TPTP Problem File: SLH0351^1.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_00187_007057__11858892_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1452 ( 705 unt; 173 typ; 0 def)
% Number of atoms : 3330 (1936 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 11269 ( 351 ~; 59 |; 317 &;9252 @)
% ( 0 <=>;1290 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 799 ( 799 >; 0 *; 0 +; 0 <<)
% Number of symbols : 166 ( 163 usr; 10 con; 0-4 aty)
% Number of variables : 3805 ( 305 ^;3230 !; 270 ?;3805 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:13:46.321
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
list_l5212752354702395664st_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__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__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (163)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__List__Olist_It__Nat__Onat_J,type,
bNF_Gr9051742241863529473st_nat: set_list_list_nat > list_nat > set_list_list_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Nat__Onat,type,
bNF_Gr1872714664788909425ft_nat: set_list_nat > nat > set_list_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__List__Olist_It__Nat__Onat_J,type,
bNF_Gr3053708287304744325st_nat: set_list_list_nat > list_list_nat > set_list_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Oenum__rgfs,type,
equiva7426478223624825838m_rgfs: nat > list_list_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Oenum__rgfs__rel,type,
equiva1432535406783100555fs_rel: nat > nat > $o ).
thf(sy_c_Equivalence__Relation__Enumeration_Orgf,type,
equiva3371634703666331078on_rgf: list_nat > $o ).
thf(sy_c_Equivalence__Relation__Enumeration_Orgf__limit,type,
equiva5889994315859557365_limit: list_nat > nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Orgf__limit__rel,type,
equiva5575797544161152836it_rel: list_nat > list_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > 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_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
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__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
append_list_list_nat: list_list_list_nat > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
append_list_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Obind_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
bind_l7796378977173581257st_nat: list_list_nat > ( list_nat > list_list_nat ) > list_list_nat ).
thf(sy_c_List_Obind_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
bind_list_nat_nat: list_list_nat > ( list_nat > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
bind_nat_list_nat: list_nat > ( nat > list_list_nat ) > list_list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Obutlast_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
butlas6429778205849610142st_nat: list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Obutlast_001t__List__Olist_It__Nat__Onat_J,type,
butlast_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Ocan__select_001t__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_Oconcat_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
concat_list_list_nat: list_l5212752354702395664st_nat > list_list_list_nat ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__Nat__Onat_J,type,
concat_list_nat: list_list_list_nat > list_list_nat ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Ocoset_001t__List__Olist_It__Nat__Onat_J,type,
coset_list_nat: list_list_nat > set_list_nat ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Ocount__list_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
count_8975921713275557620st_nat: list_list_list_nat > list_list_nat > 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_Ofilter_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
filter_list_list_nat: ( list_list_nat > $o ) > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Ofilter_001t__List__Olist_It__Nat__Onat_J,type,
filter_list_nat: ( list_nat > $o ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
filter_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Ofoldr_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__Nat__Onat,type,
foldr_2920604631923063017at_nat: ( list_list_nat > nat > nat ) > list_list_list_nat > nat > nat ).
thf(sy_c_List_Ofoldr_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
foldr_list_nat_nat: ( list_nat > nat > nat ) > list_list_nat > nat > nat ).
thf(sy_c_List_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olast_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
last_list_list_nat: list_list_list_nat > list_list_nat ).
thf(sy_c_List_Olast_001t__List__Olist_It__Nat__Onat_J,type,
last_list_nat: list_list_nat > list_nat ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
cons_list_list_nat: list_list_nat > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
nil_list_list_nat: list_list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_Ocase__list_001_Eo_001t__List__Olist_It__Nat__Onat_J,type,
case_list_o_list_nat: $o > ( list_nat > list_list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist_Ocase__list_001_Eo_001t__Nat__Onat,type,
case_list_o_nat: $o > ( nat > list_nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J_001t__List__Olist_It__Nat__Onat_J,type,
case_l9045269829974911560st_nat: list_list_list_nat > ( list_nat > list_list_nat > list_list_list_nat ) > list_list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
case_l2985193972497081272st_nat: list_list_nat > ( list_nat > list_list_nat > list_list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__Nat__Onat,type,
case_l3331202209248957608at_nat: list_list_nat > ( nat > list_nat > list_list_nat ) > list_nat > list_list_nat ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
case_l2340614614379431832at_nat: list_nat > ( nat > list_nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
map_li5769348595424326838st_nat: ( list_list_nat > list_list_list_nat ) > list_list_list_nat > list_l5212752354702395664st_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_li2855073862107769254st_nat: ( list_list_nat > list_list_nat ) > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
map_li1587113026813354006st_nat: ( list_list_nat > list_nat ) > list_list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
map_li4109527381804435110st_nat: ( list_nat > list_list_list_nat ) > list_list_nat > list_l5212752354702395664st_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_li960784813134754710st_nat: ( list_nat > list_list_nat ) > list_list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_li7225945977422193158st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
map_list_nat_nat: ( list_nat > nat ) > list_list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_na6205611841492582150st_nat: ( nat > list_list_nat ) > list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
map_nat_list_nat: ( nat > list_nat ) > list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Oset_001t__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_Otl_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
tl_list_list_nat: list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_Otl_001t__List__Olist_It__Nat__Onat_J,type,
tl_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_It__Nat__Onat_J,type,
list_ex1_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Omaps_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
maps_l5785965478274863235st_nat: ( list_nat > list_list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Omaps_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
maps_list_nat_nat: ( list_nat > list_nat ) > list_list_nat > list_nat ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
maps_nat_list_nat: ( nat > list_list_nat ) > list_nat > list_list_nat ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_On__lists_001t__List__Olist_It__Nat__Onat_J,type,
n_lists_list_nat: nat > list_list_nat > list_list_list_nat ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Onth_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
nth_list_list_nat: list_list_list_nat > nat > list_list_nat ).
thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
nth_list_nat: list_list_nat > nat > 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_Orotate1_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
rotate6412633851404001245st_nat: list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Orotate1_001t__List__Olist_It__Nat__Onat_J,type,
rotate1_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
rotate_list_list_nat: nat > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Orotate_001t__List__Olist_It__Nat__Onat_J,type,
rotate_list_nat: nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Orotate_001t__Nat__Onat,type,
rotate_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Osubseqs_001t__List__Olist_It__Nat__Onat_J,type,
subseqs_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Otake_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
take_list_list_nat: nat > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Otake_001t__List__Olist_It__Nat__Onat_J,type,
take_list_nat: nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otranspose_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
transp4696039609411287008st_nat: list_l5212752354702395664st_nat > list_l5212752354702395664st_nat ).
thf(sy_c_List_Otranspose_001t__List__Olist_It__Nat__Onat_J,type,
transpose_list_nat: list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Otranspose_001t__Nat__Onat,type,
transpose_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Otranspose__rel_001t__List__Olist_It__Nat__Onat_J,type,
transp7462230406613344747st_nat: list_list_list_nat > list_list_list_nat > $o ).
thf(sy_c_List_Otranspose__rel_001t__Nat__Onat,type,
transpose_rel_nat: list_list_nat > list_list_nat > $o ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
size_s6248950052170075156st_nat: list_list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Oord_Omax_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
max_list_list_nat: ( list_list_nat > list_list_nat > $o ) > list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Orderings_Oord_Omax_001t__List__Olist_It__Nat__Onat_J,type,
max_list_nat: ( list_nat > list_nat > $o ) > list_nat > list_nat > list_nat ).
thf(sy_c_Orderings_Oord_Omin_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
min_list_list_nat: ( list_list_nat > list_list_nat > $o ) > list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Orderings_Oord_Omin_001t__List__Olist_It__Nat__Onat_J,type,
min_list_nat: ( list_nat > list_nat > $o ) > list_nat > list_nat > list_nat ).
thf(sy_c_Orderings_Oord_Omin_001t__Nat__Onat,type,
min_nat: ( nat > nat > $o ) > nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
ord_less_list_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
ord_le1520216061033275535_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $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_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
collec5989764272469232197st_nat: ( list_list_nat > $o ) > set_list_list_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_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_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_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Stirling_Ostirling,type,
stirling: nat > nat > nat ).
thf(sy_c_Stirling_Ostirling__row,type,
stirling_row: nat > list_nat ).
thf(sy_c_Sublist_Oprefix_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
prefix_list_list_nat: list_list_list_nat > list_list_list_nat > $o ).
thf(sy_c_Sublist_Oprefix_001t__List__Olist_It__Nat__Onat_J,type,
prefix_list_nat: list_list_nat > list_list_nat > $o ).
thf(sy_c_Sublist_Oprefix_001t__Nat__Onat,type,
prefix_nat: list_nat > list_nat > $o ).
thf(sy_c_Sublist_Oprefixes_001t__List__Olist_It__Nat__Onat_J,type,
prefixes_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_Sublist_Oprefixes_001t__Nat__Onat,type,
prefixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osublists_001t__List__Olist_It__Nat__Onat_J,type,
sublists_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_Sublist_Osublists_001t__Nat__Onat,type,
sublists_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osuffixes_001t__List__Olist_It__Nat__Onat_J,type,
suffixes_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_Sublist_Osuffixes_001t__Nat__Onat,type,
suffixes_nat: list_nat > list_list_nat ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
accp_l8765801942216208055st_nat: ( list_list_list_nat > list_list_list_nat > $o ) > list_list_list_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
accp_list_list_nat: ( list_list_nat > list_list_nat > $o ) > list_list_nat > $o ).
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_Wellfounded_Oaccp_001t__Nat__Onat,type,
accp_nat: ( nat > nat > $o ) > 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_v_x,type,
x: list_nat ).
thf(sy_v_xa____,type,
xa: nat ).
thf(sy_v_xs____,type,
xs: list_nat ).
% Relevant facts (1271)
thf(fact_0_assms,axiom,
equiva3371634703666331078on_rgf @ x ).
% assms
thf(fact_1_snoc_Oprems,axiom,
equiva3371634703666331078on_rgf @ ( append_nat @ xs @ ( cons_nat @ xa @ nil_nat ) ) ).
% snoc.prems
thf(fact_2_rgf__def,axiom,
( equiva3371634703666331078on_rgf
= ( ^ [X: list_nat] :
! [Ys: list_nat,Y: nat] :
( ( prefix_nat @ ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) @ X )
=> ( ord_less_eq_nat @ Y @ ( equiva5889994315859557365_limit @ Ys ) ) ) ) ) ).
% rgf_def
thf(fact_3_snoc_OIH,axiom,
( ( equiva3371634703666331078on_rgf @ xs )
=> ( ( count_list_list_nat @ ( equiva7426478223624825838m_rgfs @ ( size_size_list_nat @ xs ) ) @ xs )
= one_one_nat ) ) ).
% snoc.IH
thf(fact_4_enum__rgfs__returns__rgfs,axiom,
! [X2: list_nat,N: nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( equiva7426478223624825838m_rgfs @ N ) ) )
=> ( equiva3371634703666331078on_rgf @ X2 ) ) ).
% enum_rgfs_returns_rgfs
thf(fact_5_rgf__imp__initial__segment,axiom,
! [Xs: list_nat] :
( ( equiva3371634703666331078on_rgf @ Xs )
=> ( ( set_nat2 @ Xs )
= ( set_ord_lessThan_nat @ ( equiva5889994315859557365_limit @ Xs ) ) ) ) ).
% rgf_imp_initial_segment
thf(fact_6_rgf__limit_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ~ ! [X3: nat,Xs2: list_nat] :
( X2
!= ( cons_nat @ X3 @ Xs2 ) ) ) ).
% rgf_limit.cases
thf(fact_7_enum__rgfs__len,axiom,
! [X2: list_nat,N: nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( equiva7426478223624825838m_rgfs @ N ) ) )
=> ( ( size_size_list_nat @ X2 )
= N ) ) ).
% enum_rgfs_len
thf(fact_8_list__induct__2__rev,axiom,
! [X2: list_list_nat,Y2: list_list_nat,P: list_list_nat > list_list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ X2 )
= ( size_s3023201423986296836st_nat @ Y2 ) )
=> ( ( P @ nil_list_nat @ nil_list_nat )
=> ( ! [X3: list_nat,Xs2: list_list_nat,Y3: list_nat,Ys2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X3 @ nil_list_nat ) ) @ ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% list_induct_2_rev
thf(fact_9_list__induct__2__rev,axiom,
! [X2: list_list_nat,Y2: list_nat,P: list_list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ X2 )
= ( size_size_list_nat @ Y2 ) )
=> ( ( P @ nil_list_nat @ nil_nat )
=> ( ! [X3: list_nat,Xs2: list_list_nat,Y3: nat,Ys2: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X3 @ nil_list_nat ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% list_induct_2_rev
thf(fact_10_list__induct__2__rev,axiom,
! [X2: list_nat,Y2: list_list_nat,P: list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ X2 )
= ( size_s3023201423986296836st_nat @ Y2 ) )
=> ( ( P @ nil_nat @ nil_list_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: list_nat,Ys2: list_list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% list_induct_2_rev
thf(fact_11_list__induct__2__rev,axiom,
! [X2: list_nat,Y2: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ X2 )
= ( size_size_list_nat @ Y2 ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% list_induct_2_rev
thf(fact_12_prefix__snoc,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,Y2: list_nat] :
( ( prefix_list_nat @ Xs @ ( append_list_nat @ Ys3 @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) )
= ( ( Xs
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) )
| ( prefix_list_nat @ Xs @ Ys3 ) ) ) ).
% prefix_snoc
thf(fact_13_prefix__snoc,axiom,
! [Xs: list_nat,Ys3: list_nat,Y2: nat] :
( ( prefix_nat @ Xs @ ( append_nat @ Ys3 @ ( cons_nat @ Y2 @ nil_nat ) ) )
= ( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ Y2 @ nil_nat ) ) )
| ( prefix_nat @ Xs @ Ys3 ) ) ) ).
% prefix_snoc
thf(fact_14_same__prefix__nil,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ Ys3 ) @ Xs )
= ( Ys3 = nil_nat ) ) ).
% same_prefix_nil
thf(fact_15_same__prefix__nil,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs @ Ys3 ) @ Xs )
= ( Ys3 = nil_list_nat ) ) ).
% same_prefix_nil
thf(fact_16_append1__eq__conv,axiom,
! [Xs: list_nat,X2: nat,Ys3: list_nat,Y2: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
= ( append_nat @ Ys3 @ ( cons_nat @ Y2 @ nil_nat ) ) )
= ( ( Xs = Ys3 )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_17_append1__eq__conv,axiom,
! [Xs: list_list_nat,X2: list_nat,Ys3: list_list_nat,Y2: list_nat] :
( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) )
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) )
= ( ( Xs = Ys3 )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_18_same__prefix__prefix,axiom,
! [Xs: list_nat,Ys3: list_nat,Zs: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ Ys3 ) @ ( append_nat @ Xs @ Zs ) )
= ( prefix_nat @ Ys3 @ Zs ) ) ).
% same_prefix_prefix
thf(fact_19_same__prefix__prefix,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,Zs: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs @ Ys3 ) @ ( append_list_nat @ Xs @ Zs ) )
= ( prefix_list_nat @ Ys3 @ Zs ) ) ).
% same_prefix_prefix
thf(fact_20_lessThan__subset__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X2 ) @ ( set_ord_lessThan_nat @ Y2 ) )
= ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_21_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys3: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys3 @ Vs ) )
= ( ( Xs = Ys3 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_22_append__eq__append__conv,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,Us: list_list_nat,Vs: list_list_nat] :
( ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
| ( ( size_s3023201423986296836st_nat @ Us )
= ( size_s3023201423986296836st_nat @ Vs ) ) )
=> ( ( ( append_list_nat @ Xs @ Us )
= ( append_list_nat @ Ys3 @ Vs ) )
= ( ( Xs = Ys3 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_23_prefix__code_I1_J,axiom,
! [Xs: list_nat] : ( prefix_nat @ nil_nat @ Xs ) ).
% prefix_code(1)
thf(fact_24_prefix__code_I1_J,axiom,
! [Xs: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ Xs ) ).
% prefix_code(1)
thf(fact_25_prefix__bot_Oextremum__unique,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
= ( A = nil_nat ) ) ).
% prefix_bot.extremum_unique
thf(fact_26_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_27_prefix__Nil,axiom,
! [Xs: list_nat] :
( ( prefix_nat @ Xs @ nil_nat )
= ( Xs = nil_nat ) ) ).
% prefix_Nil
thf(fact_28_prefix__Nil,axiom,
! [Xs: list_list_nat] :
( ( prefix_list_nat @ Xs @ nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% prefix_Nil
thf(fact_29_Cons__prefix__Cons,axiom,
! [X2: nat,Xs: list_nat,Y2: nat,Ys3: list_nat] :
( ( prefix_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys3 ) )
= ( ( X2 = Y2 )
& ( prefix_nat @ Xs @ Ys3 ) ) ) ).
% Cons_prefix_Cons
thf(fact_30_Cons__prefix__Cons,axiom,
! [X2: list_nat,Xs: list_list_nat,Y2: list_nat,Ys3: list_list_nat] :
( ( prefix_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys3 ) )
= ( ( X2 = Y2 )
& ( prefix_list_nat @ Xs @ Ys3 ) ) ) ).
% Cons_prefix_Cons
thf(fact_31_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_32_append_Oright__neutral,axiom,
! [A: list_list_nat] :
( ( append_list_nat @ A @ nil_list_nat )
= A ) ).
% append.right_neutral
thf(fact_33_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_34_append__Nil2,axiom,
! [Xs: list_list_nat] :
( ( append_list_nat @ Xs @ nil_list_nat )
= Xs ) ).
% append_Nil2
thf(fact_35_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_36_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_37_same__append__eq,axiom,
! [Xs: list_nat,Ys3: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys3 )
= ( append_nat @ Xs @ Zs ) )
= ( Ys3 = Zs ) ) ).
% same_append_eq
thf(fact_38_same__append__eq,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,Zs: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys3 )
= ( append_list_nat @ Xs @ Zs ) )
= ( Ys3 = Zs ) ) ).
% same_append_eq
thf(fact_39_append__same__eq,axiom,
! [Ys3: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys3 @ Xs )
= ( append_nat @ Zs @ Xs ) )
= ( Ys3 = Zs ) ) ).
% append_same_eq
thf(fact_40_append__same__eq,axiom,
! [Ys3: list_list_nat,Xs: list_list_nat,Zs: list_list_nat] :
( ( ( append_list_nat @ Ys3 @ Xs )
= ( append_list_nat @ Zs @ Xs ) )
= ( Ys3 = Zs ) ) ).
% append_same_eq
thf(fact_41_append__assoc,axiom,
! [Xs: list_nat,Ys3: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs @ Ys3 ) @ Zs )
= ( append_nat @ Xs @ ( append_nat @ Ys3 @ Zs ) ) ) ).
% append_assoc
thf(fact_42_append__assoc,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,Zs: list_list_nat] :
( ( append_list_nat @ ( append_list_nat @ Xs @ Ys3 ) @ Zs )
= ( append_list_nat @ Xs @ ( append_list_nat @ Ys3 @ Zs ) ) ) ).
% append_assoc
thf(fact_43_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_44_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_45_prefix__order_Odual__order_Orefl,axiom,
! [A: list_nat] : ( prefix_nat @ A @ A ) ).
% prefix_order.dual_order.refl
thf(fact_46_prefix__order_Odual__order_Orefl,axiom,
! [A: list_list_nat] : ( prefix_list_nat @ A @ A ) ).
% prefix_order.dual_order.refl
thf(fact_47_prefix__order_Oorder__refl,axiom,
! [X2: list_nat] : ( prefix_nat @ X2 @ X2 ) ).
% prefix_order.order_refl
thf(fact_48_prefix__order_Oorder__refl,axiom,
! [X2: list_list_nat] : ( prefix_list_nat @ X2 @ X2 ) ).
% prefix_order.order_refl
thf(fact_49_lessThan__eq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ( set_ord_lessThan_nat @ X2 )
= ( set_ord_lessThan_nat @ Y2 ) )
= ( X2 = Y2 ) ) ).
% lessThan_eq_iff
thf(fact_50_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( ( append_nat @ Xs @ Ys3 )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys3 = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_51_append__is__Nil__conv,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys3 )
= nil_list_nat )
= ( ( Xs = nil_list_nat )
& ( Ys3 = nil_list_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_52_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys3 ) )
= ( ( Xs = nil_nat )
& ( Ys3 = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_53_Nil__is__append__conv,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( nil_list_nat
= ( append_list_nat @ Xs @ Ys3 ) )
= ( ( Xs = nil_list_nat )
& ( Ys3 = nil_list_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_54_self__append__conv2,axiom,
! [Y2: list_nat,Xs: list_nat] :
( ( Y2
= ( append_nat @ Xs @ Y2 ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_55_self__append__conv2,axiom,
! [Y2: list_list_nat,Xs: list_list_nat] :
( ( Y2
= ( append_list_nat @ Xs @ Y2 ) )
= ( Xs = nil_list_nat ) ) ).
% self_append_conv2
thf(fact_56_append__self__conv2,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( ( append_nat @ Xs @ Ys3 )
= Ys3 )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_57_append__self__conv2,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys3 )
= Ys3 )
= ( Xs = nil_list_nat ) ) ).
% append_self_conv2
thf(fact_58_self__append__conv,axiom,
! [Y2: list_nat,Ys3: list_nat] :
( ( Y2
= ( append_nat @ Y2 @ Ys3 ) )
= ( Ys3 = nil_nat ) ) ).
% self_append_conv
thf(fact_59_self__append__conv,axiom,
! [Y2: list_list_nat,Ys3: list_list_nat] :
( ( Y2
= ( append_list_nat @ Y2 @ Ys3 ) )
= ( Ys3 = nil_list_nat ) ) ).
% self_append_conv
thf(fact_60_append__self__conv,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( ( append_nat @ Xs @ Ys3 )
= Xs )
= ( Ys3 = nil_nat ) ) ).
% append_self_conv
thf(fact_61_append__self__conv,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys3 )
= Xs )
= ( Ys3 = nil_list_nat ) ) ).
% append_self_conv
thf(fact_62_subset__code_I1_J,axiom,
! [Xs: list_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ B2 )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( member_list_nat @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_63_subset__code_I1_J,axiom,
! [Xs: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_64_set__subset__Cons,axiom,
! [Xs: list_nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_65_set__subset__Cons,axiom,
! [Xs: list_list_nat,X2: list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ ( cons_list_nat @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_66_set__mono__prefix,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( prefix_nat @ Xs @ Ys3 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys3 ) ) ) ).
% set_mono_prefix
thf(fact_67_set__mono__prefix,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( prefix_list_nat @ Xs @ Ys3 )
=> ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys3 ) ) ) ).
% set_mono_prefix
thf(fact_68_not__Cons__self2,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_69_not__Cons__self2,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( cons_list_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_70_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_71_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys3 ) )
=> ( Xs != Ys3 ) ) ).
% neq_if_length_neq
thf(fact_72_neq__if__length__neq,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
!= ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( Xs != Ys3 ) ) ).
% neq_if_length_neq
thf(fact_73_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_74_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_nat] :
( ( size_s3023201423986296836st_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_75_append__eq__append__conv2,axiom,
! [Xs: list_nat,Ys3: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs @ Ys3 )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us2: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us2 ) )
& ( ( append_nat @ Us2 @ Ys3 )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us2 )
= Zs )
& ( Ys3
= ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_76_append__eq__append__conv2,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,Zs: list_list_nat,Ts: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys3 )
= ( append_list_nat @ Zs @ Ts ) )
= ( ? [Us2: list_list_nat] :
( ( ( Xs
= ( append_list_nat @ Zs @ Us2 ) )
& ( ( append_list_nat @ Us2 @ Ys3 )
= Ts ) )
| ( ( ( append_list_nat @ Xs @ Us2 )
= Zs )
& ( Ys3
= ( append_list_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_77_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_78_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_79_Collect__mem__eq,axiom,
! [A2: set_list_nat] :
( ( collect_list_nat
@ ^ [X: list_nat] : ( member_list_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_81_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys3: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys3
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs @ Ys3 )
= ( append_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_82_append__eq__appendI,axiom,
! [Xs: list_list_nat,Xs1: list_list_nat,Zs: list_list_nat,Ys3: list_list_nat,Us: list_list_nat] :
( ( ( append_list_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys3
= ( append_list_nat @ Xs1 @ Us ) )
=> ( ( append_list_nat @ Xs @ Ys3 )
= ( append_list_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_83_prefix__same__cases,axiom,
! [Xs_1: list_nat,Ys3: list_nat,Xs_2: list_nat] :
( ( prefix_nat @ Xs_1 @ Ys3 )
=> ( ( prefix_nat @ Xs_2 @ Ys3 )
=> ( ( prefix_nat @ Xs_1 @ Xs_2 )
| ( prefix_nat @ Xs_2 @ Xs_1 ) ) ) ) ).
% prefix_same_cases
thf(fact_84_prefix__same__cases,axiom,
! [Xs_1: list_list_nat,Ys3: list_list_nat,Xs_2: list_list_nat] :
( ( prefix_list_nat @ Xs_1 @ Ys3 )
=> ( ( prefix_list_nat @ Xs_2 @ Ys3 )
=> ( ( prefix_list_nat @ Xs_1 @ Xs_2 )
| ( prefix_list_nat @ Xs_2 @ Xs_1 ) ) ) ) ).
% prefix_same_cases
thf(fact_85_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_86_prefix__order_Odual__order_Oantisym,axiom,
! [B: list_list_nat,A: list_list_nat] :
( ( prefix_list_nat @ B @ A )
=> ( ( prefix_list_nat @ A @ B )
=> ( A = B ) ) ) ).
% prefix_order.dual_order.antisym
thf(fact_87_prefix__order_Odual__order_Oeq__iff,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
= ( ^ [A3: list_nat,B3: list_nat] :
( ( prefix_nat @ B3 @ A3 )
& ( prefix_nat @ A3 @ B3 ) ) ) ) ).
% prefix_order.dual_order.eq_iff
thf(fact_88_prefix__order_Odual__order_Oeq__iff,axiom,
( ( ^ [Y4: list_list_nat,Z: list_list_nat] : ( Y4 = Z ) )
= ( ^ [A3: list_list_nat,B3: list_list_nat] :
( ( prefix_list_nat @ B3 @ A3 )
& ( prefix_list_nat @ A3 @ B3 ) ) ) ) ).
% prefix_order.dual_order.eq_iff
thf(fact_89_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_90_prefix__order_Odual__order_Otrans,axiom,
! [B: list_list_nat,A: list_list_nat,C: list_list_nat] :
( ( prefix_list_nat @ B @ A )
=> ( ( prefix_list_nat @ C @ B )
=> ( prefix_list_nat @ C @ A ) ) ) ).
% prefix_order.dual_order.trans
thf(fact_91_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_92_prefix__order_Oord__le__eq__trans,axiom,
! [A: list_list_nat,B: list_list_nat,C: list_list_nat] :
( ( prefix_list_nat @ A @ B )
=> ( ( B = C )
=> ( prefix_list_nat @ A @ C ) ) ) ).
% prefix_order.ord_le_eq_trans
thf(fact_93_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_94_prefix__order_Oord__eq__le__trans,axiom,
! [A: list_list_nat,B: list_list_nat,C: list_list_nat] :
( ( A = B )
=> ( ( prefix_list_nat @ B @ C )
=> ( prefix_list_nat @ A @ C ) ) ) ).
% prefix_order.ord_eq_le_trans
thf(fact_95_prefix__order_Oorder__antisym,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( prefix_nat @ X2 @ Y2 )
=> ( ( prefix_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% prefix_order.order_antisym
thf(fact_96_prefix__order_Oorder__antisym,axiom,
! [X2: list_list_nat,Y2: list_list_nat] :
( ( prefix_list_nat @ X2 @ Y2 )
=> ( ( prefix_list_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% prefix_order.order_antisym
thf(fact_97_prefix__order_Oorder__eq__iff,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
= ( ^ [X: list_nat,Y: list_nat] :
( ( prefix_nat @ X @ Y )
& ( prefix_nat @ Y @ X ) ) ) ) ).
% prefix_order.order_eq_iff
thf(fact_98_prefix__order_Oorder__eq__iff,axiom,
( ( ^ [Y4: list_list_nat,Z: list_list_nat] : ( Y4 = Z ) )
= ( ^ [X: list_list_nat,Y: list_list_nat] :
( ( prefix_list_nat @ X @ Y )
& ( prefix_list_nat @ Y @ X ) ) ) ) ).
