Y: nat] :
( ( prefix_nat @ Xs2 @ ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs2
= ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
| ( prefix_nat @ Xs2 @ Ys ) ) ) ).
% prefix_snoc
thf(fact_198_in__mono,axiom,
! [A2: set_list_nat,B2: set_list_nat,X: list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
=> ( ( member_list_nat @ X @ A2 )
=> ( member_list_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_199_in__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_200_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_201_subsetD,axiom,
! [A2: set_list_nat,B2: set_list_nat,C: list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
=> ( ( member_list_nat @ C @ A2 )
=> ( member_list_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_202_subsetD,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_203_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_204_equalityE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_205_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B3: set_list_nat] :
! [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
=> ( member_list_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_206_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A3 )
=> ( member_set_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_207_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_208_equalityD1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_209_equalityD2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_210_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B3: set_list_nat] :
! [T: list_nat] :
( ( member_list_nat @ T @ A3 )
=> ( member_list_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_211_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A3 )
=> ( member_set_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_212_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_213_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_214_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_215_subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_216_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_217_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_218_set__mono__prefix,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat] :
( ( prefix_list_nat @ Xs2 @ Ys )
=> ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ ( set_list_nat2 @ Ys ) ) ) ).
% set_mono_prefix
thf(fact_219_set__mono__prefix,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs2 @ Ys )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Ys ) ) ) ).
% set_mono_prefix
thf(fact_220_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_221_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_222_prefix__order_Odual__order_Oeq__iff,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
= ( ^ [A4: list_nat,B4: list_nat] :
( ( prefix_nat @ B4 @ A4 )
& ( prefix_nat @ A4 @ B4 ) ) ) ) ).
% prefix_order.dual_order.eq_iff
thf(fact_223_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_224_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_225_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_226_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_227_prefix__order_Oorder__eq__iff,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
= ( ^ [X3: list_nat,Y3: list_nat] :
( ( prefix_nat @ X3 @ Y3 )
& ( prefix_nat @ Y3 @ X3 ) ) ) ) ).
% prefix_order.order_eq_iff
thf(fact_228_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_229_prefix__order_Oorder__trans,axiom,
! [X: list_nat,Y: list_nat,Z3: list_nat] :
( ( prefix_nat @ X @ Y )
=> ( ( prefix_nat @ Y @ Z3 )
=> ( prefix_nat @ X @ Z3 ) ) ) ).
% prefix_order.order_trans
thf(fact_230_prefix__order_Oeq__refl,axiom,
! [X: list_nat,Y: list_nat] :
( ( X = Y )
=> ( prefix_nat @ X @ Y ) ) ).
% prefix_order.eq_refl
thf(fact_231_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_232_prefix__order_Oeq__iff,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
= ( ^ [A4: list_nat,B4: list_nat] :
( ( prefix_nat @ A4 @ B4 )
& ( prefix_nat @ B4 @ A4 ) ) ) ) ).
% prefix_order.eq_iff
thf(fact_233_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_234_prefix__bot_Obot__least,axiom,
! [A: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_235_prefix__bot_Obot__least,axiom,
! [A: list_nat] : ( prefix_nat @ nil_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_236_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_237_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
=> ( A = nil_nat ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_238_Nil__prefix,axiom,
! [Xs2: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ Xs2 ) ).
% Nil_prefix
thf(fact_239_Nil__prefix,axiom,
! [Xs2: list_nat] : ( prefix_nat @ nil_nat @ Xs2 ) ).
% Nil_prefix
thf(fact_240_prefixE,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat] :
( ( prefix_list_nat @ Xs2 @ Ys )
=> ~ ! [Zs: list_list_nat] :
( Ys
!= ( append_list_nat @ Xs2 @ Zs ) ) ) ).
% prefixE
thf(fact_241_prefixE,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs2 @ Ys )
=> ~ ! [Zs: list_nat] :
( Ys
!= ( append_nat @ Xs2 @ Zs ) ) ) ).
% prefixE
thf(fact_242_prefixI,axiom,
! [Ys: list_list_nat,Xs2: list_list_nat,Zs2: list_list_nat] :
( ( Ys
= ( append_list_nat @ Xs2 @ Zs2 ) )
=> ( prefix_list_nat @ Xs2 @ Ys ) ) ).
% prefixI
thf(fact_243_prefixI,axiom,
! [Ys: list_nat,Xs2: list_nat,Zs2: list_nat] :
( ( Ys
= ( append_nat @ Xs2 @ Zs2 ) )
=> ( prefix_nat @ Xs2 @ Ys ) ) ).
% prefixI
thf(fact_244_prefix__def,axiom,
( prefix_list_nat
= ( ^ [Xs4: list_list_nat,Ys3: list_list_nat] :
? [Zs3: list_list_nat] :
( Ys3
= ( append_list_nat @ Xs4 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_245_prefix__def,axiom,
( prefix_nat
= ( ^ [Xs4: list_nat,Ys3: list_nat] :
? [Zs3: list_nat] :
( Ys3
= ( append_nat @ Xs4 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_246_prefix__append,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat,Zs2: list_list_nat] :
( ( prefix_list_nat @ Xs2 @ ( append_list_nat @ Ys @ Zs2 ) )
= ( ( prefix_list_nat @ Xs2 @ Ys )
| ? [Us: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys @ Us ) )
& ( prefix_list_nat @ Us @ Zs2 ) ) ) ) ).
% prefix_append
thf(fact_247_prefix__append,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs2: list_nat] :
( ( prefix_nat @ Xs2 @ ( append_nat @ Ys @ Zs2 ) )
= ( ( prefix_nat @ Xs2 @ Ys )
| ? [Us: list_nat] :
( ( Xs2
= ( append_nat @ Ys @ Us ) )
& ( prefix_nat @ Us @ Zs2 ) ) ) ) ).
% prefix_append
thf(fact_248_prefix__prefix,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat,Zs2: list_list_nat] :
( ( prefix_list_nat @ Xs2 @ Ys )
=> ( prefix_list_nat @ Xs2 @ ( append_list_nat @ Ys @ Zs2 ) ) ) ).
% prefix_prefix
thf(fact_249_prefix__prefix,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs2: list_nat] :
( ( prefix_nat @ Xs2 @ Ys )
=> ( prefix_nat @ Xs2 @ ( append_nat @ Ys @ Zs2 ) ) ) ).
% prefix_prefix
thf(fact_250_append__prefixD,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat,Zs2: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs2 @ Ys ) @ Zs2 )
=> ( prefix_list_nat @ Xs2 @ Zs2 ) ) ).
% append_prefixD
thf(fact_251_append__prefixD,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs2: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs2 @ Ys ) @ Zs2 )
=> ( prefix_nat @ Xs2 @ Zs2 ) ) ).
% append_prefixD
thf(fact_252_subset__insertI2,axiom,
! [A2: set_set_nat,B2: set_set_nat,B: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat2 @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_253_subset__insertI2,axiom,
! [A2: set_nat,B2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_254_subset__insertI,axiom,
! [B2: set_set_nat,A: set_nat] : ( ord_le6893508408891458716et_nat @ B2 @ ( insert_set_nat2 @ A @ B2 ) ) ).
% subset_insertI
thf(fact_255_subset__insertI,axiom,
! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat2 @ A @ B2 ) ) ).
% subset_insertI
thf(fact_256_subset__insert,axiom,
! [X: list_nat,A2: set_list_nat,B2: set_list_nat] :
( ~ ( member_list_nat @ X @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ A2 @ ( insert_list_nat2 @ X @ B2 ) )
= ( ord_le6045566169113846134st_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_257_subset__insert,axiom,
! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ X @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat2 @ X @ B2 ) )
= ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_258_subset__insert,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_259_insert__mono,axiom,
! [C2: set_set_nat,D: set_set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ D )
=> ( ord_le6893508408891458716et_nat @ ( insert_set_nat2 @ A @ C2 ) @ ( insert_set_nat2 @ A @ D ) ) ) ).
% insert_mono
thf(fact_260_insert__mono,axiom,
! [C2: set_nat,D: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C2 @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ A @ C2 ) @ ( insert_nat2 @ A @ D ) ) ) ).
% insert_mono
thf(fact_261_subset__code_I1_J,axiom,
! [Xs2: list_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ B2 )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs2 ) )
=> ( member_set_nat @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_262_subset__code_I1_J,axiom,
! [Xs2: list_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ B2 )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
=> ( member_list_nat @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_263_subset__code_I1_J,axiom,
! [Xs2: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( member_nat @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_264_prefix__code_I2_J,axiom,
! [X: list_nat,Xs2: list_list_nat] :
~ ( prefix_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ nil_list_nat ) ).
% prefix_code(2)
thf(fact_265_prefix__code_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
~ ( prefix_nat @ ( cons_nat @ X @ Xs2 ) @ nil_nat ) ).
% prefix_code(2)
thf(fact_266_prefix__Cons,axiom,
! [Xs2: list_list_nat,Y: list_nat,Ys: list_list_nat] :
( ( prefix_list_nat @ Xs2 @ ( cons_list_nat @ Y @ Ys ) )
= ( ( Xs2 = nil_list_nat )
| ? [Zs3: list_list_nat] :
( ( Xs2
= ( cons_list_nat @ Y @ Zs3 ) )
& ( prefix_list_nat @ Zs3 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_267_prefix__Cons,axiom,
! [Xs2: list_nat,Y: nat,Ys: list_nat] :
( ( prefix_nat @ Xs2 @ ( cons_nat @ Y @ Ys ) )
= ( ( Xs2 = nil_nat )
| ? [Zs3: list_nat] :
( ( Xs2
= ( cons_nat @ Y @ Zs3 ) )
& ( prefix_nat @ Zs3 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_268_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 ) )
=> ! [X2: list_nat,Xs: list_list_nat] :
( ( Ls
= ( cons_list_nat @ X2 @ Xs ) )
=> ( ( X2 = A5 )
=> ( prefix_list_nat @ As @ Xs ) ) ) )
=> ~ ! [A5: list_nat] :
( ? [As: list_list_nat] :
( Ps
= ( cons_list_nat @ A5 @ As ) )
=> ! [X2: list_nat] :
( ? [Xs: list_list_nat] :
( Ls
= ( cons_list_nat @ X2 @ Xs ) )
=> ( X2 = A5 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_269_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 ) )
=> ! [X2: nat,Xs: list_nat] :
( ( Ls
= ( cons_nat @ X2 @ Xs ) )
=> ( ( X2 = A5 )
=> ( prefix_nat @ As @ Xs ) ) ) )
=> ~ ! [A5: nat] :
( ? [As: list_nat] :
( Ps
= ( cons_nat @ A5 @ As ) )
=> ! [X2: nat] :
( ? [Xs: list_nat] :
( Ls
= ( cons_nat @ X2 @ Xs ) )
=> ( X2 = A5 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_270_not__prefix__induct,axiom,
! [Ps: list_list_nat,Ls: list_list_nat,P: list_list_nat > list_list_nat > $o] :
( ~ ( prefix_list_nat @ Ps @ Ls )
=> ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_list_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: list_nat,Ys2: list_list_nat] :
( ( X2 != Y2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys2 ) ) )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: list_nat,Ys2: list_list_nat] :
( ( X2 = Y2 )
=> ( ~ ( prefix_list_nat @ Xs @ Ys2 )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys2 ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_271_not__prefix__induct,axiom,
! [Ps: list_nat,Ls: list_nat,P: list_nat > list_nat > $o] :
( ~ ( prefix_nat @ Ps @ Ls )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys2: list_nat] :
( ( X2 != Y2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys2: list_nat] :
( ( X2 = Y2 )
=> ( ~ ( prefix_nat @ Xs @ Ys2 )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_272_set__subset__Cons,axiom,
! [Xs2: list_list_nat,X: list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ ( set_list_nat2 @ ( cons_list_nat @ X @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_273_set__subset__Cons,axiom,
! [Xs2: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ ( cons_nat @ X @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_274_list__bind__cong,axiom,
! [Xs2: list_nat,Ys: list_nat,F: nat > list_nat,G: nat > list_nat] :
( ( Xs2 = Ys )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( bind_nat_nat @ Xs2 @ F )
= ( bind_nat_nat @ Ys @ G ) ) ) ) ).
% list_bind_cong
thf(fact_275_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_276_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_277_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_278_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_279_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_280_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_281_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_282_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_283_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_284_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_285_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_286_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_287_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_288_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_289_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_290_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_291_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_292_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_293_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_294_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_295_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_296_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_297_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_298_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_299_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_300_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_301_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_302_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_303_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_304_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_305_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_306_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_307_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_308_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z3 )
=> ( ord_less_eq_set_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_309_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_310_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_311_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_312_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_313_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_314_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_315_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_316_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_317_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_318_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_319_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_320_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_321_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_322_mk__disjoint__insert,axiom,
! [A: list_nat,A2: set_list_nat] :
( ( member_list_nat @ A @ A2 )
=> ? [B6: set_list_nat] :
( ( A2
= ( insert_list_nat2 @ A @ B6 ) )
& ~ ( member_list_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_323_mk__disjoint__insert,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ? [B6: set_set_nat] :
( ( A2
= ( insert_set_nat2 @ A @ B6 ) )
& ~ ( member_set_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_324_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B6: set_nat] :
( ( A2
= ( insert_nat2 @ A @ B6 ) )
& ~ ( member_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_325_insert__commute,axiom,
! [X: set_nat,Y: set_nat,A2: set_set_nat] :
( ( insert_set_nat2 @ X @ ( insert_set_nat2 @ Y @ A2 ) )
= ( insert_set_nat2 @ Y @ ( insert_set_nat2 @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_326_insert__commute,axiom,
! [X: nat,Y: nat,A2: set_nat] :
( ( insert_nat2 @ X @ ( insert_nat2 @ Y @ A2 ) )
= ( insert_nat2 @ Y @ ( insert_nat2 @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_327_insert__eq__iff,axiom,
! [A: list_nat,A2: set_list_nat,B: list_nat,B2: set_list_nat] :
( ~ ( member_list_nat @ A @ A2 )
=> ( ~ ( member_list_nat @ B @ B2 )
=> ( ( ( insert_list_nat2 @ A @ A2 )
= ( insert_list_nat2 @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_list_nat] :
( ( A2
= ( insert_list_nat2 @ B @ C3 ) )
& ~ ( member_list_nat @ B @ C3 )
& ( B2
= ( insert_list_nat2 @ A @ C3 ) )
& ~ ( member_list_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_328_insert__eq__iff,axiom,
! [A: set_nat,A2: set_set_nat,B: set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ A @ A2 )
=> ( ~ ( member_set_nat @ B @ B2 )
=> ( ( ( insert_set_nat2 @ A @ A2 )
= ( insert_set_nat2 @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_set_nat] :
( ( A2
= ( insert_set_nat2 @ B @ C3 ) )
& ~ ( member_set_nat @ B @ C3 )
& ( B2
= ( insert_set_nat2 @ A @ C3 ) )
& ~ ( member_set_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_329_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat2 @ A @ A2 )
= ( insert_nat2 @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_nat] :
( ( A2
= ( insert_nat2 @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B2
= ( insert_nat2 @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_330_insert__absorb,axiom,
! [A: list_nat,A2: set_list_nat] :
( ( member_list_nat @ A @ A2 )
=> ( ( insert_list_nat2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_331_insert__absorb,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( insert_set_nat2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_332_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_333_insert__ident,axiom,
! [X: list_nat,A2: set_list_nat,B2: set_list_nat] :
( ~ ( member_list_nat @ X @ A2 )
=> ( ~ ( member_list_nat @ X @ B2 )
=> ( ( ( insert_list_nat2 @ X @ A2 )
= ( insert_list_nat2 @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_334_insert__ident,axiom,
! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ X @ A2 )
=> ( ~ ( member_set_nat @ X @ B2 )
=> ( ( ( insert_set_nat2 @ X @ A2 )
= ( insert_set_nat2 @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_335_insert__ident,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat2 @ X @ A2 )
= ( insert_nat2 @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_336_Set_Oset__insert,axiom,
! [X: list_nat,A2: set_list_nat] :
( ( member_list_nat @ X @ A2 )
=> ~ ! [B6: set_list_nat] :
( ( A2
= ( insert_list_nat2 @ X @ B6 ) )
=> ( member_list_nat @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_337_Set_Oset__insert,axiom,
! [X: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X @ A2 )
=> ~ ! [B6: set_set_nat] :
( ( A2
= ( insert_set_nat2 @ X @ B6 ) )
=> ( member_set_nat @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_338_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B6: set_nat] :
( ( A2
= ( insert_nat2 @ X @ B6 ) )
=> ( member_nat @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_339_insertI2,axiom,
! [A: list_nat,B2: set_list_nat,B: list_nat] :
( ( member_list_nat @ A @ B2 )
=> ( member_list_nat @ A @ ( insert_list_nat2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_340_insertI2,axiom,
! [A: set_nat,B2: set_set_nat,B: set_nat] :
( ( member_set_nat @ A @ B2 )
=> ( member_set_nat @ A @ ( insert_set_nat2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_341_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat @ A @ B2 )
=> ( member_nat @ A @ ( insert_nat2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_342_insertI1,axiom,
! [A: list_nat,B2: set_list_nat] : ( member_list_nat @ A @ ( insert_list_nat2 @ A @ B2 ) ) ).