% prefix_order.order_eq_iff
thf(fact_99_prefix__order_Oantisym__conv,axiom,
! [Y2: list_nat,X2: list_nat] :
( ( prefix_nat @ Y2 @ X2 )
=> ( ( prefix_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% prefix_order.antisym_conv
thf(fact_100_prefix__order_Oantisym__conv,axiom,
! [Y2: list_list_nat,X2: list_list_nat] :
( ( prefix_list_nat @ Y2 @ X2 )
=> ( ( prefix_list_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% prefix_order.antisym_conv
thf(fact_101_prefix__order_Oorder__trans,axiom,
! [X2: list_nat,Y2: list_nat,Z2: list_nat] :
( ( prefix_nat @ X2 @ Y2 )
=> ( ( prefix_nat @ Y2 @ Z2 )
=> ( prefix_nat @ X2 @ Z2 ) ) ) ).
% prefix_order.order_trans
thf(fact_102_prefix__order_Oorder__trans,axiom,
! [X2: list_list_nat,Y2: list_list_nat,Z2: list_list_nat] :
( ( prefix_list_nat @ X2 @ Y2 )
=> ( ( prefix_list_nat @ Y2 @ Z2 )
=> ( prefix_list_nat @ X2 @ Z2 ) ) ) ).
% prefix_order.order_trans
thf(fact_103_prefix__order_Oeq__refl,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( X2 = Y2 )
=> ( prefix_nat @ X2 @ Y2 ) ) ).
% prefix_order.eq_refl
thf(fact_104_prefix__order_Oeq__refl,axiom,
! [X2: list_list_nat,Y2: list_list_nat] :
( ( X2 = Y2 )
=> ( prefix_list_nat @ X2 @ Y2 ) ) ).
% prefix_order.eq_refl
thf(fact_105_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_106_prefix__order_Oantisym,axiom,
! [A: list_list_nat,B: list_list_nat] :
( ( prefix_list_nat @ A @ B )
=> ( ( prefix_list_nat @ B @ A )
=> ( A = B ) ) ) ).
% prefix_order.antisym
thf(fact_107_prefix__order_Oeq__iff,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
= ( ^ [A3: list_nat,B3: list_nat] :
( ( prefix_nat @ A3 @ B3 )
& ( prefix_nat @ B3 @ A3 ) ) ) ) ).
% prefix_order.eq_iff
thf(fact_108_prefix__order_Oeq__iff,axiom,
( ( ^ [Y4: list_list_nat,Z: list_list_nat] : ( Y4 = Z ) )
= ( ^ [A3: list_list_nat,B3: list_list_nat] :
( ( prefix_list_nat @ A3 @ B3 )
& ( prefix_list_nat @ B3 @ A3 ) ) ) ) ).
% prefix_order.eq_iff
thf(fact_109_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_110_prefix__order_Otrans,axiom,
! [A: list_list_nat,B: list_list_nat,C: list_list_nat] :
( ( prefix_list_nat @ A @ B )
=> ( ( prefix_list_nat @ B @ C )
=> ( prefix_list_nat @ A @ C ) ) ) ).
% prefix_order.trans
thf(fact_111_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_112_list__nonempty__induct,axiom,
! [Xs: list_list_nat,P: list_list_nat > $o] :
( ( Xs != nil_list_nat )
=> ( ! [X3: list_nat] : ( P @ ( cons_list_nat @ X3 @ nil_list_nat ) )
=> ( ! [X3: list_nat,Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_list_nat @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_113_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys3: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [Y3: nat,Ys2: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys2 ) )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys3 ) ) ) ) ) ).
% list_induct2'
thf(fact_114_list__induct2_H,axiom,
! [P: list_nat > list_list_nat > $o,Xs: list_nat,Ys3: list_list_nat] :
( ( P @ nil_nat @ nil_list_nat )
=> ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_list_nat )
=> ( ! [Y3: list_nat,Ys2: list_list_nat] : ( P @ nil_nat @ ( cons_list_nat @ Y3 @ Ys2 ) )
=> ( ! [X3: nat,Xs2: list_nat,Y3: list_nat,Ys2: list_list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys3 ) ) ) ) ) ).
% list_induct2'
thf(fact_115_list__induct2_H,axiom,
! [P: list_list_nat > list_nat > $o,Xs: list_list_nat,Ys3: list_nat] :
( ( P @ nil_list_nat @ nil_nat )
=> ( ! [X3: list_nat,Xs2: list_list_nat] : ( P @ ( cons_list_nat @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [Y3: nat,Ys2: list_nat] : ( P @ nil_list_nat @ ( cons_nat @ Y3 @ Ys2 ) )
=> ( ! [X3: list_nat,Xs2: list_list_nat,Y3: nat,Ys2: list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys3 ) ) ) ) ) ).
% list_induct2'
thf(fact_116_list__induct2_H,axiom,
! [P: list_list_nat > list_list_nat > $o,Xs: list_list_nat,Ys3: list_list_nat] :
( ( P @ nil_list_nat @ nil_list_nat )
=> ( ! [X3: list_nat,Xs2: list_list_nat] : ( P @ ( cons_list_nat @ X3 @ Xs2 ) @ nil_list_nat )
=> ( ! [Y3: list_nat,Ys2: list_list_nat] : ( P @ nil_list_nat @ ( cons_list_nat @ Y3 @ Ys2 ) )
=> ( ! [X3: list_nat,Xs2: list_list_nat,Y3: list_nat,Ys2: list_list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys3 ) ) ) ) ) ).
% list_induct2'
thf(fact_117_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y: nat,Ys: list_nat] :
( Xs
= ( cons_nat @ Y @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_118_neq__Nil__conv,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
= ( ? [Y: list_nat,Ys: list_list_nat] :
( Xs
= ( cons_list_nat @ Y @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_119_remdups__adj_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X3: nat] :
( X2
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( X2
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_120_remdups__adj_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [X3: list_nat] :
( X2
!= ( cons_list_nat @ X3 @ nil_list_nat ) )
=> ~ ! [X3: list_nat,Y3: list_nat,Xs2: list_list_nat] :
( X2
!= ( cons_list_nat @ X3 @ ( cons_list_nat @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_121_transpose_Ocases,axiom,
! [X2: list_list_list_nat] :
( ( X2 != nil_list_list_nat )
=> ( ! [Xss: list_list_list_nat] :
( X2
!= ( cons_list_list_nat @ nil_list_nat @ Xss ) )
=> ~ ! [X3: list_nat,Xs2: list_list_nat,Xss: list_list_list_nat] :
( X2
!= ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_122_transpose_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_123_min__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X3: nat,Xs2: list_nat] :
( X2
!= ( cons_nat @ X3 @ Xs2 ) )
=> ( X2 = nil_nat ) ) ).
% min_list.cases
thf(fact_124_list_Oexhaust,axiom,
! [Y2: list_nat] :
( ( Y2 != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y2
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_125_list_Oexhaust,axiom,
! [Y2: list_list_nat] :
( ( Y2 != nil_list_nat )
=> ~ ! [X212: list_nat,X222: list_list_nat] :
( Y2
!= ( cons_list_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_126_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_127_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_128_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_129_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_130_set__ConsD,axiom,
! [Y2: nat,X2: nat,Xs: list_nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_nat @ Y2 @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_131_set__ConsD,axiom,
! [Y2: list_nat,X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ Y2 @ ( set_list_nat2 @ ( cons_list_nat @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_list_nat @ Y2 @ ( set_list_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_132_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_133_list_Oset__cases,axiom,
! [E: list_nat,A: list_list_nat] :
( ( member_list_nat @ E @ ( set_list_nat2 @ A ) )
=> ( ! [Z22: list_list_nat] :
( A
!= ( cons_list_nat @ E @ Z22 ) )
=> ~ ! [Z1: list_nat,Z22: list_list_nat] :
( ( A
= ( cons_list_nat @ Z1 @ Z22 ) )
=> ~ ( member_list_nat @ E @ ( set_list_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_134_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_135_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_136_list_Oset__intros_I2_J,axiom,
! [Y2: nat,X22: list_nat,X21: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ X22 ) )
=> ( member_nat @ Y2 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_137_list_Oset__intros_I2_J,axiom,
! [Y2: list_nat,X22: list_list_nat,X21: list_nat] :
( ( member_list_nat @ Y2 @ ( set_list_nat2 @ X22 ) )
=> ( member_list_nat @ Y2 @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_138_Cons__eq__appendI,axiom,
! [X2: nat,Xs1: list_nat,Ys3: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X2 @ Xs1 )
= Ys3 )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X2 @ Xs )
= ( append_nat @ Ys3 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_139_Cons__eq__appendI,axiom,
! [X2: list_nat,Xs1: list_list_nat,Ys3: list_list_nat,Xs: list_list_nat,Zs: list_list_nat] :
( ( ( cons_list_nat @ X2 @ Xs1 )
= Ys3 )
=> ( ( Xs
= ( append_list_nat @ Xs1 @ Zs ) )
=> ( ( cons_list_nat @ X2 @ Xs )
= ( append_list_nat @ Ys3 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_140_append__Cons,axiom,
! [X2: nat,Xs: list_nat,Ys3: list_nat] :
( ( append_nat @ ( cons_nat @ X2 @ Xs ) @ Ys3 )
= ( cons_nat @ X2 @ ( append_nat @ Xs @ Ys3 ) ) ) ).
% append_Cons
thf(fact_141_append__Cons,axiom,
! [X2: list_nat,Xs: list_list_nat,Ys3: list_list_nat] :
( ( append_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ Ys3 )
= ( cons_list_nat @ X2 @ ( append_list_nat @ Xs @ Ys3 ) ) ) ).
% append_Cons
thf(fact_142_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( Xs = Ys3 )
=> ( Xs
= ( append_nat @ nil_nat @ Ys3 ) ) ) ).
% eq_Nil_appendI
thf(fact_143_eq__Nil__appendI,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( Xs = Ys3 )
=> ( Xs
= ( append_list_nat @ nil_list_nat @ Ys3 ) ) ) ).
% eq_Nil_appendI
thf(fact_144_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_145_append_Oleft__neutral,axiom,
! [A: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_146_append__Nil,axiom,
! [Ys3: list_nat] :
( ( append_nat @ nil_nat @ Ys3 )
= Ys3 ) ).
% append_Nil
thf(fact_147_append__Nil,axiom,
! [Ys3: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ Ys3 )
= Ys3 ) ).
% append_Nil
thf(fact_148_Nil__prefix,axiom,
! [Xs: list_nat] : ( prefix_nat @ nil_nat @ Xs ) ).
% Nil_prefix
thf(fact_149_Nil__prefix,axiom,
! [Xs: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ Xs ) ).
% Nil_prefix
thf(fact_150_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
=> ( A = nil_nat ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_151_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_152_prefix__bot_Obot__least,axiom,
! [A: list_nat] : ( prefix_nat @ nil_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_153_prefix__bot_Obot__least,axiom,
! [A: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_154_append__prefixD,axiom,
! [Xs: list_nat,Ys3: list_nat,Zs: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ Ys3 ) @ Zs )
=> ( prefix_nat @ Xs @ Zs ) ) ).
% append_prefixD
thf(fact_155_append__prefixD,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,Zs: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs @ Ys3 ) @ Zs )
=> ( prefix_list_nat @ Xs @ Zs ) ) ).
% append_prefixD
thf(fact_156_prefix__prefix,axiom,
! [Xs: list_nat,Ys3: list_nat,Zs: list_nat] :
( ( prefix_nat @ Xs @ Ys3 )
=> ( prefix_nat @ Xs @ ( append_nat @ Ys3 @ Zs ) ) ) ).
% prefix_prefix
thf(fact_157_prefix__prefix,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,Zs: list_list_nat] :
( ( prefix_list_nat @ Xs @ Ys3 )
=> ( prefix_list_nat @ Xs @ ( append_list_nat @ Ys3 @ Zs ) ) ) ).
% prefix_prefix
thf(fact_158_prefix__append,axiom,
! [Xs: list_nat,Ys3: list_nat,Zs: list_nat] :
( ( prefix_nat @ Xs @ ( append_nat @ Ys3 @ Zs ) )
= ( ( prefix_nat @ Xs @ Ys3 )
| ? [Us2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ Us2 ) )
& ( prefix_nat @ Us2 @ Zs ) ) ) ) ).
% prefix_append
thf(fact_159_prefix__append,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,Zs: list_list_nat] :
( ( prefix_list_nat @ Xs @ ( append_list_nat @ Ys3 @ Zs ) )
= ( ( prefix_list_nat @ Xs @ Ys3 )
| ? [Us2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys3 @ Us2 ) )
& ( prefix_list_nat @ Us2 @ Zs ) ) ) ) ).
% prefix_append
thf(fact_160_prefix__def,axiom,
( prefix_nat
= ( ^ [Xs3: list_nat,Ys: list_nat] :
? [Zs2: list_nat] :
( Ys
= ( append_nat @ Xs3 @ Zs2 ) ) ) ) ).
% prefix_def
thf(fact_161_prefix__def,axiom,
( prefix_list_nat
= ( ^ [Xs3: list_list_nat,Ys: list_list_nat] :
? [Zs2: list_list_nat] :
( Ys
= ( append_list_nat @ Xs3 @ Zs2 ) ) ) ) ).
% prefix_def
thf(fact_162_prefixI,axiom,
! [Ys3: list_nat,Xs: list_nat,Zs: list_nat] :
( ( Ys3
= ( append_nat @ Xs @ Zs ) )
=> ( prefix_nat @ Xs @ Ys3 ) ) ).
% prefixI
thf(fact_163_prefixI,axiom,
! [Ys3: list_list_nat,Xs: list_list_nat,Zs: list_list_nat] :
( ( Ys3
= ( append_list_nat @ Xs @ Zs ) )
=> ( prefix_list_nat @ Xs @ Ys3 ) ) ).
% prefixI
thf(fact_164_prefixE,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( prefix_nat @ Xs @ Ys3 )
=> ~ ! [Zs3: list_nat] :
( Ys3
!= ( append_nat @ Xs @ Zs3 ) ) ) ).
% prefixE
thf(fact_165_prefixE,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( prefix_list_nat @ Xs @ Ys3 )
=> ~ ! [Zs3: list_list_nat] :
( Ys3
!= ( append_list_nat @ Xs @ Zs3 ) ) ) ).
% prefixE
thf(fact_166_list__induct4,axiom,
! [Xs: list_nat,Ys3: list_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat,Z3: nat,Zs3: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_167_list__induct4,axiom,
! [Xs: list_nat,Ys3: list_nat,Zs: list_nat,Ws: list_list_nat,P: list_nat > list_nat > list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s3023201423986296836st_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_list_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat,Z3: nat,Zs3: list_nat,W: list_nat,Ws2: list_list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_s3023201423986296836st_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_list_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_168_list__induct4,axiom,
! [Xs: list_nat,Ys3: list_nat,Zs: list_list_nat,Ws: list_nat,P: list_nat > list_nat > list_list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_list_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat,Z3: list_nat,Zs3: list_list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs3 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_list_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_169_list__induct4,axiom,
! [Xs: list_nat,Ys3: list_list_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys3 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_list_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: list_nat,Ys2: list_list_nat,Z3: nat,Zs3: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_170_list__induct4,axiom,
! [Xs: list_list_nat,Ys3: list_nat,Zs: list_nat,Ws: list_nat,P: list_list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_list_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: list_nat,Xs2: list_list_nat,Y3: nat,Ys2: list_nat,Z3: nat,Zs3: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
=> ( P @ ( cons_list_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_171_list__induct4,axiom,
! [Xs: list_nat,Ys3: list_nat,Zs: list_list_nat,Ws: list_list_nat,P: list_nat > list_nat > list_list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs )
= ( size_s3023201423986296836st_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_list_nat @ nil_list_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat,Z3: list_nat,Zs3: list_list_nat,W: list_nat,Ws2: list_list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs3 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs3 )
= ( size_s3023201423986296836st_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_list_nat @ Z3 @ Zs3 ) @ ( cons_list_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_172_list__induct4,axiom,
! [Xs: list_nat,Ys3: list_list_nat,Zs: list_nat,Ws: list_list_nat,P: list_nat > list_list_nat > list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys3 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s3023201423986296836st_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_list_nat @ nil_nat @ nil_list_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: list_nat,Ys2: list_list_nat,Z3: nat,Zs3: list_nat,W: list_nat,Ws2: list_list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_s3023201423986296836st_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_list_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_173_list__induct4,axiom,
! [Xs: list_nat,Ys3: list_list_nat,Zs: list_list_nat,Ws: list_nat,P: list_nat > list_list_nat > list_list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys3 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_list_nat @ nil_list_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: list_nat,Ys2: list_list_nat,Z3: list_nat,Zs3: list_list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs3 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) @ ( cons_list_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_174_list__induct4,axiom,
! [Xs: list_list_nat,Ys3: list_nat,Zs: list_nat,Ws: list_list_nat,P: list_list_nat > list_nat > list_nat > list_list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s3023201423986296836st_nat @ Ws ) )
=> ( ( P @ nil_list_nat @ nil_nat @ nil_nat @ nil_list_nat )
=> ( ! [X3: list_nat,Xs2: list_list_nat,Y3: nat,Ys2: list_nat,Z3: nat,Zs3: list_nat,W: list_nat,Ws2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_s3023201423986296836st_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
=> ( P @ ( cons_list_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_list_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_175_list__induct4,axiom,
! [Xs: list_list_nat,Ys3: list_nat,Zs: list_list_nat,Ws: list_nat,P: list_list_nat > list_nat > list_list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_list_nat @ nil_nat @ nil_list_nat @ nil_nat )
=> ( ! [X3: list_nat,Xs2: list_list_nat,Y3: nat,Ys2: list_nat,Z3: list_nat,Zs3: list_list_nat,W: nat,Ws2: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs3 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
=> ( P @ ( cons_list_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_list_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_176_list__induct3,axiom,
! [Xs: list_nat,Ys3: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat,Z3: nat,Zs3: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_177_list__induct3,axiom,
! [Xs: list_nat,Ys3: list_nat,Zs: list_list_nat,P: list_nat > list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_list_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat,Z3: list_nat,Zs3: list_list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_list_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_178_list__induct3,axiom,
! [Xs: list_nat,Ys3: list_list_nat,Zs: list_nat,P: list_nat > list_list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys3 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_list_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: list_nat,Ys2: list_list_nat,Z3: nat,Zs3: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_179_list__induct3,axiom,
! [Xs: list_nat,Ys3: list_list_nat,Zs: list_list_nat,P: list_nat > list_list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys3 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_list_nat @ nil_list_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: list_nat,Ys2: list_list_nat,Z3: list_nat,Zs3: list_list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) @ ( cons_list_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_180_list__induct3,axiom,
! [Xs: list_list_nat,Ys3: list_nat,Zs: list_nat,P: list_list_nat > list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_list_nat @ nil_nat @ nil_nat )
=> ( ! [X3: list_nat,Xs2: list_list_nat,Y3: nat,Ys2: list_nat,Z3: nat,Zs3: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons_list_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_181_list__induct3,axiom,
! [Xs: list_list_nat,Ys3: list_nat,Zs: list_list_nat,P: list_list_nat > list_nat > list_list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( P @ nil_list_nat @ nil_nat @ nil_list_nat )
=> ( ! [X3: list_nat,Xs2: list_list_nat,Y3: nat,Ys2: list_nat,Z3: list_nat,Zs3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons_list_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_list_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_182_list__induct3,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,Zs: list_nat,P: list_list_nat > list_list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys3 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_list_nat @ nil_list_nat @ nil_nat )
=> ( ! [X3: list_nat,Xs2: list_list_nat,Y3: list_nat,Ys2: list_list_nat,Z3: nat,Zs3: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons_list_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_183_list__induct3,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,Zs: list_list_nat,P: list_list_nat > list_list_nat > list_list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys3 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( P @ nil_list_nat @ nil_list_nat @ nil_list_nat )
=> ( ! [X3: list_nat,Xs2: list_list_nat,Y3: list_nat,Ys2: list_list_nat,Z3: list_nat,Zs3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons_list_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) @ ( cons_list_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys3 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_184_list__induct2,axiom,
! [Xs: list_nat,Ys3: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys3 ) ) ) ) ).
% list_induct2
thf(fact_185_list__induct2,axiom,
! [Xs: list_nat,Ys3: list_list_nat,P: list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( ( P @ nil_nat @ nil_list_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: list_nat,Ys2: list_list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys3 ) ) ) ) ).
% list_induct2
thf(fact_186_list__induct2,axiom,
! [Xs: list_list_nat,Ys3: list_nat,P: list_list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( P @ nil_list_nat @ nil_nat )
=> ( ! [X3: list_nat,Xs2: list_list_nat,Y3: nat,Ys2: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys3 ) ) ) ) ).
% list_induct2
thf(fact_187_list__induct2,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,P: list_list_nat > list_list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( ( P @ nil_list_nat @ nil_list_nat )
=> ( ! [X3: list_nat,Xs2: list_list_nat,Y3: list_nat,Ys2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys3 ) ) ) ) ).
% list_induct2
thf(fact_188_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_189_rev__nonempty__induct,axiom,
! [Xs: list_list_nat,P: list_list_nat > $o] :
( ( Xs != nil_list_nat )
=> ( ! [X3: list_nat] : ( P @ ( cons_list_nat @ X3 @ nil_list_nat ) )
=> ( ! [X3: list_nat,Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X3 @ nil_list_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_190_append__eq__Cons__conv,axiom,
! [Ys3: list_nat,Zs: list_nat,X2: nat,Xs: list_nat] :
( ( ( append_nat @ Ys3 @ Zs )
= ( cons_nat @ X2 @ Xs ) )
= ( ( ( Ys3 = nil_nat )
& ( Zs
= ( cons_nat @ X2 @ Xs ) ) )
| ? [Ys4: list_nat] :
( ( Ys3
= ( cons_nat @ X2 @ Ys4 ) )
& ( ( append_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_191_append__eq__Cons__conv,axiom,
! [Ys3: list_list_nat,Zs: list_list_nat,X2: list_nat,Xs: list_list_nat] :
( ( ( append_list_nat @ Ys3 @ Zs )
= ( cons_list_nat @ X2 @ Xs ) )
= ( ( ( Ys3 = nil_list_nat )
& ( Zs
= ( cons_list_nat @ X2 @ Xs ) ) )
| ? [Ys4: list_list_nat] :
( ( Ys3
= ( cons_list_nat @ X2 @ Ys4 ) )
& ( ( append_list_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_192_Cons__eq__append__conv,axiom,
! [X2: nat,Xs: list_nat,Ys3: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( append_nat @ Ys3 @ Zs ) )
= ( ( ( Ys3 = nil_nat )
& ( ( cons_nat @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_nat] :
( ( ( cons_nat @ X2 @ Ys4 )
= Ys3 )
& ( Xs
= ( append_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_193_Cons__eq__append__conv,axiom,
! [X2: list_nat,Xs: list_list_nat,Ys3: list_list_nat,Zs: list_list_nat] :
( ( ( cons_list_nat @ X2 @ Xs )
= ( append_list_nat @ Ys3 @ Zs ) )
= ( ( ( Ys3 = nil_list_nat )
& ( ( cons_list_nat @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_list_nat] :
( ( ( cons_list_nat @ X2 @ Ys4 )
= Ys3 )
& ( Xs
= ( append_list_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_194_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys2: list_nat,Y3: nat] :
( Xs
!= ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_195_rev__exhaust,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ~ ! [Ys2: list_list_nat,Y3: list_nat] :
( Xs
!= ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) ) ) ).
% rev_exhaust
thf(fact_196_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat] :
( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_197_rev__induct,axiom,
! [P: list_list_nat > $o,Xs: list_list_nat] :
( ( P @ nil_list_nat )
=> ( ! [X3: list_nat,Xs2: list_list_nat] :
( ( P @ Xs2 )
=> ( P @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X3 @ nil_list_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_198_impossible__Cons,axiom,
! [Xs: list_nat,Ys3: list_nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys3 ) )
=> ( Xs
!= ( cons_nat @ X2 @ Ys3 ) ) ) ).
% impossible_Cons
thf(fact_199_impossible__Cons,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,X2: list_nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( Xs
!= ( cons_list_nat @ X2 @ Ys3 ) ) ) ).
% impossible_Cons
thf(fact_200_not__prefix__induct,axiom,
! [Ps: list_nat,Ls: list_nat,P: list_nat > list_nat > $o] :
( ~ ( prefix_nat @ Ps @ Ls )
=> ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat] :
( ( X3 != Y3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat] :
( ( X3 = Y3 )
=> ( ~ ( prefix_nat @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_201_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 )
=> ( ! [X3: list_nat,Xs2: list_list_nat] : ( P @ ( cons_list_nat @ X3 @ Xs2 ) @ nil_list_nat )
=> ( ! [X3: list_nat,Xs2: list_list_nat,Y3: list_nat,Ys2: list_list_nat] :
( ( X3 != Y3 )
=> ( P @ ( cons_list_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) ) )
=> ( ! [X3: list_nat,Xs2: list_list_nat,Y3: list_nat,Ys2: list_list_nat] :
( ( X3 = Y3 )
=> ( ~ ( prefix_list_nat @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_202_not__prefix__cases,axiom,
! [Ps: list_nat,Ls: list_nat] :
( ~ ( prefix_nat @ Ps @ Ls )
=> ( ( ( Ps != nil_nat )
=> ( Ls != nil_nat ) )
=> ( ! [A4: nat,As: list_nat] :
( ( Ps
= ( cons_nat @ A4 @ As ) )
=> ! [X3: nat,Xs2: list_nat] :
( ( Ls
= ( cons_nat @ X3 @ Xs2 ) )
=> ( ( X3 = A4 )
=> ( prefix_nat @ As @ Xs2 ) ) ) )
=> ~ ! [A4: nat] :
( ? [As: list_nat] :
( Ps
= ( cons_nat @ A4 @ As ) )
=> ! [X3: nat] :
( ? [Xs2: list_nat] :
( Ls
= ( cons_nat @ X3 @ Xs2 ) )
=> ( X3 = A4 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_203_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 ) )
=> ( ! [A4: list_nat,As: list_list_nat] :
( ( Ps
= ( cons_list_nat @ A4 @ As ) )
=> ! [X3: list_nat,Xs2: list_list_nat] :
( ( Ls
= ( cons_list_nat @ X3 @ Xs2 ) )
=> ( ( X3 = A4 )
=> ( prefix_list_nat @ As @ Xs2 ) ) ) )
=> ~ ! [A4: list_nat] :
( ? [As: list_list_nat] :
( Ps
= ( cons_list_nat @ A4 @ As ) )
=> ! [X3: list_nat] :
( ? [Xs2: list_list_nat] :
( Ls
= ( cons_list_nat @ X3 @ Xs2 ) )
=> ( X3 = A4 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_204_prefix__Cons,axiom,
! [Xs: list_nat,Y2: nat,Ys3: list_nat] :
( ( prefix_nat @ Xs @ ( cons_nat @ Y2 @ Ys3 ) )
= ( ( Xs = nil_nat )
| ? [Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Zs2 ) )
& ( prefix_nat @ Zs2 @ Ys3 ) ) ) ) ).
% prefix_Cons
thf(fact_205_prefix__Cons,axiom,
! [Xs: list_list_nat,Y2: list_nat,Ys3: list_list_nat] :
( ( prefix_list_nat @ Xs @ ( cons_list_nat @ Y2 @ Ys3 ) )
= ( ( Xs = nil_list_nat )
| ? [Zs2: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Y2 @ Zs2 ) )
& ( prefix_list_nat @ Zs2 @ Ys3 ) ) ) ) ).
% prefix_Cons
thf(fact_206_prefix__code_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
~ ( prefix_nat @ ( cons_nat @ X2 @ Xs ) @ nil_nat ) ).
% prefix_code(2)
thf(fact_207_prefix__code_I2_J,axiom,
! [X2: list_nat,Xs: list_list_nat] :
~ ( prefix_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ nil_list_nat ) ).
% prefix_code(2)
thf(fact_208_split__list__first__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys: list_nat,X: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Y: nat] :
( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
=> ~ ( P @ Y ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_209_split__list__first__prop__iff,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ( ? [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys: list_list_nat,X: list_nat] :
( ? [Zs2: list_list_nat] :
( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Y: list_nat] :
( ( member_list_nat @ Y @ ( set_list_nat2 @ Ys ) )
=> ~ ( P @ Y ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_210_split__list__last__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys: list_nat,X: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Y: nat] :
( ( member_nat @ Y @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Y ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_211_split__list__last__prop__iff,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ( ? [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys: list_list_nat,X: list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Y: list_nat] :
( ( member_list_nat @ Y @ ( set_list_nat2 @ Zs2 ) )
=> ~ ( P @ Y ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_212_in__set__conv__decomp__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Ys ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_213_in__set__conv__decomp__first,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys: list_list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ X2 @ Zs2 ) ) )
& ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Ys ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_214_in__set__conv__decomp__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_215_in__set__conv__decomp__last,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys: list_list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ X2 @ Zs2 ) ) )
& ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_216_split__list__first__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_nat,X3: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_217_split__list__first__propE,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_list_nat,X3: list_nat] :
( ? [Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ ( set_list_nat2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_218_split__list__last__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_nat,X3: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_219_split__list__last__propE,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_list_nat,X3: list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ ( set_list_nat2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_220_split__list__first__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_nat,X3: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_221_split__list__first__prop,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_list_nat,X3: list_nat] :
( ? [Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ ( set_list_nat2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_222_split__list__last__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_nat,X3: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_223_split__list__last__prop,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_list_nat,X3: list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ ( set_list_nat2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_224_in__set__conv__decomp,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_225_in__set__conv__decomp,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys: list_list_nat,Zs2: list_list_nat] :
( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_226_append__Cons__eq__iff,axiom,
! [X2: nat,Xs: list_nat,Ys3: list_nat,Xs4: list_nat,Ys5: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat @ X2 @ ( set_nat2 @ Ys3 ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys3 ) )
= ( append_nat @ Xs4 @ ( cons_nat @ X2 @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys3 = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_227_append__Cons__eq__iff,axiom,
! [X2: list_nat,Xs: list_list_nat,Ys3: list_list_nat,Xs4: list_list_nat,Ys5: list_list_nat] :
( ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Ys3 ) )
=> ( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ Ys3 ) )
= ( append_list_nat @ Xs4 @ ( cons_list_nat @ X2 @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys3 = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_228_split__list__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_nat,X3: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_229_split__list__propE,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_list_nat,X3: list_nat] :
( ? [Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_230_split__list__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_231_split__list__first,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ? [Ys2: list_list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X2 @ Zs3 ) ) )
& ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_232_split__list__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_nat,X3: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_233_split__list__prop,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_list_nat,X3: list_nat] :
( ? [Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_234_split__list__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_235_split__list__last,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ? [Ys2: list_list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X2 @ Zs3 ) ) )
& ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_236_split__list,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_237_split__list,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ? [Ys2: list_list_nat,Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_238_prefix__length__prefix,axiom,
! [Ps: list_nat,Xs: list_nat,Qs: list_nat] :
( ( prefix_nat @ Ps @ Xs )
=> ( ( prefix_nat @ Qs @ Xs )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ps ) @ ( size_size_list_nat @ Qs ) )
=> ( prefix_nat @ Ps @ Qs ) ) ) ) ).
% prefix_length_prefix
thf(fact_239_prefix__length__prefix,axiom,
! [Ps: list_list_nat,Xs: list_list_nat,Qs: list_list_nat] :
( ( prefix_list_nat @ Ps @ Xs )
=> ( ( prefix_list_nat @ Qs @ Xs )
=> ( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Ps ) @ ( size_s3023201423986296836st_nat @ Qs ) )
=> ( prefix_list_nat @ Ps @ Qs ) ) ) ) ).
% prefix_length_prefix
thf(fact_240_prefix__length__le,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( prefix_nat @ Xs @ Ys3 )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys3 ) ) ) ).
% prefix_length_le
thf(fact_241_prefix__length__le,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( prefix_list_nat @ Xs @ Ys3 )
=> ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ ( size_s3023201423986296836st_nat @ Ys3 ) ) ) ).
% prefix_length_le
thf(fact_242_count__le__length,axiom,
! [Xs: list_nat,X2: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X2 ) @ ( size_size_list_nat @ Xs ) ) ).
% count_le_length
thf(fact_243_count__le__length,axiom,
! [Xs: list_list_nat,X2: list_nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs @ X2 ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ).
% count_le_length
thf(fact_244_same__length__different,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( Xs != Ys3 )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ? [Pre: list_nat,X3: nat,Xs5: list_nat,Y3: nat,Ys6: list_nat] :
( ( X3 != Y3 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X3 @ nil_nat ) @ Xs5 ) ) )
& ( Ys3
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_245_same__length__different,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( Xs != Ys3 )
=> ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ? [Pre: list_list_nat,X3: list_nat,Xs5: list_list_nat,Y3: list_nat,Ys6: list_list_nat] :
( ( X3 != Y3 )
& ( Xs
= ( append_list_nat @ Pre @ ( append_list_nat @ ( cons_list_nat @ X3 @ nil_list_nat ) @ Xs5 ) ) )
& ( Ys3
= ( append_list_nat @ Pre @ ( append_list_nat @ ( cons_list_nat @ Y3 @ nil_list_nat ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_246_subsetI,axiom,
! [A2: set_list_nat,B2: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
=> ( member_list_nat @ X3 @ B2 ) )
=> ( ord_le6045566169113846134st_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_247_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_248_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_249_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_250_the__elem__set,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X2 @ nil_nat ) ) )
= X2 ) ).
% the_elem_set
thf(fact_251_the__elem__set,axiom,
! [X2: list_nat] :
( ( the_elem_list_nat @ ( set_list_nat2 @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= X2 ) ).
% the_elem_set
thf(fact_252_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_253_bind__simps_I2_J,axiom,
! [X2: nat,Xs: list_nat,F: nat > list_nat] :
( ( bind_nat_nat @ ( cons_nat @ X2 @ Xs ) @ F )
= ( append_nat @ ( F @ X2 ) @ ( bind_nat_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_254_bind__simps_I2_J,axiom,
! [X2: nat,Xs: list_nat,F: nat > list_list_nat] :
( ( bind_nat_list_nat @ ( cons_nat @ X2 @ Xs ) @ F )
= ( append_list_nat @ ( F @ X2 ) @ ( bind_nat_list_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_255_bind__simps_I2_J,axiom,
! [X2: list_nat,Xs: list_list_nat,F: list_nat > list_nat] :
( ( bind_list_nat_nat @ ( cons_list_nat @ X2 @ Xs ) @ F )
= ( append_nat @ ( F @ X2 ) @ ( bind_list_nat_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_256_bind__simps_I2_J,axiom,
! [X2: list_nat,Xs: list_list_nat,F: list_nat > list_list_nat] :
( ( bind_l7796378977173581257st_nat @ ( cons_list_nat @ X2 @ Xs ) @ F )
= ( append_list_nat @ ( F @ X2 ) @ ( bind_l7796378977173581257st_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_257_rotate1__length01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_258_rotate1__length01,axiom,
! [Xs: list_list_nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_list_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_259_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_260_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_261_maps__simps_I1_J,axiom,
! [F: nat > list_nat,X2: nat,Xs: list_nat] :
( ( maps_nat_nat @ F @ ( cons_nat @ X2 @ Xs ) )
= ( append_nat @ ( F @ X2 ) @ ( maps_nat_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_262_maps__simps_I1_J,axiom,
! [F: nat > list_list_nat,X2: nat,Xs: list_nat] :
( ( maps_nat_list_nat @ F @ ( cons_nat @ X2 @ Xs ) )
= ( append_list_nat @ ( F @ X2 ) @ ( maps_nat_list_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_263_maps__simps_I1_J,axiom,
! [F: list_nat > list_nat,X2: list_nat,Xs: list_list_nat] :
( ( maps_list_nat_nat @ F @ ( cons_list_nat @ X2 @ Xs ) )
= ( append_nat @ ( F @ X2 ) @ ( maps_list_nat_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_264_maps__simps_I1_J,axiom,
! [F: list_nat > list_list_nat,X2: list_nat,Xs: list_list_nat] :
( ( maps_l5785965478274863235st_nat @ F @ ( cons_list_nat @ X2 @ Xs ) )
= ( append_list_nat @ ( F @ X2 ) @ ( maps_l5785965478274863235st_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_265_rotate1__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rotate1_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_266_rotate1__is__Nil__conv,axiom,
! [Xs: list_list_nat] :
( ( ( rotate1_list_nat @ Xs )
= nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_267_set__rotate1,axiom,
! [Xs: list_list_nat] :
( ( set_list_nat2 @ ( rotate1_list_nat @ Xs ) )
= ( set_list_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_268_set__rotate1,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( rotate1_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_269_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_270_length__rotate1,axiom,
! [Xs: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( rotate1_list_nat @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_rotate1
thf(fact_271_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_272_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_273_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_274_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_275_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_276_rotate1_Osimps_I1_J,axiom,
( ( rotate1_list_nat @ nil_list_nat )
= nil_list_nat ) ).
% rotate1.simps(1)
thf(fact_277_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_278_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_279_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_280_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_281_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_282_le__cases3,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_283_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_284_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_285_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_286_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_287_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_288_order__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_289_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_290_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_291_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_292_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_293_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_294_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_295_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_296_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_297_order__eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_298_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_299_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_300_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_301_linorder__le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_302_order__antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_303_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_304_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
! [T: list_nat] :
( ( member_list_nat @ T @ A5 )
=> ( member_list_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_305_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A5 )
=> ( member_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_306_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
! [X: list_nat] :
( ( member_list_nat @ X @ A5 )
=> ( member_list_nat @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_307_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( member_nat @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_308_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_309_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_310_in__mono,axiom,
! [A2: set_list_nat,B2: set_list_nat,X2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( member_list_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_311_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_312_subset__code_I2_J,axiom,
! [A2: set_list_nat,Ys3: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ ( coset_list_nat @ Ys3 ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Ys3 ) )
=> ~ ( member_list_nat @ X @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_313_subset__code_I2_J,axiom,
! [A2: set_nat,Ys3: list_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( coset_nat @ Ys3 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys3 ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_314_rotate1_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( rotate1_nat @ ( cons_nat @ X2 @ Xs ) )
= ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_315_rotate1_Osimps_I2_J,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( rotate1_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_316_prefixes__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( prefixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= ( append_list_nat @ ( prefixes_nat @ Xs ) @ ( cons_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ nil_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_317_prefixes__snoc,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( prefixes_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= ( append_list_list_nat @ ( prefixes_list_nat @ Xs ) @ ( cons_list_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) @ nil_list_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_318_prefixes__eq__snoc,axiom,
! [Ys3: list_list_nat,Xs: list_list_list_nat,X2: list_list_nat] :
( ( ( prefixes_list_nat @ Ys3 )
= ( append_list_list_nat @ Xs @ ( cons_list_list_nat @ X2 @ nil_list_list_nat ) ) )
= ( ( ( ( Ys3 = nil_list_nat )
& ( Xs = nil_list_list_nat ) )
| ? [Z4: list_nat,Zs2: list_list_nat] :
( ( Ys3
= ( append_list_nat @ Zs2 @ ( cons_list_nat @ Z4 @ nil_list_nat ) ) )
& ( Xs
= ( prefixes_list_nat @ Zs2 ) ) ) )
& ( X2 = Ys3 ) ) ) ).
% prefixes_eq_snoc
thf(fact_319_prefixes__eq__snoc,axiom,
! [Ys3: list_nat,Xs: list_list_nat,X2: list_nat] :
( ( ( prefixes_nat @ Ys3 )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= ( ( ( ( Ys3 = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z4: nat,Zs2: list_nat] :
( ( Ys3
= ( append_nat @ Zs2 @ ( cons_nat @ Z4 @ nil_nat ) ) )
& ( Xs
= ( prefixes_nat @ Zs2 ) ) ) )
& ( X2 = Ys3 ) ) ) ).
% prefixes_eq_snoc
thf(fact_320_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_321_sublists_Osimps_I1_J,axiom,
( ( sublists_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% sublists.simps(1)
thf(fact_322_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_323_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_324_suffixes__eq__snoc,axiom,
! [Ys3: list_list_nat,Xs: list_list_list_nat,X2: list_list_nat] :
( ( ( suffixes_list_nat @ Ys3 )
= ( append_list_list_nat @ Xs @ ( cons_list_list_nat @ X2 @ nil_list_list_nat ) ) )
= ( ( ( ( Ys3 = nil_list_nat )
& ( Xs = nil_list_list_nat ) )
| ? [Z4: list_nat,Zs2: list_list_nat] :
( ( Ys3
= ( cons_list_nat @ Z4 @ Zs2 ) )
& ( Xs
= ( suffixes_list_nat @ Zs2 ) ) ) )
& ( X2 = Ys3 ) ) ) ).
% suffixes_eq_snoc
thf(fact_325_suffixes__eq__snoc,axiom,
! [Ys3: list_nat,Xs: list_list_nat,X2: list_nat] :
( ( ( suffixes_nat @ Ys3 )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= ( ( ( ( Ys3 = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z4: nat,Zs2: list_nat] :
( ( Ys3
= ( cons_nat @ Z4 @ Zs2 ) )
& ( Xs
= ( suffixes_nat @ Zs2 ) ) ) )
& ( X2 = Ys3 ) ) ) ).
% suffixes_eq_snoc
thf(fact_326_not__in__set__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X2 @ Xs )
= ( cons_nat @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_327_not__in__set__insert,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( insert_list_nat @ X2 @ Xs )
= ( cons_list_nat @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_328_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_329_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_330_insert__Nil,axiom,
! [X2: nat] :
( ( insert_nat @ X2 @ nil_nat )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% insert_Nil
thf(fact_331_insert__Nil,axiom,
! [X2: list_nat] :
( ( insert_list_nat @ X2 @ nil_list_nat )
= ( cons_list_nat @ X2 @ nil_list_nat ) ) ).
% insert_Nil
thf(fact_332_rotate__length01,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate_nat @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_333_rotate__length01,axiom,
! [Xs: list_list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ one_one_nat )
=> ( ( rotate_list_nat @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_334_rotate__is__Nil__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( rotate_nat @ N @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate_is_Nil_conv
thf(fact_335_rotate__is__Nil__conv,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ( rotate_list_nat @ N @ Xs )
= nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% rotate_is_Nil_conv
thf(fact_336_set__rotate,axiom,
! [N: nat,Xs: list_list_nat] :
( ( set_list_nat2 @ ( rotate_list_nat @ N @ Xs ) )
= ( set_list_nat2 @ Xs ) ) ).
% set_rotate
thf(fact_337_set__rotate,axiom,
! [N: nat,Xs: list_nat] :
( ( set_nat2 @ ( rotate_nat @ N @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_rotate
thf(fact_338_length__rotate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( rotate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate
thf(fact_339_length__rotate,axiom,
! [N: nat,Xs: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( rotate_list_nat @ N @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_rotate
thf(fact_340_in__set__insert,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( insert_list_nat @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_341_in__set__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_342_in__set__prefixes,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( prefixes_nat @ Ys3 ) ) )
= ( prefix_nat @ Xs @ Ys3 ) ) ).
% in_set_prefixes
thf(fact_343_in__set__prefixes,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( member_list_list_nat @ Xs @ ( set_list_list_nat2 @ ( prefixes_list_nat @ Ys3 ) ) )
= ( prefix_list_nat @ Xs @ Ys3 ) ) ).
% in_set_prefixes
thf(fact_344_prefixes__not__Nil,axiom,
! [Xs: list_nat] :
( ( prefixes_nat @ Xs )
!= nil_list_nat ) ).
% prefixes_not_Nil
thf(fact_345_suffixes__not__Nil,axiom,
! [Xs: list_nat] :
( ( suffixes_nat @ Xs )
!= nil_list_nat ) ).
% suffixes_not_Nil
thf(fact_346_subseqs__refl,axiom,
! [Xs: list_nat] : ( member_list_nat @ Xs @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ).
% subseqs_refl
thf(fact_347_rotate__append,axiom,
! [L: list_nat,Q: list_nat] :
( ( rotate_nat @ ( size_size_list_nat @ L ) @ ( append_nat @ L @ Q ) )
= ( append_nat @ Q @ L ) ) ).
% rotate_append
thf(fact_348_rotate__append,axiom,
! [L: list_list_nat,Q: list_list_nat] :
( ( rotate_list_nat @ ( size_s3023201423986296836st_nat @ L ) @ ( append_list_nat @ L @ Q ) )
= ( append_list_nat @ Q @ L ) ) ).
% rotate_append
thf(fact_349_Cons__in__subseqsD,axiom,
! [Y2: nat,Ys3: list_nat,Xs: list_nat] :
( ( member_list_nat @ ( cons_nat @ Y2 @ Ys3 ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
=> ( member_list_nat @ Ys3 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_350_Cons__in__subseqsD,axiom,
! [Y2: list_nat,Ys3: list_list_nat,Xs: list_list_nat] :
( ( member_list_list_nat @ ( cons_list_nat @ Y2 @ Ys3 ) @ ( set_list_list_nat2 @ ( subseqs_list_nat @ Xs ) ) )
=> ( member_list_list_nat @ Ys3 @ ( set_list_list_nat2 @ ( subseqs_list_nat @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_351_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat @ X @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_352_List_Oinsert__def,axiom,
( insert_list_nat
= ( ^ [X: list_nat,Xs3: list_list_nat] : ( if_list_list_nat @ ( member_list_nat @ X @ ( set_list_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_list_nat @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_353_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_354_prefixes_Osimps_I1_J,axiom,
( ( prefixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_355_suffixes_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( suffixes_nat @ ( cons_nat @ X2 @ Xs ) )
= ( append_list_nat @ ( suffixes_nat @ Xs ) @ ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ nil_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_356_suffixes_Osimps_I2_J,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( suffixes_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= ( append_list_list_nat @ ( suffixes_list_nat @ Xs ) @ ( cons_list_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ nil_list_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_357_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_358_suffixes_Osimps_I1_J,axiom,
( ( suffixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_359_in__set__product__lists__length,axiom,
! [Xs: list_nat,Xss2: list_list_nat] :
( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss2 ) ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_360_in__set__product__lists__length,axiom,
! [Xs: list_list_nat,Xss2: list_list_list_nat] :
( ( member_list_list_nat @ Xs @ ( set_list_list_nat2 @ ( produc6783906451316923569st_nat @ Xss2 ) ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s6248950052170075156st_nat @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_361_concat__eq__append__conv,axiom,
! [Xss2: list_list_list_nat,Ys3: list_list_nat,Zs: list_list_nat] :
( ( ( concat_list_nat @ Xss2 )
= ( append_list_nat @ Ys3 @ Zs ) )
= ( ( ( Xss2 = nil_list_list_nat )
=> ( ( Ys3 = nil_list_nat )
& ( Zs = nil_list_nat ) ) )
& ( ( Xss2 != nil_list_list_nat )
=> ? [Xss1: list_list_list_nat,Xs3: list_list_nat,Xs6: list_list_nat,Xss22: list_list_list_nat] :
( ( Xss2
= ( append_list_list_nat @ Xss1 @ ( cons_list_list_nat @ ( append_list_nat @ Xs3 @ Xs6 ) @ Xss22 ) ) )
& ( Ys3
= ( append_list_nat @ ( concat_list_nat @ Xss1 ) @ Xs3 ) )
& ( Zs
= ( append_list_nat @ Xs6 @ ( concat_list_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_362_concat__eq__append__conv,axiom,
! [Xss2: list_list_nat,Ys3: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys3 @ Zs ) )
= ( ( ( Xss2 = nil_list_nat )
=> ( ( Ys3 = nil_nat )
& ( Zs = nil_nat ) ) )
& ( ( Xss2 != nil_list_nat )
=> ? [Xss1: list_list_nat,Xs3: list_nat,Xs6: list_nat,Xss22: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs3 @ Xs6 ) @ Xss22 ) ) )
& ( Ys3
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs3 ) )
& ( Zs
= ( append_nat @ Xs6 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_363_concat__eq__appendD,axiom,
! [Xss2: list_list_list_nat,Ys3: list_list_nat,Zs: list_list_nat] :
( ( ( concat_list_nat @ Xss2 )
= ( append_list_nat @ Ys3 @ Zs ) )
=> ( ( Xss2 != nil_list_list_nat )
=> ? [Xss12: list_list_list_nat,Xs2: list_list_nat,Xs5: list_list_nat,Xss23: list_list_list_nat] :
( ( Xss2
= ( append_list_list_nat @ Xss12 @ ( cons_list_list_nat @ ( append_list_nat @ Xs2 @ Xs5 ) @ Xss23 ) ) )
& ( Ys3
= ( append_list_nat @ ( concat_list_nat @ Xss12 ) @ Xs2 ) )
& ( Zs
= ( append_list_nat @ Xs5 @ ( concat_list_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_364_concat__eq__appendD,axiom,
! [Xss2: list_list_nat,Ys3: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys3 @ Zs ) )
=> ( ( Xss2 != nil_list_nat )
=> ? [Xss12: list_list_nat,Xs2: list_nat,Xs5: list_nat,Xss23: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs2 @ Xs5 ) @ Xss23 ) ) )
& ( Ys3
= ( append_nat @ ( concat_nat @ Xss12 ) @ Xs2 ) )
& ( Zs
= ( append_nat @ Xs5 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_365_prefix__bot_Omin__bot2,axiom,
! [X2: list_nat] :
( ( min_list_nat @ prefix_nat @ X2 @ nil_nat )
= nil_nat ) ).
% prefix_bot.min_bot2
thf(fact_366_prefix__bot_Omin__bot2,axiom,
! [X2: list_list_nat] :
( ( min_list_list_nat @ prefix_list_nat @ X2 @ nil_list_nat )
= nil_list_nat ) ).
% prefix_bot.min_bot2
thf(fact_367_prefix__bot_Omax__bot2,axiom,
! [X2: list_nat] :
( ( max_list_nat @ prefix_nat @ X2 @ nil_nat )
= X2 ) ).
% prefix_bot.max_bot2
thf(fact_368_prefix__bot_Omax__bot2,axiom,
! [X2: list_list_nat] :
( ( max_list_list_nat @ prefix_list_nat @ X2 @ nil_list_nat )
= X2 ) ).
% prefix_bot.max_bot2
thf(fact_369_prefix__bot_Omin__bot,axiom,
! [X2: list_nat] :
( ( min_list_nat @ prefix_nat @ nil_nat @ X2 )
= nil_nat ) ).
% prefix_bot.min_bot
thf(fact_370_prefix__bot_Omin__bot,axiom,
! [X2: list_list_nat] :
( ( min_list_list_nat @ prefix_list_nat @ nil_list_nat @ X2 )
= nil_list_nat ) ).
% prefix_bot.min_bot
thf(fact_371_prefix__bot_Omax__bot,axiom,
! [X2: list_nat] :
( ( max_list_nat @ prefix_nat @ nil_nat @ X2 )
= X2 ) ).
% prefix_bot.max_bot
thf(fact_372_prefix__bot_Omax__bot,axiom,
! [X2: list_list_nat] :
( ( max_list_list_nat @ prefix_list_nat @ nil_list_nat @ X2 )
= X2 ) ).
% prefix_bot.max_bot
thf(fact_373_length__n__lists__elem,axiom,
! [Ys3: list_nat,N: nat,Xs: list_nat] :
( ( member_list_nat @ Ys3 @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys3 )
= N ) ) ).
% length_n_lists_elem
thf(fact_374_length__n__lists__elem,axiom,
! [Ys3: list_list_nat,N: nat,Xs: list_list_nat] :
( ( member_list_list_nat @ Ys3 @ ( set_list_list_nat2 @ ( n_lists_list_nat @ N @ Xs ) ) )
=> ( ( size_s3023201423986296836st_nat @ Ys3 )
= N ) ) ).
% length_n_lists_elem
thf(fact_375_member__remove,axiom,
! [X2: list_nat,Y2: list_nat,A2: set_list_nat] :
( ( member_list_nat @ X2 @ ( remove_list_nat @ Y2 @ A2 ) )
= ( ( member_list_nat @ X2 @ A2 )
& ( X2 != Y2 ) ) ) ).
% member_remove
thf(fact_376_member__remove,axiom,
! [X2: nat,Y2: nat,A2: set_nat] :
( ( member_nat @ X2 @ ( remove_nat @ Y2 @ A2 ) )
= ( ( member_nat @ X2 @ A2 )
& ( X2 != Y2 ) ) ) ).
% member_remove
thf(fact_377_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_list_nat] :
( ( nil_list_nat
= ( concat_list_nat @ Xss2 ) )
= ( ! [X: list_list_nat] :
( ( member_list_list_nat @ X @ ( set_list_list_nat2 @ Xss2 ) )
=> ( X = nil_list_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_378_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_nat] :
( ( nil_nat
= ( concat_nat @ Xss2 ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xss2 ) )
=> ( X = nil_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_379_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_list_nat] :
( ( ( concat_list_nat @ Xss2 )
= nil_list_nat )
= ( ! [X: list_list_nat] :
( ( member_list_list_nat @ X @ ( set_list_list_nat2 @ Xss2 ) )
=> ( X = nil_list_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_380_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_nat] :
( ( ( concat_nat @ Xss2 )
= nil_nat )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xss2 ) )
=> ( X = nil_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_381_concat__append,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( concat_nat @ ( append_list_nat @ Xs @ Ys3 ) )
= ( append_nat @ ( concat_nat @ Xs ) @ ( concat_nat @ Ys3 ) ) ) ).
% concat_append
thf(fact_382_concat__append,axiom,
! [Xs: list_list_list_nat,Ys3: list_list_list_nat] :
( ( concat_list_nat @ ( append_list_list_nat @ Xs @ Ys3 ) )
= ( append_list_nat @ ( concat_list_nat @ Xs ) @ ( concat_list_nat @ Ys3 ) ) ) ).
% concat_append
thf(fact_383_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_384_prefix__order_Omax__def,axiom,
! [A: list_list_nat,B: list_list_nat] :
( ( ( prefix_list_nat @ A @ B )
=> ( ( max_list_list_nat @ prefix_list_nat @ A @ B )
= B ) )
& ( ~ ( prefix_list_nat @ A @ B )
=> ( ( max_list_list_nat @ prefix_list_nat @ A @ B )
= A ) ) ) ).
% prefix_order.max_def
thf(fact_385_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_386_prefix__order_Omin__def,axiom,
! [A: list_list_nat,B: list_list_nat] :
( ( ( prefix_list_nat @ A @ B )
=> ( ( min_list_list_nat @ prefix_list_nat @ A @ B )
= A ) )
& ( ~ ( prefix_list_nat @ A @ B )
=> ( ( min_list_list_nat @ prefix_list_nat @ A @ B )
= B ) ) ) ).
% prefix_order.min_def
thf(fact_387_concat_Osimps_I2_J,axiom,
! [X2: list_list_nat,Xs: list_list_list_nat] :
( ( concat_list_nat @ ( cons_list_list_nat @ X2 @ Xs ) )
= ( append_list_nat @ X2 @ ( concat_list_nat @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_388_concat_Osimps_I2_J,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( concat_nat @ ( cons_list_nat @ X2 @ Xs ) )
= ( append_nat @ X2 @ ( concat_nat @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_389_concat_Osimps_I1_J,axiom,
( ( concat_list_nat @ nil_list_list_nat )
= nil_list_nat ) ).
% concat.simps(1)
thf(fact_390_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_391_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_list_nat @ N @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_list_nat @ N @ nil_list_nat )
= nil_list_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_392_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_393_length__prefixes,axiom,
! [Xs: list_list_nat] :
( ( size_s6248950052170075156st_nat @ ( prefixes_list_nat @ Xs ) )
= ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ one_one_nat ) ) ).
% length_prefixes
thf(fact_394_length__prefixes,axiom,
! [Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( prefixes_nat @ Xs ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_prefixes
thf(fact_395_n__lists_Osimps_I1_J,axiom,
! [Xs: list_list_nat] :
( ( n_lists_list_nat @ zero_zero_nat @ Xs )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_396_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_397_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_398_list__ex1__simps_I1_J,axiom,
! [P: list_nat > $o] :
~ ( list_ex1_list_nat @ P @ nil_list_nat ) ).