% insertI1
thf(fact_343_insertI1,axiom,
! [A: set_nat,B2: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat2 @ A @ B2 ) ) ).
% insertI1
thf(fact_344_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat2 @ A @ B2 ) ) ).
% insertI1
thf(fact_345_insertE,axiom,
! [A: list_nat,B: list_nat,A2: set_list_nat] :
( ( member_list_nat @ A @ ( insert_list_nat2 @ B @ A2 ) )
=> ( ( A != B )
=> ( member_list_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_346_insertE,axiom,
! [A: set_nat,B: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ ( insert_set_nat2 @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_347_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat2 @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_348_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_349_List_Oinsert__def,axiom,
( insert_set_nat
= ( ^ [X3: set_nat,Xs4: list_set_nat] : ( if_list_set_nat @ ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs4 ) ) @ Xs4 @ ( cons_set_nat @ X3 @ Xs4 ) ) ) ) ).
% List.insert_def
thf(fact_350_List_Oinsert__def,axiom,
( insert_list_nat
= ( ^ [X3: list_nat,Xs4: list_list_nat] : ( if_list_list_nat @ ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs4 ) ) @ Xs4 @ ( cons_list_nat @ X3 @ Xs4 ) ) ) ) ).
% List.insert_def
thf(fact_351_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X3: nat,Xs4: list_nat] : ( if_list_nat @ ( member_nat @ X3 @ ( set_nat2 @ Xs4 ) ) @ Xs4 @ ( cons_nat @ X3 @ Xs4 ) ) ) ) ).
% List.insert_def
thf(fact_352_prefixes__snoc,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( prefixes_list_nat @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( append_list_list_nat @ ( prefixes_list_nat @ Xs2 ) @ ( cons_list_list_nat @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) ) @ nil_list_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_353_prefixes__snoc,axiom,
! [Xs2: list_nat,X: nat] :
( ( prefixes_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
= ( append_list_nat @ ( prefixes_nat @ Xs2 ) @ ( cons_list_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) @ nil_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_354_insert__subsetI,axiom,
! [X: list_nat,A2: set_list_nat,X5: set_list_nat] :
( ( member_list_nat @ X @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ X5 @ A2 )
=> ( ord_le6045566169113846134st_nat @ ( insert_list_nat2 @ X @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_355_insert__subsetI,axiom,
! [X: set_nat,A2: set_set_nat,X5: set_set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X5 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( insert_set_nat2 @ X @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_356_insert__subsetI,axiom,
! [X: nat,A2: set_nat,X5: set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ X5 @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_357_prefixes__eq__snoc,axiom,
! [Ys: list_list_nat,Xs2: list_list_list_nat,X: list_list_nat] :
( ( ( prefixes_list_nat @ Ys )
= ( append_list_list_nat @ Xs2 @ ( cons_list_list_nat @ X @ nil_list_list_nat ) ) )
= ( ( ( ( Ys = nil_list_nat )
& ( Xs2 = nil_list_list_nat ) )
| ? [Z4: list_nat,Zs3: list_list_nat] :
( ( Ys
= ( append_list_nat @ Zs3 @ ( cons_list_nat @ Z4 @ nil_list_nat ) ) )
& ( Xs2
= ( prefixes_list_nat @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_358_prefixes__eq__snoc,axiom,
! [Ys: list_nat,Xs2: list_list_nat,X: list_nat] :
( ( ( prefixes_nat @ Ys )
= ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs2 = nil_list_nat ) )
| ? [Z4: nat,Zs3: list_nat] :
( ( Ys
= ( append_nat @ Zs3 @ ( cons_nat @ Z4 @ nil_nat ) ) )
& ( Xs2
= ( prefixes_nat @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_359_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_360_sublists_Osimps_I1_J,axiom,
( ( sublists_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% sublists.simps(1)
thf(fact_361_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_362_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_363_suffixes__eq__snoc,axiom,
! [Ys: list_list_nat,Xs2: list_list_list_nat,X: list_list_nat] :
( ( ( suffixes_list_nat @ Ys )
= ( append_list_list_nat @ Xs2 @ ( cons_list_list_nat @ X @ nil_list_list_nat ) ) )
= ( ( ( ( Ys = nil_list_nat )
& ( Xs2 = nil_list_list_nat ) )
| ? [Z4: list_nat,Zs3: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z4 @ Zs3 ) )
& ( Xs2
= ( suffixes_list_nat @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_364_suffixes__eq__snoc,axiom,
! [Ys: list_nat,Xs2: list_list_nat,X: list_nat] :
( ( ( suffixes_nat @ Ys )
= ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs2 = nil_list_nat ) )
| ? [Z4: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z4 @ Zs3 ) )
& ( Xs2
= ( suffixes_nat @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_365_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_366_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_367_maps__simps_I1_J,axiom,
! [F: nat > list_list_nat,X: nat,Xs2: list_nat] :
( ( maps_nat_list_nat @ F @ ( cons_nat @ X @ Xs2 ) )
= ( append_list_nat @ ( F @ X ) @ ( maps_nat_list_nat @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_368_maps__simps_I1_J,axiom,
! [F: list_nat > list_nat,X: list_nat,Xs2: list_list_nat] :
( ( maps_list_nat_nat @ F @ ( cons_list_nat @ X @ Xs2 ) )
= ( append_nat @ ( F @ X ) @ ( maps_list_nat_nat @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_369_maps__simps_I1_J,axiom,
! [F: list_nat > list_list_nat,X: list_nat,Xs2: list_list_nat] :
( ( maps_l5785965478274863235st_nat @ F @ ( cons_list_nat @ X @ Xs2 ) )
= ( append_list_nat @ ( F @ X ) @ ( maps_l5785965478274863235st_nat @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_370_maps__simps_I1_J,axiom,
! [F: nat > list_nat,X: nat,Xs2: list_nat] :
( ( maps_nat_nat @ F @ ( cons_nat @ X @ Xs2 ) )
= ( append_nat @ ( F @ X ) @ ( maps_nat_nat @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_371_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_372_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_373_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_374_prefix__bot_Omax__bot,axiom,
! [X: list_nat] :
( ( max_list_nat @ prefix_nat @ nil_nat @ X )
= X ) ).
% prefix_bot.max_bot
thf(fact_375_in__set__prefixes,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( member_list_nat @ Xs2 @ ( set_list_nat2 @ ( prefixes_nat @ Ys ) ) )
= ( prefix_nat @ Xs2 @ Ys ) ) ).
% in_set_prefixes
thf(fact_376_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_377_prefix__bot_Omax__bot2,axiom,
! [X: list_nat] :
( ( max_list_nat @ prefix_nat @ X @ nil_nat )
= X ) ).
% prefix_bot.max_bot2
thf(fact_378_subseqs__refl,axiom,
! [Xs2: list_nat] : ( member_list_nat @ Xs2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) ) ).
% subseqs_refl
thf(fact_379_ord_Omax__def,axiom,
( max_list_nat
= ( ^ [Less_eq: list_nat > list_nat > $o,A4: list_nat,B4: list_nat] : ( if_list_nat @ ( Less_eq @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% ord.max_def
thf(fact_380_ord_Omax_Ocong,axiom,
max_list_nat = max_list_nat ).
% ord.max.cong
thf(fact_381_prefixes__not__Nil,axiom,
! [Xs2: list_nat] :
( ( prefixes_nat @ Xs2 )
!= nil_list_nat ) ).
% prefixes_not_Nil
thf(fact_382_suffixes__not__Nil,axiom,
! [Xs2: list_nat] :
( ( suffixes_nat @ Xs2 )
!= nil_list_nat ) ).
% suffixes_not_Nil
thf(fact_383_Cons__in__subseqsD,axiom,
! [Y: list_nat,Ys: list_list_nat,Xs2: list_list_nat] :
( ( member_list_list_nat @ ( cons_list_nat @ Y @ Ys ) @ ( set_list_list_nat2 @ ( subseqs_list_nat @ Xs2 ) ) )
=> ( member_list_list_nat @ Ys @ ( set_list_list_nat2 @ ( subseqs_list_nat @ Xs2 ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_384_Cons__in__subseqsD,axiom,
! [Y: nat,Ys: list_nat,Xs2: list_nat] :
( ( member_list_nat @ ( cons_nat @ Y @ Ys ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) )
=> ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_385_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_386_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_387_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_388_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_389_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_390_subset__code_I2_J,axiom,
! [A2: set_set_nat,Ys: list_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( coset_set_nat @ Ys ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( set_set_nat2 @ Ys ) )
=> ~ ( member_set_nat @ X3 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_391_subset__code_I2_J,axiom,
! [A2: set_list_nat,Ys: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ ( coset_list_nat @ Ys ) )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Ys ) )
=> ~ ( member_list_nat @ X3 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_392_subset__code_I2_J,axiom,
! [A2: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( coset_nat @ Ys ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_393_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_394_prefixes_Osimps_I1_J,axiom,
( ( prefixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_395_suffixes_Osimps_I2_J,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( suffixes_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= ( append_list_list_nat @ ( suffixes_list_nat @ Xs2 ) @ ( cons_list_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ nil_list_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_396_suffixes_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( suffixes_nat @ ( cons_nat @ X @ Xs2 ) )
= ( append_list_nat @ ( suffixes_nat @ Xs2 ) @ ( cons_list_nat @ ( cons_nat @ X @ Xs2 ) @ nil_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_397_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_398_suffixes_Osimps_I1_J,axiom,
( ( suffixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_399_concat__eq__append__conv,axiom,
! [Xss2: list_list_list_nat,Ys: list_list_nat,Zs2: list_list_nat] :
( ( ( concat_list_nat @ Xss2 )
= ( append_list_nat @ Ys @ Zs2 ) )
= ( ( ( Xss2 = nil_list_list_nat )
=> ( ( Ys = nil_list_nat )
& ( Zs2 = nil_list_nat ) ) )
& ( ( Xss2 != nil_list_list_nat )
=> ? [Xss1: list_list_list_nat,Xs4: 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 @ Xs4 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_list_nat @ ( concat_list_nat @ Xss1 ) @ Xs4 ) )
& ( Zs2
= ( append_list_nat @ Xs5 @ ( concat_list_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_400_concat__eq__append__conv,axiom,
! [Xss2: list_list_nat,Ys: list_nat,Zs2: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys @ Zs2 ) )
= ( ( ( Xss2 = nil_list_nat )
=> ( ( Ys = nil_nat )
& ( Zs2 = nil_nat ) ) )
& ( ( Xss2 != nil_list_nat )
=> ? [Xss1: list_list_nat,Xs4: list_nat,Xs5: list_nat,Xss22: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs4 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs4 ) )
& ( Zs2
= ( append_nat @ Xs5 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_401_remove__code_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( remove_nat @ X @ ( coset_nat @ Xs2 ) )
= ( coset_nat @ ( insert_nat @ X @ Xs2 ) ) ) ).
% remove_code(2)
thf(fact_402_concat__eq__appendD,axiom,
! [Xss2: list_list_list_nat,Ys: list_list_nat,Zs2: list_list_nat] :
( ( ( concat_list_nat @ Xss2 )
= ( append_list_nat @ Ys @ Zs2 ) )
=> ( ( Xss2 != nil_list_list_nat )
=> ? [Xss12: list_list_list_nat,Xs: 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 @ Xs @ Xs6 ) @ Xss23 ) ) )
& ( Ys
= ( append_list_nat @ ( concat_list_nat @ Xss12 ) @ Xs ) )
& ( Zs2
= ( append_list_nat @ Xs6 @ ( concat_list_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_403_concat__eq__appendD,axiom,
! [Xss2: list_list_nat,Ys: list_nat,Zs2: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys @ Zs2 ) )
=> ( ( Xss2 != nil_list_nat )
=> ? [Xss12: list_list_nat,Xs: list_nat,Xs6: list_nat,Xss23: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs @ Xs6 ) @ Xss23 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss12 ) @ Xs ) )
& ( Zs2
= ( append_nat @ Xs6 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_404_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_405_prefix__bot_Omin__bot2,axiom,
! [X: list_nat] :
( ( min_list_nat @ prefix_nat @ X @ nil_nat )
= nil_nat ) ).
% prefix_bot.min_bot2
thf(fact_406_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_407_prefix__bot_Omin__bot,axiom,
! [X: list_nat] :
( ( min_list_nat @ prefix_nat @ nil_nat @ X )
= nil_nat ) ).
% prefix_bot.min_bot
thf(fact_408_list__ex1__simps_I1_J,axiom,
! [P: list_nat > $o] :
~ ( list_ex1_list_nat @ P @ nil_list_nat ) ).
% list_ex1_simps(1)
thf(fact_409_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_410_butlast__snoc,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( butlast_list_nat @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= Xs2 ) ).
% butlast_snoc
thf(fact_411_butlast__snoc,axiom,
! [Xs2: list_nat,X: nat] :
( ( butlast_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
= Xs2 ) ).
% butlast_snoc
thf(fact_412_sublists_Osimps_I2_J,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( sublists_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= ( append_list_list_nat @ ( sublists_list_nat @ Xs2 ) @ ( map_li2855073862107769254st_nat @ ( cons_list_nat @ X ) @ ( prefixes_list_nat @ Xs2 ) ) ) ) ).
% sublists.simps(2)
thf(fact_413_sublists_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( sublists_nat @ ( cons_nat @ X @ Xs2 ) )
= ( append_list_nat @ ( sublists_nat @ Xs2 ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs2 ) ) ) ) ).