% list_ex1_simps(1)
thf(fact_399_butlast__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_400_butlast__snoc,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( butlast_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_401_sublists_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( sublists_nat @ ( cons_nat @ X2 @ Xs ) )
= ( append_list_nat @ ( sublists_nat @ Xs ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X2 ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_402_sublists_Osimps_I2_J,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( sublists_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= ( append_list_list_nat @ ( sublists_list_nat @ Xs ) @ ( map_li2855073862107769254st_nat @ ( cons_list_nat @ X2 ) @ ( prefixes_list_nat @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_403_length__suffixes,axiom,
! [Xs: list_list_nat] :
( ( size_s6248950052170075156st_nat @ ( suffixes_list_nat @ Xs ) )
= ( suc @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_suffixes
thf(fact_404_length__suffixes,axiom,
! [Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( suffixes_nat @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_suffixes
thf(fact_405_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_406_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_407_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_408_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_409_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_410_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_411_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_412_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_413_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_414_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_415_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_416_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y2 ) )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_417_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_418_map__is__Nil__conv,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_419_map__is__Nil__conv,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_420_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_421_map__is__Nil__conv,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( ( map_li960784813134754710st_nat @ F @ Xs )
= nil_list_list_nat )
= ( Xs = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_422_map__is__Nil__conv,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( ( map_nat_list_nat @ F @ Xs )
= nil_list_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_423_Nil__is__map__conv,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( nil_nat
= ( map_list_nat_nat @ F @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_424_Nil__is__map__conv,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat] :
( ( nil_list_nat
= ( map_li7225945977422193158st_nat @ F @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_425_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_426_Nil__is__map__conv,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( nil_list_list_nat
= ( map_li960784813134754710st_nat @ F @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_427_Nil__is__map__conv,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( nil_list_nat
= ( map_nat_list_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_428_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_429_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_430_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_431_list_Omap__disc__iff,axiom,
! [F: list_nat > list_list_nat,A: list_list_nat] :
( ( ( map_li960784813134754710st_nat @ F @ A )
= nil_list_list_nat )
= ( A = nil_list_nat ) ) ).
% list.map_disc_iff
thf(fact_432_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_433_map__eq__conv,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat,G: list_nat > list_list_nat] :
( ( ( map_li960784813134754710st_nat @ F @ Xs )
= ( map_li960784813134754710st_nat @ G @ Xs ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_434_map__eq__conv,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_435_map__eq__conv,axiom,
! [F: nat > list_nat,Xs: list_nat,G: nat > list_nat] :
( ( ( map_nat_list_nat @ F @ Xs )
= ( map_nat_list_nat @ G @ Xs ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_436_length__map,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( size_s6248950052170075156st_nat @ ( map_li960784813134754710st_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_437_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_438_length__map,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( size_size_list_nat @ ( map_list_nat_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_439_length__map,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( map_nat_list_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_440_length__map,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_441_map__append,axiom,
! [F: list_nat > nat,Xs: list_list_nat,Ys3: list_list_nat] :
( ( map_list_nat_nat @ F @ ( append_list_nat @ Xs @ Ys3 ) )
= ( append_nat @ ( map_list_nat_nat @ F @ Xs ) @ ( map_list_nat_nat @ F @ Ys3 ) ) ) ).
% map_append
thf(fact_442_map__append,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,Ys3: list_list_nat] :
( ( map_li7225945977422193158st_nat @ F @ ( append_list_nat @ Xs @ Ys3 ) )
= ( append_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) @ ( map_li7225945977422193158st_nat @ F @ Ys3 ) ) ) ).
% map_append
thf(fact_443_map__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys3: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys3 ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys3 ) ) ) ).
% map_append
thf(fact_444_map__append,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat,Ys3: list_list_nat] :
( ( map_li960784813134754710st_nat @ F @ ( append_list_nat @ Xs @ Ys3 ) )
= ( append_list_list_nat @ ( map_li960784813134754710st_nat @ F @ Xs ) @ ( map_li960784813134754710st_nat @ F @ Ys3 ) ) ) ).
% map_append
thf(fact_445_map__append,axiom,
! [F: nat > list_nat,Xs: list_nat,Ys3: list_nat] :
( ( map_nat_list_nat @ F @ ( append_nat @ Xs @ Ys3 ) )
= ( append_list_nat @ ( map_nat_list_nat @ F @ Xs ) @ ( map_nat_list_nat @ F @ Ys3 ) ) ) ).
% map_append
thf(fact_446_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_447_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_448_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_449_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_450_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_451_length__0__conv,axiom,
! [Xs: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_nat ) ) ).
% length_0_conv
thf(fact_452_length__append,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ Ys3 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys3 ) ) ) ).
% length_append
thf(fact_453_length__append,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( append_list_nat @ Xs @ Ys3 ) )
= ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ ( size_s3023201423986296836st_nat @ Ys3 ) ) ) ).
% length_append
thf(fact_454_count__notin,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( count_list_nat @ Xs @ X2 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_455_count__notin,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( count_list_list_nat @ Xs @ X2 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_456_count__list__append,axiom,
! [Xs: list_nat,Ys3: list_nat,X2: nat] :
( ( count_list_nat @ ( append_nat @ Xs @ Ys3 ) @ X2 )
= ( plus_plus_nat @ ( count_list_nat @ Xs @ X2 ) @ ( count_list_nat @ Ys3 @ X2 ) ) ) ).
% count_list_append
thf(fact_457_count__list__append,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,X2: list_nat] :
( ( count_list_list_nat @ ( append_list_nat @ Xs @ Ys3 ) @ X2 )
= ( plus_plus_nat @ ( count_list_list_nat @ Xs @ X2 ) @ ( count_list_list_nat @ Ys3 @ X2 ) ) ) ).
% count_list_append
thf(fact_458_list_Osize_I4_J,axiom,
! [X21: nat,X22: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_459_list_Osize_I4_J,axiom,
! [X21: list_nat,X22: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( cons_list_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_460_map__concat,axiom,
! [F: list_nat > list_nat,Xs: list_list_list_nat] :
( ( map_li7225945977422193158st_nat @ F @ ( concat_list_nat @ Xs ) )
= ( concat_list_nat @ ( map_li2855073862107769254st_nat @ ( map_li7225945977422193158st_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_461_map__concat,axiom,
! [F: nat > nat,Xs: list_list_nat] :
( ( map_nat_nat @ F @ ( concat_nat @ Xs ) )
= ( concat_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_462_map__concat,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_list_nat] :
( ( map_li960784813134754710st_nat @ F @ ( concat_list_nat @ Xs ) )
= ( concat_list_list_nat @ ( map_li5769348595424326838st_nat @ ( map_li960784813134754710st_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_463_map__concat,axiom,
! [F: nat > list_nat,Xs: list_list_nat] :
( ( map_nat_list_nat @ F @ ( concat_nat @ Xs ) )
= ( concat_list_nat @ ( map_li960784813134754710st_nat @ ( map_nat_list_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_464_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_465_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_466_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_467_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_468_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_469_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_470_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_471_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_472_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_473_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_474_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_475_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_476_enum__rgfs_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ~ ! [N2: nat] :
( X2
!= ( suc @ N2 ) ) ) ).
% enum_rgfs.cases
thf(fact_477_map__butlast,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( map_nat_nat @ F @ ( butlast_nat @ Xs ) )
= ( butlast_nat @ ( map_nat_nat @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_478_map__butlast,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( map_li960784813134754710st_nat @ F @ ( butlast_list_nat @ Xs ) )
= ( butlas6429778205849610142st_nat @ ( map_li960784813134754710st_nat @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_479_map__butlast,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( map_nat_list_nat @ F @ ( butlast_nat @ Xs ) )
= ( butlast_list_nat @ ( map_nat_list_nat @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_480_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_481_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_482_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_483_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_484_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_485_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_486_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_487_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_488_map__eq__Cons__conv,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat,Y2: list_list_nat,Ys3: list_list_list_nat] :
( ( ( map_li960784813134754710st_nat @ F @ Xs )
= ( cons_list_list_nat @ Y2 @ Ys3 ) )
= ( ? [Z4: list_nat,Zs2: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y2 )
& ( ( map_li960784813134754710st_nat @ F @ Zs2 )
= Ys3 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_489_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y2: nat,Ys3: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys3 ) )
= ( ? [Z4: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y2 )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys3 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_490_map__eq__Cons__conv,axiom,
! [F: list_nat > nat,Xs: list_list_nat,Y2: nat,Ys3: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys3 ) )
= ( ? [Z4: list_nat,Zs2: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y2 )
& ( ( map_list_nat_nat @ F @ Zs2 )
= Ys3 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_491_map__eq__Cons__conv,axiom,
! [F: nat > list_nat,Xs: list_nat,Y2: list_nat,Ys3: list_list_nat] :
( ( ( map_nat_list_nat @ F @ Xs )
= ( cons_list_nat @ Y2 @ Ys3 ) )
= ( ? [Z4: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y2 )
& ( ( map_nat_list_nat @ F @ Zs2 )
= Ys3 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_492_map__eq__Cons__conv,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,Y2: list_nat,Ys3: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= ( cons_list_nat @ Y2 @ Ys3 ) )
= ( ? [Z4: list_nat,Zs2: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y2 )
& ( ( map_li7225945977422193158st_nat @ F @ Zs2 )
= Ys3 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_493_Cons__eq__map__conv,axiom,
! [X2: list_list_nat,Xs: list_list_list_nat,F: list_nat > list_list_nat,Ys3: list_list_nat] :
( ( ( cons_list_list_nat @ X2 @ Xs )
= ( map_li960784813134754710st_nat @ F @ Ys3 ) )
= ( ? [Z4: list_nat,Zs2: list_list_nat] :
( ( Ys3
= ( cons_list_nat @ Z4 @ Zs2 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_li960784813134754710st_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_494_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys3: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys3 ) )
= ( ? [Z4: nat,Zs2: list_nat] :
( ( Ys3
= ( cons_nat @ Z4 @ Zs2 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_495_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: list_nat > nat,Ys3: list_list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_list_nat_nat @ F @ Ys3 ) )
= ( ? [Z4: list_nat,Zs2: list_list_nat] :
( ( Ys3
= ( cons_list_nat @ Z4 @ Zs2 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_list_nat_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_496_Cons__eq__map__conv,axiom,
! [X2: list_nat,Xs: list_list_nat,F: nat > list_nat,Ys3: list_nat] :
( ( ( cons_list_nat @ X2 @ Xs )
= ( map_nat_list_nat @ F @ Ys3 ) )
= ( ? [Z4: nat,Zs2: list_nat] :
( ( Ys3
= ( cons_nat @ Z4 @ Zs2 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_nat_list_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_497_Cons__eq__map__conv,axiom,
! [X2: list_nat,Xs: list_list_nat,F: list_nat > list_nat,Ys3: list_list_nat] :
( ( ( cons_list_nat @ X2 @ Xs )
= ( map_li7225945977422193158st_nat @ F @ Ys3 ) )
= ( ? [Z4: list_nat,Zs2: list_list_nat] :
( ( Ys3
= ( cons_list_nat @ Z4 @ Zs2 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_li7225945977422193158st_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_498_map__eq__Cons__D,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat,Y2: list_list_nat,Ys3: list_list_list_nat] :
( ( ( map_li960784813134754710st_nat @ F @ Xs )
= ( cons_list_list_nat @ Y2 @ Ys3 ) )
=> ? [Z3: list_nat,Zs3: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_li960784813134754710st_nat @ F @ Zs3 )
= Ys3 ) ) ) ).
% map_eq_Cons_D
thf(fact_499_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y2: nat,Ys3: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys3 ) )
=> ? [Z3: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys3 ) ) ) ).
% map_eq_Cons_D
thf(fact_500_map__eq__Cons__D,axiom,
! [F: list_nat > nat,Xs: list_list_nat,Y2: nat,Ys3: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys3 ) )
=> ? [Z3: list_nat,Zs3: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_list_nat_nat @ F @ Zs3 )
= Ys3 ) ) ) ).
% map_eq_Cons_D
thf(fact_501_map__eq__Cons__D,axiom,
! [F: nat > list_nat,Xs: list_nat,Y2: list_nat,Ys3: list_list_nat] :
( ( ( map_nat_list_nat @ F @ Xs )
= ( cons_list_nat @ Y2 @ Ys3 ) )
=> ? [Z3: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_nat_list_nat @ F @ Zs3 )
= Ys3 ) ) ) ).
% map_eq_Cons_D
thf(fact_502_map__eq__Cons__D,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,Y2: list_nat,Ys3: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= ( cons_list_nat @ Y2 @ Ys3 ) )
=> ? [Z3: list_nat,Zs3: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_li7225945977422193158st_nat @ F @ Zs3 )
= Ys3 ) ) ) ).
% map_eq_Cons_D
thf(fact_503_Cons__eq__map__D,axiom,
! [X2: list_list_nat,Xs: list_list_list_nat,F: list_nat > list_list_nat,Ys3: list_list_nat] :
( ( ( cons_list_list_nat @ X2 @ Xs )
= ( map_li960784813134754710st_nat @ F @ Ys3 ) )
=> ? [Z3: list_nat,Zs3: list_list_nat] :
( ( Ys3
= ( cons_list_nat @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_li960784813134754710st_nat @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_504_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys3: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys3 ) )
=> ? [Z3: nat,Zs3: list_nat] :
( ( Ys3
= ( cons_nat @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_505_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: list_nat > nat,Ys3: list_list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_list_nat_nat @ F @ Ys3 ) )
=> ? [Z3: list_nat,Zs3: list_list_nat] :
( ( Ys3
= ( cons_list_nat @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_list_nat_nat @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_506_Cons__eq__map__D,axiom,
! [X2: list_nat,Xs: list_list_nat,F: nat > list_nat,Ys3: list_nat] :
( ( ( cons_list_nat @ X2 @ Xs )
= ( map_nat_list_nat @ F @ Ys3 ) )
=> ? [Z3: nat,Zs3: list_nat] :
( ( Ys3
= ( cons_nat @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_list_nat @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_507_Cons__eq__map__D,axiom,
! [X2: list_nat,Xs: list_list_nat,F: list_nat > list_nat,Ys3: list_list_nat] :
( ( ( cons_list_nat @ X2 @ Xs )
= ( map_li7225945977422193158st_nat @ F @ Ys3 ) )
=> ? [Z3: list_nat,Zs3: list_list_nat] :
( ( Ys3
= ( cons_list_nat @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_li7225945977422193158st_nat @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_508_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_509_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_510_list_Osimps_I9_J,axiom,
! [F: list_nat > list_list_nat,X21: list_nat,X22: list_list_nat] :
( ( map_li960784813134754710st_nat @ F @ ( cons_list_nat @ X21 @ X22 ) )
= ( cons_list_list_nat @ ( F @ X21 ) @ ( map_li960784813134754710st_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_511_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_512_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_513_list_Osimps_I8_J,axiom,
! [F: list_nat > nat] :
( ( map_list_nat_nat @ F @ nil_list_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_514_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_515_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_516_list_Osimps_I8_J,axiom,
! [F: list_nat > list_list_nat] :
( ( map_li960784813134754710st_nat @ F @ nil_list_nat )
= nil_list_list_nat ) ).
% list.simps(8)
thf(fact_517_list_Osimps_I8_J,axiom,
! [F: nat > list_nat] :
( ( map_nat_list_nat @ F @ nil_nat )
= nil_list_nat ) ).
% list.simps(8)
thf(fact_518_ex__map__conv,axiom,
! [Ys3: list_list_list_nat,F: list_nat > list_list_nat] :
( ( ? [Xs3: list_list_nat] :
( Ys3
= ( map_li960784813134754710st_nat @ F @ Xs3 ) ) )
= ( ! [X: list_list_nat] :
( ( member_list_list_nat @ X @ ( set_list_list_nat2 @ Ys3 ) )
=> ? [Y: list_nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_519_ex__map__conv,axiom,
! [Ys3: list_list_nat,F: nat > list_nat] :
( ( ? [Xs3: list_nat] :
( Ys3
= ( map_nat_list_nat @ F @ Xs3 ) ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Ys3 ) )
=> ? [Y: nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_520_ex__map__conv,axiom,
! [Ys3: list_nat,F: nat > nat] :
( ( ? [Xs3: list_nat] :
( Ys3
= ( map_nat_nat @ F @ Xs3 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys3 ) )
=> ? [Y: nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_521_map__cong,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,F: list_nat > list_list_nat,G: list_nat > list_list_nat] :
( ( Xs = Ys3 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Ys3 ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li960784813134754710st_nat @ F @ Xs )
= ( map_li960784813134754710st_nat @ G @ Ys3 ) ) ) ) ).
% map_cong
thf(fact_522_map__cong,axiom,
! [Xs: list_nat,Ys3: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs = Ys3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys3 ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys3 ) ) ) ) ).
% map_cong
thf(fact_523_map__cong,axiom,
! [Xs: list_nat,Ys3: list_nat,F: nat > list_nat,G: nat > list_nat] :
( ( Xs = Ys3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys3 ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_list_nat @ F @ Xs )
= ( map_nat_list_nat @ G @ Ys3 ) ) ) ) ).
% map_cong
thf(fact_524_map__idI,axiom,
! [Xs: list_list_nat,F: list_nat > list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_525_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_526_map__ext,axiom,
! [Xs: list_list_nat,F: list_nat > list_list_nat,G: list_nat > list_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li960784813134754710st_nat @ F @ Xs )
= ( map_li960784813134754710st_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_527_map__ext,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_528_map__ext,axiom,
! [Xs: list_nat,F: nat > list_nat,G: nat > list_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_list_nat @ F @ Xs )
= ( map_nat_list_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_529_list_Omap__ident__strong,axiom,
! [T2: list_list_nat,F: list_nat > list_nat] :
( ! [Z3: list_nat] :
( ( member_list_nat @ Z3 @ ( set_list_nat2 @ T2 ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_530_list_Omap__ident__strong,axiom,
! [T2: list_nat,F: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ T2 ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_nat_nat @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_531_list_Oinj__map__strong,axiom,
! [X2: list_list_nat,Xa2: list_list_nat,F: list_nat > list_list_nat,Fa: list_nat > list_list_nat] :
( ! [Z3: list_nat,Za: list_nat] :
( ( member_list_nat @ Z3 @ ( set_list_nat2 @ X2 ) )
=> ( ( member_list_nat @ Za @ ( set_list_nat2 @ Xa2 ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_li960784813134754710st_nat @ F @ X2 )
= ( map_li960784813134754710st_nat @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_532_list_Oinj__map__strong,axiom,
! [X2: list_nat,Xa2: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z3: nat,Za: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa2 ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_533_list_Oinj__map__strong,axiom,
! [X2: list_nat,Xa2: list_nat,F: nat > list_nat,Fa: nat > list_nat] :
( ! [Z3: nat,Za: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa2 ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_nat_list_nat @ F @ X2 )
= ( map_nat_list_nat @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_534_list_Omap__cong0,axiom,
! [X2: list_list_nat,F: list_nat > list_list_nat,G: list_nat > list_list_nat] :
( ! [Z3: list_nat] :
( ( member_list_nat @ Z3 @ ( set_list_nat2 @ X2 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_li960784813134754710st_nat @ F @ X2 )
= ( map_li960784813134754710st_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_535_list_Omap__cong0,axiom,
! [X2: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_536_list_Omap__cong0,axiom,
! [X2: list_nat,F: nat > list_nat,G: nat > list_nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_list_nat @ F @ X2 )
= ( map_nat_list_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_537_list_Omap__cong,axiom,
! [X2: list_list_nat,Ya: list_list_nat,F: list_nat > list_list_nat,G: list_nat > list_list_nat] :
( ( X2 = Ya )
=> ( ! [Z3: list_nat] :
( ( member_list_nat @ Z3 @ ( set_list_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_li960784813134754710st_nat @ F @ X2 )
= ( map_li960784813134754710st_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_538_list_Omap__cong,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X2 = Ya )
=> ( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_539_list_Omap__cong,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > list_nat,G: nat > list_nat] :
( ( X2 = Ya )
=> ( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_list_nat @ F @ X2 )
= ( map_nat_list_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_540_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys3: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys3 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_541_map__eq__imp__length__eq,axiom,
! [F: nat > list_nat,Xs: list_nat,G: nat > list_nat,Ys3: list_nat] :
( ( ( map_nat_list_nat @ F @ Xs )
= ( map_nat_list_nat @ G @ Ys3 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_542_map__eq__imp__length__eq,axiom,
! [F: nat > list_list_nat,Xs: list_nat,G: list_nat > list_list_nat,Ys3: list_list_nat] :
( ( ( map_na6205611841492582150st_nat @ F @ Xs )
= ( map_li960784813134754710st_nat @ G @ Ys3 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_543_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: list_nat > nat,Ys3: list_list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_list_nat_nat @ G @ Ys3 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_544_map__eq__imp__length__eq,axiom,
! [F: nat > list_nat,Xs: list_nat,G: list_nat > list_nat,Ys3: list_list_nat] :
( ( ( map_nat_list_nat @ F @ Xs )
= ( map_li7225945977422193158st_nat @ G @ Ys3 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_545_map__eq__imp__length__eq,axiom,
! [F: list_nat > nat,Xs: list_list_nat,G: nat > nat,Ys3: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys3 ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_546_map__eq__imp__length__eq,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,G: nat > list_nat,Ys3: list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= ( map_nat_list_nat @ G @ Ys3 ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_547_map__eq__imp__length__eq,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat,G: nat > list_list_nat,Ys3: list_nat] :
( ( ( map_li960784813134754710st_nat @ F @ Xs )
= ( map_na6205611841492582150st_nat @ G @ Ys3 ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_548_map__eq__imp__length__eq,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat,G: list_nat > list_list_nat,Ys3: list_list_nat] :
( ( ( map_li960784813134754710st_nat @ F @ Xs )
= ( map_li960784813134754710st_nat @ G @ Ys3 ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_549_map__eq__append__conv,axiom,
! [F: list_nat > nat,Xs: list_list_nat,Ys3: list_nat,Zs: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( append_nat @ Ys3 @ Zs ) )
= ( ? [Us2: list_list_nat,Vs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Us2 @ Vs2 ) )
& ( Ys3
= ( map_list_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_list_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_550_map__eq__append__conv,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,Ys3: list_list_nat,Zs: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= ( append_list_nat @ Ys3 @ Zs ) )
= ( ? [Us2: list_list_nat,Vs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Us2 @ Vs2 ) )
& ( Ys3
= ( map_li7225945977422193158st_nat @ F @ Us2 ) )
& ( Zs
= ( map_li7225945977422193158st_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_551_map__eq__append__conv,axiom,
! [F: nat > nat,Xs: list_nat,Ys3: list_nat,Zs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( append_nat @ Ys3 @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys3
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_552_map__eq__append__conv,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat,Ys3: list_list_list_nat,Zs: list_list_list_nat] :
( ( ( map_li960784813134754710st_nat @ F @ Xs )
= ( append_list_list_nat @ Ys3 @ Zs ) )
= ( ? [Us2: list_list_nat,Vs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Us2 @ Vs2 ) )
& ( Ys3
= ( map_li960784813134754710st_nat @ F @ Us2 ) )
& ( Zs
= ( map_li960784813134754710st_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_553_map__eq__append__conv,axiom,
! [F: nat > list_nat,Xs: list_nat,Ys3: list_list_nat,Zs: list_list_nat] :
( ( ( map_nat_list_nat @ F @ Xs )
= ( append_list_nat @ Ys3 @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys3
= ( map_nat_list_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_list_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_554_append__eq__map__conv,axiom,
! [Ys3: list_nat,Zs: list_nat,F: list_nat > nat,Xs: list_list_nat] :
( ( ( append_nat @ Ys3 @ Zs )
= ( map_list_nat_nat @ F @ Xs ) )
= ( ? [Us2: list_list_nat,Vs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Us2 @ Vs2 ) )
& ( Ys3
= ( map_list_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_list_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_555_append__eq__map__conv,axiom,
! [Ys3: list_list_nat,Zs: list_list_nat,F: list_nat > list_nat,Xs: list_list_nat] :
( ( ( append_list_nat @ Ys3 @ Zs )
= ( map_li7225945977422193158st_nat @ F @ Xs ) )
= ( ? [Us2: list_list_nat,Vs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Us2 @ Vs2 ) )
& ( Ys3
= ( map_li7225945977422193158st_nat @ F @ Us2 ) )
& ( Zs
= ( map_li7225945977422193158st_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_556_append__eq__map__conv,axiom,
! [Ys3: list_nat,Zs: list_nat,F: nat > nat,Xs: list_nat] :
( ( ( append_nat @ Ys3 @ Zs )
= ( map_nat_nat @ F @ Xs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys3
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_557_append__eq__map__conv,axiom,
! [Ys3: list_list_list_nat,Zs: list_list_list_nat,F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( ( append_list_list_nat @ Ys3 @ Zs )
= ( map_li960784813134754710st_nat @ F @ Xs ) )
= ( ? [Us2: list_list_nat,Vs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Us2 @ Vs2 ) )
& ( Ys3
= ( map_li960784813134754710st_nat @ F @ Us2 ) )
& ( Zs
= ( map_li960784813134754710st_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_558_append__eq__map__conv,axiom,
! [Ys3: list_list_nat,Zs: list_list_nat,F: nat > list_nat,Xs: list_nat] :
( ( ( append_list_nat @ Ys3 @ Zs )
= ( map_nat_list_nat @ F @ Xs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys3
= ( map_nat_list_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_list_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_559_prefix__map__rightE,axiom,
! [Xs: list_list_list_nat,F: list_nat > list_list_nat,Ys3: list_list_nat] :
( ( prefix_list_list_nat @ Xs @ ( map_li960784813134754710st_nat @ F @ Ys3 ) )
=> ? [Xs5: list_list_nat] :
( ( prefix_list_nat @ Xs5 @ Ys3 )
& ( Xs
= ( map_li960784813134754710st_nat @ F @ Xs5 ) ) ) ) ).
% prefix_map_rightE
thf(fact_560_prefix__map__rightE,axiom,
! [Xs: list_nat,F: nat > nat,Ys3: list_nat] :
( ( prefix_nat @ Xs @ ( map_nat_nat @ F @ Ys3 ) )
=> ? [Xs5: list_nat] :
( ( prefix_nat @ Xs5 @ Ys3 )
& ( Xs
= ( map_nat_nat @ F @ Xs5 ) ) ) ) ).
% prefix_map_rightE
thf(fact_561_prefix__map__rightE,axiom,
! [Xs: list_nat,F: list_nat > nat,Ys3: list_list_nat] :
( ( prefix_nat @ Xs @ ( map_list_nat_nat @ F @ Ys3 ) )
=> ? [Xs5: list_list_nat] :
( ( prefix_list_nat @ Xs5 @ Ys3 )
& ( Xs
= ( map_list_nat_nat @ F @ Xs5 ) ) ) ) ).
% prefix_map_rightE
thf(fact_562_prefix__map__rightE,axiom,
! [Xs: list_list_nat,F: nat > list_nat,Ys3: list_nat] :
( ( prefix_list_nat @ Xs @ ( map_nat_list_nat @ F @ Ys3 ) )
=> ? [Xs5: list_nat] :
( ( prefix_nat @ Xs5 @ Ys3 )
& ( Xs
= ( map_nat_list_nat @ F @ Xs5 ) ) ) ) ).
% prefix_map_rightE
thf(fact_563_prefix__map__rightE,axiom,
! [Xs: list_list_nat,F: list_nat > list_nat,Ys3: list_list_nat] :
( ( prefix_list_nat @ Xs @ ( map_li7225945977422193158st_nat @ F @ Ys3 ) )
=> ? [Xs5: list_list_nat] :
( ( prefix_list_nat @ Xs5 @ Ys3 )
& ( Xs
= ( map_li7225945977422193158st_nat @ F @ Xs5 ) ) ) ) ).
% prefix_map_rightE
thf(fact_564_map__mono__prefix,axiom,
! [Xs: list_nat,Ys3: list_nat,F: nat > nat] :
( ( prefix_nat @ Xs @ Ys3 )
=> ( prefix_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys3 ) ) ) ).
% map_mono_prefix
thf(fact_565_map__mono__prefix,axiom,
! [Xs: list_nat,Ys3: list_nat,F: nat > list_nat] :
( ( prefix_nat @ Xs @ Ys3 )
=> ( prefix_list_nat @ ( map_nat_list_nat @ F @ Xs ) @ ( map_nat_list_nat @ F @ Ys3 ) ) ) ).
% map_mono_prefix
thf(fact_566_map__mono__prefix,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,F: list_nat > list_list_nat] :
( ( prefix_list_nat @ Xs @ Ys3 )
=> ( prefix_list_list_nat @ ( map_li960784813134754710st_nat @ F @ Xs ) @ ( map_li960784813134754710st_nat @ F @ Ys3 ) ) ) ).
% map_mono_prefix
thf(fact_567_map__mono__prefix,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,F: list_nat > nat] :
( ( prefix_list_nat @ Xs @ Ys3 )
=> ( prefix_nat @ ( map_list_nat_nat @ F @ Xs ) @ ( map_list_nat_nat @ F @ Ys3 ) ) ) ).
% map_mono_prefix
thf(fact_568_map__mono__prefix,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,F: list_nat > list_nat] :
( ( prefix_list_nat @ Xs @ Ys3 )
=> ( prefix_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) @ ( map_li7225945977422193158st_nat @ F @ Ys3 ) ) ) ).
% map_mono_prefix
thf(fact_569_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_570_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_571_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C2: nat] :
( B3
= ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_572_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_573_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_574_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_575_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_576_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_577_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_578_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_579_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_580_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_581_rotate__map,axiom,
! [N: nat,F: nat > nat,Xs: list_nat] :
( ( rotate_nat @ N @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( rotate_nat @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_582_rotate__map,axiom,
! [N: nat,F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( rotate_list_list_nat @ N @ ( map_li960784813134754710st_nat @ F @ Xs ) )
= ( map_li960784813134754710st_nat @ F @ ( rotate_list_nat @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_583_rotate__map,axiom,
! [N: nat,F: nat > list_nat,Xs: list_nat] :
( ( rotate_list_nat @ N @ ( map_nat_list_nat @ F @ Xs ) )
= ( map_nat_list_nat @ F @ ( rotate_nat @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_584_rotate1__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( rotate1_nat @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( rotate1_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_585_rotate1__map,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( rotate6412633851404001245st_nat @ ( map_li960784813134754710st_nat @ F @ Xs ) )
= ( map_li960784813134754710st_nat @ F @ ( rotate1_list_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_586_rotate1__map,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( rotate1_list_nat @ ( map_nat_list_nat @ F @ Xs ) )
= ( map_nat_list_nat @ F @ ( rotate1_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_587_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_588_butlast_Osimps_I1_J,axiom,
( ( butlast_list_nat @ nil_list_nat )
= nil_list_nat ) ).
% butlast.simps(1)
thf(fact_589_in__set__butlastD,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( butlast_list_nat @ Xs ) ) )
=> ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_590_in__set__butlastD,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
=> ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_591_prefixeq__butlast,axiom,
! [Xs: list_nat] : ( prefix_nat @ ( butlast_nat @ Xs ) @ Xs ) ).
% prefixeq_butlast
thf(fact_592_prefixeq__butlast,axiom,
! [Xs: list_list_nat] : ( prefix_list_nat @ ( butlast_list_nat @ Xs ) @ Xs ) ).
% prefixeq_butlast
thf(fact_593_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y: nat,Ys: list_nat] :
( ( Xs
= ( cons_nat @ Y @ Ys ) )
& ( ( size_size_list_nat @ Ys )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_594_length__Suc__conv,axiom,
! [Xs: list_list_nat,N: nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y: list_nat,Ys: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Y @ Ys ) )
& ( ( size_s3023201423986296836st_nat @ Ys )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_595_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y: nat,Ys: list_nat] :
( ( Xs
= ( cons_nat @ Y @ Ys ) )
& ( ( size_size_list_nat @ Ys )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_596_Suc__length__conv,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ( suc @ N )
= ( size_s3023201423986296836st_nat @ Xs ) )
= ( ? [Y: list_nat,Ys: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Y @ Ys ) )
& ( ( size_s3023201423986296836st_nat @ Ys )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_597_count__list__map__ge,axiom,
! [Xs: list_nat,X2: nat,F: nat > nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X2 ) @ ( count_list_nat @ ( map_nat_nat @ F @ Xs ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_598_count__list__map__ge,axiom,
! [Xs: list_nat,X2: nat,F: nat > list_nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X2 ) @ ( count_list_list_nat @ ( map_nat_list_nat @ F @ Xs ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_599_count__list__map__ge,axiom,
! [Xs: list_list_nat,X2: list_nat,F: list_nat > list_list_nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs @ X2 ) @ ( count_8975921713275557620st_nat @ ( map_li960784813134754710st_nat @ F @ Xs ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_600_count__list__map__ge,axiom,
! [Xs: list_list_nat,X2: list_nat,F: list_nat > list_nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs @ X2 ) @ ( count_list_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_601_prefix__order_Olift__Suc__antimono__le,axiom,
! [F: nat > list_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( prefix_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( prefix_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% prefix_order.lift_Suc_antimono_le
thf(fact_602_prefix__order_Olift__Suc__antimono__le,axiom,
! [F: nat > list_list_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( prefix_list_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( prefix_list_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% prefix_order.lift_Suc_antimono_le
thf(fact_603_prefix__order_Olift__Suc__mono__le,axiom,
! [F: nat > list_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( prefix_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( prefix_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% prefix_order.lift_Suc_mono_le
thf(fact_604_prefix__order_Olift__Suc__mono__le,axiom,
! [F: nat > list_list_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( prefix_list_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( prefix_list_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% prefix_order.lift_Suc_mono_le
thf(fact_605_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_606_list_Osize_I3_J,axiom,
( ( size_s3023201423986296836st_nat @ nil_list_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_607_butlast_Osimps_I2_J,axiom,
! [Xs: list_nat,X2: nat] :
( ( ( Xs = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X2 @ Xs ) )
= nil_nat ) )
& ( ( Xs != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( butlast_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_608_butlast_Osimps_I2_J,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( ( Xs = nil_list_nat )
=> ( ( butlast_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= nil_list_nat ) )
& ( ( Xs != nil_list_nat )
=> ( ( butlast_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= ( cons_list_nat @ X2 @ ( butlast_list_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_609_List_Obind__def,axiom,
( bind_l7796378977173581257st_nat
= ( ^ [Xs3: list_list_nat,F2: list_nat > list_list_nat] : ( concat_list_nat @ ( map_li960784813134754710st_nat @ F2 @ Xs3 ) ) ) ) ).
% List.bind_def
thf(fact_610_List_Obind__def,axiom,
( bind_nat_nat
= ( ^ [Xs3: list_nat,F2: nat > list_nat] : ( concat_nat @ ( map_nat_list_nat @ F2 @ Xs3 ) ) ) ) ).
% List.bind_def
thf(fact_611_butlast__append,axiom,
! [Ys3: list_nat,Xs: list_nat] :
( ( ( Ys3 = nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys3 ) )
= ( butlast_nat @ Xs ) ) )
& ( ( Ys3 != nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys3 ) )
= ( append_nat @ Xs @ ( butlast_nat @ Ys3 ) ) ) ) ) ).
% butlast_append
thf(fact_612_butlast__append,axiom,
! [Ys3: list_list_nat,Xs: list_list_nat] :
( ( ( Ys3 = nil_list_nat )
=> ( ( butlast_list_nat @ ( append_list_nat @ Xs @ Ys3 ) )
= ( butlast_list_nat @ Xs ) ) )
& ( ( Ys3 != nil_list_nat )
=> ( ( butlast_list_nat @ ( append_list_nat @ Xs @ Ys3 ) )
= ( append_list_nat @ Xs @ ( butlast_list_nat @ Ys3 ) ) ) ) ) ).
% butlast_append
thf(fact_613_in__set__butlast__appendI,axiom,
! [X2: list_nat,Xs: list_list_nat,Ys3: list_list_nat] :
( ( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( butlast_list_nat @ Xs ) ) )
| ( member_list_nat @ X2 @ ( set_list_nat2 @ ( butlast_list_nat @ Ys3 ) ) ) )
=> ( member_list_nat @ X2 @ ( set_list_nat2 @ ( butlast_list_nat @ ( append_list_nat @ Xs @ Ys3 ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_614_in__set__butlast__appendI,axiom,
! [X2: nat,Xs: list_nat,Ys3: list_nat] :
( ( ( member_nat @ X2 @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
| ( member_nat @ X2 @ ( set_nat2 @ ( butlast_nat @ Ys3 ) ) ) )
=> ( member_nat @ X2 @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs @ Ys3 ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_615_count__list_Osimps_I1_J,axiom,
! [Y2: nat] :
( ( count_list_nat @ nil_nat @ Y2 )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_616_count__list_Osimps_I1_J,axiom,
! [Y2: list_nat] :
( ( count_list_list_nat @ nil_list_nat @ Y2 )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_617_maps__def,axiom,
( maps_l5785965478274863235st_nat
= ( ^ [F2: list_nat > list_list_nat,Xs3: list_list_nat] : ( concat_list_nat @ ( map_li960784813134754710st_nat @ F2 @ Xs3 ) ) ) ) ).
% maps_def
thf(fact_618_maps__def,axiom,
( maps_nat_nat
= ( ^ [F2: nat > list_nat,Xs3: list_nat] : ( concat_nat @ ( map_nat_list_nat @ F2 @ Xs3 ) ) ) ) ).
% maps_def
thf(fact_619_concat__map__maps,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( concat_list_nat @ ( map_li960784813134754710st_nat @ F @ Xs ) )
= ( maps_l5785965478274863235st_nat @ F @ Xs ) ) ).
% concat_map_maps
thf(fact_620_concat__map__maps,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( concat_nat @ ( map_nat_list_nat @ F @ Xs ) )
= ( maps_nat_nat @ F @ Xs ) ) ).
% concat_map_maps
thf(fact_621_count__list__0__iff,axiom,
! [Xs: list_nat,X2: nat] :
( ( ( count_list_nat @ Xs @ X2 )
= zero_zero_nat )
= ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_622_count__list__0__iff,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( ( count_list_list_nat @ Xs @ X2 )
= zero_zero_nat )
= ( ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_623_list__ex1__iff,axiom,
( list_ex1_list_nat
= ( ^ [P2: list_nat > $o,Xs3: list_list_nat] :
? [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs3 ) )
& ( P2 @ X )
& ! [Y: list_nat] :
( ( ( member_list_nat @ Y @ ( set_list_nat2 @ Xs3 ) )
& ( P2 @ Y ) )
=> ( Y = X ) ) ) ) ) ).
% list_ex1_iff
thf(fact_624_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P2: nat > $o,Xs3: list_nat] :
? [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs3 ) )
& ( P2 @ X )
& ! [Y: nat] :
( ( ( member_nat @ Y @ ( set_nat2 @ Xs3 ) )
& ( P2 @ Y ) )
=> ( Y = X ) ) ) ) ) ).
% list_ex1_iff
thf(fact_625_rgf__limit_Osimps_I1_J,axiom,
( ( equiva5889994315859557365_limit @ nil_nat )
= zero_zero_nat ) ).
% rgf_limit.simps(1)
thf(fact_626_enum__rgfs_Osimps_I1_J,axiom,
( ( equiva7426478223624825838m_rgfs @ zero_zero_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% enum_rgfs.simps(1)
thf(fact_627_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X: nat,Ys: list_nat] :
( ( Xs
= ( cons_nat @ X @ Ys ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_628_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s3023201423986296836st_nat @ Xs ) )
= ( ? [X: list_nat,Ys: list_list_nat] :
( ( Xs
= ( cons_list_nat @ X @ Ys ) )
& ( ord_less_eq_nat @ N @ ( size_s3023201423986296836st_nat @ Ys ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_629_count__list_Osimps_I2_J,axiom,
! [X2: nat,Y2: nat,Xs: list_nat] :
( ( ( X2 = Y2 )
=> ( ( count_list_nat @ ( cons_nat @ X2 @ Xs ) @ Y2 )
= ( plus_plus_nat @ ( count_list_nat @ Xs @ Y2 ) @ one_one_nat ) ) )
& ( ( X2 != Y2 )
=> ( ( count_list_nat @ ( cons_nat @ X2 @ Xs ) @ Y2 )
= ( count_list_nat @ Xs @ Y2 ) ) ) ) ).
% count_list.simps(2)
thf(fact_630_count__list_Osimps_I2_J,axiom,
! [X2: list_nat,Y2: list_nat,Xs: list_list_nat] :
( ( ( X2 = Y2 )
=> ( ( count_list_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ Y2 )
= ( plus_plus_nat @ ( count_list_list_nat @ Xs @ Y2 ) @ one_one_nat ) ) )
& ( ( X2 != Y2 )
=> ( ( count_list_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ Y2 )
= ( count_list_list_nat @ Xs @ Y2 ) ) ) ) ).
% count_list.simps(2)
thf(fact_631_prefixes_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( prefixes_nat @ ( cons_nat @ X2 @ Xs ) )
= ( cons_list_nat @ nil_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X2 ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_632_prefixes_Osimps_I2_J,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( prefixes_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= ( cons_list_list_nat @ nil_list_nat @ ( map_li2855073862107769254st_nat @ ( cons_list_nat @ X2 ) @ ( prefixes_list_nat @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_633_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y: nat,Ys: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_634_length__Suc__conv__rev,axiom,
! [Xs: list_list_nat,N: nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y: list_nat,Ys: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ Y @ nil_list_nat ) ) )
& ( ( size_s3023201423986296836st_nat @ Ys )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_635_nat__add__left__cancel__le,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_636_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_637_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_638_Suc__le__mono,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M3 ) )
= ( ord_less_eq_nat @ N @ M3 ) ) ).
% Suc_le_mono
thf(fact_639_length__append__singleton,axiom,
! [Xs: list_nat,X2: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_640_length__append__singleton,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( size_s3023201423986296836st_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= ( suc @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_641_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_642_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_643_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_644_size__neq__size__imp__neq,axiom,
! [X2: char,Y2: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_645_size__neq__size__imp__neq,axiom,
! [X2: list_list_nat,Y2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ X2 )
!= ( size_s3023201423986296836st_nat @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_646_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_647_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_648_eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( M3 = N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% eq_imp_le
thf(fact_649_le__antisym,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( ord_less_eq_nat @ N @ M3 )
=> ( M3 = N ) ) ) ).
% le_antisym
thf(fact_650_nat__le__linear,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
| ( ord_less_eq_nat @ N @ M3 ) ) ).
% nat_le_linear
thf(fact_651_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_652_Suc__leD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% Suc_leD
thf(fact_653_le__SucE,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M3 @ N )
=> ( M3
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_654_le__SucI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ M3 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_655_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M2: nat] :
( M4
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_656_le__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M3 @ N )
| ( M3
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_657_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_658_not__less__eq__eq,axiom,
! [M3: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M3 ) ) ).
% not_less_eq_eq
thf(fact_659_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_660_nat__induct__at__least,axiom,
! [M3: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( P @ M3 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_661_transitive__stepwise__le,axiom,
! [M3: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M3 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_662_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_663_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_664_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_665_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_666_add__leE,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M3 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_667_le__add1,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M3 ) ) ).
% le_add1
thf(fact_668_le__add2,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M3 @ N ) ) ).
% le_add2
thf(fact_669_add__leD1,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% add_leD1
thf(fact_670_add__leD2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_671_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_672_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_673_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_674_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_675_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_676_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus_nat @ M6 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_677_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_678_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_679_length__Cons,axiom,
! [X2: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X2 @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_680_length__Cons,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( cons_list_nat @ X2 @ Xs ) )
= ( suc @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_681_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_682_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_683_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_684_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_685_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_686_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_687_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_688_rgf__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( equiva3371634703666331078on_rgf @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= ( ( equiva3371634703666331078on_rgf @ Xs )
& ( ord_less_nat @ X2 @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ Xs ) @ one_one_nat ) ) ) ) ).
% rgf_snoc
thf(fact_689_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_690_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_691_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_692_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_693_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_694_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_695_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_696_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_697_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_698_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_699_length__greater__0__conv,axiom,
! [Xs: list_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) )
= ( Xs != nil_list_nat ) ) ).
% length_greater_0_conv
thf(fact_700_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_701_less__imp__neq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_702_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_703_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_704_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_705_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_706_antisym__conv3,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_nat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_707_linorder__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_708_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_709_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_710_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P2: nat > $o] :
? [N4: nat] :
( ( P2 @ N4 )
& ! [M6: nat] :
( ( ord_less_nat @ M6 @ N4 )
=> ~ ( P2 @ M6 ) ) ) ) ) ).
% exists_least_iff
thf(fact_711_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_712_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_713_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_714_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_715_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_716_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_717_linorder__neqE,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_718_order__less__asym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_719_linorder__neq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
= ( ( ord_less_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_720_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_721_order__less__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_722_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_723_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_724_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_725_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_726_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_727_order__less__not__sym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_728_order__less__imp__triv,axiom,
! [X2: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_729_linorder__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_730_order__less__imp__not__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_731_order__less__imp__not__eq2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_732_order__less__imp__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_733_lessThan__strict__subset__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M3 ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_734_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_735_linorder__le__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_736_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_737_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_738_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_739_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_740_order__less__le__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_741_order__le__less__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_742_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_743_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_744_order__less__imp__le,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_745_linorder__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_746_linorder__not__le,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_747_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_748_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_749_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_750_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_751_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_752_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_753_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_754_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_755_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_756_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_757_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_758_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_759_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_760_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_761_not__le__imp__less,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ord_less_nat @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_762_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_763_antisym__conv2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_764_antisym__conv1,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_765_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_766_leI,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% leI
thf(fact_767_leD,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y2 ) ) ).
% leD
thf(fact_768_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_769_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_770_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_771_gr__implies__not__zero,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_772_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_773_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_774_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_775_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_776_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_777_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_778_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_779_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_780_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_781_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_782_can__select__def,axiom,
( can_select_list_nat
= ( ^ [P2: list_nat > $o,A5: set_list_nat] :
? [X: list_nat] :
( ( member_list_nat @ X @ A5 )
& ( P2 @ X )
& ! [Y: list_nat] :
( ( ( member_list_nat @ Y @ A5 )
& ( P2 @ Y ) )
=> ( Y = X ) ) ) ) ) ).
% can_select_def
thf(fact_783_can__select__def,axiom,
( can_select_nat
= ( ^ [P2: nat > $o,A5: set_nat] :
? [X: nat] :
( ( member_nat @ X @ A5 )
& ( P2 @ X )
& ! [Y: nat] :
( ( ( member_nat @ Y @ A5 )
& ( P2 @ Y ) )
=> ( Y = X ) ) ) ) ) ).
% can_select_def
thf(fact_784_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys7: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys7 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys7 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_785_length__induct,axiom,
! [P: list_list_nat > $o,Xs: list_list_nat] :
( ! [Xs2: list_list_nat] :
( ! [Ys7: list_list_nat] :
( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ Ys7 ) @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( P @ Ys7 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_786_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_787_le__neq__implies__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( M3 != N )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_788_less__or__eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( ( ord_less_nat @ M3 @ N )
| ( M3 = N ) )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_789_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N4: nat] :
( ( ord_less_nat @ M6 @ N4 )
| ( M6 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_790_less__imp__le__nat,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_imp_le_nat
thf(fact_791_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N4: nat] :
( ( ord_less_eq_nat @ M6 @ N4 )
& ( M6 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_792_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_793_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_794_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_795_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_796_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_797_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_798_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_799_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_800_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_801_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_802_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_803_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_804_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_805_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_806_Suc__leI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( suc @ M3 ) @ N ) ) ).
% Suc_leI
thf(fact_807_Suc__le__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
= ( ord_less_nat @ M3 @ N ) ) ).
% Suc_le_eq
thf(fact_808_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_809_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_810_Suc__le__lessD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
=> ( ord_less_nat @ M3 @ N ) ) ).
% Suc_le_lessD
thf(fact_811_le__less__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M3 ) )
= ( N = M3 ) ) ) ).
% le_less_Suc_eq
thf(fact_812_less__Suc__eq__le,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_Suc_eq_le
thf(fact_813_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_814_le__imp__less__Suc,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_nat @ M3 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_815_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_816_mono__nat__linear__lb,axiom,
! [F: nat > nat,M3: nat,K: nat] :
( ! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( F @ M2 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K ) @ ( F @ ( plus_plus_nat @ M3 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_817_rgf__limit__ge,axiom,
! [Y2: nat,Xs: list_nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ Y2 @ ( equiva5889994315859557365_limit @ Xs ) ) ) ).
% rgf_limit_ge
thf(fact_818_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_819_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_820_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_821_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_822_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_823_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_824_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_825_length__pos__if__in__set,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_826_length__pos__if__in__set,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_827_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_828_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_829_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_830_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_831_psubsetD,axiom,
! [A2: set_list_nat,B2: set_list_nat,C: list_nat] :
( ( ord_le1190675801316882794st_nat @ A2 @ B2 )
=> ( ( member_list_nat @ C @ A2 )
=> ( member_list_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_832_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_833_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M3: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K3 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K3 ) ) )
=> ( P @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_834_SuccD,axiom,
! [K: nat,Kl: set_list_nat,Kl2: list_nat] :
( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) )
=> ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl ) ) ).
% SuccD
thf(fact_835_SuccD,axiom,
! [K: list_nat,Kl: set_list_list_nat,Kl2: list_list_nat] :
( ( member_list_nat @ K @ ( bNF_Gr3053708287304744325st_nat @ Kl @ Kl2 ) )
=> ( member_list_list_nat @ ( append_list_nat @ Kl2 @ ( cons_list_nat @ K @ nil_list_nat ) ) @ Kl ) ) ).
% SuccD
thf(fact_836_SuccI,axiom,
! [Kl2: list_nat,K: nat,Kl: set_list_nat] :
( ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl )
=> ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_837_SuccI,axiom,
! [Kl2: list_list_nat,K: list_nat,Kl: set_list_list_nat] :
( ( member_list_list_nat @ ( append_list_nat @ Kl2 @ ( cons_list_nat @ K @ nil_list_nat ) ) @ Kl )
=> ( member_list_nat @ K @ ( bNF_Gr3053708287304744325st_nat @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_838_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_839_atMost__eq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ( set_ord_atMost_nat @ X2 )
= ( set_ord_atMost_nat @ Y2 ) )
= ( X2 = Y2 ) ) ).
% atMost_eq_iff
thf(fact_840_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_841_atMost__subset__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X2 ) @ ( set_ord_atMost_nat @ Y2 ) )
= ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_842_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_843_Iic__subset__Iio__iff,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_844_empty__Shift,axiom,
! [Kl: set_list_nat,K: nat] :
( ( member_list_nat @ nil_nat @ Kl )
=> ( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ nil_nat ) )
=> ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_845_empty__Shift,axiom,
! [Kl: set_list_list_nat,K: list_nat] :
( ( member_list_list_nat @ nil_list_nat @ Kl )
=> ( ( member_list_nat @ K @ ( bNF_Gr3053708287304744325st_nat @ Kl @ nil_list_nat ) )
=> ( member_list_list_nat @ nil_list_nat @ ( bNF_Gr9051742241863529473st_nat @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_846_Succ__Shift,axiom,
! [Kl: set_list_nat,K: nat,Kl2: list_nat] :
( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) @ Kl2 )
= ( bNF_Gr6352880689984616693cc_nat @ Kl @ ( cons_nat @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_847_Succ__Shift,axiom,
! [Kl: set_list_list_nat,K: list_nat,Kl2: list_list_nat] :
( ( bNF_Gr3053708287304744325st_nat @ ( bNF_Gr9051742241863529473st_nat @ Kl @ K ) @ Kl2 )
= ( bNF_Gr3053708287304744325st_nat @ Kl @ ( cons_list_nat @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_848_rgf__limit__snoc,axiom,
! [X2: list_nat,Y2: nat] :
( ( equiva5889994315859557365_limit @ ( append_nat @ X2 @ ( cons_nat @ Y2 @ nil_nat ) ) )
= ( ord_max_nat @ ( plus_plus_nat @ Y2 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ X2 ) ) ) ).
% rgf_limit_snoc
thf(fact_849_max__0__1_I2_J,axiom,
( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
= one_one_nat ) ).
% max_0_1(2)
thf(fact_850_max__0__1_I1_J,axiom,
( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
= one_one_nat ) ).
% max_0_1(1)
thf(fact_851_max__add__distrib__left,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ X2 @ Y2 ) @ Z2 )
= ( ord_max_nat @ ( plus_plus_nat @ X2 @ Z2 ) @ ( plus_plus_nat @ Y2 @ Z2 ) ) ) ).
% max_add_distrib_left
thf(fact_852_max__add__distrib__right,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( plus_plus_nat @ X2 @ ( ord_max_nat @ Y2 @ Z2 ) )
= ( ord_max_nat @ ( plus_plus_nat @ X2 @ Y2 ) @ ( plus_plus_nat @ X2 @ Z2 ) ) ) ).
% max_add_distrib_right
thf(fact_853_max__absorb2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_max_nat @ X2 @ Y2 )
= Y2 ) ) ).
% max_absorb2
thf(fact_854_max__absorb1,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_max_nat @ X2 @ Y2 )
= X2 ) ) ).
% max_absorb1
thf(fact_855_max__def,axiom,
( ord_max_nat
= ( ^ [A3: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% max_def
thf(fact_856_ShiftD,axiom,
! [Kl2: list_nat,Kl: set_list_nat,K: nat] :
( ( member_list_nat @ Kl2 @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) )
=> ( member_list_nat @ ( cons_nat @ K @ Kl2 ) @ Kl ) ) ).
% ShiftD
thf(fact_857_ShiftD,axiom,
! [Kl2: list_list_nat,Kl: set_list_list_nat,K: list_nat] :
( ( member_list_list_nat @ Kl2 @ ( bNF_Gr9051742241863529473st_nat @ Kl @ K ) )
=> ( member_list_list_nat @ ( cons_list_nat @ K @ Kl2 ) @ Kl ) ) ).
% ShiftD
thf(fact_858_rgf__limit_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( equiva5889994315859557365_limit @ ( cons_nat @ X2 @ Xs ) )
= ( ord_max_nat @ ( plus_plus_nat @ X2 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs ) ) ) ).
% rgf_limit.simps(2)
thf(fact_859_rgf__limit_Oelims,axiom,
! [X2: list_nat,Y2: nat] :
( ( ( equiva5889994315859557365_limit @ X2 )
= Y2 )
=> ( ( ( X2 = nil_nat )
=> ( Y2 != zero_zero_nat ) )
=> ~ ! [X3: nat,Xs2: list_nat] :
( ( X2
= ( cons_nat @ X3 @ Xs2 ) )
=> ( Y2
!= ( ord_max_nat @ ( plus_plus_nat @ X3 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs2 ) ) ) ) ) ) ).
% rgf_limit.elims
thf(fact_860_max_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_861_max_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_862_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_863_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_864_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_865_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_max_nat @ A3 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_866_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_max_nat @ A3 @ B3 )
= A3 ) ) ) ).
% max.absorb_iff1
thf(fact_867_le__max__iff__disj,axiom,
! [Z2: nat,X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X2 @ Y2 ) )
= ( ( ord_less_eq_nat @ Z2 @ X2 )
| ( ord_less_eq_nat @ Z2 @ Y2 ) ) ) ).
% le_max_iff_disj
thf(fact_868_max_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded2
thf(fact_869_max_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded1
thf(fact_870_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( A3
= ( ord_max_nat @ A3 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_871_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_872_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_873_max_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_max_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% max.orderI
thf(fact_874_max_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( ord_max_nat @ A @ B ) ) ) ).
% max.orderE
thf(fact_875_max_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_876_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D2: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_877_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_878_pinf_I6_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_879_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_880_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_881_minf_I8_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ~ ( ord_less_eq_nat @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_882_minf_I6_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_883_pinf_I8_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_884_rgf__limit_Opelims,axiom,
! [X2: list_nat,Y2: nat] :
( ( ( equiva5889994315859557365_limit @ X2 )
= Y2 )
=> ( ( accp_list_nat @ equiva5575797544161152836it_rel @ X2 )
=> ( ( ( X2 = nil_nat )
=> ( ( Y2 = zero_zero_nat )
=> ~ ( accp_list_nat @ equiva5575797544161152836it_rel @ nil_nat ) ) )
=> ~ ! [X3: nat,Xs2: list_nat] :
( ( X2
= ( cons_nat @ X3 @ Xs2 ) )
=> ( ( Y2
= ( ord_max_nat @ ( plus_plus_nat @ X3 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs2 ) ) )
=> ~ ( accp_list_nat @ equiva5575797544161152836it_rel @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).
% rgf_limit.pelims
thf(fact_885_append__butlast__last__id,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( append_nat @ ( butlast_nat @ Xs ) @ ( cons_nat @ ( last_nat @ Xs ) @ nil_nat ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_886_append__butlast__last__id,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( append_list_nat @ ( butlast_list_nat @ Xs ) @ ( cons_list_nat @ ( last_list_nat @ Xs ) @ nil_list_nat ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_887_last__appendR,axiom,
! [Ys3: list_nat,Xs: list_nat] :
( ( Ys3 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys3 ) )
= ( last_nat @ Ys3 ) ) ) ).