% sublists.simps(2)
thf(fact_414_member__remove,axiom,
! [X: list_nat,Y: list_nat,A2: set_list_nat] :
( ( member_list_nat @ X @ ( remove_list_nat @ Y @ A2 ) )
= ( ( member_list_nat @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_415_member__remove,axiom,
! [X: set_nat,Y: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X @ ( remove_set_nat @ Y @ A2 ) )
= ( ( member_set_nat @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_416_member__remove,axiom,
! [X: nat,Y: nat,A2: set_nat] :
( ( member_nat @ X @ ( remove_nat @ Y @ A2 ) )
= ( ( member_nat @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_417_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_418_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_419_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_420_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_421_Nil__is__map__conv,axiom,
! [F: list_nat > nat,Xs2: list_list_nat] :
( ( nil_nat
= ( map_list_nat_nat @ F @ Xs2 ) )
= ( Xs2 = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_422_Nil__is__map__conv,axiom,
! [F: nat > list_nat,Xs2: list_nat] :
( ( nil_list_nat
= ( map_nat_list_nat @ F @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_423_Nil__is__map__conv,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat] :
( ( nil_list_nat
= ( map_li7225945977422193158st_nat @ F @ Xs2 ) )
= ( Xs2 = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_424_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_425_map__is__Nil__conv,axiom,
! [F: list_nat > nat,Xs2: list_list_nat] :
( ( ( map_list_nat_nat @ F @ Xs2 )
= nil_nat )
= ( Xs2 = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_426_map__is__Nil__conv,axiom,
! [F: nat > list_nat,Xs2: list_nat] :
( ( ( map_nat_list_nat @ F @ Xs2 )
= nil_list_nat )
= ( Xs2 = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_427_map__is__Nil__conv,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs2 )
= nil_list_nat )
= ( Xs2 = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_428_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_429_map__eq__conv,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Xs2 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_430_map__eq__conv,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat,G: list_nat > list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs2 )
= ( map_li7225945977422193158st_nat @ G @ Xs2 ) )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_431_map__append,axiom,
! [F: nat > list_nat,Xs2: list_nat,Ys: list_nat] :
( ( map_nat_list_nat @ F @ ( append_nat @ Xs2 @ Ys ) )
= ( append_list_nat @ ( map_nat_list_nat @ F @ Xs2 ) @ ( map_nat_list_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_432_map__append,axiom,
! [F: list_nat > nat,Xs2: list_list_nat,Ys: list_list_nat] :
( ( map_list_nat_nat @ F @ ( append_list_nat @ Xs2 @ Ys ) )
= ( append_nat @ ( map_list_nat_nat @ F @ Xs2 ) @ ( map_list_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_433_map__append,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat,Ys: list_list_nat] :
( ( map_li7225945977422193158st_nat @ F @ ( append_list_nat @ Xs2 @ Ys ) )
= ( append_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) @ ( map_li7225945977422193158st_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_434_map__append,axiom,
! [F: nat > nat,Xs2: list_nat,Ys: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs2 @ Ys ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_435_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_list_nat] :
( ( ( concat_list_nat @ Xss2 )
= nil_list_nat )
= ( ! [X3: list_list_nat] :
( ( member_list_list_nat @ X3 @ ( set_list_list_nat2 @ Xss2 ) )
=> ( X3 = nil_list_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_436_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_nat] :
( ( ( concat_nat @ Xss2 )
= nil_nat )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xss2 ) )
=> ( X3 = nil_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_437_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_list_nat] :
( ( nil_list_nat
= ( concat_list_nat @ Xss2 ) )
= ( ! [X3: list_list_nat] :
( ( member_list_list_nat @ X3 @ ( set_list_list_nat2 @ Xss2 ) )
=> ( X3 = nil_list_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_438_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_nat] :
( ( nil_nat
= ( concat_nat @ Xss2 ) )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xss2 ) )
=> ( X3 = nil_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_439_concat__append,axiom,
! [Xs2: list_list_list_nat,Ys: list_list_list_nat] :
( ( concat_list_nat @ ( append_list_list_nat @ Xs2 @ Ys ) )
= ( append_list_nat @ ( concat_list_nat @ Xs2 ) @ ( concat_list_nat @ Ys ) ) ) ).
% concat_append
thf(fact_440_concat__append,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat] :
( ( concat_nat @ ( append_list_nat @ Xs2 @ Ys ) )
= ( append_nat @ ( concat_nat @ Xs2 ) @ ( concat_nat @ Ys ) ) ) ).
% concat_append
thf(fact_441_List_Obind__def,axiom,
( bind_nat_nat
= ( ^ [Xs4: list_nat,F2: nat > list_nat] : ( concat_nat @ ( map_nat_list_nat @ F2 @ Xs4 ) ) ) ) ).
% List.bind_def
thf(fact_442_List_Obind__def,axiom,
( bind_list_nat_nat
= ( ^ [Xs4: list_list_nat,F2: list_nat > list_nat] : ( concat_nat @ ( map_li7225945977422193158st_nat @ F2 @ Xs4 ) ) ) ) ).
% List.bind_def
thf(fact_443_map__concat,axiom,
! [F: list_nat > list_nat,Xs2: list_list_list_nat] :
( ( map_li7225945977422193158st_nat @ F @ ( concat_list_nat @ Xs2 ) )
= ( concat_list_nat @ ( map_li2855073862107769254st_nat @ ( map_li7225945977422193158st_nat @ F ) @ Xs2 ) ) ) ).
% map_concat
thf(fact_444_map__concat,axiom,
! [F: nat > nat,Xs2: list_list_nat] :
( ( map_nat_nat @ F @ ( concat_nat @ Xs2 ) )
= ( concat_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs2 ) ) ) ).
% map_concat
thf(fact_445_map__butlast,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat] :
( ( map_li7225945977422193158st_nat @ F @ ( butlast_list_nat @ Xs2 ) )
= ( butlast_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) ) ) ).
% map_butlast
thf(fact_446_map__butlast,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( map_nat_nat @ F @ ( butlast_nat @ Xs2 ) )
= ( butlast_nat @ ( map_nat_nat @ F @ Xs2 ) ) ) ).
% map_butlast
thf(fact_447_ord_Omin_Ocong,axiom,
min_list_nat = min_list_nat ).
% ord.min.cong
thf(fact_448_ord_Omin__def,axiom,
( min_list_nat
= ( ^ [Less_eq: list_nat > list_nat > $o,A4: list_nat,B4: list_nat] : ( if_list_nat @ ( Less_eq @ A4 @ B4 ) @ A4 @ B4 ) ) ) ).
% ord.min_def
thf(fact_449_maps__def,axiom,
( maps_nat_nat
= ( ^ [F2: nat > list_nat,Xs4: list_nat] : ( concat_nat @ ( map_nat_list_nat @ F2 @ Xs4 ) ) ) ) ).
% maps_def
thf(fact_450_maps__def,axiom,
( maps_list_nat_nat
= ( ^ [F2: list_nat > list_nat,Xs4: list_list_nat] : ( concat_nat @ ( map_li7225945977422193158st_nat @ F2 @ Xs4 ) ) ) ) ).
% maps_def
thf(fact_451_concat__map__maps,axiom,
! [F: nat > list_nat,Xs2: list_nat] :
( ( concat_nat @ ( map_nat_list_nat @ F @ Xs2 ) )
= ( maps_nat_nat @ F @ Xs2 ) ) ).
% concat_map_maps
thf(fact_452_concat__map__maps,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat] :
( ( concat_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) )
= ( maps_list_nat_nat @ F @ Xs2 ) ) ).
% concat_map_maps
thf(fact_453_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_454_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_455_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_456_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_457_Cons__eq__map__D,axiom,
! [X: nat,Xs2: list_nat,F: list_nat > nat,Ys: list_list_nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( map_list_nat_nat @ F @ Ys ) )
=> ? [Z5: list_nat,Zs: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z5 @ Zs ) )
& ( X
= ( F @ Z5 ) )
& ( Xs2
= ( map_list_nat_nat @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_458_Cons__eq__map__D,axiom,
! [X: list_nat,Xs2: list_list_nat,F: nat > list_nat,Ys: list_nat] :
( ( ( cons_list_nat @ X @ Xs2 )
= ( map_nat_list_nat @ F @ Ys ) )
=> ? [Z5: nat,Zs: list_nat] :
( ( Ys
= ( cons_nat @ Z5 @ Zs ) )
& ( X
= ( F @ Z5 ) )
& ( Xs2
= ( map_nat_list_nat @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_459_Cons__eq__map__D,axiom,
! [X: list_nat,Xs2: list_list_nat,F: list_nat > list_nat,Ys: list_list_nat] :
( ( ( cons_list_nat @ X @ Xs2 )
= ( map_li7225945977422193158st_nat @ F @ Ys ) )
=> ? [Z5: list_nat,Zs: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z5 @ Zs ) )
& ( X
= ( F @ Z5 ) )
& ( Xs2
= ( map_li7225945977422193158st_nat @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_460_Cons__eq__map__D,axiom,
! [X: nat,Xs2: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z5: nat,Zs: list_nat] :
( ( Ys
= ( cons_nat @ Z5 @ Zs ) )
& ( X
= ( F @ Z5 ) )
& ( Xs2
= ( map_nat_nat @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_461_map__eq__Cons__D,axiom,
! [F: list_nat > nat,Xs2: list_list_nat,Y: nat,Ys: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs2 )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z5: list_nat,Zs: list_list_nat] :
( ( Xs2
= ( cons_list_nat @ Z5 @ Zs ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_list_nat_nat @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_462_map__eq__Cons__D,axiom,
! [F: nat > list_nat,Xs2: list_nat,Y: list_nat,Ys: list_list_nat] :
( ( ( map_nat_list_nat @ F @ Xs2 )
= ( cons_list_nat @ Y @ Ys ) )
=> ? [Z5: nat,Zs: list_nat] :
( ( Xs2
= ( cons_nat @ Z5 @ Zs ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_nat_list_nat @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_463_map__eq__Cons__D,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat,Y: list_nat,Ys: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs2 )
= ( cons_list_nat @ Y @ Ys ) )
=> ? [Z5: list_nat,Zs: list_list_nat] :
( ( Xs2
= ( cons_list_nat @ Z5 @ Zs ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_li7225945977422193158st_nat @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_464_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs2: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z5: nat,Zs: list_nat] :
( ( Xs2
= ( cons_nat @ Z5 @ Zs ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_nat_nat @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_465_Cons__eq__map__conv,axiom,
! [X: nat,Xs2: list_nat,F: list_nat > nat,Ys: list_list_nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( map_list_nat_nat @ F @ Ys ) )
= ( ? [Z4: list_nat,Zs3: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z4 @ Zs3 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs2
= ( map_list_nat_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_466_Cons__eq__map__conv,axiom,
! [X: list_nat,Xs2: list_list_nat,F: nat > list_nat,Ys: list_nat] :
( ( ( cons_list_nat @ X @ Xs2 )
= ( map_nat_list_nat @ F @ Ys ) )
= ( ? [Z4: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z4 @ Zs3 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs2
= ( map_nat_list_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_467_Cons__eq__map__conv,axiom,
! [X: list_nat,Xs2: list_list_nat,F: list_nat > list_nat,Ys: list_list_nat] :
( ( ( cons_list_nat @ X @ Xs2 )
= ( map_li7225945977422193158st_nat @ F @ Ys ) )
= ( ? [Z4: list_nat,Zs3: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z4 @ Zs3 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs2
= ( map_li7225945977422193158st_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_468_Cons__eq__map__conv,axiom,
! [X: nat,Xs2: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z4: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z4 @ Zs3 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs2
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_469_map__eq__Cons__conv,axiom,
! [F: list_nat > nat,Xs2: list_list_nat,Y: nat,Ys: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs2 )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z4: list_nat,Zs3: list_list_nat] :
( ( Xs2
= ( cons_list_nat @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_list_nat_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_470_map__eq__Cons__conv,axiom,
! [F: nat > list_nat,Xs2: list_nat,Y: list_nat,Ys: list_list_nat] :
( ( ( map_nat_list_nat @ F @ Xs2 )
= ( cons_list_nat @ Y @ Ys ) )
= ( ? [Z4: nat,Zs3: list_nat] :
( ( Xs2
= ( cons_nat @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_nat_list_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_471_map__eq__Cons__conv,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat,Y: list_nat,Ys: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs2 )
= ( cons_list_nat @ Y @ Ys ) )
= ( ? [Z4: list_nat,Zs3: list_list_nat] :
( ( Xs2
= ( cons_list_nat @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_li7225945977422193158st_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_472_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs2: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z4: nat,Zs3: list_nat] :
( ( Xs2
= ( cons_nat @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_473_list_Osimps_I8_J,axiom,
! [F: nat > list_nat] :
( ( map_nat_list_nat @ F @ nil_nat )
= nil_list_nat ) ).
% list.simps(8)
thf(fact_474_list_Osimps_I8_J,axiom,
! [F: list_nat > nat] :
( ( map_list_nat_nat @ F @ nil_list_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_475_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_476_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_477_list_Omap__cong,axiom,
! [X: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X = Ya )
=> ( ! [Z5: nat] :
( ( member_nat @ Z5 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z5 )
= ( G @ Z5 ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_478_list_Omap__cong,axiom,
! [X: list_list_nat,Ya: list_list_nat,F: list_nat > list_nat,G: list_nat > list_nat] :
( ( X = Ya )
=> ( ! [Z5: list_nat] :
( ( member_list_nat @ Z5 @ ( set_list_nat2 @ Ya ) )
=> ( ( F @ Z5 )
= ( G @ Z5 ) ) )
=> ( ( map_li7225945977422193158st_nat @ F @ X )
= ( map_li7225945977422193158st_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_479_list_Omap__cong0,axiom,
! [X: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z5: nat] :
( ( member_nat @ Z5 @ ( set_nat2 @ X ) )
=> ( ( F @ Z5 )
= ( G @ Z5 ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_480_list_Omap__cong0,axiom,
! [X: list_list_nat,F: list_nat > list_nat,G: list_nat > list_nat] :
( ! [Z5: list_nat] :
( ( member_list_nat @ Z5 @ ( set_list_nat2 @ X ) )
=> ( ( F @ Z5 )
= ( G @ Z5 ) ) )
=> ( ( map_li7225945977422193158st_nat @ F @ X )
= ( map_li7225945977422193158st_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_481_list_Oinj__map__strong,axiom,
! [X: list_nat,Xa2: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z5: nat,Za: nat] :
( ( member_nat @ Z5 @ ( set_nat2 @ X ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa2 ) )
=> ( ( ( F @ Z5 )
= ( Fa @ Za ) )
=> ( Z5 = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_482_list_Oinj__map__strong,axiom,
! [X: list_list_nat,Xa2: list_list_nat,F: list_nat > list_nat,Fa: list_nat > list_nat] :
( ! [Z5: list_nat,Za: list_nat] :
( ( member_list_nat @ Z5 @ ( set_list_nat2 @ X ) )
=> ( ( member_list_nat @ Za @ ( set_list_nat2 @ Xa2 ) )
=> ( ( ( F @ Z5 )
= ( Fa @ Za ) )
=> ( Z5 = Za ) ) ) )
=> ( ( ( map_li7225945977422193158st_nat @ F @ X )
= ( map_li7225945977422193158st_nat @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_483_list_Omap__ident__strong,axiom,
! [T2: list_set_nat,F: set_nat > set_nat] :
( ! [Z5: set_nat] :
( ( member_set_nat @ Z5 @ ( set_set_nat2 @ T2 ) )
=> ( ( F @ Z5 )
= Z5 ) )
=> ( ( map_set_nat_set_nat @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_484_list_Omap__ident__strong,axiom,
! [T2: list_list_nat,F: list_nat > list_nat] :
( ! [Z5: list_nat] :
( ( member_list_nat @ Z5 @ ( set_list_nat2 @ T2 ) )
=> ( ( F @ Z5 )
= Z5 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_485_list_Omap__ident__strong,axiom,
! [T2: list_nat,F: nat > nat] :
( ! [Z5: nat] :
( ( member_nat @ Z5 @ ( set_nat2 @ T2 ) )
=> ( ( F @ Z5 )
= Z5 ) )
=> ( ( map_nat_nat @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_486_map__ext,axiom,
! [Xs2: list_nat,F: nat > nat,G: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_487_map__ext,axiom,
! [Xs2: list_list_nat,F: list_nat > list_nat,G: list_nat > list_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs2 )
= ( map_li7225945977422193158st_nat @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_488_map__idI,axiom,
! [Xs2: list_set_nat,F: set_nat > set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs2 ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_set_nat_set_nat @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_489_map__idI,axiom,
! [Xs2: list_list_nat,F: list_nat > list_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_490_map__idI,axiom,
! [Xs2: list_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_491_map__cong,axiom,
! [Xs2: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs2 = Ys )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_492_map__cong,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat,F: list_nat > list_nat,G: list_nat > list_nat] :
( ( Xs2 = Ys )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Ys ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs2 )
= ( map_li7225945977422193158st_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_493_ex__map__conv,axiom,
! [Ys: list_nat,F: nat > nat] :
( ( ? [Xs4: list_nat] :
( Ys
= ( map_nat_nat @ F @ Xs4 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ? [Y3: nat] :
( X3
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_494_ex__map__conv,axiom,
! [Ys: list_list_nat,F: list_nat > list_nat] :
( ( ? [Xs4: list_list_nat] :
( Ys
= ( map_li7225945977422193158st_nat @ F @ Xs4 ) ) )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Ys ) )
=> ? [Y3: list_nat] :
( X3
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_495_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs2: list_nat,F: list_nat > nat,Xs2: list_list_nat] :
( ( ( append_nat @ Ys @ Zs2 )
= ( map_list_nat_nat @ F @ Xs2 ) )
= ( ? [Us: list_list_nat,Vs: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Us @ Vs ) )
& ( Ys
= ( map_list_nat_nat @ F @ Us ) )
& ( Zs2
= ( map_list_nat_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_496_append__eq__map__conv,axiom,
! [Ys: list_list_nat,Zs2: list_list_nat,F: nat > list_nat,Xs2: list_nat] :
( ( ( append_list_nat @ Ys @ Zs2 )
= ( map_nat_list_nat @ F @ Xs2 ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs2
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_nat_list_nat @ F @ Us ) )
& ( Zs2
= ( map_nat_list_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_497_append__eq__map__conv,axiom,
! [Ys: list_list_nat,Zs2: list_list_nat,F: list_nat > list_nat,Xs2: list_list_nat] :
( ( ( append_list_nat @ Ys @ Zs2 )
= ( map_li7225945977422193158st_nat @ F @ Xs2 ) )
= ( ? [Us: list_list_nat,Vs: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Us @ Vs ) )
& ( Ys
= ( map_li7225945977422193158st_nat @ F @ Us ) )
& ( Zs2
= ( map_li7225945977422193158st_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_498_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs2: list_nat,F: nat > nat,Xs2: list_nat] :
( ( ( append_nat @ Ys @ Zs2 )
= ( map_nat_nat @ F @ Xs2 ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs2
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_nat_nat @ F @ Us ) )
& ( Zs2
= ( map_nat_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_499_map__eq__append__conv,axiom,
! [F: list_nat > nat,Xs2: list_list_nat,Ys: list_nat,Zs2: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs2 )
= ( append_nat @ Ys @ Zs2 ) )
= ( ? [Us: list_list_nat,Vs: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Us @ Vs ) )
& ( Ys
= ( map_list_nat_nat @ F @ Us ) )
& ( Zs2
= ( map_list_nat_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_500_map__eq__append__conv,axiom,
! [F: nat > list_nat,Xs2: list_nat,Ys: list_list_nat,Zs2: list_list_nat] :
( ( ( map_nat_list_nat @ F @ Xs2 )
= ( append_list_nat @ Ys @ Zs2 ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs2
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_nat_list_nat @ F @ Us ) )
& ( Zs2
= ( map_nat_list_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_501_map__eq__append__conv,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat,Ys: list_list_nat,Zs2: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs2 )
= ( append_list_nat @ Ys @ Zs2 ) )
= ( ? [Us: list_list_nat,Vs: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Us @ Vs ) )
& ( Ys
= ( map_li7225945977422193158st_nat @ F @ Us ) )
& ( Zs2
= ( map_li7225945977422193158st_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_502_map__eq__append__conv,axiom,
! [F: nat > nat,Xs2: list_nat,Ys: list_nat,Zs2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( append_nat @ Ys @ Zs2 ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs2
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_nat_nat @ F @ Us ) )
& ( Zs2
= ( map_nat_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_503_prefix__map__rightE,axiom,
! [Xs2: list_list_nat,F: list_nat > list_nat,Ys: list_list_nat] :
( ( prefix_list_nat @ Xs2 @ ( map_li7225945977422193158st_nat @ F @ Ys ) )
=> ? [Xs6: list_list_nat] :
( ( prefix_list_nat @ Xs6 @ Ys )
& ( Xs2
= ( map_li7225945977422193158st_nat @ F @ Xs6 ) ) ) ) ).
% prefix_map_rightE
thf(fact_504_prefix__map__rightE,axiom,
! [Xs2: list_nat,F: nat > nat,Ys: list_nat] :
( ( prefix_nat @ Xs2 @ ( map_nat_nat @ F @ Ys ) )
=> ? [Xs6: list_nat] :
( ( prefix_nat @ Xs6 @ Ys )
& ( Xs2
= ( map_nat_nat @ F @ Xs6 ) ) ) ) ).
% prefix_map_rightE
thf(fact_505_map__mono__prefix,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat,F: list_nat > list_nat] :
( ( prefix_list_nat @ Xs2 @ Ys )
=> ( prefix_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) @ ( map_li7225945977422193158st_nat @ F @ Ys ) ) ) ).
% map_mono_prefix
thf(fact_506_map__mono__prefix,axiom,
! [Xs2: list_nat,Ys: list_nat,F: nat > nat] :
( ( prefix_nat @ Xs2 @ Ys )
=> ( prefix_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_mono_prefix
thf(fact_507_butlast_Osimps_I1_J,axiom,
( ( butlast_list_nat @ nil_list_nat )
= nil_list_nat ) ).
% butlast.simps(1)
thf(fact_508_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_509_in__set__butlastD,axiom,
! [X: set_nat,Xs2: list_set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ ( butlast_set_nat @ Xs2 ) ) )
=> ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) ) ) ).
% in_set_butlastD
thf(fact_510_in__set__butlastD,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ ( butlast_list_nat @ Xs2 ) ) )
=> ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) ) ) ).
% in_set_butlastD
thf(fact_511_in__set__butlastD,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs2 ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).
% in_set_butlastD
thf(fact_512_prefixeq__butlast,axiom,
! [Xs2: list_nat] : ( prefix_nat @ ( butlast_nat @ Xs2 ) @ Xs2 ) ).
% prefixeq_butlast
thf(fact_513_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_514_concat_Osimps_I2_J,axiom,
! [X: list_list_nat,Xs2: list_list_list_nat] :
( ( concat_list_nat @ ( cons_list_list_nat @ X @ Xs2 ) )
= ( append_list_nat @ X @ ( concat_list_nat @ Xs2 ) ) ) ).
% concat.simps(2)
thf(fact_515_concat_Osimps_I2_J,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( concat_nat @ ( cons_list_nat @ X @ Xs2 ) )
= ( append_nat @ X @ ( concat_nat @ Xs2 ) ) ) ).
% concat.simps(2)
thf(fact_516_concat_Osimps_I1_J,axiom,
( ( concat_list_nat @ nil_list_list_nat )
= nil_list_nat ) ).
% concat.simps(1)
thf(fact_517_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_518_butlast_Osimps_I2_J,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( ( Xs2 = nil_list_nat )
=> ( ( butlast_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= nil_list_nat ) )
& ( ( Xs2 != nil_list_nat )
=> ( ( butlast_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= ( cons_list_nat @ X @ ( butlast_list_nat @ Xs2 ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_519_butlast_Osimps_I2_J,axiom,
! [Xs2: list_nat,X: nat] :
( ( ( Xs2 = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs2 ) )
= nil_nat ) )
& ( ( Xs2 != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ X @ ( butlast_nat @ Xs2 ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_520_butlast__append,axiom,
! [Ys: list_list_nat,Xs2: list_list_nat] :
( ( ( Ys = nil_list_nat )
=> ( ( butlast_list_nat @ ( append_list_nat @ Xs2 @ Ys ) )
= ( butlast_list_nat @ Xs2 ) ) )
& ( ( Ys != nil_list_nat )
=> ( ( butlast_list_nat @ ( append_list_nat @ Xs2 @ Ys ) )
= ( append_list_nat @ Xs2 @ ( butlast_list_nat @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_521_butlast__append,axiom,
! [Ys: list_nat,Xs2: list_nat] :
( ( ( Ys = nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs2 @ Ys ) )
= ( butlast_nat @ Xs2 ) ) )
& ( ( Ys != nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs2 @ Ys ) )
= ( append_nat @ Xs2 @ ( butlast_nat @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_522_in__set__butlast__appendI,axiom,
! [X: set_nat,Xs2: list_set_nat,Ys: list_set_nat] :
( ( ( member_set_nat @ X @ ( set_set_nat2 @ ( butlast_set_nat @ Xs2 ) ) )
| ( member_set_nat @ X @ ( set_set_nat2 @ ( butlast_set_nat @ Ys ) ) ) )
=> ( member_set_nat @ X @ ( set_set_nat2 @ ( butlast_set_nat @ ( append_set_nat @ Xs2 @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_523_in__set__butlast__appendI,axiom,
! [X: list_nat,Xs2: list_list_nat,Ys: list_list_nat] :
( ( ( member_list_nat @ X @ ( set_list_nat2 @ ( butlast_list_nat @ Xs2 ) ) )
| ( member_list_nat @ X @ ( set_list_nat2 @ ( butlast_list_nat @ Ys ) ) ) )
=> ( member_list_nat @ X @ ( set_list_nat2 @ ( butlast_list_nat @ ( append_list_nat @ Xs2 @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_524_in__set__butlast__appendI,axiom,
! [X: nat,Xs2: list_nat,Ys: list_nat] :
( ( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs2 ) ) )
| ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Ys ) ) ) )
=> ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs2 @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_525_list__ex1__iff,axiom,
( list_ex1_set_nat
= ( ^ [P2: set_nat > $o,Xs4: list_set_nat] :
? [X3: set_nat] :
( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs4 ) )
& ( P2 @ X3 )
& ! [Y3: set_nat] :
( ( ( member_set_nat @ Y3 @ ( set_set_nat2 @ Xs4 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_526_list__ex1__iff,axiom,
( list_ex1_list_nat
= ( ^ [P2: list_nat > $o,Xs4: list_list_nat] :
? [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs4 ) )
& ( P2 @ X3 )
& ! [Y3: list_nat] :
( ( ( member_list_nat @ Y3 @ ( set_list_nat2 @ Xs4 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_527_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P2: nat > $o,Xs4: list_nat] :
? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs4 ) )
& ( P2 @ X3 )
& ! [Y3: nat] :
( ( ( member_nat @ Y3 @ ( set_nat2 @ Xs4 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_528_prefixes_Osimps_I2_J,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( prefixes_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= ( cons_list_list_nat @ nil_list_nat @ ( map_li2855073862107769254st_nat @ ( cons_list_nat @ X ) @ ( prefixes_list_nat @ Xs2 ) ) ) ) ).
% prefixes.simps(2)
thf(fact_529_prefixes_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( prefixes_nat @ ( cons_nat @ X @ Xs2 ) )
= ( cons_list_nat @ nil_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs2 ) ) ) ) ).
% prefixes.simps(2)
thf(fact_530_can__select__set__list__ex1,axiom,
! [P: list_nat > $o,A2: list_list_nat] :
( ( can_select_list_nat @ P @ ( set_list_nat2 @ A2 ) )
= ( list_ex1_list_nat @ P @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_531_can__select__set__list__ex1,axiom,
! [P: nat > $o,A2: list_nat] :
( ( can_select_nat @ P @ ( set_nat2 @ A2 ) )
= ( list_ex1_nat @ P @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_532_append__butlast__last__id,axiom,
! [Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( append_list_nat @ ( butlast_list_nat @ Xs2 ) @ ( cons_list_nat @ ( last_list_nat @ Xs2 ) @ nil_list_nat ) )
= Xs2 ) ) ).
% append_butlast_last_id
thf(fact_533_append__butlast__last__id,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( append_nat @ ( butlast_nat @ Xs2 ) @ ( cons_nat @ ( last_nat @ Xs2 ) @ nil_nat ) )
= Xs2 ) ) ).
% append_butlast_last_id
thf(fact_534_map__eq__map__tailrec,axiom,
map_li7225945977422193158st_nat = map_ta9145449198693458768st_nat ).
% map_eq_map_tailrec
thf(fact_535_map__eq__map__tailrec,axiom,
map_nat_nat = map_tailrec_nat_nat ).
% map_eq_map_tailrec
thf(fact_536_snoc__eq__iff__butlast,axiom,
! [Xs2: list_list_nat,X: list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) )
= Ys )
= ( ( Ys != nil_list_nat )
& ( ( butlast_list_nat @ Ys )
= Xs2 )
& ( ( last_list_nat @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_537_snoc__eq__iff__butlast,axiom,
! [Xs2: list_nat,X: nat,Ys: list_nat] :
( ( ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) )
= Ys )
= ( ( Ys != nil_nat )
& ( ( butlast_nat @ Ys )
= Xs2 )
& ( ( last_nat @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_538_subset__subseqs,axiom,
! [X5: set_list_nat,Xs2: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ X5 @ ( set_list_nat2 @ Xs2 ) )
=> ( member_set_list_nat @ X5 @ ( image_4375647060027540749st_nat @ set_list_nat2 @ ( set_list_list_nat2 @ ( subseqs_list_nat @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_539_subset__subseqs,axiom,
! [X5: set_nat,Xs2: list_nat] :
( ( ord_less_eq_set_nat @ X5 @ ( set_nat2 @ Xs2 ) )
=> ( member_set_nat @ X5 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_540_last__snoc,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( last_list_nat @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= X ) ).
% last_snoc
thf(fact_541_last__snoc,axiom,
! [Xs2: list_nat,X: nat] :
( ( last_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% last_snoc
thf(fact_542__092_060open_062_123_O_O_060max_A_Ix_A_L_A1_J_A_Irgf__limit_Axs_J_125_A_061_A_123_O_O_060rgf__limit_A_Ixs_A_064_A_091x_093_J_125_092_060close_062,axiom,
( ( set_ord_lessThan_nat @ ( ord_max_nat @ ( plus_plus_nat @ x @ one_one_nat ) @ ( equiva5889994315859557365_limit @ xsa ) ) )
= ( set_ord_lessThan_nat @ ( equiva5889994315859557365_limit @ ( append_nat @ xsa @ ( cons_nat @ x @ nil_nat ) ) ) ) ) ).
% \<open>{..<max (x + 1) (rgf_limit xs)} = {..<rgf_limit (xs @ [x])}\<close>
thf(fact_543_image__eqI,axiom,
! [B: list_nat,F: nat > list_nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_list_nat @ B @ ( image_nat_list_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_544_image__eqI,axiom,
! [B: set_nat,F: nat > set_nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_545_image__eqI,axiom,
! [B: nat,F: list_nat > nat,X: list_nat,A2: set_list_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_list_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_list_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_546_image__eqI,axiom,
! [B: list_nat,F: list_nat > list_nat,X: list_nat,A2: set_list_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_list_nat @ X @ A2 )
=> ( member_list_nat @ B @ ( image_7976474329151083847st_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_547_image__eqI,axiom,
! [B: set_nat,F: list_nat > set_nat,X: list_nat,A2: set_list_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_list_nat @ X @ A2 )
=> ( member_set_nat @ B @ ( image_1775855109352712557et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_548_image__eqI,axiom,
! [B: nat,F: set_nat > nat,X: set_nat,A2: set_set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_set_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_set_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_549_image__eqI,axiom,
! [B: list_nat,F: set_nat > list_nat,X: set_nat,A2: set_set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_set_nat @ X @ A2 )
=> ( member_list_nat @ B @ ( image_5426082062715393517st_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_550_image__eqI,axiom,
! [B: set_nat,F: set_nat > set_nat,X: set_nat,A2: set_set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_551_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_552_last__prefixes,axiom,
! [Xs2: list_nat] :
( ( last_list_nat @ ( prefixes_nat @ Xs2 ) )
= Xs2 ) ).
% last_prefixes
thf(fact_553_last__suffixes,axiom,
! [Xs2: list_nat] :
( ( last_list_nat @ ( suffixes_nat @ Xs2 ) )
= Xs2 ) ).