% last_appendR
thf(fact_888_last__appendR,axiom,
! [Ys3: list_list_nat,Xs: list_list_nat] :
( ( Ys3 != nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs @ Ys3 ) )
= ( last_list_nat @ Ys3 ) ) ) ).
% last_appendR
thf(fact_889_last__appendL,axiom,
! [Ys3: list_nat,Xs: list_nat] :
( ( Ys3 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys3 ) )
= ( last_nat @ Xs ) ) ) ).
% last_appendL
thf(fact_890_last__appendL,axiom,
! [Ys3: list_list_nat,Xs: list_list_nat] :
( ( Ys3 = nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs @ Ys3 ) )
= ( last_list_nat @ Xs ) ) ) ).
% last_appendL
thf(fact_891_last__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= X2 ) ).
% last_snoc
thf(fact_892_last__snoc,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( last_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= X2 ) ).
% last_snoc
thf(fact_893_last__ConsR,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_894_last__ConsR,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( Xs != nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= ( last_list_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_895_last__ConsL,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_896_last__ConsL,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( Xs = nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_897_last_Osimps,axiom,
! [Xs: list_nat,X2: nat] :
( ( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= ( last_nat @ Xs ) ) ) ) ).
% last.simps
thf(fact_898_last_Osimps,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( ( Xs = nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= ( last_list_nat @ Xs ) ) ) ) ).
% last.simps
thf(fact_899_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_900_last__in__set,axiom,
! [As2: list_nat] :
( ( As2 != nil_nat )
=> ( member_nat @ ( last_nat @ As2 ) @ ( set_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_901_longest__common__suffix,axiom,
! [Xs: list_nat,Ys3: list_nat] :
? [Ss: list_nat,Xs5: list_nat,Ys6: list_nat] :
( ( Xs
= ( append_nat @ Xs5 @ Ss ) )
& ( Ys3
= ( append_nat @ Ys6 @ Ss ) )
& ( ( Xs5 = nil_nat )
| ( Ys6 = nil_nat )
| ( ( last_nat @ Xs5 )
!= ( last_nat @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_902_longest__common__suffix,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
? [Ss: list_list_nat,Xs5: list_list_nat,Ys6: list_list_nat] :
( ( Xs
= ( append_list_nat @ Xs5 @ Ss ) )
& ( Ys3
= ( append_list_nat @ Ys6 @ Ss ) )
& ( ( Xs5 = nil_list_nat )
| ( Ys6 = nil_list_nat )
| ( ( last_list_nat @ Xs5 )
!= ( last_list_nat @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_903_last__append,axiom,
! [Ys3: list_nat,Xs: list_nat] :
( ( ( Ys3 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys3 ) )
= ( last_nat @ Xs ) ) )
& ( ( Ys3 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys3 ) )
= ( last_nat @ Ys3 ) ) ) ) ).
% last_append
thf(fact_904_last__append,axiom,
! [Ys3: list_list_nat,Xs: list_list_nat] :
( ( ( Ys3 = nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs @ Ys3 ) )
= ( last_list_nat @ Xs ) ) )
& ( ( Ys3 != nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs @ Ys3 ) )
= ( last_list_nat @ Ys3 ) ) ) ) ).
% last_append
thf(fact_905_last__map,axiom,
! [Xs: list_list_nat,F: list_nat > nat] :
( ( Xs != nil_list_nat )
=> ( ( last_nat @ ( map_list_nat_nat @ F @ Xs ) )
= ( F @ ( last_list_nat @ Xs ) ) ) ) ).
% last_map
thf(fact_906_last__map,axiom,
! [Xs: list_nat,F: nat > nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( map_nat_nat @ F @ Xs ) )
= ( F @ ( last_nat @ Xs ) ) ) ) ).
% last_map
thf(fact_907_last__map,axiom,
! [Xs: list_list_nat,F: list_nat > list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( last_list_list_nat @ ( map_li960784813134754710st_nat @ F @ Xs ) )
= ( F @ ( last_list_nat @ Xs ) ) ) ) ).
% last_map
thf(fact_908_last__map,axiom,
! [Xs: list_nat,F: nat > list_nat] :
( ( Xs != nil_nat )
=> ( ( last_list_nat @ ( map_nat_list_nat @ F @ Xs ) )
= ( F @ ( last_nat @ Xs ) ) ) ) ).
% last_map
thf(fact_909_snoc__eq__iff__butlast,axiom,
! [Xs: list_nat,X2: nat,Ys3: list_nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
= Ys3 )
= ( ( Ys3 != nil_nat )
& ( ( butlast_nat @ Ys3 )
= Xs )
& ( ( last_nat @ Ys3 )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_910_snoc__eq__iff__butlast,axiom,
! [Xs: list_list_nat,X2: list_nat,Ys3: list_list_nat] :
( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) )
= Ys3 )
= ( ( Ys3 != nil_list_nat )
& ( ( butlast_list_nat @ Ys3 )
= Xs )
& ( ( last_list_nat @ Ys3 )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_911_append__one__prefix,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( prefix_nat @ Xs @ Ys3 )
=> ( ( ord_less_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys3 ) )
=> ( prefix_nat @ ( append_nat @ Xs @ ( cons_nat @ ( nth_nat @ Ys3 @ ( size_size_list_nat @ Xs ) ) @ nil_nat ) ) @ Ys3 ) ) ) ).
% append_one_prefix
thf(fact_912_append__one__prefix,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( prefix_list_nat @ Xs @ Ys3 )
=> ( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( prefix_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ ( nth_list_nat @ Ys3 @ ( size_s3023201423986296836st_nat @ Xs ) ) @ nil_list_nat ) ) @ Ys3 ) ) ) ).
% append_one_prefix
thf(fact_913_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_914_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_915_nth__Cons__Suc,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_916_nth__Cons__Suc,axiom,
! [X2: list_nat,Xs: list_list_nat,N: nat] :
( ( nth_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_list_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_917_nth__Cons__0,axiom,
! [X2: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_918_nth__Cons__0,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( nth_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_919_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
=> ( ( nth_nat @ ( upt @ I @ J ) @ K )
= ( plus_plus_nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_920_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_921_nth__append__length,axiom,
! [Xs: list_nat,X2: nat,Ys3: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys3 ) ) @ ( size_size_list_nat @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_922_nth__append__length,axiom,
! [Xs: list_list_nat,X2: list_nat,Ys3: list_list_nat] :
( ( nth_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ Ys3 ) ) @ ( size_s3023201423986296836st_nat @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_923_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_924_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_list_nat @ ( map_nat_list_nat @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_925_nth__map,axiom,
! [N: nat,Xs: list_list_nat,F: list_nat > nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_nat @ ( map_list_nat_nat @ F @ Xs ) @ N )
= ( F @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_926_nth__map,axiom,
! [N: nat,Xs: list_list_nat,F: list_nat > list_list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_list_nat @ ( map_li960784813134754710st_nat @ F @ Xs ) @ N )
= ( F @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_927_nth__append__length__plus,axiom,
! [Xs: list_nat,Ys3: list_nat,N: nat] :
( ( nth_nat @ ( append_nat @ Xs @ Ys3 ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
= ( nth_nat @ Ys3 @ N ) ) ).
% nth_append_length_plus
thf(fact_928_nth__append__length__plus,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,N: nat] :
( ( nth_list_nat @ ( append_list_nat @ Xs @ Ys3 ) @ ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) )
= ( nth_list_nat @ Ys3 @ N ) ) ).
% nth_append_length_plus
thf(fact_929_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_930_map__Suc__upt,axiom,
! [M3: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M3 @ N ) )
= ( upt @ ( suc @ M3 ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_931_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
= ( ^ [Xs3: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I4 )
= ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_932_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_list_nat,Z: list_list_nat] : ( Y4 = Z ) )
= ( ^ [Xs3: list_list_nat,Ys: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs3 )
= ( size_s3023201423986296836st_nat @ Ys ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs3 ) )
=> ( ( nth_list_nat @ Xs3 @ I4 )
= ( nth_list_nat @ Ys @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_933_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X6: nat] : ( P @ I4 @ X6 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_nat @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_934_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X6: list_nat] : ( P @ I4 @ X6 ) ) )
= ( ? [Xs3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_list_nat @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_935_nth__equalityI,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys3 @ I2 ) ) )
=> ( Xs = Ys3 ) ) ) ).
% nth_equalityI
thf(fact_936_nth__equalityI,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ Xs @ I2 )
= ( nth_list_nat @ Ys3 @ I2 ) ) )
=> ( Xs = Ys3 ) ) ) ).
% nth_equalityI
thf(fact_937_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_938_upt__conv__Cons__Cons,axiom,
! [M3: nat,N: nat,Ns: list_nat,Q: nat] :
( ( ( cons_nat @ M3 @ ( cons_nat @ N @ Ns ) )
= ( upt @ M3 @ Q ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M3 ) @ Q ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_939_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_940_all__set__conv__all__nth,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( P @ X ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P @ ( nth_list_nat @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_941_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X2: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_942_all__nth__imp__all__set,axiom,
! [Xs: list_list_nat,P: list_nat > $o,X2: list_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P @ ( nth_list_nat @ Xs @ I2 ) ) )
=> ( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_943_in__set__conv__nth,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I4 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_944_in__set__conv__nth,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ I4 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_945_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_946_list__ball__nth,axiom,
! [N: nat,Xs: list_list_nat,P: list_nat > $o] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_947_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_948_nth__mem,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( member_list_nat @ ( nth_list_nat @ Xs @ N ) @ ( set_list_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_949_map__equality__iff,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys3: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys3 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys3 ) )
=> ( ( F @ ( nth_nat @ Xs @ I4 ) )
= ( G @ ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_950_map__equality__iff,axiom,
! [F: nat > list_nat,Xs: list_nat,G: nat > list_nat,Ys3: list_nat] :
( ( ( map_nat_list_nat @ F @ Xs )
= ( map_nat_list_nat @ G @ Ys3 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys3 ) )
=> ( ( F @ ( nth_nat @ Xs @ I4 ) )
= ( G @ ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_951_map__equality__iff,axiom,
! [F: nat > list_list_nat,Xs: list_nat,G: list_nat > list_list_nat,Ys3: list_list_nat] :
( ( ( map_na6205611841492582150st_nat @ F @ Xs )
= ( map_li960784813134754710st_nat @ G @ Ys3 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( ( F @ ( nth_nat @ Xs @ I4 ) )
= ( G @ ( nth_list_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_952_map__equality__iff,axiom,
! [F: nat > nat,Xs: list_nat,G: list_nat > nat,Ys3: list_list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_list_nat_nat @ G @ Ys3 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( ( F @ ( nth_nat @ Xs @ I4 ) )
= ( G @ ( nth_list_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_953_map__equality__iff,axiom,
! [F: nat > list_nat,Xs: list_nat,G: list_nat > list_nat,Ys3: list_list_nat] :
( ( ( map_nat_list_nat @ F @ Xs )
= ( map_li7225945977422193158st_nat @ G @ Ys3 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( ( F @ ( nth_nat @ Xs @ I4 ) )
= ( G @ ( nth_list_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_954_map__equality__iff,axiom,
! [F: list_nat > nat,Xs: list_list_nat,G: nat > nat,Ys3: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys3 ) )
= ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys3 ) )
=> ( ( F @ ( nth_list_nat @ Xs @ I4 ) )
= ( G @ ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_955_map__equality__iff,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,G: nat > list_nat,Ys3: list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= ( map_nat_list_nat @ G @ Ys3 ) )
= ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys3 ) )
=> ( ( F @ ( nth_list_nat @ Xs @ I4 ) )
= ( G @ ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_956_map__equality__iff,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat,G: nat > list_list_nat,Ys3: list_nat] :
( ( ( map_li960784813134754710st_nat @ F @ Xs )
= ( map_na6205611841492582150st_nat @ G @ Ys3 ) )
= ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys3 ) )
=> ( ( F @ ( nth_list_nat @ Xs @ I4 ) )
= ( G @ ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_957_map__equality__iff,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat,G: list_nat > list_list_nat,Ys3: list_list_nat] :
( ( ( map_li960784813134754710st_nat @ F @ Xs )
= ( map_li960784813134754710st_nat @ G @ Ys3 ) )
= ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( ( F @ ( nth_list_nat @ Xs @ I4 ) )
= ( G @ ( nth_list_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_958_nth__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs ) ) )
=> ( ( nth_nat @ ( butlast_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_959_nth__butlast,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ ( butlast_list_nat @ Xs ) ) )
=> ( ( nth_list_nat @ ( butlast_list_nat @ Xs ) @ N )
= ( nth_list_nat @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_960_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_961_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
= ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_962_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N4 ) ) ) ) ).
% atLeast_upt
thf(fact_963_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X2: nat,Xs: list_nat] :
( ( ( upt @ I @ J )
= ( cons_nat @ X2 @ Xs ) )
= ( ( ord_less_nat @ I @ J )
& ( I = X2 )
& ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_964_upt__rec,axiom,
( upt
= ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_965_atMost__upto,axiom,
( set_ord_atMost_nat
= ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N4 ) ) ) ) ) ).
% atMost_upto
thf(fact_966_nth__equal__first__eq,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_967_nth__equal__first__eq,axiom,
! [X2: list_nat,Xs: list_list_nat,N: nat] :
( ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ( nth_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_968_upt__rec__numeral,axiom,
! [M3: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M3 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M3 ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_969_enum__rgfs_Oelims,axiom,
! [X2: nat,Y2: list_list_nat] :
( ( ( equiva7426478223624825838m_rgfs @ X2 )
= Y2 )
=> ( ( ( X2 = zero_zero_nat )
=> ( Y2
!= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ N2 ) )
=> ( Y2
!= ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ^ [X: list_nat] :
( map_nat_list_nat
@ ^ [Y: nat] : ( append_nat @ X @ ( cons_nat @ Y @ nil_nat ) )
@ ( upt @ zero_zero_nat @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ X ) @ one_one_nat ) ) )
@ ( equiva7426478223624825838m_rgfs @ N2 ) ) ) ) ) ) ) ).
% enum_rgfs.elims
thf(fact_970_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ ( suc @ I ) @ Xs )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ nil_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_971_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( take_list_nat @ ( suc @ I ) @ Xs )
= ( append_list_nat @ ( take_list_nat @ I @ Xs ) @ ( cons_list_nat @ ( nth_list_nat @ Xs @ I ) @ nil_list_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_972_numeral__eq__iff,axiom,
! [M3: num,N: num] :
( ( ( numeral_numeral_nat @ M3 )
= ( numeral_numeral_nat @ N ) )
= ( M3 = N ) ) ).
% numeral_eq_iff
thf(fact_973_map__ident,axiom,
( ( map_nat_nat
@ ^ [X: nat] : X )
= ( ^ [Xs3: list_nat] : Xs3 ) ) ).
% map_ident
thf(fact_974_numeral__le__iff,axiom,
! [M3: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M3 @ N ) ) ).
% numeral_le_iff
thf(fact_975_numeral__less__iff,axiom,
! [M3: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M3 @ N ) ) ).
% numeral_less_iff
thf(fact_976_add__numeral__left,axiom,
! [V: num,W2: num,Z2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_977_numeral__plus__numeral,axiom,
! [M3: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M3 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_978_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ V ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_979_max__0__1_I4_J,axiom,
! [X2: num] :
( ( ord_max_nat @ ( numeral_numeral_nat @ X2 ) @ zero_zero_nat )
= ( numeral_numeral_nat @ X2 ) ) ).
% max_0_1(4)
thf(fact_980_max__0__1_I3_J,axiom,
! [X2: num] :
( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X2 ) )
= ( numeral_numeral_nat @ X2 ) ) ).
% max_0_1(3)
thf(fact_981_max__0__1_I5_J,axiom,
! [X2: num] :
( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X2 ) )
= ( numeral_numeral_nat @ X2 ) ) ).
% max_0_1(5)
thf(fact_982_max__0__1_I6_J,axiom,
! [X2: num] :
( ( ord_max_nat @ ( numeral_numeral_nat @ X2 ) @ one_one_nat )
= ( numeral_numeral_nat @ X2 ) ) ).
% max_0_1(6)
thf(fact_983_take__Suc__Cons,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( take_nat @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_984_take__Suc__Cons,axiom,
! [N: nat,X2: list_nat,Xs: list_list_nat] :
( ( take_list_nat @ ( suc @ N ) @ ( cons_list_nat @ X2 @ Xs ) )
= ( cons_list_nat @ X2 @ ( take_list_nat @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_985_take__eq__Nil2,axiom,
! [N: nat,Xs: list_nat] :
( ( nil_nat
= ( take_nat @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil2
thf(fact_986_take__eq__Nil2,axiom,
! [N: nat,Xs: list_list_nat] :
( ( nil_list_nat
= ( take_list_nat @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_list_nat ) ) ) ).
% take_eq_Nil2
thf(fact_987_take__eq__Nil,axiom,
! [N: nat,Xs: list_nat] :
( ( ( take_nat @ N @ Xs )
= nil_nat )
= ( ( N = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil
thf(fact_988_take__eq__Nil,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ( take_list_nat @ N @ Xs )
= nil_list_nat )
= ( ( N = zero_zero_nat )
| ( Xs = nil_list_nat ) ) ) ).
% take_eq_Nil
thf(fact_989_take0,axiom,
( ( take_nat @ zero_zero_nat )
= ( ^ [Xs3: list_nat] : nil_nat ) ) ).
% take0
thf(fact_990_take0,axiom,
( ( take_list_nat @ zero_zero_nat )
= ( ^ [Xs3: list_list_nat] : nil_list_nat ) ) ).
% take0
thf(fact_991_take__all__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ( take_nat @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_992_take__all__iff,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ( take_list_nat @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_993_take__all,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( take_nat @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_994_take__all,axiom,
! [Xs: list_list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N )
=> ( ( take_list_nat @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_995_take__upt,axiom,
! [I: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M3 ) @ N )
=> ( ( take_nat @ M3 @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus_nat @ I @ M3 ) ) ) ) ).
% take_upt
thf(fact_996_nth__take,axiom,
! [I: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( take_nat @ N @ Xs ) @ I )
= ( nth_nat @ Xs @ I ) ) ) ).
% nth_take
thf(fact_997_concat__map__singleton,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( concat_list_list_nat
@ ( map_li4109527381804435110st_nat
@ ^ [X: list_nat] : ( cons_list_list_nat @ ( F @ X ) @ nil_list_list_nat )
@ Xs ) )
= ( map_li960784813134754710st_nat @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_998_concat__map__singleton,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( concat_nat
@ ( map_nat_list_nat
@ ^ [X: nat] : ( cons_nat @ ( F @ X ) @ nil_nat )
@ Xs ) )
= ( map_nat_nat @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_999_concat__map__singleton,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( concat_list_nat
@ ( map_na6205611841492582150st_nat
@ ^ [X: nat] : ( cons_list_nat @ ( F @ X ) @ nil_list_nat )
@ Xs ) )
= ( map_nat_list_nat @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_1000_concat__map__singleton,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat] :
( ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ^ [X: list_nat] : ( cons_list_nat @ ( F @ X ) @ nil_list_nat )
@ Xs ) )
= ( map_li7225945977422193158st_nat @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_1001_suffixes__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( suffixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= ( cons_list_nat @ nil_nat
@ ( map_li7225945977422193158st_nat
@ ^ [Ys: list_nat] : ( append_nat @ Ys @ ( cons_nat @ X2 @ nil_nat ) )
@ ( suffixes_nat @ Xs ) ) ) ) ).
% suffixes_snoc
thf(fact_1002_suffixes__snoc,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( suffixes_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= ( cons_list_list_nat @ nil_list_nat
@ ( map_li2855073862107769254st_nat
@ ^ [Ys: list_list_nat] : ( append_list_nat @ Ys @ ( cons_list_nat @ X2 @ nil_list_nat ) )
@ ( suffixes_list_nat @ Xs ) ) ) ) ).
% suffixes_snoc
thf(fact_1003_map__add__upt,axiom,
! [N: nat,M3: nat] :
( ( map_nat_nat
@ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
@ ( upt @ zero_zero_nat @ M3 ) )
= ( upt @ N @ ( plus_plus_nat @ M3 @ N ) ) ) ).
% map_add_upt
thf(fact_1004_map__upt__Suc,axiom,
! [F: nat > nat,N: nat] :
( ( map_nat_nat @ F @ ( upt @ zero_zero_nat @ ( suc @ N ) ) )
= ( cons_nat @ ( F @ zero_zero_nat )
@ ( map_nat_nat
@ ^ [I4: nat] : ( F @ ( suc @ I4 ) )
@ ( upt @ zero_zero_nat @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_1005_map__upt__Suc,axiom,
! [F: nat > list_nat,N: nat] :
( ( map_nat_list_nat @ F @ ( upt @ zero_zero_nat @ ( suc @ N ) ) )
= ( cons_list_nat @ ( F @ zero_zero_nat )
@ ( map_nat_list_nat
@ ^ [I4: nat] : ( F @ ( suc @ I4 ) )
@ ( upt @ zero_zero_nat @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_1006_map__nth,axiom,
! [Xs: list_nat] :
( ( map_nat_nat @ ( nth_nat @ Xs ) @ ( upt @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) )
= Xs ) ).
% map_nth
thf(fact_1007_map__nth,axiom,
! [Xs: list_list_nat] :
( ( map_nat_list_nat @ ( nth_list_nat @ Xs ) @ ( upt @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) )
= Xs ) ).
% map_nth
thf(fact_1008_max__def__raw,axiom,
( ord_max_nat
= ( ^ [A3: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% max_def_raw
thf(fact_1009_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X: nat] : ( ord_less_eq_nat @ X @ U2 ) ) ) ) ).
% atMost_def
thf(fact_1010_Shift__def,axiom,
( bNF_Gr1872714664788909425ft_nat
= ( ^ [Kl3: set_list_nat,K2: nat] :
( collect_list_nat
@ ^ [Kl4: list_nat] : ( member_list_nat @ ( cons_nat @ K2 @ Kl4 ) @ Kl3 ) ) ) ) ).
% Shift_def
thf(fact_1011_Shift__def,axiom,
( bNF_Gr9051742241863529473st_nat
= ( ^ [Kl3: set_list_list_nat,K2: list_nat] :
( collec5989764272469232197st_nat
@ ^ [Kl4: list_list_nat] : ( member_list_list_nat @ ( cons_list_nat @ K2 @ Kl4 ) @ Kl3 ) ) ) ) ).
% Shift_def
thf(fact_1012_set__prefixes__eq,axiom,
! [Xs: list_nat] :
( ( set_list_nat2 @ ( prefixes_nat @ Xs ) )
= ( collect_list_nat
@ ^ [Ys: list_nat] : ( prefix_nat @ Ys @ Xs ) ) ) ).
% set_prefixes_eq
thf(fact_1013_set__prefixes__eq,axiom,
! [Xs: list_list_nat] :
( ( set_list_list_nat2 @ ( prefixes_list_nat @ Xs ) )
= ( collec5989764272469232197st_nat
@ ^ [Ys: list_list_nat] : ( prefix_list_nat @ Ys @ Xs ) ) ) ).
% set_prefixes_eq
thf(fact_1014_list_Omap__ident,axiom,
! [T2: list_nat] :
( ( map_nat_nat
@ ^ [X: nat] : X
@ T2 )
= T2 ) ).
% list.map_ident
thf(fact_1015_subset__Collect__iff,axiom,
! [B2: set_list_nat,A2: set_list_nat,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ B2
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A2 )
& ( P @ X ) ) ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ B2 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1016_subset__Collect__iff,axiom,
! [B2: set_nat,A2: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ B2
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1017_subset__CollectI,axiom,
! [B2: set_list_nat,A2: set_list_nat,Q2: list_nat > $o,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ B2 )
=> ( ( Q2 @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ B2 )
& ( Q2 @ X ) ) )
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A2 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1018_subset__CollectI,axiom,
! [B2: set_nat,A2: set_nat,Q2: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( Q2 @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ B2 )
& ( Q2 @ X ) ) )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1019_take__Nil,axiom,
! [N: nat] :
( ( take_nat @ N @ nil_nat )
= nil_nat ) ).
% take_Nil
thf(fact_1020_take__Nil,axiom,
! [N: nat] :
( ( take_list_nat @ N @ nil_list_nat )
= nil_list_nat ) ).
% take_Nil
thf(fact_1021_in__set__takeD,axiom,
! [X2: list_nat,N: nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( take_list_nat @ N @ Xs ) ) )
=> ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1022_in__set__takeD,axiom,
! [X2: nat,N: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) )
=> ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1023_take__map,axiom,
! [N: nat,F: nat > nat,Xs: list_nat] :
( ( take_nat @ N @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( take_nat @ N @ Xs ) ) ) ).
% take_map
thf(fact_1024_take__map,axiom,
! [N: nat,F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( take_list_list_nat @ N @ ( map_li960784813134754710st_nat @ F @ Xs ) )
= ( map_li960784813134754710st_nat @ F @ ( take_list_nat @ N @ Xs ) ) ) ).
% take_map
thf(fact_1025_take__map,axiom,
! [N: nat,F: nat > list_nat,Xs: list_nat] :
( ( take_list_nat @ N @ ( map_nat_list_nat @ F @ Xs ) )
= ( map_nat_list_nat @ F @ ( take_nat @ N @ Xs ) ) ) ).
% take_map
thf(fact_1026_take__is__prefix,axiom,
! [N: nat,Xs: list_nat] : ( prefix_nat @ ( take_nat @ N @ Xs ) @ Xs ) ).
% take_is_prefix
thf(fact_1027_take__is__prefix,axiom,
! [N: nat,Xs: list_list_nat] : ( prefix_list_nat @ ( take_list_nat @ N @ Xs ) @ Xs ) ).
% take_is_prefix
thf(fact_1028_take__equalityI,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ! [I2: nat] :
( ( take_nat @ I2 @ Xs )
= ( take_nat @ I2 @ Ys3 ) )
=> ( Xs = Ys3 ) ) ).
% take_equalityI
thf(fact_1029_less__eq__set__def,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
( ord_le1520216061033275535_nat_o
@ ^ [X: list_nat] : ( member_list_nat @ X @ A5 )
@ ^ [X: list_nat] : ( member_list_nat @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1030_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1031_Collect__subset,axiom,
! [A2: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1032_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1033_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_1034_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X: nat] : ( ord_less_nat @ X @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_1035_less__set__def,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
( ord_less_list_nat_o
@ ^ [X: list_nat] : ( member_list_nat @ X @ A5 )
@ ^ [X: list_nat] : ( member_list_nat @ X @ B5 ) ) ) ) ).
% less_set_def
thf(fact_1036_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ord_less_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).
% less_set_def
thf(fact_1037_dual__min,axiom,
( ( min_nat
@ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X ) )
= ord_max_nat ) ).
% dual_min
thf(fact_1038_take__0,axiom,
! [Xs: list_nat] :
( ( take_nat @ zero_zero_nat @ Xs )
= nil_nat ) ).
% take_0
thf(fact_1039_take__0,axiom,
! [Xs: list_list_nat] :
( ( take_list_nat @ zero_zero_nat @ Xs )
= nil_list_nat ) ).
% take_0
thf(fact_1040_set__take__subset,axiom,
! [N: nat,Xs: list_list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( take_list_nat @ N @ Xs ) ) @ ( set_list_nat2 @ Xs ) ) ).
% set_take_subset
thf(fact_1041_set__take__subset,axiom,
! [N: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_take_subset
thf(fact_1042_Succ__def,axiom,
( bNF_Gr6352880689984616693cc_nat
= ( ^ [Kl3: set_list_nat,Kl4: list_nat] :
( collect_nat
@ ^ [K2: nat] : ( member_list_nat @ ( append_nat @ Kl4 @ ( cons_nat @ K2 @ nil_nat ) ) @ Kl3 ) ) ) ) ).
% Succ_def
thf(fact_1043_Succ__def,axiom,
( bNF_Gr3053708287304744325st_nat
= ( ^ [Kl3: set_list_list_nat,Kl4: list_list_nat] :
( collect_list_nat
@ ^ [K2: list_nat] : ( member_list_list_nat @ ( append_list_nat @ Kl4 @ ( cons_list_nat @ K2 @ nil_list_nat ) ) @ Kl3 ) ) ) ) ).