% last_suffixes
thf(fact_554_image__insert,axiom,
! [F: list_nat > set_nat,A: list_nat,B2: set_list_nat] :
( ( image_1775855109352712557et_nat @ F @ ( insert_list_nat2 @ A @ B2 ) )
= ( insert_set_nat2 @ ( F @ A ) @ ( image_1775855109352712557et_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_555_image__insert,axiom,
! [F: nat > set_nat,A: nat,B2: set_nat] :
( ( image_nat_set_nat @ F @ ( insert_nat2 @ A @ B2 ) )
= ( insert_set_nat2 @ ( F @ A ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_556_image__insert,axiom,
! [F: set_nat > nat,A: set_nat,B2: set_set_nat] :
( ( image_set_nat_nat @ F @ ( insert_set_nat2 @ A @ B2 ) )
= ( insert_nat2 @ ( F @ A ) @ ( image_set_nat_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_557_image__insert,axiom,
! [F: set_nat > set_nat,A: set_nat,B2: set_set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( insert_set_nat2 @ A @ B2 ) )
= ( insert_set_nat2 @ ( F @ A ) @ ( image_7916887816326733075et_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_558_image__insert,axiom,
! [F: nat > nat,A: nat,B2: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat2 @ A @ B2 ) )
= ( insert_nat2 @ ( F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_559_insert__image,axiom,
! [X: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( insert_set_nat2 @ ( F @ X ) @ ( image_nat_set_nat @ F @ A2 ) )
= ( image_nat_set_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_560_insert__image,axiom,
! [X: list_nat,A2: set_list_nat,F: list_nat > nat] :
( ( member_list_nat @ X @ A2 )
=> ( ( insert_nat2 @ ( F @ X ) @ ( image_list_nat_nat @ F @ A2 ) )
= ( image_list_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_561_insert__image,axiom,
! [X: list_nat,A2: set_list_nat,F: list_nat > set_nat] :
( ( member_list_nat @ X @ A2 )
=> ( ( insert_set_nat2 @ ( F @ X ) @ ( image_1775855109352712557et_nat @ F @ A2 ) )
= ( image_1775855109352712557et_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_562_insert__image,axiom,
! [X: set_nat,A2: set_set_nat,F: set_nat > nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( insert_nat2 @ ( F @ X ) @ ( image_set_nat_nat @ F @ A2 ) )
= ( image_set_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_563_insert__image,axiom,
! [X: set_nat,A2: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( insert_set_nat2 @ ( F @ X ) @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_564_insert__image,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( insert_nat2 @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) )
= ( image_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_565__092_060open_062insert_Ax_A_123_O_O_060rgf__limit_Axs_125_A_061_A_123_O_O_060max_A_Ix_A_L_A1_J_A_Irgf__limit_Axs_J_125_092_060close_062,axiom,
( ( insert_nat2 @ x @ ( set_ord_lessThan_nat @ ( equiva5889994315859557365_limit @ xsa ) ) )
= ( set_ord_lessThan_nat @ ( ord_max_nat @ ( plus_plus_nat @ x @ one_one_nat ) @ ( equiva5889994315859557365_limit @ xsa ) ) ) ) ).
% \<open>insert x {..<rgf_limit xs} = {..<max (x + 1) (rgf_limit xs)}\<close>
thf(fact_566_list_Oset__map,axiom,
! [F: set_nat > set_nat,V: list_set_nat] :
( ( set_set_nat2 @ ( map_set_nat_set_nat @ F @ V ) )
= ( image_7916887816326733075et_nat @ F @ ( set_set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_567_list_Oset__map,axiom,
! [F: list_nat > set_nat,V: list_list_nat] :
( ( set_set_nat2 @ ( map_list_nat_set_nat @ F @ V ) )
= ( image_1775855109352712557et_nat @ F @ ( set_list_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_568_list_Oset__map,axiom,
! [F: list_nat > nat,V: list_list_nat] :
( ( set_nat2 @ ( map_list_nat_nat @ F @ V ) )
= ( image_list_nat_nat @ F @ ( set_list_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_569_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_570_list_Oset__map,axiom,
! [F: list_nat > list_nat,V: list_list_nat] :
( ( set_list_nat2 @ ( map_li7225945977422193158st_nat @ F @ V ) )
= ( image_7976474329151083847st_nat @ F @ ( set_list_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_571_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_572_last__appendR,axiom,
! [Ys: list_list_nat,Xs2: list_list_nat] :
( ( Ys != nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs2 @ Ys ) )
= ( last_list_nat @ Ys ) ) ) ).
% last_appendR
thf(fact_573_last__appendR,axiom,
! [Ys: list_nat,Xs2: list_nat] :
( ( Ys != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs2 @ Ys ) )
= ( last_nat @ Ys ) ) ) ).
% last_appendR
thf(fact_574_last__appendL,axiom,
! [Ys: list_list_nat,Xs2: list_list_nat] :
( ( Ys = nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs2 @ Ys ) )
= ( last_list_nat @ Xs2 ) ) ) ).
% last_appendL
thf(fact_575_last__appendL,axiom,
! [Ys: list_nat,Xs2: list_nat] :
( ( Ys = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs2 @ Ys ) )
= ( last_nat @ Xs2 ) ) ) ).
% last_appendL
thf(fact_576_rgf__limit_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( equiva5889994315859557365_limit @ ( cons_nat @ X @ Xs2 ) )
= ( ord_max_nat @ ( plus_plus_nat @ X @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs2 ) ) ) ).
% rgf_limit.simps(2)
thf(fact_577_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: list_nat,F: nat > list_nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_list_nat @ B @ ( image_nat_list_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_578_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: set_nat,F: nat > set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_579_rev__image__eqI,axiom,
! [X: list_nat,A2: set_list_nat,B: nat,F: list_nat > nat] :
( ( member_list_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_list_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_580_rev__image__eqI,axiom,
! [X: list_nat,A2: set_list_nat,B: list_nat,F: list_nat > list_nat] :
( ( member_list_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_list_nat @ B @ ( image_7976474329151083847st_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_581_rev__image__eqI,axiom,
! [X: list_nat,A2: set_list_nat,B: set_nat,F: list_nat > set_nat] :
( ( member_list_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_nat @ B @ ( image_1775855109352712557et_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_582_rev__image__eqI,axiom,
! [X: set_nat,A2: set_set_nat,B: nat,F: set_nat > nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_set_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_583_rev__image__eqI,axiom,
! [X: set_nat,A2: set_set_nat,B: list_nat,F: set_nat > list_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_list_nat @ B @ ( image_5426082062715393517st_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_584_rev__image__eqI,axiom,
! [X: set_nat,A2: set_set_nat,B: set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_585_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_586_ball__imageD,axiom,
! [F: list_nat > set_nat,A2: set_list_nat,P: set_nat > $o] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( image_1775855109352712557et_nat @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_587_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_588_ball__imageD,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P: set_nat > $o] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_589_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_590_image__cong,axiom,
! [M: set_list_nat,N: set_list_nat,F: list_nat > set_nat,G: list_nat > set_nat] :
( ( M = N )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_1775855109352712557et_nat @ F @ M )
= ( image_1775855109352712557et_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_591_image__cong,axiom,
! [M: set_set_nat,N: set_set_nat,F: set_nat > set_nat,G: set_nat > set_nat] :
( ( M = N )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_7916887816326733075et_nat @ F @ M )
= ( image_7916887816326733075et_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_592_bex__imageD,axiom,
! [F: list_nat > set_nat,A2: set_list_nat,P: set_nat > $o] :
( ? [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_1775855109352712557et_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X2: list_nat] :
( ( member_list_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_593_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_594_bex__imageD,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P: set_nat > $o] :
( ? [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_595_image__iff,axiom,
! [Z3: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_596_image__iff,axiom,
! [Z3: set_nat,F: list_nat > set_nat,A2: set_list_nat] :
( ( member_set_nat @ Z3 @ ( image_1775855109352712557et_nat @ F @ A2 ) )
= ( ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_597_image__iff,axiom,
! [Z3: set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( member_set_nat @ Z3 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_598_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > list_nat] :
( ( member_nat @ X @ A2 )
=> ( member_list_nat @ ( F @ X ) @ ( image_nat_list_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_599_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ ( image_nat_set_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_600_imageI,axiom,
! [X: list_nat,A2: set_list_nat,F: list_nat > nat] :
( ( member_list_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_list_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_601_imageI,axiom,
! [X: list_nat,A2: set_list_nat,F: list_nat > list_nat] :
( ( member_list_nat @ X @ A2 )
=> ( member_list_nat @ ( F @ X ) @ ( image_7976474329151083847st_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_602_imageI,axiom,
! [X: list_nat,A2: set_list_nat,F: list_nat > set_nat] :
( ( member_list_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ ( image_1775855109352712557et_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_603_imageI,axiom,
! [X: set_nat,A2: set_set_nat,F: set_nat > nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_set_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_604_imageI,axiom,
! [X: set_nat,A2: set_set_nat,F: set_nat > list_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_list_nat @ ( F @ X ) @ ( image_5426082062715393517st_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_605_imageI,axiom,
! [X: set_nat,A2: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_606_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_607_max__def,axiom,
( ord_max_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% max_def
thf(fact_608_max__def,axiom,
( ord_max_nat
= ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% max_def
thf(fact_609_max__absorb1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_max_set_nat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_610_max__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_max_nat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_611_max__absorb2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_max_set_nat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_612_max__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_max_nat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_613_subset__image__iff,axiom,
! [B2: set_set_nat,F: list_nat > set_nat,A2: set_list_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_1775855109352712557et_nat @ F @ A2 ) )
= ( ? [AA: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ AA @ A2 )
& ( B2
= ( image_1775855109352712557et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_614_subset__image__iff,axiom,
! [B2: set_set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B2
= ( image_7916887816326733075et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_615_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_616_image__subset__iff,axiom,
! [F: list_nat > set_nat,A2: set_list_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_1775855109352712557et_nat @ F @ A2 ) @ B2 )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
=> ( member_set_nat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_617_image__subset__iff,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ B2 )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( member_set_nat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_618_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_619_subset__imageE,axiom,
! [B2: set_set_nat,F: list_nat > set_nat,A2: set_list_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_1775855109352712557et_nat @ F @ A2 ) )
=> ~ ! [C4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ C4 @ A2 )
=> ( B2
!= ( image_1775855109352712557et_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_620_subset__imageE,axiom,
! [B2: set_set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ~ ! [C4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C4 @ A2 )
=> ( B2
!= ( image_7916887816326733075et_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_621_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
=> ( B2
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_622_image__subsetI,axiom,
! [A2: set_nat,F: nat > list_nat,B2: set_list_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_list_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_le6045566169113846134st_nat @ ( image_nat_list_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_623_image__subsetI,axiom,
! [A2: set_nat,F: nat > set_nat,B2: set_set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_624_image__subsetI,axiom,
! [A2: set_list_nat,F: list_nat > list_nat,B2: set_list_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A2 )
=> ( member_list_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_le6045566169113846134st_nat @ ( image_7976474329151083847st_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_625_image__subsetI,axiom,
! [A2: set_list_nat,F: list_nat > set_nat,B2: set_set_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_1775855109352712557et_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_626_image__subsetI,axiom,
! [A2: set_set_nat,F: set_nat > list_nat,B2: set_list_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_list_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_le6045566169113846134st_nat @ ( image_5426082062715393517st_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_627_image__subsetI,axiom,
! [A2: set_set_nat,F: set_nat > set_nat,B2: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_628_image__subsetI,axiom,
! [A2: set_list_nat,F: list_nat > nat,B2: set_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_list_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_629_image__subsetI,axiom,
! [A2: set_set_nat,F: set_nat > nat,B2: set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_630_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_631_image__mono,axiom,
! [A2: set_list_nat,B2: set_list_nat,F: list_nat > set_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_1775855109352712557et_nat @ F @ A2 ) @ ( image_1775855109352712557et_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_632_image__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ ( image_7916887816326733075et_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_633_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_634_can__select__def,axiom,
( can_select_list_nat
= ( ^ [P2: list_nat > $o,A3: set_list_nat] :
? [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
& ( P2 @ X3 )
& ! [Y3: list_nat] :
( ( ( member_list_nat @ Y3 @ A3 )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_635_can__select__def,axiom,
( can_select_set_nat
= ( ^ [P2: set_nat > $o,A3: set_set_nat] :
? [X3: set_nat] :
( ( member_set_nat @ X3 @ A3 )
& ( P2 @ X3 )
& ! [Y3: set_nat] :
( ( ( member_set_nat @ Y3 @ A3 )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_636_can__select__def,axiom,
( can_select_nat
= ( ^ [P2: nat > $o,A3: set_nat] :
? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( P2 @ X3 )
& ! [Y3: nat] :
( ( ( member_nat @ Y3 @ A3 )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_637_rgf__limit__snoc,axiom,
! [X: list_nat,Y: nat] :
( ( equiva5889994315859557365_limit @ ( append_nat @ X @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ord_max_nat @ ( plus_plus_nat @ Y @ one_one_nat ) @ ( equiva5889994315859557365_limit @ X ) ) ) ).
% rgf_limit_snoc
thf(fact_638_image__set,axiom,
! [F: set_nat > set_nat,Xs2: list_set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( set_set_nat2 @ Xs2 ) )
= ( set_set_nat2 @ ( map_set_nat_set_nat @ F @ Xs2 ) ) ) ).
% image_set
thf(fact_639_image__set,axiom,
! [F: nat > list_nat,Xs2: list_nat] :
( ( image_nat_list_nat @ F @ ( set_nat2 @ Xs2 ) )
= ( set_list_nat2 @ ( map_nat_list_nat @ F @ Xs2 ) ) ) ).
% image_set
thf(fact_640_image__set,axiom,
! [F: list_nat > set_nat,Xs2: list_list_nat] :
( ( image_1775855109352712557et_nat @ F @ ( set_list_nat2 @ Xs2 ) )
= ( set_set_nat2 @ ( map_list_nat_set_nat @ F @ Xs2 ) ) ) ).
% image_set
thf(fact_641_image__set,axiom,
! [F: list_nat > nat,Xs2: list_list_nat] :
( ( image_list_nat_nat @ F @ ( set_list_nat2 @ Xs2 ) )
= ( set_nat2 @ ( map_list_nat_nat @ F @ Xs2 ) ) ) ).
% image_set
thf(fact_642_image__set,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat] :
( ( image_7976474329151083847st_nat @ F @ ( set_list_nat2 @ Xs2 ) )
= ( set_list_nat2 @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) ) ) ).
% image_set
thf(fact_643_image__set,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( image_nat_nat @ F @ ( set_nat2 @ Xs2 ) )
= ( set_nat2 @ ( map_nat_nat @ F @ Xs2 ) ) ) ).
% image_set
thf(fact_644_last__ConsR,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= ( last_list_nat @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_645_last__ConsR,axiom,
! [Xs2: list_nat,X: nat] :
( ( Xs2 != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= ( last_nat @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_646_last__ConsL,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( Xs2 = nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= X ) ) ).
% last_ConsL
thf(fact_647_last__ConsL,axiom,
! [Xs2: list_nat,X: nat] :
( ( Xs2 = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= X ) ) ).
% last_ConsL
thf(fact_648_last_Osimps,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( ( Xs2 = nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= X ) )
& ( ( Xs2 != nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= ( last_list_nat @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_649_last_Osimps,axiom,
! [Xs2: list_nat,X: nat] :
( ( ( Xs2 = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= X ) )
& ( ( Xs2 != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= ( last_nat @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_650_last__in__set,axiom,
! [As2: list_set_nat] :
( ( As2 != nil_set_nat )
=> ( member_set_nat @ ( last_set_nat @ As2 ) @ ( set_set_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_651_last__in__set,axiom,
! [As2: list_list_nat] :
( ( As2 != nil_list_nat )
=> ( member_list_nat @ ( last_list_nat @ As2 ) @ ( set_list_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_652_last__in__set,axiom,
! [As2: list_nat] :
( ( As2 != nil_nat )
=> ( member_nat @ ( last_nat @ As2 ) @ ( set_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_653_longest__common__suffix,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat] :
? [Ss: list_list_nat,Xs6: list_list_nat,Ys6: list_list_nat] :
( ( Xs2
= ( 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_654_longest__common__suffix,axiom,
! [Xs2: list_nat,Ys: list_nat] :
? [Ss: list_nat,Xs6: list_nat,Ys6: list_nat] :
( ( Xs2
= ( 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_655_last__append,axiom,
! [Ys: list_list_nat,Xs2: list_list_nat] :
( ( ( Ys = nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs2 @ Ys ) )
= ( last_list_nat @ Xs2 ) ) )
& ( ( Ys != nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs2 @ Ys ) )
= ( last_list_nat @ Ys ) ) ) ) ).
% last_append
thf(fact_656_last__append,axiom,
! [Ys: list_nat,Xs2: list_nat] :
( ( ( Ys = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs2 @ Ys ) )
= ( last_nat @ Xs2 ) ) )
& ( ( Ys != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs2 @ Ys ) )
= ( last_nat @ Ys ) ) ) ) ).
% last_append
thf(fact_657_last__map,axiom,
! [Xs2: list_list_nat,F: list_nat > nat] :
( ( Xs2 != nil_list_nat )
=> ( ( last_nat @ ( map_list_nat_nat @ F @ Xs2 ) )
= ( F @ ( last_list_nat @ Xs2 ) ) ) ) ).
% last_map
thf(fact_658_last__map,axiom,
! [Xs2: list_list_nat,F: list_nat > list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( last_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) )
= ( F @ ( last_list_nat @ Xs2 ) ) ) ) ).
% last_map
thf(fact_659_last__map,axiom,
! [Xs2: list_nat,F: nat > nat] :
( ( Xs2 != nil_nat )
=> ( ( last_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( F @ ( last_nat @ Xs2 ) ) ) ) ).
% last_map
thf(fact_660_nat__add__left__cancel__le,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_661_max_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_662_max_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_663_max_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_664_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_665_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_666_max_Oidem,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ A )
= A ) ).
% max.idem
thf(fact_667_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_668_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_669_max_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ B )
= ( ord_max_nat @ A @ B ) ) ).
% max.right_idem
thf(fact_670_max_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( ord_max_nat @ A @ ( ord_max_nat @ A @ B ) )
= ( ord_max_nat @ A @ B ) ) ).
% max.left_idem
thf(fact_671_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_672_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_673_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_674_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_675_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_676_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_677_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_678_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_679_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_680_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_681_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_682_nat__le__linear,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
| ( ord_less_eq_nat @ N2 @ M3 ) ) ).
% nat_le_linear
thf(fact_683_le__antisym,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M3 )
=> ( M3 = N2 ) ) ) ).
% le_antisym
thf(fact_684_eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( M3 = N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% eq_imp_le
thf(fact_685_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_686_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_687_max_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_max_nat @ B @ ( ord_max_nat @ A @ C ) )
= ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% max.left_commute
thf(fact_688_max_Ocommute,axiom,
( ord_max_nat
= ( ^ [A4: nat,B4: nat] : ( ord_max_nat @ B4 @ A4 ) ) ) ).
% max.commute
thf(fact_689_max_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ C )
= ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% max.assoc
thf(fact_690_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_691_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_692_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_693_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_694_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_695_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C5: nat] :
( B
!= ( plus_plus_nat @ A @ C5 ) ) ) ).
% less_eqE
thf(fact_696_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_697_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C6: nat] :
( B4
= ( plus_plus_nat @ A4 @ C6 ) ) ) ) ).
% le_iff_add
thf(fact_698_add__le__imp__le__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_imp_le_left
thf(fact_699_add__le__imp__le__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_imp_le_right
thf(fact_700_max_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_701_max_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_702_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_max_nat @ A4 @ B4 )
= B4 ) ) ) ).
% max.absorb_iff2
thf(fact_703_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_max_nat @ A4 @ B4 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_704_le__max__iff__disj,axiom,
! [Z3: nat,X: nat,Y: nat] :
( ( ord_less_eq_nat @ Z3 @ ( ord_max_nat @ X @ Y ) )
= ( ( ord_less_eq_nat @ Z3 @ X )
| ( ord_less_eq_nat @ Z3 @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_705_max_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded2
thf(fact_706_max_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded1
thf(fact_707_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( ord_max_nat @ A4 @ B4 ) ) ) ) ).
% max.order_iff
thf(fact_708_max_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_709_max_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% max.boundedE
thf(fact_710_max_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_max_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% max.orderI
thf(fact_711_max_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( ord_max_nat @ A @ B ) ) ) ).
% max.orderE
thf(fact_712_max_Omono,axiom,
! [C: nat,A: nat,D2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D2 @ B )
=> ( ord_less_eq_nat @ ( ord_max_nat @ C @ D2 ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_713_add__leE,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M3 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_714_le__add1,axiom,
! [N2: nat,M3: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M3 ) ) ).
% le_add1
thf(fact_715_le__add2,axiom,
! [N2: nat,M3: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M3 @ N2 ) ) ).
% le_add2
thf(fact_716_add__leD1,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% add_leD1
thf(fact_717_add__leD2,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_718_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_719_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_720_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_721_trans__le__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_le_add1
thf(fact_722_trans__le__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_le_add2
thf(fact_723_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_724_max__add__distrib__right,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z3 ) )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z3 ) ) ) ).
% max_add_distrib_right
thf(fact_725_max__add__distrib__left,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z3 )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Z3 ) @ ( plus_plus_nat @ Y @ Z3 ) ) ) ).
% max_add_distrib_left
thf(fact_726_nat__add__max__right,axiom,
! [M3: nat,N2: nat,Q2: nat] :
( ( plus_plus_nat @ M3 @ ( ord_max_nat @ N2 @ Q2 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M3 @ N2 ) @ ( plus_plus_nat @ M3 @ Q2 ) ) ) ).
% nat_add_max_right
thf(fact_727_nat__add__max__left,axiom,
! [M3: nat,N2: nat,Q2: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M3 @ N2 ) @ Q2 )
= ( ord_max_nat @ ( plus_plus_nat @ M3 @ Q2 ) @ ( plus_plus_nat @ N2 @ Q2 ) ) ) ).
% nat_add_max_left
thf(fact_728_rgf__snoc,axiom,
! [Xs2: list_nat,X: nat] :
( ( equiva3371634703666331078on_rgf @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
= ( ( equiva3371634703666331078on_rgf @ Xs2 )
& ( ord_less_nat @ X @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ Xs2 ) @ one_one_nat ) ) ) ) ).
% rgf_snoc
thf(fact_729_rgf__limit_Oelims,axiom,
! [X: list_nat,Y: nat] :
( ( ( equiva5889994315859557365_limit @ X )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != zero_zero_nat ) )
=> ~ ! [X2: nat,Xs: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs ) )
=> ( Y
!= ( ord_max_nat @ ( plus_plus_nat @ X2 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs ) ) ) ) ) ) ).
% rgf_limit.elims
thf(fact_730_all__subset__image,axiom,
! [F: list_nat > set_nat,A2: set_list_nat,P: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_1775855109352712557et_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B3 @ A2 )
=> ( P @ ( image_1775855109352712557et_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_731_all__subset__image,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( P @ ( image_7916887816326733075et_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_732_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_733_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_734_subseqs__powset,axiom,
! [Xs2: list_list_nat] :
( ( image_4375647060027540749st_nat @ set_list_nat2 @ ( set_list_list_nat2 @ ( subseqs_list_nat @ Xs2 ) ) )
= ( pow_list_nat @ ( set_list_nat2 @ Xs2 ) ) ) ).
% subseqs_powset
thf(fact_735_subseqs__powset,axiom,
! [Xs2: list_nat] :
( ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) )
= ( pow_nat @ ( set_nat2 @ Xs2 ) ) ) ).
% subseqs_powset
thf(fact_736_count__list_Osimps_I2_J,axiom,
! [X: list_nat,Y: list_nat,Xs2: list_list_nat] :
( ( ( X = Y )
=> ( ( count_list_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ Y )
= ( plus_plus_nat @ ( count_list_list_nat @ Xs2 @ Y ) @ one_one_nat ) ) )
& ( ( X != Y )
=> ( ( count_list_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ Y )
= ( count_list_list_nat @ Xs2 @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_737_count__list_Osimps_I2_J,axiom,
! [X: nat,Y: nat,Xs2: list_nat] :
( ( ( X = Y )
=> ( ( count_list_nat @ ( cons_nat @ X @ Xs2 ) @ Y )
= ( plus_plus_nat @ ( count_list_nat @ Xs2 @ Y ) @ one_one_nat ) ) )
& ( ( X != Y )
=> ( ( count_list_nat @ ( cons_nat @ X @ Xs2 ) @ Y )
= ( count_list_nat @ Xs2 @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_738_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_739_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_740_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_741_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_742_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_743_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_744_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_745_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_746_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_747_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_748_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_749_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_750_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_751_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_752_add__is__0,axiom,
! [M3: nat,N2: nat] :
( ( ( plus_plus_nat @ M3 @ N2 )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_753_nat__add__left__cancel__less,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_754_max__less__iff__conj,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z3 )
= ( ( ord_less_nat @ X @ Z3 )
& ( ord_less_nat @ Y @ Z3 ) ) ) ).
% max_less_iff_conj
thf(fact_755_max_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_756_max_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_757_lessThan__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
= ( ord_less_set_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_758_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_759_Pow__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( member_set_nat @ A2 @ ( pow_nat @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Pow_iff
thf(fact_760_PowI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( member_set_nat @ A2 @ ( pow_nat @ B2 ) ) ) ).
% PowI
thf(fact_761_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_max_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_762_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_763_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_764_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_765_max__0L,axiom,
! [N2: nat] :
( ( ord_max_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% max_0L
thf(fact_766_max__0R,axiom,
! [N2: nat] :
( ( ord_max_nat @ N2 @ zero_zero_nat )
= N2 ) ).
% max_0R
thf(fact_767_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_768_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_769_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_770_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_771_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_772_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_773_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_774_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_775_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_776_max__0__1_I2_J,axiom,
( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
= one_one_nat ) ).
% max_0_1(2)
thf(fact_777_max__0__1_I1_J,axiom,
( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
= one_one_nat ) ).
% max_0_1(1)
thf(fact_778_add__gr__0,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_779_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_780_count__notin,axiom,
! [X: set_nat,Xs2: list_set_nat] :
( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
=> ( ( count_list_set_nat @ Xs2 @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_781_count__notin,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( ( count_list_list_nat @ Xs2 @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_782_count__notin,axiom,
! [X: nat,Xs2: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( count_list_nat @ Xs2 @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_783_count__list__append,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat,X: list_nat] :
( ( count_list_list_nat @ ( append_list_nat @ Xs2 @ Ys ) @ X )
= ( plus_plus_nat @ ( count_list_list_nat @ Xs2 @ X ) @ ( count_list_list_nat @ Ys @ X ) ) ) ).
% count_list_append
thf(fact_784_count__list__append,axiom,
! [Xs2: list_nat,Ys: list_nat,X: nat] :
( ( count_list_nat @ ( append_nat @ Xs2 @ Ys ) @ X )
= ( plus_plus_nat @ ( count_list_nat @ Xs2 @ X ) @ ( count_list_nat @ Ys @ X ) ) ) ).
% count_list_append
thf(fact_785_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_786_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_787_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C5: nat] :
( ( B
= ( plus_plus_nat @ A @ C5 ) )
=> ( C5 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_788_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_789_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_790_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K3 )
=> ~ ( P @ I2 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_791_count__list__0__iff,axiom,
! [Xs2: list_set_nat,X: set_nat] :
( ( ( count_list_set_nat @ Xs2 @ X )
= zero_zero_nat )
= ( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) ) ) ) ).
% count_list_0_iff
thf(fact_792_count__list__0__iff,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( ( count_list_list_nat @ Xs2 @ X )
= zero_zero_nat )
= ( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) ) ) ) ).
% count_list_0_iff
thf(fact_793_count__list__0__iff,axiom,
! [Xs2: list_nat,X: nat] :
( ( ( count_list_nat @ Xs2 @ X )
= zero_zero_nat )
= ( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ) ).
% count_list_0_iff
thf(fact_794_order__less__imp__not__less,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ~ ( ord_less_set_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_795_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_796_order__less__imp__not__eq2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_797_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_798_order__less__imp__not__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_799_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_800_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_801_order__less__imp__triv,axiom,
! [X: set_nat,Y: set_nat,P: $o] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_802_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_803_order__less__not__sym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ~ ( ord_less_set_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_804_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_805_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_806_order__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_807_order__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_808_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_809_order__less__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_810_order__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_811_order__less__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_812_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_813_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_814_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_815_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_816_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_817_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_818_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_819_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_820_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_821_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_822_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_823_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_824_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_825_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_826_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_827_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P2: nat > $o] :
? [N4: nat] :
( ( P2 @ N4 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ~ ( P2 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_828_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_829_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_830_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_831_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_832_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_833_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_834_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_835_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_836_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_837_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_838_count__list_Osimps_I1_J,axiom,
! [Y: nat] :
( ( count_list_nat @ nil_nat @ Y )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_839_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_840_lessThan__strict__subset__iff,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M3 ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_841_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_842_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_843_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_844_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_845_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_846_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_847_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_848_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_849_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_850_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_851_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_852_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_853_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_854_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_855_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_856_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_857_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_858_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_859_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_860_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_861_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_862_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_863_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_864_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_865_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_866_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_867_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_868_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_869_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_870_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_871_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_872_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_873_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_874_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_875_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_876_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_877_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_878_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_879_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_880_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_881_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_882_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_883_add__less__imp__less__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_imp_less_right
thf(fact_884_add__less__imp__less__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_imp_less_left
thf(fact_885_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_886_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_887_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_888_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_889_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_890_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_891_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_892_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_893_le__neq__implies__less,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( M3 != N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_894_less__or__eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_895_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_896_less__imp__le__nat,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_897_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_898_less__add__eq__less,axiom,
! [K: nat,L: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M3 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M3 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_899_trans__less__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_less_add2
thf(fact_900_trans__less__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_less_add1
thf(fact_901_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_902_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_903_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_904_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_905_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_906_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_907_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_908_max_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_909_max_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_910_max_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( A4
= ( ord_max_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% max.strict_order_iff
thf(fact_911_max_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_912_less__max__iff__disj,axiom,
! [Z3: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z3 @ ( ord_max_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z3 @ X )
| ( ord_less_nat @ Z3 @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_913_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_914_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_915_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_916_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_917_add__eq__self__zero,axiom,
! [M3: nat,N2: nat] :
( ( ( plus_plus_nat @ M3 @ N2 )
= M3 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_918_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_919_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_920_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_921_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_922_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_923_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_924_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_925_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_926_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_927_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_928_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_929_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_930_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_931_mono__nat__linear__lb,axiom,
! [F: nat > nat,M3: nat,K: nat] :
( ! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ord_less_nat @ ( F @ M2 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K ) @ ( F @ ( plus_plus_nat @ M3 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_932_rgf__limit__ge,axiom,
! [Y: nat,Xs2: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_nat @ Y @ ( equiva5889994315859557365_limit @ Xs2 ) ) ) ).
% rgf_limit_ge
thf(fact_933_rgf__limit_Osimps_I1_J,axiom,
( ( equiva5889994315859557365_limit @ nil_nat )
= zero_zero_nat ) ).