% Succ_def
thf(fact_1044_n__lists_Osimps_I2_J,axiom,
! [N: nat,Xs: list_nat] :
( ( n_lists_nat @ ( suc @ N ) @ Xs )
= ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ^ [Ys: list_nat] :
( map_nat_list_nat
@ ^ [Y: nat] : ( cons_nat @ Y @ Ys )
@ Xs )
@ ( n_lists_nat @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_1045_n__lists_Osimps_I2_J,axiom,
! [N: nat,Xs: list_list_nat] :
( ( n_lists_list_nat @ ( suc @ N ) @ Xs )
= ( concat_list_list_nat
@ ( map_li5769348595424326838st_nat
@ ^ [Ys: list_list_nat] :
( map_li960784813134754710st_nat
@ ^ [Y: list_nat] : ( cons_list_nat @ Y @ Ys )
@ Xs )
@ ( n_lists_list_nat @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_1046_product__lists_Osimps_I2_J,axiom,
! [Xs: list_list_nat,Xss2: list_list_list_nat] :
( ( produc6783906451316923569st_nat @ ( cons_list_list_nat @ Xs @ Xss2 ) )
= ( concat_list_list_nat
@ ( map_li4109527381804435110st_nat
@ ^ [X: list_nat] : ( map_li2855073862107769254st_nat @ ( cons_list_nat @ X ) @ ( produc6783906451316923569st_nat @ Xss2 ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_1047_product__lists_Osimps_I2_J,axiom,
! [Xs: list_nat,Xss2: list_list_nat] :
( ( product_lists_nat @ ( cons_list_nat @ Xs @ Xss2 ) )
= ( concat_list_nat
@ ( map_na6205611841492582150st_nat
@ ^ [X: nat] : ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( product_lists_nat @ Xss2 ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_1048_subseqs_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( subseqs_nat @ ( cons_nat @ X2 @ Xs ) )
= ( append_list_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X2 ) @ ( subseqs_nat @ Xs ) ) @ ( subseqs_nat @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_1049_subseqs_Osimps_I2_J,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( subseqs_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= ( append_list_list_nat @ ( map_li2855073862107769254st_nat @ ( cons_list_nat @ X2 ) @ ( subseqs_list_nat @ Xs ) ) @ ( subseqs_list_nat @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_1050_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_1051_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_1052_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_1053_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_1054_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_1055_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_1056_one__plus__numeral__commute,axiom,
! [X2: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X2 ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_1057_set__n__lists,axiom,
! [N: nat,Xs: list_nat] :
( ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) )
= ( collect_list_nat
@ ^ [Ys: list_nat] :
( ( ( size_size_list_nat @ Ys )
= N )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Ys ) @ ( set_nat2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_1058_set__n__lists,axiom,
! [N: nat,Xs: list_list_nat] :
( ( set_list_list_nat2 @ ( n_lists_list_nat @ N @ Xs ) )
= ( collec5989764272469232197st_nat
@ ^ [Ys: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Ys )
= N )
& ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Ys ) @ ( set_list_nat2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_1059_set__take__subset__set__take,axiom,
! [M3: nat,N: nat,Xs: list_list_nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( take_list_nat @ M3 @ Xs ) ) @ ( set_list_nat2 @ ( take_list_nat @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1060_set__take__subset__set__take,axiom,
! [M3: nat,N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ M3 @ Xs ) ) @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1061_take__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ N @ ( butlast_nat @ Xs ) )
= ( take_nat @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_1062_take__butlast,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( take_list_nat @ N @ ( butlast_list_nat @ Xs ) )
= ( take_list_nat @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_1063_nth__take__lemma,axiom,
! [K: nat,Xs: list_nat,Ys3: list_nat] :
( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys3 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys3 @ I2 ) ) )
=> ( ( take_nat @ K @ Xs )
= ( take_nat @ K @ Ys3 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_1064_nth__take__lemma,axiom,
! [K: nat,Xs: list_list_nat,Ys3: list_list_nat] :
( ( ord_less_eq_nat @ K @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_list_nat @ Xs @ I2 )
= ( nth_list_nat @ Ys3 @ I2 ) ) )
=> ( ( take_list_nat @ K @ Xs )
= ( take_list_nat @ K @ Ys3 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_1065_enum__rgfs_Osimps_I2_J,axiom,
! [N: nat] :
( ( equiva7426478223624825838m_rgfs @ ( suc @ N ) )
= ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ^ [X: list_nat] :
( map_nat_list_nat
@ ^ [Y: nat] : ( append_nat @ X @ ( cons_nat @ Y @ nil_nat ) )
@ ( upt @ zero_zero_nat @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ X ) @ one_one_nat ) ) )
@ ( equiva7426478223624825838m_rgfs @ N ) ) ) ) ).
% enum_rgfs.simps(2)
thf(fact_1066_enum__rgfs_Opelims,axiom,
! [X2: nat,Y2: list_list_nat] :
( ( ( equiva7426478223624825838m_rgfs @ X2 )
= Y2 )
=> ( ( accp_nat @ equiva1432535406783100555fs_rel @ X2 )
=> ( ( ( X2 = zero_zero_nat )
=> ( ( Y2
= ( cons_list_nat @ nil_nat @ nil_list_nat ) )
=> ~ ( accp_nat @ equiva1432535406783100555fs_rel @ zero_zero_nat ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ N2 ) )
=> ( ( Y2
= ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ^ [X: list_nat] :
( map_nat_list_nat
@ ^ [Y: nat] : ( append_nat @ X @ ( cons_nat @ Y @ nil_nat ) )
@ ( upt @ zero_zero_nat @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ X ) @ one_one_nat ) ) )
@ ( equiva7426478223624825838m_rgfs @ N2 ) ) ) )
=> ~ ( accp_nat @ equiva1432535406783100555fs_rel @ ( suc @ N2 ) ) ) ) ) ) ) ).
% enum_rgfs.pelims
thf(fact_1067_transpose__rectangle,axiom,
! [Xs: list_list_list_nat,N: nat] :
( ( ( Xs = nil_list_list_nat )
=> ( N = zero_zero_nat ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s6248950052170075156st_nat @ Xs ) )
=> ( ( size_s3023201423986296836st_nat @ ( nth_list_list_nat @ Xs @ I2 ) )
= N ) )
=> ( ( transpose_list_nat @ Xs )
= ( map_na6205611841492582150st_nat
@ ^ [I4: nat] :
( map_nat_list_nat
@ ^ [J3: nat] : ( nth_list_nat @ ( nth_list_list_nat @ Xs @ J3 ) @ I4 )
@ ( upt @ zero_zero_nat @ ( size_s6248950052170075156st_nat @ Xs ) ) )
@ ( upt @ zero_zero_nat @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_1068_transpose__rectangle,axiom,
! [Xs: list_list_nat,N: nat] :
( ( ( Xs = nil_list_nat )
=> ( N = zero_zero_nat ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
= N ) )
=> ( ( transpose_nat @ Xs )
= ( map_nat_list_nat
@ ^ [I4: nat] :
( map_nat_nat
@ ^ [J3: nat] : ( nth_nat @ ( nth_list_nat @ Xs @ J3 ) @ I4 )
@ ( upt @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) )
@ ( upt @ zero_zero_nat @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_1069_transpose_Osimps_I1_J,axiom,
( ( transpose_nat @ nil_list_nat )
= nil_list_nat ) ).
% transpose.simps(1)
thf(fact_1070_transpose_Osimps_I2_J,axiom,
! [Xss2: list_list_list_nat] :
( ( transpose_list_nat @ ( cons_list_list_nat @ nil_list_nat @ Xss2 ) )
= ( transpose_list_nat @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_1071_transpose_Osimps_I2_J,axiom,
! [Xss2: list_list_nat] :
( ( transpose_nat @ ( cons_list_nat @ nil_nat @ Xss2 ) )
= ( transpose_nat @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_1072_transpose__map__map,axiom,
! [F: nat > nat,Xs: list_list_nat] :
( ( transpose_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) )
= ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ ( transpose_nat @ Xs ) ) ) ).
% transpose_map_map
thf(fact_1073_transpose__map__map,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_list_nat] :
( ( transp4696039609411287008st_nat @ ( map_li5769348595424326838st_nat @ ( map_li960784813134754710st_nat @ F ) @ Xs ) )
= ( map_li5769348595424326838st_nat @ ( map_li960784813134754710st_nat @ F ) @ ( transpose_list_nat @ Xs ) ) ) ).
% transpose_map_map
thf(fact_1074_transpose__map__map,axiom,
! [F: nat > list_nat,Xs: list_list_nat] :
( ( transpose_list_nat @ ( map_li960784813134754710st_nat @ ( map_nat_list_nat @ F ) @ Xs ) )
= ( map_li960784813134754710st_nat @ ( map_nat_list_nat @ F ) @ ( transpose_nat @ Xs ) ) ) ).
% transpose_map_map
thf(fact_1075_transpose_Oelims,axiom,
! [X2: list_list_list_nat,Y2: list_list_list_nat] :
( ( ( transpose_list_nat @ X2 )
= Y2 )
=> ( ( ( X2 = nil_list_list_nat )
=> ( Y2 != nil_list_list_nat ) )
=> ( ! [Xss: list_list_list_nat] :
( ( X2
= ( cons_list_list_nat @ nil_list_nat @ Xss ) )
=> ( Y2
!= ( transpose_list_nat @ Xss ) ) )
=> ~ ! [X3: list_nat,Xs2: list_list_nat,Xss: list_list_list_nat] :
( ( X2
= ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ Xss ) )
=> ( Y2
!= ( cons_list_list_nat
@ ( cons_list_nat @ X3
@ ( concat_list_nat
@ ( map_li2855073862107769254st_nat
@ ( case_l2985193972497081272st_nat @ nil_list_nat
@ ^ [H: list_nat,T: list_list_nat] : ( cons_list_nat @ H @ nil_list_nat ) )
@ Xss ) ) )
@ ( transpose_list_nat
@ ( cons_list_list_nat @ Xs2
@ ( concat_list_list_nat
@ ( map_li5769348595424326838st_nat
@ ( case_l9045269829974911560st_nat @ nil_list_list_nat
@ ^ [H: list_nat,T: list_list_nat] : ( cons_list_list_nat @ T @ nil_list_list_nat ) )
@ Xss ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_1076_transpose_Oelims,axiom,
! [X2: list_list_nat,Y2: list_list_nat] :
( ( ( transpose_nat @ X2 )
= Y2 )
=> ( ( ( X2 = nil_list_nat )
=> ( Y2 != nil_list_nat ) )
=> ( ! [Xss: list_list_nat] :
( ( X2
= ( cons_list_nat @ nil_nat @ Xss ) )
=> ( Y2
!= ( transpose_nat @ Xss ) ) )
=> ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
( ( X2
= ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) )
=> ( Y2
!= ( cons_list_nat
@ ( cons_nat @ X3
@ ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H: nat,T: list_nat] : ( cons_nat @ H @ nil_nat ) )
@ Xss ) ) )
@ ( transpose_nat
@ ( cons_list_nat @ Xs2
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: nat,T: list_nat] : ( cons_list_nat @ T @ nil_list_nat ) )
@ Xss ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_1077_transpose_Osimps_I3_J,axiom,
! [X2: list_nat,Xs: list_list_nat,Xss2: list_list_list_nat] :
( ( transpose_list_nat @ ( cons_list_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ Xss2 ) )
= ( cons_list_list_nat
@ ( cons_list_nat @ X2
@ ( concat_list_nat
@ ( map_li2855073862107769254st_nat
@ ( case_l2985193972497081272st_nat @ nil_list_nat
@ ^ [H: list_nat,T: list_list_nat] : ( cons_list_nat @ H @ nil_list_nat ) )
@ Xss2 ) ) )
@ ( transpose_list_nat
@ ( cons_list_list_nat @ Xs
@ ( concat_list_list_nat
@ ( map_li5769348595424326838st_nat
@ ( case_l9045269829974911560st_nat @ nil_list_list_nat
@ ^ [H: list_nat,T: list_list_nat] : ( cons_list_list_nat @ T @ nil_list_list_nat ) )
@ Xss2 ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_1078_transpose_Osimps_I3_J,axiom,
! [X2: nat,Xs: list_nat,Xss2: list_list_nat] :
( ( transpose_nat @ ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss2 ) )
= ( cons_list_nat
@ ( cons_nat @ X2
@ ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H: nat,T: list_nat] : ( cons_nat @ H @ nil_nat ) )
@ Xss2 ) ) )
@ ( transpose_nat
@ ( cons_list_nat @ Xs
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: nat,T: list_nat] : ( cons_list_nat @ T @ nil_list_nat ) )
@ Xss2 ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_1079_transpose__empty,axiom,
! [Xs: list_list_list_nat] :
( ( ( transpose_list_nat @ Xs )
= nil_list_list_nat )
= ( ! [X: list_list_nat] :
( ( member_list_list_nat @ X @ ( set_list_list_nat2 @ Xs ) )
=> ( X = nil_list_nat ) ) ) ) ).
% transpose_empty
thf(fact_1080_transpose__empty,axiom,
! [Xs: list_list_nat] :
( ( ( transpose_nat @ Xs )
= nil_list_nat )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( X = nil_nat ) ) ) ) ).
% transpose_empty
thf(fact_1081_pred__subset__eq,axiom,
! [R: set_list_nat,S: set_list_nat] :
( ( ord_le1520216061033275535_nat_o
@ ^ [X: list_nat] : ( member_list_nat @ X @ R )
@ ^ [X: list_nat] : ( member_list_nat @ X @ S ) )
= ( ord_le6045566169113846134st_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1082_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1083_transpose_Opsimps_I3_J,axiom,
! [X2: list_nat,Xs: list_list_nat,Xss2: list_list_list_nat] :
( ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ ( cons_list_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ Xss2 ) )
=> ( ( transpose_list_nat @ ( cons_list_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ Xss2 ) )
= ( cons_list_list_nat
@ ( cons_list_nat @ X2
@ ( concat_list_nat
@ ( map_li2855073862107769254st_nat
@ ( case_l2985193972497081272st_nat @ nil_list_nat
@ ^ [H: list_nat,T: list_list_nat] : ( cons_list_nat @ H @ nil_list_nat ) )
@ Xss2 ) ) )
@ ( transpose_list_nat
@ ( cons_list_list_nat @ Xs
@ ( concat_list_list_nat
@ ( map_li5769348595424326838st_nat
@ ( case_l9045269829974911560st_nat @ nil_list_list_nat
@ ^ [H: list_nat,T: list_list_nat] : ( cons_list_list_nat @ T @ nil_list_list_nat ) )
@ Xss2 ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_1084_transpose_Opsimps_I3_J,axiom,
! [X2: nat,Xs: list_nat,Xss2: list_list_nat] :
( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss2 ) )
=> ( ( transpose_nat @ ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss2 ) )
= ( cons_list_nat
@ ( cons_nat @ X2
@ ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H: nat,T: list_nat] : ( cons_nat @ H @ nil_nat ) )
@ Xss2 ) ) )
@ ( transpose_nat
@ ( cons_list_nat @ Xs
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: nat,T: list_nat] : ( cons_list_nat @ T @ nil_list_nat ) )
@ Xss2 ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_1085_transpose_Opelims,axiom,
! [X2: list_list_list_nat,Y2: list_list_list_nat] :
( ( ( transpose_list_nat @ X2 )
= Y2 )
=> ( ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ X2 )
=> ( ( ( X2 = nil_list_list_nat )
=> ( ( Y2 = nil_list_list_nat )
=> ~ ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ nil_list_list_nat ) ) )
=> ( ! [Xss: list_list_list_nat] :
( ( X2
= ( cons_list_list_nat @ nil_list_nat @ Xss ) )
=> ( ( Y2
= ( transpose_list_nat @ Xss ) )
=> ~ ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ ( cons_list_list_nat @ nil_list_nat @ Xss ) ) ) )
=> ~ ! [X3: list_nat,Xs2: list_list_nat,Xss: list_list_list_nat] :
( ( X2
= ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ Xss ) )
=> ( ( Y2
= ( cons_list_list_nat
@ ( cons_list_nat @ X3
@ ( concat_list_nat
@ ( map_li2855073862107769254st_nat
@ ( case_l2985193972497081272st_nat @ nil_list_nat
@ ^ [H: list_nat,T: list_list_nat] : ( cons_list_nat @ H @ nil_list_nat ) )
@ Xss ) ) )
@ ( transpose_list_nat
@ ( cons_list_list_nat @ Xs2
@ ( concat_list_list_nat
@ ( map_li5769348595424326838st_nat
@ ( case_l9045269829974911560st_nat @ nil_list_list_nat
@ ^ [H: list_nat,T: list_list_nat] : ( cons_list_list_nat @ T @ nil_list_list_nat ) )
@ Xss ) ) ) ) ) )
=> ~ ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ) ) ) ) ).
% transpose.pelims
thf(fact_1086_transpose_Opelims,axiom,
! [X2: list_list_nat,Y2: list_list_nat] :
( ( ( transpose_nat @ X2 )
= Y2 )
=> ( ( accp_list_list_nat @ transpose_rel_nat @ X2 )
=> ( ( ( X2 = nil_list_nat )
=> ( ( Y2 = nil_list_nat )
=> ~ ( accp_list_list_nat @ transpose_rel_nat @ nil_list_nat ) ) )
=> ( ! [Xss: list_list_nat] :
( ( X2
= ( cons_list_nat @ nil_nat @ Xss ) )
=> ( ( Y2
= ( transpose_nat @ Xss ) )
=> ~ ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ nil_nat @ Xss ) ) ) )
=> ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
( ( X2
= ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) )
=> ( ( Y2
= ( cons_list_nat
@ ( cons_nat @ X3
@ ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H: nat,T: list_nat] : ( cons_nat @ H @ nil_nat ) )
@ Xss ) ) )
@ ( transpose_nat
@ ( cons_list_nat @ Xs2
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: nat,T: list_nat] : ( cons_list_nat @ T @ nil_list_nat ) )
@ Xss ) ) ) ) ) )
=> ~ ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ) ) ) ) ).
% transpose.pelims
thf(fact_1087_transpose_Opsimps_I2_J,axiom,
! [Xss2: list_list_list_nat] :
( ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ ( cons_list_list_nat @ nil_list_nat @ Xss2 ) )
=> ( ( transpose_list_nat @ ( cons_list_list_nat @ nil_list_nat @ Xss2 ) )
= ( transpose_list_nat @ Xss2 ) ) ) ).
% transpose.psimps(2)
thf(fact_1088_transpose_Opsimps_I2_J,axiom,
! [Xss2: list_list_nat] :
( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ nil_nat @ Xss2 ) )
=> ( ( transpose_nat @ ( cons_list_nat @ nil_nat @ Xss2 ) )
= ( transpose_nat @ Xss2 ) ) ) ).
% transpose.psimps(2)
thf(fact_1089_transpose_Opsimps_I1_J,axiom,
( ( accp_list_list_nat @ transpose_rel_nat @ nil_list_nat )
=> ( ( transpose_nat @ nil_list_nat )
= nil_list_nat ) ) ).
% transpose.psimps(1)
thf(fact_1090_list_Odisc__eq__case_I2_J,axiom,
! [List: list_nat] :
( ( List != nil_nat )
= ( case_list_o_nat @ $false
@ ^ [Uu: nat,Uv: list_nat] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_1091_list_Odisc__eq__case_I2_J,axiom,
! [List: list_list_nat] :
( ( List != nil_list_nat )
= ( case_list_o_list_nat @ $false
@ ^ [Uu: list_nat,Uv: list_list_nat] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_1092_list_Odisc__eq__case_I1_J,axiom,
! [List: list_nat] :
( ( List = nil_nat )
= ( case_list_o_nat @ $true
@ ^ [Uu: nat,Uv: list_nat] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_1093_list_Odisc__eq__case_I1_J,axiom,
! [List: list_list_nat] :
( ( List = nil_list_nat )
= ( case_list_o_list_nat @ $true
@ ^ [Uu: list_nat,Uv: list_list_nat] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_1094_transpose_Opinduct,axiom,
! [A0: list_list_list_nat,P: list_list_list_nat > $o] :
( ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ A0 )
=> ( ( ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ nil_list_list_nat )
=> ( P @ nil_list_list_nat ) )
=> ( ! [Xss: list_list_list_nat] :
( ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ ( cons_list_list_nat @ nil_list_nat @ Xss ) )
=> ( ( P @ Xss )
=> ( P @ ( cons_list_list_nat @ nil_list_nat @ Xss ) ) ) )
=> ( ! [X3: list_nat,Xs2: list_list_nat,Xss: list_list_list_nat] :
( ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ Xss ) )
=> ( ( P
@ ( cons_list_list_nat @ Xs2
@ ( concat_list_list_nat
@ ( map_li5769348595424326838st_nat
@ ( case_l9045269829974911560st_nat @ nil_list_list_nat
@ ^ [H: list_nat,T: list_list_nat] : ( cons_list_list_nat @ T @ nil_list_list_nat ) )
@ Xss ) ) ) )
=> ( P @ ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ Xss ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_1095_transpose_Opinduct,axiom,
! [A0: list_list_nat,P: list_list_nat > $o] :
( ( accp_list_list_nat @ transpose_rel_nat @ A0 )
=> ( ( ( accp_list_list_nat @ transpose_rel_nat @ nil_list_nat )
=> ( P @ nil_list_nat ) )
=> ( ! [Xss: list_list_nat] :
( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ nil_nat @ Xss ) )
=> ( ( P @ Xss )
=> ( P @ ( cons_list_nat @ nil_nat @ Xss ) ) ) )
=> ( ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) )
=> ( ( P
@ ( cons_list_nat @ Xs2
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: nat,T: list_nat] : ( cons_list_nat @ T @ nil_list_nat ) )
@ Xss ) ) ) )
=> ( P @ ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_1096_nth__transpose,axiom,
! [I: nat,Xs: list_list_list_nat] :
( ( ord_less_nat @ I @ ( size_s6248950052170075156st_nat @ ( transpose_list_nat @ Xs ) ) )
=> ( ( nth_list_list_nat @ ( transpose_list_nat @ Xs ) @ I )
= ( map_li1587113026813354006st_nat
@ ^ [Xs3: list_list_nat] : ( nth_list_nat @ Xs3 @ I )
@ ( filter_list_list_nat
@ ^ [Ys: list_list_nat] : ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ys ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_1097_nth__transpose,axiom,
! [I: nat,Xs: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs ) ) )
=> ( ( nth_list_nat @ ( transpose_nat @ Xs ) @ I )
= ( map_list_nat_nat
@ ^ [Xs3: list_nat] : ( nth_nat @ Xs3 @ I )
@ ( filter_list_nat
@ ^ [Ys: list_nat] : ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_1098_length__transpose,axiom,
! [Xs: list_list_list_nat] :
( ( size_s6248950052170075156st_nat @ ( transpose_list_nat @ Xs ) )
= ( foldr_2920604631923063017at_nat
@ ^ [Xs3: list_list_nat] : ( ord_max_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) )
@ Xs
@ zero_zero_nat ) ) ).
% length_transpose
thf(fact_1099_length__transpose,axiom,
! [Xs: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs ) )
= ( foldr_list_nat_nat
@ ^ [Xs3: list_nat] : ( ord_max_nat @ ( size_size_list_nat @ Xs3 ) )
@ Xs
@ zero_zero_nat ) ) ).
% length_transpose
thf(fact_1100_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
@ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
@ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_1101_filter__True,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( filter_list_nat @ P @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_1102_filter__True,axiom,
! [Xs: list_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( filter_nat @ P @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_1103_filter__append,axiom,
! [P: nat > $o,Xs: list_nat,Ys3: list_nat] :
( ( filter_nat @ P @ ( append_nat @ Xs @ Ys3 ) )
= ( append_nat @ ( filter_nat @ P @ Xs ) @ ( filter_nat @ P @ Ys3 ) ) ) ).
% filter_append
thf(fact_1104_filter__append,axiom,
! [P: list_nat > $o,Xs: list_list_nat,Ys3: list_list_nat] :
( ( filter_list_nat @ P @ ( append_list_nat @ Xs @ Ys3 ) )
= ( append_list_nat @ ( filter_list_nat @ P @ Xs ) @ ( filter_list_nat @ P @ Ys3 ) ) ) ).
% filter_append
thf(fact_1105_set__filter,axiom,
! [P: list_nat > $o,Xs: list_list_nat] :
( ( set_list_nat2 @ ( filter_list_nat @ P @ Xs ) )
= ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
& ( P @ X ) ) ) ) ).
% set_filter
thf(fact_1106_set__filter,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( set_nat2 @ ( filter_nat @ P @ Xs ) )
= ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
& ( P @ X ) ) ) ) ).
% set_filter
thf(fact_1107_filter__False,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ~ ( P @ X3 ) )
=> ( ( filter_list_nat @ P @ Xs )
= nil_list_nat ) ) ).
% filter_False
thf(fact_1108_filter__False,axiom,
! [Xs: list_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ~ ( P @ X3 ) )
=> ( ( filter_nat @ P @ Xs )
= nil_nat ) ) ).
% filter_False
thf(fact_1109_filter__mono__prefix,axiom,
! [Xs: list_nat,Ys3: list_nat,P: nat > $o] :
( ( prefix_nat @ Xs @ Ys3 )
=> ( prefix_nat @ ( filter_nat @ P @ Xs ) @ ( filter_nat @ P @ Ys3 ) ) ) ).
% filter_mono_prefix
thf(fact_1110_filter__mono__prefix,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,P: list_nat > $o] :
( ( prefix_list_nat @ Xs @ Ys3 )
=> ( prefix_list_nat @ ( filter_list_nat @ P @ Xs ) @ ( filter_list_nat @ P @ Ys3 ) ) ) ).
% filter_mono_prefix
thf(fact_1111_filter__id__conv,axiom,
! [P: list_nat > $o,Xs: list_list_nat] :
( ( ( filter_list_nat @ P @ Xs )
= Xs )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( P @ X ) ) ) ) ).
% filter_id_conv
thf(fact_1112_filter__id__conv,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( ( filter_nat @ P @ Xs )
= Xs )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) ) ) ).
% filter_id_conv
thf(fact_1113_filter__cong,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat,P: list_nat > $o,Q2: list_nat > $o] :
( ( Xs = Ys3 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Ys3 ) )
=> ( ( P @ X3 )
= ( Q2 @ X3 ) ) )
=> ( ( filter_list_nat @ P @ Xs )
= ( filter_list_nat @ Q2 @ Ys3 ) ) ) ) ).
% filter_cong
thf(fact_1114_filter__cong,axiom,
! [Xs: list_nat,Ys3: list_nat,P: nat > $o,Q2: nat > $o] :
( ( Xs = Ys3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys3 ) )
=> ( ( P @ X3 )
= ( Q2 @ X3 ) ) )
=> ( ( filter_nat @ P @ Xs )
= ( filter_nat @ Q2 @ Ys3 ) ) ) ) ).
% filter_cong
thf(fact_1115_filter_Osimps_I1_J,axiom,
! [P: nat > $o] :
( ( filter_nat @ P @ nil_nat )
= nil_nat ) ).
% filter.simps(1)
thf(fact_1116_filter_Osimps_I1_J,axiom,
! [P: list_nat > $o] :
( ( filter_list_nat @ P @ nil_list_nat )
= nil_list_nat ) ).
% filter.simps(1)
thf(fact_1117_filter_Osimps_I2_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( ( P @ X2 )
=> ( ( filter_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( filter_nat @ P @ Xs ) ) ) )
& ( ~ ( P @ X2 )
=> ( ( filter_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( filter_nat @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_1118_filter_Osimps_I2_J,axiom,
! [P: list_nat > $o,X2: list_nat,Xs: list_list_nat] :
( ( ( P @ X2 )
=> ( ( filter_list_nat @ P @ ( cons_list_nat @ X2 @ Xs ) )
= ( cons_list_nat @ X2 @ ( filter_list_nat @ P @ Xs ) ) ) )
& ( ~ ( P @ X2 )
=> ( ( filter_list_nat @ P @ ( cons_list_nat @ X2 @ Xs ) )
= ( filter_list_nat @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_1119_filter__empty__conv,axiom,
! [P: list_nat > $o,Xs: list_list_nat] :
( ( ( filter_list_nat @ P @ Xs )
= nil_list_nat )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ~ ( P @ X ) ) ) ) ).
% filter_empty_conv
thf(fact_1120_filter__empty__conv,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( ( filter_nat @ P @ Xs )
= nil_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ~ ( P @ X ) ) ) ) ).
% filter_empty_conv
thf(fact_1121_empty__filter__conv,axiom,
! [P: list_nat > $o,Xs: list_list_nat] :
( ( nil_list_nat
= ( filter_list_nat @ P @ Xs ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ~ ( P @ X ) ) ) ) ).
% empty_filter_conv
thf(fact_1122_empty__filter__conv,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( nil_nat
= ( filter_nat @ P @ Xs ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ~ ( P @ X ) ) ) ) ).
% empty_filter_conv
thf(fact_1123_sum__length__filter__compl,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( plus_plus_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs ) )
@ ( size_size_list_nat
@ ( filter_nat
@ ^ [X: nat] :
~ ( P @ X )
@ Xs ) ) )
= ( size_size_list_nat @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_1124_sum__length__filter__compl,axiom,
! [P: list_nat > $o,Xs: list_list_nat] :
( ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ ( filter_list_nat @ P @ Xs ) )
@ ( size_s3023201423986296836st_nat
@ ( filter_list_nat
@ ^ [X: list_nat] :
~ ( P @ X )
@ Xs ) ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_1125_filter__concat,axiom,
! [P4: list_nat > $o,Xs: list_list_list_nat] :
( ( filter_list_nat @ P4 @ ( concat_list_nat @ Xs ) )
= ( concat_list_nat @ ( map_li2855073862107769254st_nat @ ( filter_list_nat @ P4 ) @ Xs ) ) ) ).
% filter_concat
thf(fact_1126_count__list__expand,axiom,
( count_list_nat
= ( ^ [Xs3: list_nat,X: nat] :
( size_size_list_nat
@ ( filter_nat
@ ( ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ X )
@ Xs3 ) ) ) ) ).
% count_list_expand
thf(fact_1127_count__list__expand,axiom,
( count_list_list_nat
= ( ^ [Xs3: list_list_nat,X: list_nat] :
( size_s3023201423986296836st_nat
@ ( filter_list_nat
@ ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z )
@ X )
@ Xs3 ) ) ) ) ).
% count_list_expand
thf(fact_1128_filter__is__subset,axiom,
! [P: list_nat > $o,Xs: list_list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( filter_list_nat @ P @ Xs ) ) @ ( set_list_nat2 @ Xs ) ) ).
% filter_is_subset
thf(fact_1129_filter__is__subset,axiom,
! [P: nat > $o,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( filter_nat @ P @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% filter_is_subset
thf(fact_1130_length__filter__le,axiom,
! [P: nat > $o,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% length_filter_le
thf(fact_1131_length__filter__le,axiom,
! [P: list_nat > $o,Xs: list_list_nat] : ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ ( filter_list_nat @ P @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_filter_le
thf(fact_1132_Cons__eq__filterD,axiom,
! [X2: nat,Xs: list_nat,P: nat > $o,Ys3: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( filter_nat @ P @ Ys3 ) )
=> ? [Us3: list_nat,Vs3: list_nat] :
( ( Ys3
= ( append_nat @ Us3 @ ( cons_nat @ X2 @ Vs3 ) ) )
& ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Us3 ) )
=> ~ ( P @ X4 ) )
& ( P @ X2 )
& ( Xs
= ( filter_nat @ P @ Vs3 ) ) ) ) ).
% Cons_eq_filterD
thf(fact_1133_Cons__eq__filterD,axiom,
! [X2: list_nat,Xs: list_list_nat,P: list_nat > $o,Ys3: list_list_nat] :
( ( ( cons_list_nat @ X2 @ Xs )
= ( filter_list_nat @ P @ Ys3 ) )
=> ? [Us3: list_list_nat,Vs3: list_list_nat] :
( ( Ys3
= ( append_list_nat @ Us3 @ ( cons_list_nat @ X2 @ Vs3 ) ) )
& ! [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Us3 ) )
=> ~ ( P @ X4 ) )
& ( P @ X2 )
& ( Xs
= ( filter_list_nat @ P @ Vs3 ) ) ) ) ).
% Cons_eq_filterD
thf(fact_1134_filter__eq__ConsD,axiom,
! [P: nat > $o,Ys3: list_nat,X2: nat,Xs: list_nat] :
( ( ( filter_nat @ P @ Ys3 )
= ( cons_nat @ X2 @ Xs ) )
=> ? [Us3: list_nat,Vs3: list_nat] :
( ( Ys3
= ( append_nat @ Us3 @ ( cons_nat @ X2 @ Vs3 ) ) )
& ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Us3 ) )
=> ~ ( P @ X4 ) )
& ( P @ X2 )
& ( Xs
= ( filter_nat @ P @ Vs3 ) ) ) ) ).
% filter_eq_ConsD
thf(fact_1135_filter__eq__ConsD,axiom,
! [P: list_nat > $o,Ys3: list_list_nat,X2: list_nat,Xs: list_list_nat] :
( ( ( filter_list_nat @ P @ Ys3 )
= ( cons_list_nat @ X2 @ Xs ) )
=> ? [Us3: list_list_nat,Vs3: list_list_nat] :
( ( Ys3
= ( append_list_nat @ Us3 @ ( cons_list_nat @ X2 @ Vs3 ) ) )
& ! [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Us3 ) )
=> ~ ( P @ X4 ) )
& ( P @ X2 )
& ( Xs
= ( filter_list_nat @ P @ Vs3 ) ) ) ) ).
% filter_eq_ConsD
thf(fact_1136_Cons__eq__filter__iff,axiom,
! [X2: nat,Xs: list_nat,P: nat > $o,Ys3: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( filter_nat @ P @ Ys3 ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Ys3
= ( append_nat @ Us2 @ ( cons_nat @ X2 @ Vs2 ) ) )
& ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Us2 ) )
=> ~ ( P @ X ) )
& ( P @ X2 )
& ( Xs
= ( filter_nat @ P @ Vs2 ) ) ) ) ) ).
% Cons_eq_filter_iff
thf(fact_1137_Cons__eq__filter__iff,axiom,
! [X2: list_nat,Xs: list_list_nat,P: list_nat > $o,Ys3: list_list_nat] :
( ( ( cons_list_nat @ X2 @ Xs )
= ( filter_list_nat @ P @ Ys3 ) )
= ( ? [Us2: list_list_nat,Vs2: list_list_nat] :
( ( Ys3
= ( append_list_nat @ Us2 @ ( cons_list_nat @ X2 @ Vs2 ) ) )
& ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Us2 ) )
=> ~ ( P @ X ) )
& ( P @ X2 )
& ( Xs
= ( filter_list_nat @ P @ Vs2 ) ) ) ) ) ).
% Cons_eq_filter_iff
thf(fact_1138_filter__eq__Cons__iff,axiom,
! [P: nat > $o,Ys3: list_nat,X2: nat,Xs: list_nat] :
( ( ( filter_nat @ P @ Ys3 )
= ( cons_nat @ X2 @ Xs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Ys3
= ( append_nat @ Us2 @ ( cons_nat @ X2 @ Vs2 ) ) )
& ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Us2 ) )
=> ~ ( P @ X ) )
& ( P @ X2 )
& ( Xs
= ( filter_nat @ P @ Vs2 ) ) ) ) ) ).
% filter_eq_Cons_iff
thf(fact_1139_filter__eq__Cons__iff,axiom,
! [P: list_nat > $o,Ys3: list_list_nat,X2: list_nat,Xs: list_list_nat] :
( ( ( filter_list_nat @ P @ Ys3 )
= ( cons_list_nat @ X2 @ Xs ) )
= ( ? [Us2: list_list_nat,Vs2: list_list_nat] :
( ( Ys3
= ( append_list_nat @ Us2 @ ( cons_list_nat @ X2 @ Vs2 ) ) )
& ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Us2 ) )
=> ~ ( P @ X ) )
& ( P @ X2 )
& ( Xs
= ( filter_list_nat @ P @ Vs2 ) ) ) ) ) ).
% filter_eq_Cons_iff
thf(fact_1140_length__filter__less,axiom,
! [X2: nat,Xs: list_nat,P: nat > $o] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ~ ( P @ X2 )
=> ( ord_less_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ) ).
% length_filter_less
thf(fact_1141_length__filter__less,axiom,
! [X2: list_nat,Xs: list_list_nat,P: list_nat > $o] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ~ ( P @ X2 )
=> ( ord_less_nat @ ( size_s3023201423986296836st_nat @ ( filter_list_nat @ P @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ).
% length_filter_less
thf(fact_1142_transpose__max__length,axiom,
! [Xs: list_list_nat] :
( ( foldr_list_nat_nat
@ ^ [Xs3: list_nat] : ( ord_max_nat @ ( size_size_list_nat @ Xs3 ) )
@ ( transpose_nat @ Xs )
@ zero_zero_nat )
= ( size_s3023201423986296836st_nat
@ ( filter_list_nat
@ ^ [X: list_nat] : ( X != nil_nat )
@ Xs ) ) ) ).
% transpose_max_length
thf(fact_1143_transpose__max__length,axiom,
! [Xs: list_list_list_nat] :
( ( foldr_2920604631923063017at_nat
@ ^ [Xs3: list_list_nat] : ( ord_max_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) )
@ ( transpose_list_nat @ Xs )
@ zero_zero_nat )
= ( size_s6248950052170075156st_nat
@ ( filter_list_list_nat
@ ^ [X: list_list_nat] : ( X != nil_list_nat )
@ Xs ) ) ) ).
% transpose_max_length
thf(fact_1144_transpose__aux__filter__tail,axiom,
! [Xss2: list_list_list_nat] :
( ( concat_list_list_nat
@ ( map_li5769348595424326838st_nat
@ ( case_l9045269829974911560st_nat @ nil_list_list_nat
@ ^ [H: list_nat,T: list_list_nat] : ( cons_list_list_nat @ T @ nil_list_list_nat ) )
@ Xss2 ) )
= ( map_li2855073862107769254st_nat @ tl_list_nat
@ ( filter_list_list_nat
@ ^ [Ys: list_list_nat] : ( Ys != nil_list_nat )
@ Xss2 ) ) ) ).
% transpose_aux_filter_tail
thf(fact_1145_transpose__aux__filter__tail,axiom,
! [Xss2: list_list_nat] :
( ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: nat,T: list_nat] : ( cons_list_nat @ T @ nil_list_nat ) )
@ Xss2 ) )
= ( map_li7225945977422193158st_nat @ tl_nat
@ ( filter_list_nat
@ ^ [Ys: list_nat] : ( Ys != nil_nat )
@ Xss2 ) ) ) ).
% transpose_aux_filter_tail
thf(fact_1146_transpose__aux__max,axiom,
! [Xs: list_nat,Xss2: list_list_nat] :
( ( ord_max_nat @ ( suc @ ( size_size_list_nat @ Xs ) )
@ ( foldr_list_nat_nat
@ ^ [Xs3: list_nat] : ( ord_max_nat @ ( size_size_list_nat @ Xs3 ) )
@ Xss2
@ zero_zero_nat ) )
= ( suc
@ ( ord_max_nat @ ( size_size_list_nat @ Xs )
@ ( foldr_list_nat_nat
@ ^ [X: list_nat] : ( ord_max_nat @ ( minus_minus_nat @ ( size_size_list_nat @ X ) @ ( suc @ zero_zero_nat ) ) )
@ ( filter_list_nat
@ ^ [Ys: list_nat] : ( Ys != nil_nat )
@ Xss2 )
@ zero_zero_nat ) ) ) ) ).
% transpose_aux_max
thf(fact_1147_transpose__aux__max,axiom,
! [Xs: list_nat,Xss2: list_list_list_nat] :
( ( ord_max_nat @ ( suc @ ( size_size_list_nat @ Xs ) )
@ ( foldr_2920604631923063017at_nat
@ ^ [Xs3: list_list_nat] : ( ord_max_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) )
@ Xss2
@ zero_zero_nat ) )
= ( suc
@ ( ord_max_nat @ ( size_size_list_nat @ Xs )
@ ( foldr_2920604631923063017at_nat
@ ^ [X: list_list_nat] : ( ord_max_nat @ ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ X ) @ ( suc @ zero_zero_nat ) ) )
@ ( filter_list_list_nat
@ ^ [Ys: list_list_nat] : ( Ys != nil_list_nat )
@ Xss2 )
@ zero_zero_nat ) ) ) ) ).
% transpose_aux_max
thf(fact_1148_transpose__aux__max,axiom,
! [Xs: list_list_nat,Xss2: list_list_nat] :
( ( ord_max_nat @ ( suc @ ( size_s3023201423986296836st_nat @ Xs ) )
@ ( foldr_list_nat_nat
@ ^ [Xs3: list_nat] : ( ord_max_nat @ ( size_size_list_nat @ Xs3 ) )
@ Xss2
@ zero_zero_nat ) )
= ( suc
@ ( ord_max_nat @ ( size_s3023201423986296836st_nat @ Xs )
@ ( foldr_list_nat_nat
@ ^ [X: list_nat] : ( ord_max_nat @ ( minus_minus_nat @ ( size_size_list_nat @ X ) @ ( suc @ zero_zero_nat ) ) )
@ ( filter_list_nat
@ ^ [Ys: list_nat] : ( Ys != nil_nat )
@ Xss2 )
@ zero_zero_nat ) ) ) ) ).
% transpose_aux_max
thf(fact_1149_transpose__aux__max,axiom,
! [Xs: list_list_nat,Xss2: list_list_list_nat] :
( ( ord_max_nat @ ( suc @ ( size_s3023201423986296836st_nat @ Xs ) )
@ ( foldr_2920604631923063017at_nat
@ ^ [Xs3: list_list_nat] : ( ord_max_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) )
@ Xss2
@ zero_zero_nat ) )
= ( suc
@ ( ord_max_nat @ ( size_s3023201423986296836st_nat @ Xs )
@ ( foldr_2920604631923063017at_nat
@ ^ [X: list_list_nat] : ( ord_max_nat @ ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ X ) @ ( suc @ zero_zero_nat ) ) )
@ ( filter_list_list_nat
@ ^ [Ys: list_list_nat] : ( Ys != nil_list_nat )
@ Xss2 )
@ zero_zero_nat ) ) ) ) ).
% transpose_aux_max
thf(fact_1150_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1151_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_1152_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1153_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_1154_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_1155_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_1156_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_1157_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1158_tl__upt,axiom,
! [M3: nat,N: nat] :
( ( tl_nat @ ( upt @ M3 @ N ) )
= ( upt @ ( suc @ M3 ) @ N ) ) ).
% tl_upt
thf(fact_1159_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_1160_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_1161_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_1162_diff__is__0__eq,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% diff_is_0_eq
thf(fact_1163_diff__is__0__eq_H,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1164_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_1165_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_1166_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_1167_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1168_tl__append2,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( Xs != nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys3 ) )
= ( append_nat @ ( tl_nat @ Xs ) @ Ys3 ) ) ) ).
% tl_append2
thf(fact_1169_tl__append2,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( tl_list_nat @ ( append_list_nat @ Xs @ Ys3 ) )
= ( append_list_nat @ ( tl_list_nat @ Xs ) @ Ys3 ) ) ) ).
% tl_append2
thf(fact_1170_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_1171_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1172_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1173_take__append,axiom,
! [N: nat,Xs: list_nat,Ys3: list_nat] :
( ( take_nat @ N @ ( append_nat @ Xs @ Ys3 ) )
= ( append_nat @ ( take_nat @ N @ Xs ) @ ( take_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys3 ) ) ) ).
% take_append
thf(fact_1174_take__append,axiom,
! [N: nat,Xs: list_list_nat,Ys3: list_list_nat] :
( ( take_list_nat @ N @ ( append_list_nat @ Xs @ Ys3 ) )
= ( append_list_nat @ ( take_list_nat @ N @ Xs ) @ ( take_list_nat @ ( minus_minus_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) ) @ Ys3 ) ) ) ).
% take_append
thf(fact_1175_last__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( last_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% last_upt
thf(fact_1176_length__tl,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( tl_nat @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_1177_length__tl,axiom,
! [Xs: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( tl_list_nat @ Xs ) )
= ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_1178_length__butlast,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( butlast_nat @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_1179_length__butlast,axiom,
! [Xs: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( butlast_list_nat @ Xs ) )
= ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_1180_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1181_nth__Cons__numeral,axiom,
! [X2: nat,Xs: list_nat,V: num] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_1182_nth__Cons__numeral,axiom,
! [X2: list_nat,Xs: list_list_nat,V: num] :
( ( nth_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_list_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_1183_take__Cons__numeral,axiom,
! [V: num,X2: nat,Xs: list_nat] :
( ( take_nat @ ( numeral_numeral_nat @ V ) @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( take_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_1184_take__Cons__numeral,axiom,
! [V: num,X2: list_nat,Xs: list_list_nat] :
( ( take_list_nat @ ( numeral_numeral_nat @ V ) @ ( cons_list_nat @ X2 @ Xs ) )
= ( cons_list_nat @ X2 @ ( take_list_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_1185_nth__Cons__pos,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1186_nth__Cons__pos,axiom,
! [N: nat,X2: list_nat,Xs: list_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ N )
= ( nth_list_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1187_list_Omap__sel_I2_J,axiom,
! [A: list_list_nat,F: list_nat > nat] :
( ( A != nil_list_nat )
=> ( ( tl_nat @ ( map_list_nat_nat @ F @ A ) )
= ( map_list_nat_nat @ F @ ( tl_list_nat @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_1188_list_Omap__sel_I2_J,axiom,
! [A: list_nat,F: nat > nat] :
( ( A != nil_nat )
=> ( ( tl_nat @ ( map_nat_nat @ F @ A ) )
= ( map_nat_nat @ F @ ( tl_nat @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_1189_list_Omap__sel_I2_J,axiom,
! [A: list_list_nat,F: list_nat > list_list_nat] :
( ( A != nil_list_nat )
=> ( ( tl_list_list_nat @ ( map_li960784813134754710st_nat @ F @ A ) )
= ( map_li960784813134754710st_nat @ F @ ( tl_list_nat @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_1190_list_Omap__sel_I2_J,axiom,
! [A: list_nat,F: nat > list_nat] :
( ( A != nil_nat )
=> ( ( tl_list_nat @ ( map_nat_list_nat @ F @ A ) )
= ( map_nat_list_nat @ F @ ( tl_nat @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_1191_map__tl,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( map_nat_nat @ F @ ( tl_nat @ Xs ) )
= ( tl_nat @ ( map_nat_nat @ F @ Xs ) ) ) ).
% map_tl
thf(fact_1192_map__tl,axiom,
! [F: list_nat > list_list_nat,Xs: list_list_nat] :
( ( map_li960784813134754710st_nat @ F @ ( tl_list_nat @ Xs ) )
= ( tl_list_list_nat @ ( map_li960784813134754710st_nat @ F @ Xs ) ) ) ).
% map_tl
thf(fact_1193_map__tl,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( map_nat_list_nat @ F @ ( tl_nat @ Xs ) )
= ( tl_list_nat @ ( map_nat_list_nat @ F @ Xs ) ) ) ).
% map_tl
thf(fact_1194_tl__take,axiom,
! [N: nat,Xs: list_nat] :
( ( tl_nat @ ( take_nat @ N @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( tl_nat @ Xs ) ) ) ).
% tl_take
thf(fact_1195_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_1196_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_1197_diff__le__mono2,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_1198_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_1199_diff__le__self,axiom,
! [M3: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ).
% diff_le_self
thf(fact_1200_diff__le__mono,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1201_Nat_Odiff__diff__eq,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M3 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1202_le__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1203_eq__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M3 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M3 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1204_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_1205_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_1206_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_1207_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_1208_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_1209_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_1210_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_1211_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_1212_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_1213_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_1214_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1215_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1216_Nil__tl,axiom,
! [Xs: list_nat] :
( ( nil_nat
= ( tl_nat @ Xs ) )
= ( ( Xs = nil_nat )
| ? [X: nat] :
( Xs
= ( cons_nat @ X @ nil_nat ) ) ) ) ).
% Nil_tl
thf(fact_1217_Nil__tl,axiom,
! [Xs: list_list_nat] :
( ( nil_list_nat
= ( tl_list_nat @ Xs ) )
= ( ( Xs = nil_list_nat )
| ? [X: list_nat] :
( Xs
= ( cons_list_nat @ X @ nil_list_nat ) ) ) ) ).
% Nil_tl
thf(fact_1218_tl__Nil,axiom,
! [Xs: list_nat] :
( ( ( tl_nat @ Xs )
= nil_nat )
= ( ( Xs = nil_nat )
| ? [X: nat] :
( Xs
= ( cons_nat @ X @ nil_nat ) ) ) ) ).
% tl_Nil
thf(fact_1219_tl__Nil,axiom,
! [Xs: list_list_nat] :
( ( ( tl_list_nat @ Xs )
= nil_list_nat )
= ( ( Xs = nil_list_nat )
| ? [X: list_nat] :
( Xs
= ( cons_list_nat @ X @ nil_list_nat ) ) ) ) ).
% tl_Nil
thf(fact_1220_list_Oset__sel_I2_J,axiom,
! [A: list_list_nat,X2: list_nat] :
( ( A != nil_list_nat )
=> ( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( tl_list_nat @ A ) ) )
=> ( member_list_nat @ X2 @ ( set_list_nat2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1221_list_Oset__sel_I2_J,axiom,
! [A: list_nat,X2: nat] :
( ( A != nil_nat )
=> ( ( member_nat @ X2 @ ( set_nat2 @ ( tl_nat @ A ) ) )
=> ( member_nat @ X2 @ ( set_nat2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1222_Suc__diff__le,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ N @ M3 )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N )
= ( suc @ ( minus_minus_nat @ M3 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1223_tl__append__if,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( ( Xs = nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys3 ) )
= ( tl_nat @ Ys3 ) ) )
& ( ( Xs != nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys3 ) )
= ( append_nat @ ( tl_nat @ Xs ) @ Ys3 ) ) ) ) ).
% tl_append_if
thf(fact_1224_tl__append__if,axiom,
! [Xs: list_list_nat,Ys3: list_list_nat] :
( ( ( Xs = nil_list_nat )
=> ( ( tl_list_nat @ ( append_list_nat @ Xs @ Ys3 ) )
= ( tl_list_nat @ Ys3 ) ) )
& ( ( Xs != nil_list_nat )
=> ( ( tl_list_nat @ ( append_list_nat @ Xs @ Ys3 ) )
= ( append_list_nat @ ( tl_list_nat @ Xs ) @ Ys3 ) ) ) ) ).
% tl_append_if
thf(fact_1225_less__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M3 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1226_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_1227_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_1228_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_1229_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_1230_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_1231_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_1232_diff__Suc__eq__diff__pred,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ M3 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1233_butlast__tl,axiom,
! [Xs: list_nat] :
( ( butlast_nat @ ( tl_nat @ Xs ) )
= ( tl_nat @ ( butlast_nat @ Xs ) ) ) ).
% butlast_tl
thf(fact_1234_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_1235_list_Osel_I2_J,axiom,
( ( tl_nat @ nil_nat )
= nil_nat ) ).
% list.sel(2)
thf(fact_1236_list_Osel_I2_J,axiom,
( ( tl_list_nat @ nil_list_nat )
= nil_list_nat ) ).
% list.sel(2)
thf(fact_1237_list_Osel_I3_J,axiom,
! [X21: nat,X22: list_nat] :
( ( tl_nat @ ( cons_nat @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_1238_list_Osel_I3_J,axiom,
! [X21: list_nat,X22: list_list_nat] :
( ( tl_list_nat @ ( cons_list_nat @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_1239_take__tl,axiom,
! [N: nat,Xs: list_nat] :
( ( take_nat @ N @ ( tl_nat @ Xs ) )
= ( tl_nat @ ( take_nat @ ( suc @ N ) @ Xs ) ) ) ).
% take_tl
thf(fact_1240_last__tl,axiom,
! [Xs: list_nat] :
( ( ( Xs = nil_nat )
| ( ( tl_nat @ Xs )
!= nil_nat ) )
=> ( ( last_nat @ ( tl_nat @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_tl
thf(fact_1241_last__tl,axiom,
! [Xs: list_list_nat] :
( ( ( Xs = nil_list_nat )
| ( ( tl_list_nat @ Xs )
!= nil_list_nat ) )
=> ( ( last_list_nat @ ( tl_list_nat @ Xs ) )
= ( last_list_nat @ Xs ) ) ) ).
% last_tl
thf(fact_1242_tl__def,axiom,
( tl_nat
= ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [X213: nat,X223: list_nat] : X223 ) ) ).
% tl_def
thf(fact_1243_tl__def,axiom,
( tl_list_nat
= ( case_l2985193972497081272st_nat @ nil_list_nat
@ ^ [X213: list_nat,X223: list_list_nat] : X223 ) ) ).
% tl_def
thf(fact_1244_tl__append,axiom,
! [Xs: list_nat,Ys3: list_nat] :
( ( tl_nat @ ( append_nat @ Xs @ Ys3 ) )
= ( case_l2340614614379431832at_nat @ ( tl_nat @ Ys3 )
@ ^ [Z4: nat,Zs2: list_nat] : ( append_nat @ Zs2 @ Ys3 )
@ Xs ) ) ).
% tl_append
thf(fact_1245_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_1246_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N )
= ( minus_minus_nat @ M3 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1247_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1248_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M6: nat,N4: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_1249_map__decr__upt,axiom,
! [M3: nat,N: nat] :
( ( map_nat_nat
@ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M3 ) @ ( suc @ N ) ) )
= ( upt @ M3 @ N ) ) ).
% map_decr_upt
thf(fact_1250_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( hd_nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_1251_stirling__row__code_I1_J,axiom,
( ( stirling_row @ zero_zero_nat )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% stirling_row_code(1)
thf(fact_1252_length__stirling__row,axiom,
! [N: nat] :
( ( size_size_list_nat @ ( stirling_row @ N ) )
= ( suc @ N ) ) ).
% length_stirling_row
thf(fact_1253_stirling__row__nonempty,axiom,
! [N: nat] :
( ( stirling_row @ N )
!= nil_nat ) ).
% stirling_row_nonempty
thf(fact_1254_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_1255_nat__mult__eq__1__iff,axiom,
! [M3: nat,N: nat] :
( ( ( times_times_nat @ M3 @ N )
= one_one_nat )
= ( ( M3 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1256_nat__1__eq__mult__iff,axiom,
! [M3: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M3 @ N ) )
= ( ( M3 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1257_stirling__same,axiom,
! [N: nat] :
( ( stirling @ N @ N )
= one_one_nat ) ).
% stirling_same
thf(fact_1258_one__le__mult__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M3 @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M3 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1259_mult__le__cancel2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M3 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1260_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_1261_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_1262_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_1263_le__square,axiom,
! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ).
% le_square
thf(fact_1264_le__cube,axiom,
! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ) ).
% le_cube
thf(fact_1265_Suc__mult__le__cancel1,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M3 ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1266_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1267_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1268_mult__eq__self__implies__10,axiom,
! [M3: nat,N: nat] :
( ( M3
= ( times_times_nat @ M3 @ N ) )
=> ( ( N = one_one_nat )
| ( M3 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1269_stirling_Osimps_I1_J,axiom,
( ( stirling @ zero_zero_nat @ zero_zero_nat )
= one_one_nat ) ).
% stirling.simps(1)
thf(fact_1270_stirling_Oelims,axiom,
! [X2: nat,Xa2: nat,Y2: nat] :
( ( ( stirling @ X2 @ Xa2 )
= Y2 )
=> ( ( ( X2 = zero_zero_nat )
=> ( ( Xa2 = zero_zero_nat )
=> ( Y2 != one_one_nat ) ) )
=> ( ( ( X2 = zero_zero_nat )
=> ( ? [K3: nat] :
( Xa2
= ( suc @ K3 ) )
=> ( Y2 != zero_zero_nat ) ) )
=> ( ( ? [N2: nat] :
( X2
= ( suc @ N2 ) )
=> ( ( Xa2 = zero_zero_nat )
=> ( Y2 != zero_zero_nat ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ N2 ) )
=> ! [K3: nat] :
( ( Xa2
= ( suc @ K3 ) )
=> ( Y2
!= ( plus_plus_nat @ ( times_times_nat @ N2 @ ( stirling @ N2 @ ( suc @ K3 ) ) ) @ ( stirling @ N2 @ K3 ) ) ) ) ) ) ) ) ) ).
% stirling.elims
% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( if_list_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( if_list_nat @ $true @ X2 @ Y2 )
= X2 ) ).
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,
! [X2: list_list_nat,Y2: list_list_nat] :
( ( if_list_list_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
! [X2: list_list_nat,Y2: list_list_nat] :
( ( if_list_list_nat @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
equiva3371634703666331078on_rgf @ xs ).
%------------------------------------------------------------------------------