% rgf_limit.simps(1)
thf(fact_934_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_935_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_936_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_937_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_938_n__lists_Osimps_I1_J,axiom,
! [Xs2: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs2 )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_939_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_940_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_941_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_942_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_943_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_944_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_945_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_946_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M3: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N2 @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ! [I2: nat] :
( ( ord_less_nat @ K3 @ I2 )
=> ( P @ I2 ) )
=> ( P @ K3 ) ) )
=> ( P @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_947_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
= ( P @ B5 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B5: nat] :
( ( P @ A5 @ B5 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_948_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_949_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_950_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C5: nat] :
( ( ord_less_eq_nat @ A @ C5 )
& ( ord_less_eq_nat @ C5 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C5 ) )
=> ( P @ X4 ) )
& ! [D3: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D3 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D3 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_951_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_952_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_953_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
= ( ord_less_nat @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_954_minf_I8_J,axiom,
! [T2: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ~ ( ord_less_eq_nat @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_955_minf_I6_J,axiom,
! [T2: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ord_less_eq_nat @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_956_pinf_I8_J,axiom,
! [T2: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ord_less_eq_nat @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_957_pinf_I6_J,axiom,
! [T2: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_958_rgf__limit_Opelims,axiom,
! [X: list_nat,Y: nat] :
( ( ( equiva5889994315859557365_limit @ X )
= Y )
=> ( ( accp_list_nat @ equiva5575797544161152836it_rel @ X )
=> ( ( ( X = nil_nat )
=> ( ( Y = zero_zero_nat )
=> ~ ( accp_list_nat @ equiva5575797544161152836it_rel @ nil_nat ) ) )
=> ~ ! [X2: nat,Xs: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs ) )
=> ( ( Y
= ( ord_max_nat @ ( plus_plus_nat @ X2 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs ) ) )
=> ~ ( accp_list_nat @ equiva5575797544161152836it_rel @ ( cons_nat @ X2 @ Xs ) ) ) ) ) ) ) ).
% rgf_limit.pelims
thf(fact_959_stirling__row__code_I1_J,axiom,
( ( stirling_row @ zero_zero_nat )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% stirling_row_code(1)
thf(fact_960_stirling__row__nonempty,axiom,
! [N2: nat] :
( ( stirling_row @ N2 )
!= nil_nat ) ).
% stirling_row_nonempty
thf(fact_961_stirling__row__aux_Osimps_I1_J,axiom,
! [N2: nat,Y: nat] :
( ( stirling_row_aux_nat @ N2 @ Y @ nil_nat )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% stirling_row_aux.simps(1)
thf(fact_962_comm__append__is__replicate,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( Ys != nil_nat )
=> ( ( ( append_nat @ Xs2 @ Ys )
= ( append_nat @ Ys @ Xs2 ) )
=> ? [N3: nat,Zs: list_nat] :
( ( ord_less_nat @ one_one_nat @ N3 )
& ( ( concat_nat @ ( replicate_list_nat @ N3 @ Zs ) )
= ( append_nat @ Xs2 @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_963_nat__1__eq__mult__iff,axiom,
! [M3: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M3 @ N2 ) )
= ( ( M3 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_964_nat__mult__eq__1__iff,axiom,
! [M3: nat,N2: nat] :
( ( ( times_times_nat @ M3 @ N2 )
= one_one_nat )
= ( ( M3 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_965_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_966_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_967_mult__le__cancel2,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_968_replicate__empty,axiom,
! [N2: nat,X: nat] :
( ( ( replicate_nat @ N2 @ X )
= nil_nat )
= ( N2 = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_969_empty__replicate,axiom,
! [N2: nat,X: nat] :
( ( nil_nat
= ( replicate_nat @ N2 @ X ) )
= ( N2 = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_970_in__set__replicate,axiom,
! [X: nat,N2: nat,Y: nat] :
( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ Y ) ) )
= ( ( X = Y )
& ( N2 != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_971_Bex__set__replicate,axiom,
! [N2: nat,A: nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
& ( P @ X3 ) ) )
= ( ( P @ A )
& ( N2 != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_972_Ball__set__replicate,axiom,
! [N2: nat,A: nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
=> ( P @ X3 ) ) )
= ( ( P @ A )
| ( N2 = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_973_concat__replicate__trivial,axiom,
! [I: nat] :
( ( concat_nat @ ( replicate_list_nat @ I @ nil_nat ) )
= nil_nat ) ).
% concat_replicate_trivial
thf(fact_974_stirling__row__aux_Osimps_I2_J,axiom,
! [N2: nat,Y: nat,X: nat,Xs2: list_nat] :
( ( stirling_row_aux_nat @ N2 @ Y @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ X ) ) @ ( stirling_row_aux_nat @ N2 @ X @ Xs2 ) ) ) ).
% stirling_row_aux.simps(2)
thf(fact_975_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( A != B )
& ( C != D2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D2 ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_976_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z3: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z3 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z3 ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z3 ) ) ) ).
% crossproduct_eq
thf(fact_977_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_978_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_979_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_980_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_981_append__replicate__commute,axiom,
! [N2: nat,X: nat,K: nat] :
( ( append_nat @ ( replicate_nat @ N2 @ X ) @ ( replicate_nat @ K @ X ) )
= ( append_nat @ ( replicate_nat @ K @ X ) @ ( replicate_nat @ N2 @ X ) ) ) ).
% append_replicate_commute
thf(fact_982_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_983_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_984_le__cube,axiom,
! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ) ).
% le_cube
thf(fact_985_le__square,axiom,
! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ).
% le_square
thf(fact_986_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_987_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_988_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_989_add__mult__distrib,axiom,
! [M3: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M3 @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_990_add__mult__distrib2,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M3 @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_991_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_992_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_993_nat__mult__max__left,axiom,
! [M3: nat,N2: nat,Q2: nat] :
( ( times_times_nat @ ( ord_max_nat @ M3 @ N2 ) @ Q2 )
= ( ord_max_nat @ ( times_times_nat @ M3 @ Q2 ) @ ( times_times_nat @ N2 @ Q2 ) ) ) ).
% nat_mult_max_left
thf(fact_994_nat__mult__max__right,axiom,
! [M3: nat,N2: nat,Q2: nat] :
( ( times_times_nat @ M3 @ ( ord_max_nat @ N2 @ Q2 ) )
= ( ord_max_nat @ ( times_times_nat @ M3 @ N2 ) @ ( times_times_nat @ M3 @ Q2 ) ) ) ).
% nat_mult_max_right
thf(fact_995_replicate__0,axiom,
! [X: nat] :
( ( replicate_nat @ zero_zero_nat @ X )
= nil_nat ) ).
% replicate_0
thf(fact_996_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_997_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_998_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_999_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1000_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1001_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_1002_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1003_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1004_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1005_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_1006_replicate__app__Cons__same,axiom,
! [N2: nat,X: nat,Xs2: list_nat] :
( ( append_nat @ ( replicate_nat @ N2 @ X ) @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ X @ ( append_nat @ ( replicate_nat @ N2 @ X ) @ Xs2 ) ) ) ).
% replicate_app_Cons_same
thf(fact_1007_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C: nat,D2: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D2 ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1008_less__1__mult,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ M3 )
=> ( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M3 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_1009_replicate__add,axiom,
! [N2: nat,M3: nat,X: nat] :
( ( replicate_nat @ ( plus_plus_nat @ N2 @ M3 ) @ X )
= ( append_nat @ ( replicate_nat @ N2 @ X ) @ ( replicate_nat @ M3 @ X ) ) ) ).
% replicate_add
thf(fact_1010_comm__append__are__replicate,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= ( append_nat @ Ys @ Xs2 ) )
=> ? [M2: nat,N3: nat,Zs: list_nat] :
( ( ( concat_nat @ ( replicate_list_nat @ M2 @ Zs ) )
= Xs2 )
& ( ( concat_nat @ ( replicate_list_nat @ N3 @ Zs ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_1011_mult__eq__self__implies__10,axiom,
! [M3: nat,N2: nat] :
( ( M3
= ( times_times_nat @ M3 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M3 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1012_replicate__append__same,axiom,
! [I: nat,X: nat] :
( ( append_nat @ ( replicate_nat @ I @ X ) @ ( cons_nat @ X @ nil_nat ) )
= ( cons_nat @ X @ ( replicate_nat @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_1013_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1014_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1015_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1016_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1017_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1018_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1019_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1020_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1021_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1022_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1023_nat__mult__le__cancel__disj,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1024_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1025_nat__mult__le__cancel1,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1026_power__decreasing__iff,axiom,
! [B: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M3 ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M3 ) ) ) ) ).
% power_decreasing_iff
thf(fact_1027_power__one,axiom,
! [N2: nat] :
( ( power_power_nat @ one_one_nat @ N2 )
= one_one_nat ) ).
% power_one
thf(fact_1028_power__inject__exp,axiom,
! [A: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M3 )
= ( power_power_nat @ A @ N2 ) )
= ( M3 = N2 ) ) ) ).
% power_inject_exp
thf(fact_1029_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_1030_power__strict__decreasing__iff,axiom,
! [B: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M3 ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M3 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1031_power__mono__iff,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_1032_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_1033_power__add,axiom,
! [A: nat,M3: nat,N2: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M3 @ N2 ) )
= ( times_times_nat @ ( power_power_nat @ A @ M3 ) @ ( power_power_nat @ A @ N2 ) ) ) ).
% power_add
thf(fact_1034_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N2: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_1035_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_1036_one__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_1037_zero__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_1038_power__mono,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_1039_power__less__imp__less__base,axiom,
! [A: nat,N2: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1040_power__le__one,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_1041_power__gt1__lemma,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_gt1_lemma
thf(fact_1042_power__less__power__Suc,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_less_power_Suc
thf(fact_1043_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= one_one_nat ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_1044_power__increasing,axiom,
! [N2: nat,N: nat,A: nat] :
( ( ord_less_eq_nat @ N2 @ N )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_increasing
thf(fact_1045_power__strict__increasing,axiom,
! [N2: nat,N: nat,A: nat] :
( ( ord_less_nat @ N2 @ N )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_strict_increasing
thf(fact_1046_power__less__imp__less__exp,axiom,
! [A: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M3 ) @ ( power_power_nat @ A @ N2 ) )
=> ( ord_less_nat @ M3 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_1047_power__Suc__less,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_Suc_less
thf(fact_1048_power__decreasing,axiom,
! [N2: nat,N: nat,A: nat] :
( ( ord_less_eq_nat @ N2 @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_1049_power__strict__decreasing,axiom,
! [N2: nat,N: nat,A: nat] :
( ( ord_less_nat @ N2 @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1050_power__eq__imp__eq__base,axiom,
! [A: nat,N2: nat,B: nat] :
( ( ( power_power_nat @ A @ N2 )
= ( power_power_nat @ B @ N2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1051_power__eq__iff__eq__base,axiom,
! [N2: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N2 )
= ( power_power_nat @ B @ N2 ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1052_power__le__imp__le__exp,axiom,
! [A: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M3 ) @ ( power_power_nat @ A @ N2 ) )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_1053_self__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% self_le_power
thf(fact_1054_one__less__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% one_less_power
thf(fact_1055_power__strict__mono,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ) ).
% power_strict_mono
thf(fact_1056_set__replicate,axiom,
! [N2: nat,X: nat] :
( ( N2 != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N2 @ X ) )
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ).
% set_replicate
thf(fact_1057_Cons__replicate__eq,axiom,
! [X: nat,Xs2: list_nat,N2: nat,Y: nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( replicate_nat @ N2 @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N2 )
& ( Xs2
= ( replicate_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_1058_power__minus__mult,axiom,
! [N2: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N2 ) ) ) ).
% power_minus_mult
thf(fact_1059_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_1060_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_1061_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_1062_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_1063_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1064_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1065_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1066_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1067_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1068_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1069_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1070_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_1071_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_1072_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_1073_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1074_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1075_max__bot2,axiom,
! [X: set_nat] :
( ( ord_max_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% max_bot2
thf(fact_1076_max__bot2,axiom,
! [X: nat] :
( ( ord_max_nat @ X @ bot_bot_nat )
= X ) ).
% max_bot2
thf(fact_1077_max__bot,axiom,
! [X: set_nat] :
( ( ord_max_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% max_bot
thf(fact_1078_max__bot,axiom,
! [X: nat] :
( ( ord_max_nat @ bot_bot_nat @ X )
= X ) ).
% max_bot
thf(fact_1079_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_1080_Pow__empty,axiom,
( ( pow_nat @ bot_bot_set_nat )
= ( insert_set_nat2 @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_empty
thf(fact_1081_Pow__singleton__iff,axiom,
! [X5: set_nat,Y6: set_nat] :
( ( ( pow_nat @ X5 )
= ( insert_set_nat2 @ Y6 @ bot_bot_set_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
& ( Y6 = bot_bot_set_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_1082_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1083_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1084_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1085_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat2 @ B @ bot_bot_set_nat )
= ( insert_nat2 @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1086_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat2 @ A @ A2 )
= ( insert_nat2 @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1087_set__empty,axiom,
! [Xs2: list_nat] :
( ( ( set_nat2 @ Xs2 )
= bot_bot_set_nat )
= ( Xs2 = nil_nat ) ) ).
% set_empty
thf(fact_1088_set__empty2,axiom,
! [Xs2: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% set_empty2
thf(fact_1089_diff__is__0__eq_H,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( minus_minus_nat @ M3 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1090_diff__is__0__eq,axiom,
! [M3: nat,N2: nat] :
( ( ( minus_minus_nat @ M3 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1091_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1092_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1093_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1094_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_1095_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_1096_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1097_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1098_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_1099_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1100_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1101_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1102_Iio__eq__empty__iff,axiom,
! [N2: nat] :
( ( ( set_ord_lessThan_nat @ N2 )
= bot_bot_set_nat )
= ( N2 = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_1103_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_1104_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_1105_equals0I,axiom,
! [A2: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_1106_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_1107_Pow__set_I1_J,axiom,
( ( pow_nat @ ( set_nat2 @ nil_nat ) )
= ( insert_set_nat2 @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_set(1)
thf(fact_1108_Pow__bottom,axiom,
! [B2: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B2 ) ) ).
% Pow_bottom
thf(fact_1109_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1110_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1111_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( insert_nat2 @ A @ ( insert_nat2 @ B @ bot_bot_set_nat ) )
= ( insert_nat2 @ C @ ( insert_nat2 @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1112_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat2 @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_1113_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat2 @ A @ bot_bot_set_nat )
= ( insert_nat2 @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_1114_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1115_diff__add__inverse2,axiom,
! [M3: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N2 ) @ N2 )
= M3 ) ).
% diff_add_inverse2
thf(fact_1116_diff__add__inverse,axiom,
! [N2: nat,M3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M3 ) @ N2 )
= M3 ) ).
% diff_add_inverse
thf(fact_1117_diff__cancel2,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ).
% diff_cancel2
thf(fact_1118_Nat_Odiff__cancel,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1119_diff__le__mono2,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_1120_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1121_diff__le__self,axiom,
! [M3: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N2 ) @ M3 ) ).
% diff_le_self
thf(fact_1122_diff__le__mono,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1123_Nat_Odiff__diff__eq,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1124_le__diff__iff,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1125_eq__diff__iff,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M3 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M3 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1126_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1127_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1128_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_1129_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1130_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_1131_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_1132_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1133_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1134_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1135_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1136_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1137_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1138_add__diff__inverse__nat,axiom,
! [M3: nat,N2: nat] :
( ~ ( ord_less_nat @ M3 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M3 @ N2 ) )
= M3 ) ) ).
% add_diff_inverse_nat
thf(fact_1139_less__diff__iff,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M3 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1140_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1141_diff__add__0,axiom,
! [N2: nat,M3: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M3 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1142_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1143_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1144_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1145_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1146_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1147_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1148_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1149_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1150_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1151_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_1152_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_1153_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N2: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1154_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1155_subset__singletonD,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1156_subset__singleton__iff,axiom,
! [X5: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_1157_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_1158_nat__minus__add__max,axiom,
! [N2: nat,M3: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N2 @ M3 ) @ M3 )
= ( ord_max_nat @ N2 @ M3 ) ) ).
% nat_minus_add_max
thf(fact_1159_lessThan__empty__iff,axiom,
! [N2: nat] :
( ( ( set_ord_lessThan_nat @ N2 )
= bot_bot_set_nat )
= ( N2 = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1160_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1161_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1162_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1163_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M3 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M3 )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1164_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M3 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( M3
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1165_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M3 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M3 ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_1166_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M3 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ M3 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1167_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M3 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M3 ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1168_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M3 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ M3 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1169_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M3 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_nat @ M3 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1170_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M3 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M3 ) @ N2 ) ) ) ).
% nat_less_add_iff1
thf(fact_1171_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1172_set__replicate__conv__if,axiom,
! [N2: nat,X: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N2 @ X ) )
= bot_bot_set_nat ) )
& ( ( N2 != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N2 @ X ) )
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).
% set_replicate_conv_if
thf(fact_1173_power__eq__if,axiom,
( power_power_nat
= ( ^ [P4: nat,M4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P4 @ ( power_power_nat @ P4 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1174_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1175_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat2 @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1176_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_1177_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_1178_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_1179_Diff__insert0,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ B2 ) )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1180_insert__Diff1,axiom,
! [X: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1181_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_1182_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat2 @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
= ( insert_nat2 @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_1183_single__Diff__lessThan,axiom,
! [K: nat] :
( ( minus_minus_set_nat @ ( insert_nat2 @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
= ( insert_nat2 @ K @ bot_bot_set_nat ) ) ).
% single_Diff_lessThan
thf(fact_1184_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_1185_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1186_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1187_Diff__insert__absorb,axiom,
! [X: nat,A2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A2 ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1188_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_1189_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat2 @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1190_Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_1191_insert__Diff__if,axiom,
! [X: nat,B2: set_nat,A2: set_nat] :
( ( ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) )
& ( ~ ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A2 ) @ B2 )
= ( insert_nat2 @ X @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1192_subset__Diff__insert,axiom,
! [A2: set_nat,B2: set_nat,X: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat2 @ X @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C2 ) )
& ~ ( member_nat @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1193_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1194_Diff__single__insert,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1195_subset__insert__iff,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X @ B2 ) )
= ( ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1196_in__image__insert__iff,axiom,
! [B2: set_set_nat,X: nat,A2: set_nat] :
( ! [C4: set_nat] :
( ( member_set_nat @ C4 @ B2 )
=> ~ ( member_nat @ X @ C4 ) )
=> ( ( member_set_nat @ A2 @ ( image_7916887816326733075et_nat @ ( insert_nat2 @ X ) @ B2 ) )
= ( ( member_nat @ X @ A2 )
& ( member_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1197_remove__def,axiom,
( remove_nat
= ( ^ [X3: nat,A3: set_nat] : ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_1198_psubset__insert__iff,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat2 @ X @ B2 ) )
= ( ( ( member_nat @ X @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) )
& ( ~ ( member_nat @ X @ B2 )
=> ( ( ( member_nat @ X @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1199_is__singletonE,axiom,
! [A2: set_nat] :
( ( is_singleton_nat @ A2 )
=> ~ ! [X2: nat] :
( A2
!= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_1200_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X3: nat] :
( A3
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_1201_elim__singleton,axiom,
! [X: nat,S2: nat,X7: nat,T2: nat] :
( ( ( member_nat @ X @ ( insert_nat2 @ S2 @ bot_bot_set_nat ) )
& ( member_nat @ X7 @ ( insert_nat2 @ T2 @ bot_bot_set_nat ) ) )
=> ( ( X = S2 )
& ( X7 = T2 ) ) ) ).
% elim_singleton
thf(fact_1202_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1203_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1204_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1205_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_1206_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_1207_set__removeAll,axiom,
! [X: nat,Xs2: list_nat] :
( ( set_nat2 @ ( removeAll_nat @ X @ Xs2 ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs2 ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ).
% set_removeAll
thf(fact_1208_Gcd__0__iff,axiom,
! [A2: set_nat] :
( ( ( gcd_Gcd_nat @ A2 )
= zero_zero_nat )
= ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% Gcd_0_iff
thf(fact_1209_removeAll__id,axiom,
! [X: nat,Xs2: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( removeAll_nat @ X @ Xs2 )
= Xs2 ) ) ).
% removeAll_id
thf(fact_1210_removeAll__append,axiom,
! [X: nat,Xs2: list_nat,Ys: list_nat] :
( ( removeAll_nat @ X @ ( append_nat @ Xs2 @ Ys ) )
= ( append_nat @ ( removeAll_nat @ X @ Xs2 ) @ ( removeAll_nat @ X @ Ys ) ) ) ).
% removeAll_append
thf(fact_1211_removeAll_Osimps_I2_J,axiom,
! [X: nat,Y: nat,Xs2: list_nat] :
( ( ( X = Y )
=> ( ( removeAll_nat @ X @ ( cons_nat @ Y @ Xs2 ) )
= ( removeAll_nat @ X @ Xs2 ) ) )
& ( ( X != Y )
=> ( ( removeAll_nat @ X @ ( cons_nat @ Y @ Xs2 ) )
= ( cons_nat @ Y @ ( removeAll_nat @ X @ Xs2 ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_1212_removeAll_Osimps_I1_J,axiom,
! [X: nat] :
( ( removeAll_nat @ X @ nil_nat )
= nil_nat ) ).
% removeAll.simps(1)
thf(fact_1213_Gcd__nat__eq__one,axiom,
! [N: set_nat] :
( ( member_nat @ one_one_nat @ N )
=> ( ( gcd_Gcd_nat @ N )
= one_one_nat ) ) ).
% Gcd_nat_eq_one
thf(fact_1214_Gcd__1,axiom,
! [A2: set_nat] :
( ( member_nat @ one_one_nat @ A2 )
=> ( ( gcd_Gcd_nat @ A2 )
= one_one_nat ) ) ).
% Gcd_1
thf(fact_1215_insert__code_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( insert_nat2 @ X @ ( coset_nat @ Xs2 ) )
= ( coset_nat @ ( removeAll_nat @ X @ Xs2 ) ) ) ).
% insert_code(2)
thf(fact_1216_remove__code_I1_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( remove_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( set_nat2 @ ( removeAll_nat @ X @ Xs2 ) ) ) ).
% remove_code(1)
thf(fact_1217_card__insert__le__m1,axiom,
! [N2: nat,Y: set_nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat2 @ X @ Y ) ) @ N2 ) ) ) ).
% card_insert_le_m1
thf(fact_1218_nth__Cons__pos,axiom,
! [N2: nat,X: nat,Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N2 )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1219_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_1220_nth__Cons__0,axiom,
! [X: nat,Xs2: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_1221_nth__replicate,axiom,
! [I: nat,N2: nat,X: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( ( nth_nat @ ( replicate_nat @ N2 @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_1222_card__Diff__insert,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ A @ B2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1223_card__1__singletonE,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= one_one_nat )
=> ~ ! [X2: nat] :
( A2
!= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_1224_is__singleton__altdef,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( ( finite_card_nat @ A3 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_1225_card__insert__le,axiom,
! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( insert_nat2 @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_1226_nth__Cons_H,axiom,
! [N2: nat,X: nat,Xs2: list_nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N2 )
= X ) )
& ( ( N2 != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N2 )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1227_card__Diff1__le,axiom,
! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_1228_nth__non__equal__first__eq,axiom,
! [X: nat,Y: nat,Xs2: list_nat,N2: nat] :
( ( X != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N2 )
= Y )
= ( ( ( nth_nat @ Xs2 @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1229_card__Diff__singleton__if,axiom,
! [X: nat,A2: set_nat] :
( ( ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1230_card__Diff__singleton,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1231_append__one__prefix,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs2 @ Ys )
=> ( ( ord_less_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) )
=> ( prefix_nat @ ( append_nat @ Xs2 @ ( cons_nat @ ( nth_nat @ Ys @ ( size_size_list_nat @ Xs2 ) ) @ nil_nat ) ) @ Ys ) ) ) ).
% append_one_prefix
thf(fact_1232_stirling__code,axiom,
( stirling
= ( ^ [N4: nat,K2: nat] : ( if_nat @ ( K2 = zero_zero_nat ) @ ( if_nat @ ( N4 = zero_zero_nat ) @ one_one_nat @ zero_zero_nat ) @ ( if_nat @ ( ord_less_nat @ N4 @ K2 ) @ zero_zero_nat @ ( if_nat @ ( K2 = N4 ) @ one_one_nat @ ( nth_nat @ ( stirling_row @ N4 ) @ K2 ) ) ) ) ) ) ).
% stirling_code
thf(fact_1233_append__eq__append__conv,axiom,
! [Xs2: list_nat,Ys: list_nat,Us2: list_nat,Vs2: list_nat] :
( ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us2 )
= ( size_size_list_nat @ Vs2 ) ) )
=> ( ( ( append_nat @ Xs2 @ Us2 )
= ( append_nat @ Ys @ Vs2 ) )
= ( ( Xs2 = Ys )
& ( Us2 = Vs2 ) ) ) ) ).
% append_eq_append_conv
thf(fact_1234_stirling__same,axiom,
! [N2: nat] :
( ( stirling @ N2 @ N2 )
= one_one_nat ) ).
% stirling_same
thf(fact_1235_length__0__conv,axiom,
! [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_nat ) ) ).
% length_0_conv
thf(fact_1236_length__append,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs2 @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_append
thf(fact_1237_length__greater__0__conv,axiom,
! [Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) )
= ( Xs2 != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_1238_nth__append__length,axiom,
! [Xs2: list_nat,X: nat,Ys: list_nat] :
( ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_1239_nth__map,axiom,
! [N2: nat,Xs2: list_nat,F: nat > nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs2 ) @ N2 )
= ( F @ ( nth_nat @ Xs2 @ N2 ) ) ) ) ).
% nth_map
thf(fact_1240_nth__append__length__plus,axiom,
! [Xs2: list_nat,Ys: list_nat,N2: nat] :
( ( nth_nat @ ( append_nat @ Xs2 @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ N2 ) )
= ( nth_nat @ Ys @ N2 ) ) ).
% nth_append_length_plus
thf(fact_1241_card__length,axiom,
! [Xs2: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ).
% card_length
thf(fact_1242_nth__equalityI,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_1243_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X8: nat] : ( P @ I4 @ X8 ) ) )
= ( ? [Xs4: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_nat @ Xs4 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1244_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
= ( ^ [Xs4: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs4 ) )
=> ( ( nth_nat @ Xs4 @ I4 )
= ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1245_all__set__conv__all__nth,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X3 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1246_all__nth__imp__all__set,axiom,
! [Xs2: list_nat,P: nat > $o,X: nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I3 ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1247_in__set__conv__nth,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1248_list__ball__nth,axiom,
! [N2: nat,Xs2: list_nat,P: nat > $o] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_nat @ Xs2 @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_1249_nth__mem,axiom,
! [N2: nat,Xs2: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ ( nth_nat @ Xs2 @ N2 ) @ ( set_nat2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1250_stirling_Osimps_I1_J,axiom,
( ( stirling @ zero_zero_nat @ zero_zero_nat )
= one_one_nat ) ).
% stirling.simps(1)
thf(fact_1251_prefix__length__prefix,axiom,
! [Ps: list_nat,Xs2: list_nat,Qs: list_nat] :
( ( prefix_nat @ Ps @ Xs2 )
=> ( ( prefix_nat @ Qs @ Xs2 )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ps ) @ ( size_size_list_nat @ Qs ) )
=> ( prefix_nat @ Ps @ Qs ) ) ) ) ).
% prefix_length_prefix
thf(fact_1252_prefix__length__le,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs2 @ Ys )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% prefix_length_le
thf(fact_1253_impossible__Cons,axiom,
! [Xs2: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs2
!= ( cons_nat @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1254_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1255_list__induct4,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs2: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys2: list_nat,Z5: nat,Zs: list_nat,W2: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z5 @ Zs ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1256_list__induct3,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs2: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys2: list_nat,Z5: nat,Zs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs @ Ys2 @ Zs )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z5 @ Zs ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_1257_list__induct2,axiom,
! [Xs2: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1258_replicate__eqI,axiom,
! [Xs2: list_nat,N2: nat,X: nat] :
( ( ( size_size_list_nat @ Xs2 )
= N2 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ Xs2 ) )
=> ( Y2 = X ) )
=> ( Xs2
= ( replicate_nat @ N2 @ X ) ) ) ) ).
% replicate_eqI
thf(fact_1259_replicate__length__same,axiom,
! [Xs2: list_nat,X: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( X2 = X ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_1260_length__pos__if__in__set,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1261_same__length__different,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( Xs2 != Ys )
=> ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ? [Pre: list_nat,X2: nat,Xs6: list_nat,Y2: nat,Ys6: list_nat] :
( ( X2 != Y2 )
& ( Xs2
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X2 @ nil_nat ) @ Xs6 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y2 @ nil_nat ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_1262_list__induct__2__rev,axiom,
! [X: list_nat,Y: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_1263_Longest__common__prefix__unique,axiom,
! [L2: set_list_nat] :
( ( L2 != bot_bot_set_list_nat )
=> ? [X2: list_nat] :
( ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ L2 )
=> ( prefix_nat @ X2 @ Xa ) )
& ! [Qs2: list_nat] :
( ! [Xa3: list_nat] :
( ( member_list_nat @ Xa3 @ L2 )
=> ( prefix_nat @ Qs2 @ Xa3 ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Qs2 ) @ ( size_size_list_nat @ X2 ) ) )
& ! [Y5: list_nat] :
( ( ! [Xa3: list_nat] :
( ( member_list_nat @ Xa3 @ L2 )
=> ( prefix_nat @ Y5 @ Xa3 ) )
& ! [Qs3: list_nat] :
( ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ L2 )
=> ( prefix_nat @ Qs3 @ Xa ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Qs3 ) @ ( size_size_list_nat @ Y5 ) ) ) )
=> ( Y5 = X2 ) ) ) ) ).
% Longest_common_prefix_unique
thf(fact_1264_Longest__common__prefix__ex,axiom,
! [L2: set_list_nat] :
( ( L2 != bot_bot_set_list_nat )
=> ? [Ps2: list_nat] :
( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ L2 )
=> ( prefix_nat @ Ps2 @ X4 ) )
& ! [Qs2: list_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ L2 )
=> ( prefix_nat @ Qs2 @ X2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Qs2 ) @ ( size_size_list_nat @ Ps2 ) ) ) ) ) ).
% Longest_common_prefix_ex
thf(fact_1265_nth__butlast,axiom,
! [N2: nat,Xs2: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ ( butlast_nat @ Xs2 ) ) )
=> ( ( nth_nat @ ( butlast_nat @ Xs2 ) @ N2 )
= ( nth_nat @ Xs2 @ N2 ) ) ) ).
% nth_butlast
thf(fact_1266_length__removeAll__less,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_removeAll_less
thf(fact_1267_nth__stirling__row,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( nth_nat @ ( stirling_row @ N2 ) @ K )
= ( stirling @ N2 @ K ) ) ) ).
% nth_stirling_row
thf(fact_1268_nth__append,axiom,
! [N2: nat,Xs2: list_nat,Ys: list_nat] :
( ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( append_nat @ Xs2 @ Ys ) @ N2 )
= ( nth_nat @ Xs2 @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( append_nat @ Xs2 @ Ys ) @ N2 )
= ( nth_nat @ Ys @ ( minus_minus_nat @ N2 @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_1269_nth__equal__first__eq,axiom,
! [X: nat,Xs2: list_nat,N2: nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( ord_less_eq_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N2 )
= X )
= ( N2 = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1270_last__conv__nth,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( last_nat @ Xs2 )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_1271_card__set__1__iff__replicate,axiom,
! [Xs2: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs2 ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs2 != nil_nat )
& ? [X3: nat] :
( Xs2
= ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X3 ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_1272_rotate1__fixpoint__card,axiom,
! [Xs2: list_nat] :
( ( ( rotate1_nat @ Xs2 )
= Xs2 )
=> ( ( Xs2 = nil_nat )
| ( ( finite_card_nat @ ( set_nat2 @ Xs2 ) )
= one_one_nat ) ) ) ).
% rotate1_fixpoint_card
% Helper facts (11)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $true @ X @ Y )
= X ) ).
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_2_1_If_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [X: list_set_nat,Y: list_set_nat] :
( ( if_list_set_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [X: list_set_nat,Y: list_set_nat] :
( ( if_list_set_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $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 @ ( append_nat @ xsa @ ( cons_nat @ x @ nil_nat ) ) )
= ( set_ord_lessThan_nat @ ( equiva5889994315859557365_limit @ ( append_nat @ xsa @ ( cons_nat @ x @ nil_nat ) ) ) ) ) ).
%------------------------------------------------------------------------------